text
stringlengths 59
500k
| subset
stringclasses 6
values |
|---|---|
\begin{document}
\title[Cosimplicial meromorphic functions cohomology on complex manifolds]
{Cosimplicial meromorphic functions cohomology on complex manifolds}
\author{A. Zuevsky}
\address{Institute of Mathematics \\ Czech Academy of Sciences\\ Zitna 25, Prague\\ Czech Republic}
\email{[email protected]}
\begin{abstract}
Developing ideas of \cite{Fei},
we introduce canonical cosimplicial cohomology of meromorphic functions
for infinite-dimensional Lie algebra formal series
with prescribed analytic behavior
on domains of a complex manifold $M$.
Graded differential cohomology of a sheaf of Lie algebras $\mathcal G$
via the cosimplicial cohomology of $\mathcal G$-formal series for any covering by Stein spaces on $M$ is computed.
A relation between cosimplicial cohomology (on a special set of open domains of $M$)
of formal series of an infinite-dimensional Lie algebra $\mathcal G$
and singular cohomology of auxiliary manifold associated to a $\mathcal G$-module is found.
Finally, multiple applications in conformal field theory,
deformation theory, and in the theory
of foliations are proposed.
AMS Classification: 53C12, 57R20, 17B69
\end{abstract}
\keywords{Meromorphic functions, cosimplicial cohomology, complex manifolds}
\vskip12pt
\maketitle
\section{Introduction}
The continuous cohomology of Lie algebras of ${C}^{\infty}$-vector fields \cite{BS, Fuk, FF1988} has proven to be a subject of great geometrical interest.
There exists the natural problem of calculating the continuous cohomology of
meromorphic structures on
complex manifolds \cite{Fei, wag, Khor, Kaw, Hae, BS}.
In \cite{Fei} Feigin obtained various results concerning (co)homologies of certain Lie algebras associated to a complex curve $M$.
For the
Hodge decomposition of the tangent bundle complexification of $M$
corresponding
Lie bracket in the space of holomorphic
vector fields extends to a differential Lie superalgebra structure on the Dolbeault complex.
This
is called the differential Lie superalgebra $\Gamma(Lie(M))$ of holomorphic vector fields on $M$.
The
Lie algebra of holomorphic vector fields $Lie_{D}(M)$
is defined as the cosimplicial object in the
category of Lie algebras obtained from a covering
of $M$ by associating to any $i_1<i_2< \ldots <i_l$,
the Lie algebra of holomorphic vector fields $Lie( U_{i_1} \cap \ldots \cap U_{i_l})$.
In \cite{Fei} the author calculates the continuous (co)homologies
with coefficients in certain one-dimensional representations $\tau_{c_{p,q}}$ of these Lie (super)algebras
where $c_{p,q}$ denotes the value of the central charge for corresponding Virasoro algebra.
The main
result
states that
$H_0(Lie_D(M), \tau_{c_{p,q}} )$ is isomorphic to $H(M,p,q)$, where
the representation $\tau_{c_{p,q}}$
is derived from a vacuum representation of the Virasoro algebra,
and $H(M,p,q)$ is the modular functor for the minimal conformal field theory \cite{DiMaSe}.
The algebra
$H^*_c(\Gamma(Lie(M)))$ of continuous cohomologies acts naturally on $H_*(Lie_D(M),\tau_{c_{p,q}})$,
and the dual space is a free $H^*_c(\Gamma(Lie(M)))$-module
with generators in degree zero.
The paper \cite{wag}
continues the work of Feigin
\cite{Fei} and Kawazumi
\cite{Kaw}
on the Gelfand-Fuks cohomology of the Lie algebra of holomorphic vector fields on complex manifolds.
To enrich the cohomological structure, one has to involve
cosimplicial and graded differential Lie algebras
well known in Kodaira-Spencer deformation theory.
The idea to use cosimplicial spaces to study the cohomology of mapping spaces
goes back at least to
Anderson \cite{A},
and it was further developed
in \cite{BS}.
In \cite{wag}
they compute the
corresponding cohomologies
for arbitrary complex
manifolds up to calculation of
cohomology of
sections spaces of complex bundles on extra manifolds.
The results obtained are
very similar to the results of Haefliger \cite{Hae} and \cite{BS}
in the case of ${C}^{\infty}$ vector fields.
Following constructions of \cite{Fei}
applications in conformal
field theory (for Riemann surfaces), deformation theory, and foliation theory were proposed.
In addition to that, in \cite{wag}
the Quillen functor scheme was used for
the sheaf of holomorphic
vector fields
on a complex manifold, and its fine resolution was given by the sheaf
of $d\bar{z}$-forms with
values in holomorphic vector fields, the sheaf of Kodaira-Spencer algebras.
Let $M$ be a smooth compact manifold and $Vect(M)$ be the Lie algebra of vector fields on $M$.
Bott and Segal \cite{BS}
proved that the Gelfand-Fuks cohomology $H^*(Vect(M))$
is isomorphic to the singular cohomology $H^*(E)$ of the space $E$ of continuous cross sections of a certain
fibre bundle $\mathcal E$ over $M$.
Authors of \cite{PT, Sm} continued to use advanced topological methods for more general cosimplicial spaces of maps.
The main purposes of this paper are:
to compute the cosimplicial version of cohomology of
meromorphic functions with prescribed analytic behavior on domains of arbitrary complex manifolds,
and to find relations with
other types of cohomologies.
We also propose applications in conformal field theory, deformation theory, cohomology and characteristic
classes of foliations on smooth manifolds.
As it was demonstrated in \cite{wag}, the ordinary
cohomology of vector fields on complex manifolds turns to be not the most effective and general one.
In order to avoid trivialization
and reveal a richer cohomological structure
of complex manifolds cohomology, one has to
treat \cite{Fei} holomorphic vector fields as a sheaf rather than taking
global sections.
Inspite results in previous approaches, it is desirable to
find a way to enrich cohomological structure which motivates
construction of more refined cohomology description for non-commutative
algebraic structures.
The idea of meromorphic function cosimplician cohomology for complex manifolds was outlined in \cite{Fei} in
conformal field theory form (for Riemann surfaces)
and is developing in this paper.
In particular,
we study relations of the sheaf
of meromorphic functions associated to certain Lie algebras
to the sheaf $\mathfrak{g}$
of vector valued
differential forms.
\section{Meromorphic functions with prescribed analytic behavior}
In this section the space underlying cohomological complexes is defined in terms of meromorphic
functions with certain properties \cite{H2, Huang}, in particular, prescribed analytic behavior.
In order to shorthand expressions and notations, we call such functions mero functions.
Mero functions depend implicitly on an infinite number of non-commutative parameters via
Lie-algebra valued formal series in several complex variables.
\subsection{Algebraic completion of the space of formal $\mathcal G({\bf z})$-module valued series}
In this subsection we describe the algebraic completion of the space of an infinite-dimensional
Lie algebra $\mathcal G$ module-valued formal series.
In the whole body of the paper we use the notation
${\bf y}_{(n, 1), \ldots, (n, k)}$, ${\bf y}_n=(y_1, \ldots, y_n)$, for $n \ge 0$
sets of $k_i$, $1 \le i \le n$ of $k_i$ variables.
If $n$ and $k_i$ are not explicitly specified we write ${\bf y}$.
Let $\mathcal G$ be an infinite-dimensional Lie algebra
generated by $g_i$, $i \in \mathbb{Z}$.
Let $\mathcal G({\bf z})$ be the space of power series in several complex formal variables.
In order to define a specific space of meromorphic functions associated to $\mathcal G$
and satisfying certain properties mentioned later on,
we have to (as in \cite{Huang}) work with the algebraic completion of a $\mathcal G$-module $W$ \cite{Huang}.
In particular, for that purpose, we have to consider
elements of a $\mathcal G$-module $W$ with inserted
exponentials of the grading operator $L(0)$, i.e., of the form
$\sum\limits_{n \in \mathbb{Z}} z^{-n-1} \; a^{L(0)} v_n \; b^{L(0)} v$.
For general $a$, $b$, $z \in \mathbb{C}^\times$,
such elements
do not satisfy the properties (as elements of $W$ do) needed to construct a closed theory of meromophic functions with
prescribed properties.
Thus we have to extend $W$ algebraically (and analytically), i.e., include extra elements
to make the structure of $W$ compatible with the descending filtration with respect to the grading subspaces,
and analytic properties with respect to formal parameter $z$.
Thus we end up with are elements of the algebraic completion $\overline{W} = \prod_{n\in \mathbb{Z}} W(n)$, of $W$,
that has the structure that is complete in the
topology determined by the filtration and the residue pairing above.
Recall that
$W'=\coprod_{n\in \mathbb{Z}}W_{(n)}^{*}$.
For a infinite-dimensional Lie algebra $\mathcal G$-module $W$,
let $G$ be the algebraic completion of the space of $W$-valued formal series
$G=\prod_{n\in \mathbb{C}} W_{(n)}=(W')^{*}$,
endowed with a complex grading (with respect to a grading operator $K_G$).
We assume that on the space of formal series associated to $\mathcal G$,
there exists a non-degenerate bilinear pairing $( ., .)$,
and matrix elements of $G$ elements are given by this pairing.
In addition to that, we consider an analytical extension with respect to the pairing
${\rm Res}_z ( ., . )$ for $G$.
\subsection{Meromorphic functions with non-commutative parameters}
\label{functional}
In this subsection we give axiomatic definition of meromorphic functions with specific properties.
Let $M$ be an $n$-dimensional smooth complex manifold.
Let ${\bf p}_l$ be a set of $l$ points on $M$.
For each $p_i$, $1 \le i \le l$, let $U_i$
be an open domain surrounding $p_i$.
Let us identify formal variables ${\bf z}_{n,l}$ with $l$ sets of $n$ local coordinates on $\mathcal{U}_l$.
In this paper we consider
meromorphic functions
of several complex variables
defined on sets of open
domains of $M$ with local coordinates ${\bf z}_{l}$
which are extandable to meromorphic functions
on larger domains on $M$. We denote such extensions by $R(f({\bf z}_{l}))$.
Denote by $F_{n}\mathbb C$ the
configuration space of $l \ge 1$ ordered coordinates in $\mathbb C^{ln}$,
\[
F_{ln}\mathbb C=\{{\bf z}_l \in \mathbb C^{ln}\;|\; z_{i, l} \ne z_{j, l'}, i\ne j\}.
\]
We assume that there exists a non-degenerate bilinear pairing $( .,. )$ on $G$, and
denote by $\widetilde G$ the space dual with respect to this pairing.
In order to work with objects having coordinate invariant formulation \cite{BZF},
for a set of $G$-elements ${\bf g}_{l}$
we consider converging
meromorphic
functions $f({\bf x}_{l})$ of ${\bf z}_{l} \in F_{ln}\mathbb C$,
with ${\bf x}_{l}= ({\bf g}_{l}, {\bf z}_{l} {\bf dz}_{l})$,
where ${\bf z}_{l}$ are multiplied by corresponding differentials ${\bf dz}_{l}$.
\begin{definition}
For arbitrary $\vartheta \in \widetilde G$,
we call
a map linear in ${\bf g}_l$ and ${\bf z}_l$,
\begin{equation}
F: {\bf x}_l
\mapsto
\label{deff}
R(\vartheta, f({\bf x}_l )),
\end{equation}
a meromorphic function in ${\bf z}_l$
with the only possible poles at
$z_{i, l}=z_{j, l'}$, $i\ne j$.
Abusing notations, we denote $F({\bf x}_l)= R(\vartheta, f({\bf x}_l))$.
\end{definition}
\begin{definition}
We define left action of the permutation group $S_{ln}$ on $F({\bf z}_l)$ by
\[
\newcommand{\bfzq}{{\bf z}_l}
\sigma(F)({\bf x}_l)=F({\bf g}_l, {\bf z}_{\sigma(i)}).
\]
\end{definition}
In particular, meromorphic functions described above can be realized as $R(( ., .))$
of the bilinear pairing.
\subsection{Conditions on meromorphic functions}
Let ${\bf z}_l \in F_{ln}\mathbb{C}$.
Denote by $T_G$ the translation operator \cite{K}.
We define now extra conditions leading to the definition of restricted
meromorphic functions.
\begin{definition}
\label{tupo1}
Denote by $(T_G)_i$ the operator acting on the $i$-th entry.
We then define the action of partial derivatives on an element $F({\bf x}_l)$
\begin{eqnarray}
\label{cond1}
\partial_{z_i} F({\bf x}_l) &=& F((T_G)_i \; {\bf g}_l, {\bf z}_l),
\nonumber \\
\sum\limits_{i \ge 1} \partial_{z_i} F({\bf x}_l)
&=& T_{G} F({\bf x}_l),
\end{eqnarray}
\end{definition}
and call it $T_{G}$-derivative property.
\begin{definition}
\label{tupo2}
For $z \in \mathbb{C}$, let
\begin{eqnarray}
\label{ldir1}
e^{zT_G} F ({\bf x}_l)
= F({\bf g}_l, {\bf z}_l +z).
\end{eqnarray}
Let
${\rm Ins}_i(A)$ denotes the operator of multiplication by $A \in \mathbb{C}$ at the $i$-th position. Then we define
\begin{equation}
\label{expansion-fn}
F({\bf g}_l, {\rm Ins}_i(z) \; {\bf z}_l)=
F( {\rm Ins}_i (e^{zT_G}) \; {\bf g}_l, {\bf z}_l),
\end{equation}
are equal as power series expansions in $z$, in particular,
absolutely convergent
on the open disk $|z|<\min_{i\ne j}\{|z_{i, l}-z_{j, l'}|\}$.
\end{definition}
\begin{definition}
A meromorphic function has $K_G$-property
if for $z\in \mathbb{C}^{\times}$ satisfies
$z{\bf z}_l \in F_{ln}\mathbb{C}$,
\begin{eqnarray}
\label{loconj}
z^{K_G } F ({\bf x}_l) =
F (z^{K_G} {\bf g}_l,
z\; {\bf z}_l).
\end{eqnarray}
\end{definition}
\subsection{Meromorphic functions with prescribed analytical behavior}
In this subsection we give the definition of meromorphic functions with prescribed analytical behavior
on a domain of complex manifold $M$ of dimension $n$.
To shorthand, we call such functions mero functions.
We denote by $P_{k}: G \to G_{(k)}$, $k \in \mathbb{C}$,
the projection of $G$ on $G_{(k)}$.
For each element $g_i \in G$, and $x_i=(g_i, z)$, $z\in \mathbb{C}$ let us associate a formal series
$W_{g_i}(z)= W(x_i)=
\sum\limits_{k \in \mathbb{C} } g_{i} \; z^{k} \; dz $, $i \in \mathbb{Z}$.
Following \cite{Huang}, we formulate
\begin{definition}
\label{defcomp}
We assume that there exist positive integers $\beta(g_{l', i}, g_{l", j})$
depending only on $g_{l', i}$, $g_{l'', j} \in G$ for
$i$, $j=1, \dots, (l+k)n $, $k \ge 0$, $i\ne j$, $ 1 \le l', l'' \le n$.
Let
${\bf l}_n$ be a partition of $(l+ k)n
=\sum\limits_{i \ge 1} l_i$, and $k_i=l_{1}+\cdots +l_{i-1}$.
For $\zeta_i \in \mathbb{C}$,
define
$h_i
=F
({\bf W}_{ { \bf g}_{ k_i+{\bf l}_i }} (
{\bf z}_{k_i + {\bf l}_i }- \zeta_i ))$,
for $i=1, \dots, ln$.
We then call a meromorphic function $F$ satisfying properties \eqref{cond1}--
\eqref{loconj},
a meromorphic function with prescribed analytical behavior, of a mero function, if
under the following conditions on domains,
\[
|z_{k_i+p} -\zeta_{i}|
+ |z_{k_j+q}-\zeta_{j}|< |\zeta_{i} -\zeta_{j}|,
\]
for $i$, $j=1, \dots, k$, $i\ne j$, and for $p=1, \dots$, $l_i$, $q=1$, $\dots$, $l_j$,
the function
$\sum\limits_{ {\bf r}_n \in \mathbb{Z}^n}
F( {\bf P_{r_{i}} h_i}; (\zeta)_{l})$,
is absolutely convergent to an analytically extension
in ${\bf z}_{l+k}$, independently of complex parameters $(\zeta)_{l}$,
with the only possible poles on the diagonal of ${\bf z}_{l+k}$
of order less than or equal to $\beta(g_{l',i}, g_{l'', j})$.
In addition to that, for ${\bf g}_{l+k}\in G$, the series
$\sum_{q\in \mathbb{C}}$
$F( {\bf W(g_k}$, ${\bf P}_q ( {\bf W(g}_{l+k}, {\bf z}_k), {\bf z}_{ k + {\bf l} }))$,
is absolutely convergent when $z_{i}\ne z_{j}$, $i\ne j$
$|z_{i}|>|z_{s}|>0$, for $i=1, \dots, k$ and
$s=k+1, \dots, l+k$ and the sum can be analytically extended to a
meromorphic function
in ${\bf z}_{l+k}$ with the only possible poles at
$z_{i}=z_{j}$ of orders less than or equal to
$\beta(g_{l', i}, g_{l'', j})$.
\end{definition}
For $m \in \ensuremath{\mathbb {N}}$ and $1\le p \le m-1$,
let $J_{m; p}$ be the set of elements of
$S_{m}$ which preserve the order of the first $p$ numbers and the order of the last
$m-p$ numbers, that is,
$$J_{m, p}=\{\sigma\in S_{m}\;|\;\sigma(1)<\cdots <\sigma(p),\; \sigma(p+1)<\cdots <\sigma(m)\}.$$
Let $J_{m; p}^{-1}=\{\sigma\;|\; \sigma\in J_{m; p}\}$.
In addition to that, for some meromorphic functions require the property:
\begin{equation}
\label{shushu}
\sum_{\sigma\in J_{ln; p}^{-1}}(-1)^{|\sigma|}
\sigma(
F ({\bf g}_{\sigma(i)}, {\bf z}_l)
)=0.
\end{equation}
Finally, we formulate
\begin{definition}
\label{poyma}
We define the space $\Theta(ln, k, U)$ of all $l$ complex $n$-variable restricted
meromorphic functions
with prescribed analytical behavior
on a $F_{ln}\mathbb{C}$-domain $U \subset M$ and
satisfying
$T_G$- and $K_G$-properties \eqref{cond1}--
\eqref{loconj},
(definition \eqref{defcomp},
and \eqref{shushu}).
\end{definition}
\section{Properties of cosimplicial double complex spaces}
\label{cohass}
In this section we define the double complexes of mero
function cohomology
on a complex manifold $M$ of complex dimension $n$.
\subsection{Spaces of cosimplicial double complexes}
In \cite{GF} the original approach to cohomology of vector fields of manifolds
was initiated.
Another approach to cohomology
of the Lie algebra of vector fields on a manifold in the cosimplicial setup we find
in \cite{Fei, wag}.
Let $\mathcal{U}$ be a covering $\left\{ U_j \right\}$ on $M$,
and ${\bf z}_{l, j}$ be
$l$ sets of local complex coordinates on each domain $U_j$
around $l$ points ${\bf p}_{l, j}$.
For a set of $G$-elements
${\bf g_l}$,
and differentials ${\bf dz}_{l, j}$,
we consider
${\bf x}_{l, j}= ({\bf g}_l, {\bf z}_{l, j} \;{\bf dz}_{l, j})$.
\begin{definition}
For a domain $U \subset M$, and
$l$, $k \ge 0$,
we denote by ${C}^{l}( \Theta(ln, 1), U)$
the space of all mero
functions $\Theta(ln, 1)$ with prescribed analytic behavior with respect to
$l$ sets of $n$ complex coordinates introduced on $U$.
\end{definition}
\begin{remark}
Note that according to our construction, $M$ can be infinite-dimensional. Thus, in that case,
we consider $l$ infinite sets of complex coordinates.
The set of $ln$ $G$-elements
${\bf g_{l} }$
plays the role of non-commutative parameters in our cohomological construction.
\end{remark}
Using the standard method of defining canonical (i.e., independent of the choice
of covering $\mathcal{U}$) cosimplicial object \cite{Fei, wag},
we consider mero
functions
$F ( {\bf x}_{l, j} )$ on $\mathcal{U}$,
and give the following definition of a general cosimplicial double complex for $\mathcal G$.
\begin{definition}
Choose a covering $\mathcal{U}=\left\{U_i, i \in I\right\}$ of $n$-dimensional complex manifold $M$.
Let us associate to any subset
$\left\{i_1< \cdots < i_k\right\}$ of $I$,
the space of restricted meromorphic functions
converging on the intersection
$\left\{U_{i_1}\cap \ldots \cap U_{i_k}\right\}$. Let us introduce the space
\begin{equation}
\label{ourbi-complex}
{C}^{l}_{k }(G, \mathcal{U}) =
C^{l}
\left( \Theta(ln, kn), \; \bigcap_
{
i_1 \le \ldots \le i_k, k \ge 0 } U_{ i_k} \right).
\end{equation}
We call this space a cosimplicial cohomology object in the category of algebras of
mero functions on $M$.
\end{definition}
\subsection{Co-boundary operators}
\label{coboundaryoperator}
Let us take
${C}_{k}^{0} = G$.
Then we have
\[
{C}_{k}^{l} \subset {C}_{k-1}^{l},
\]
when lower index is zero the sequence terminates.
We also define
\[
C_{\infty}^{n}= \bigcap_{m\in \ensuremath{\mathbb {N}}}C_{m}^{n}.
\]
We organize elements of ${\bf x}_{(n,l)} =\left({\bf x}_{n, 1} \ldots, {\bf x}_{n,l} \right)$
as $l$ groups by $n$ elements.
Denote by ${\bf x}_{(n,\widehat {i}, \ldots, \widehat{j}) }$, $1 \le i, j \le l$,
the set ${\bf x}_{(n, l)}$ with ${\bf x}_{n, i}, \ldots, {\bf x}_{n,j}$ omitted.
For
$F \in {C}_{k }^{l}$,
we define
the operator $D^{l }_{k }$ by
\begin{eqnarray}
\label{hatdelta}
{D}^{l }_{k } F({\bf x}_{(n,l)}) &=& T_1(W_W({\bf x}_{n, 1} )).F({\bf x}_{(n, \widehat {1}) })
\nonumber \\
&+&\sum_{i=1}^{l }(-1)^{i}
T_i( W_V({\bf x}_{n, i}) )T_{i+1}(W_V({\bf x}_{n, i+1})).
F\left( {\bf x}_{(n, \widehat{i}, \widehat{i+1} ) }
\right)
\nonumber \\
&+&(-1)^{l+1}
T_1(W_W({\bf x}_{n, l+1})).F\left( {\bf x}_{n, \widehat {l+1}} \right),
\end{eqnarray}
where $T_i(\gamma).$ denotes insertion of $\gamma$ at $i$-th position of $F$.
In \cite{Huang} we find the construction of double chain-cochain complex for a space of
meromorphic functions compatible with a few formal series as in definition \ref{defcomp}.
In particular, (c.f. Proposition 4.1), the chain condition for such double complex is proven.
Here we use that construction to prove the chain property for the operator \eqref{hatdelta}
\begin{proposition}
\label{cochainprop}
The operator \eqref{hatdelta}
forms the double chain complex
\begin{equation}
\label{conde}
D^{l }_{k }: {C}_{k }^{l }
\to {C}_{k-1 }^{l+1},
\end{equation}
\begin{equation*}
D^{l+1 }_{k-1 } \circ {D}^{l }_{k }=0,
\end{equation*}
on the spaces \eqref{ourbi-complex}.
\end{proposition}
\begin{definition}
According to this proposition,
one defines the
$(l, k)$-th mero
function cosimplicial cohomology $H^{l}_{k, \; cos}(\mathcal G, \mathcal{U})$ of $M$
to be
$H_{k, \; cos}^l(\mathcal G, \mathcal{U}) ={\rm Ker} \;
D^l_k/\mbox{\rm Im}\; D^{l-1}_{k+1 }$.
\end{definition}
\begin{proof}
We use the proof of Proposition 4.1 of \cite{Huang}.
In that proof, for the case $n=1$, it was shown that the coboundary operator
$D^l_k$ acting on elements of $C^l_k$
brings about elements of $C^{l+1}_{k-1}$.
Namely, for $n=1$ and $F \in \Theta(l+1, m)$ one has
\begin{eqnarray*}
D^{l}_{m} F\left(x_1, \ldots, x_{l+1} \right)
&=& W_W(x_1) \; F(x_2, \ldots, x_{l+1})
\nonumber \\
&+& \sum_{i=1}^{l}(-1)^{i}
F(x_{1}, \ldots, x_{i-1},
\; W_{V}(v_{i}, z_{i}-\zeta_{i}) W_{V}(v_{i+1}, z_{i+1}-\zeta_{i}), \;
\nonumber \\
&& \quad\quad\quad\quad
x_{i+2}, \ldots, x_{l+1})
\ + (-1)^{l+1} W_W(x_{l+1}) \;
F(x_{1}, \ldots, x_{l}),
\end{eqnarray*}
By Proposition 2.8
of \cite{Huang},
$D^{l}_{k}F$ is
compatible with $(k-1)$ formal series as in definition \ref{defcomp} ,and has the $T_G$-derivative \eqref{cond1}
property and the $K_G$-conjugation \eqref{ldir1} properties according to definitions \ref{tupo1} and \ref{tupo2}.
So $D^{l}_{k}F \in
C_{k-1}^{l+1}$ and $D^l_{k}$ is indeed a map whose image is in
${C}_{k-1}^{l+1}$.
The chain property for $D^l_k$ was proven also.
Applying inductively Proposition 4.1 of \cite{Huang} to our setup and increasing recursively for $n\ge 0$,
we obtain the result of our proposition.
\end{proof}
\section{Sheaf formulation of cosimplicial cohomology}
In this section we generalize the construction of \cite{GF, Fei, wag} and
replace the algebra of holomorphic vector field used in \cite{wag}
with $G$-valued series for an infinite-dimensional
Lie algebra $\mathfrak g$ in the sheaf-theoretical formulation.
In particular we prove a generalization of Theorem 4 of \cite{wag} and
relate the graded differential cohomology of the sheaf of $G$-valued formal series
and the cosimplicial cohomology of the sheaf of mero functions over Stein sets on $M$.
Let us start with some general definitions needed for further explanations.
We understand from sections that $G$-valued series for $\mathfrak g$ are formalized via
the space of mero functions.
Thus it makes sense to formulate the definition of the sheaf of graded differential
Lie algebras of $G$-valued formal series for $\mathfrak g$ simultaneously with
the definition of the sheaf of mero functions.
Moreover, as we will see in one of next subsections, there exists an equivalence of the cohomology of
the sheaf of graded differential
Lie algebras of $G$-valued formal series for $\mathfrak g$
and cosimiplicial cohomology of a complex for a sheaf of $G$-valued series for $\mathfrak g$.
To any $U$ of $\mathcal{U}$ of $M$ we can assign the set $\Theta(ln, kn, U)$ of mero functions on $U$.
The restriction maps are then just given by
restricting a mero function on $U$ to a smaller open subset $\widetilde{U} \subset U$,
which, according to definition \ref{defcomp} is again a mero function.
We then immediately check the presheaf axioms.
\subsection{Coherent sheaf of mero functions}
$M$ is endowed with a sheaf
of rings $\mathcal O_X$, the sheaf of holomorphic functions or regular functions,
and coherent sheaves are defined as a full subcategory of the category
of
$\mathcal O_X$-modules (that is, sheaves of
$\mathcal O_X$-modules).
We are able
localize mero functions
to open subsets
$U \subset M$.
The presheaf of mero functions
can be glued to global data.
We formulate the following
Denote by ${\mathcal F}_M$ the coherent sheaf of mero
functions on $M$, and consider ${\mathcal F}_M$-modules.
The sheaf of mero functions on $M$ is defined by
and ${\mathcal F}_M$-modules.
As in \cite{wag} we transfer to the sheaf setup of $G$-valued series and mero functions.
Let us denote by ${\mathcal O}_M$ the coherent sheaf of
holomorphic functions on $M$ and by ${\mathcal E}_M$ the sheaf of
$C^{\infty}$ functions on $M$.
We denote by $\ensuremath{\mathcal{F}}_M$ the sheaf of mero functions
on $M$, and
by ${\mathfrak f}_M$ the sheaf of $\ensuremath{\mathcal{F}}_M$-modules.
Denote by ${\mathcal G}({\bf z})$ the sheaf of Lie-algebra $\mathfrak g$ $n$-parameter ${\bf z}$ formal series.
It can be represented via by ${\mathcal F}_M$-modules for the sheaf of mero
functions.
Note that the sheaf $\mathcal G({\bf z})$ naturally induces the sheaf ${\mathcal E}_M$.
Let ${\mathfrak F}$ a sheaf of ${\mathcal O}_X$-modules which are Lie algebras.
Let
$(\mathfrak{F},\bar{\partial})$ a sheaf of differential graded Lie algebras which are ${\mathcal E}_M$-modules.
We denote by $\ensuremath{\Gamma}(\mathfrak{F})$, $\ensuremath{\Gamma}(X,\mathfrak{F})$ or
$\mathfrak{F}(X)$ the differential graded Lie algebras of global sections of the sheaf
$\mathfrak{F}$.
As in \cite{wag},
we can associate to $\mathfrak{F}$ resp. to
$(\mathfrak{F},\bar{\partial})$ sheaves of differential graded coalgebras
$C_*(\mathfrak{F})$, $C_{*,dg}(\mathfrak{F})$, $H_*(\mathfrak{F})$ and
$H_{*,dg}(\mathfrak{F})$ where the last two carry the trivial
differential.
In the same way, we have sheaves of differential graded
algebras $C^*_{cont}(\mathfrak{F})$, $C_{dg}^*(\mathfrak{F})$,
$H^*_{cont}(\mathfrak{F})$ and $H_{dg}^*(\mathfrak{F})$.
Furthermore, we have differential graded coalgebras $C_*(\ensuremath{\Gamma}(\mathfrak{g}))$,
$C_{*,dg}(\ensuremath{\Gamma}(\mathfrak{F}))$, $H_*(\ensuremath{\Gamma}(\mathfrak{F}))$ and
$H_{*,dg}(\ensuremath{\Gamma}(\mathfrak{F}))$, and the corresponding algebras.
For a sheaf $\ensuremath{\mathcal{A}}$
on $M$, the sheaf cohomology groups
$H^{i}(\ensuremath{\mathcal{A}}, M)$
for integers $i$
are defined as the right derived functors of the functor of global sections,
$\ensuremath{\mathcal{A}} \mapsto \ensuremath{\mathcal{A}}(X)$.
As a result,
${\displaystyle H^{i}(\ensuremath{\mathcal{A}}, M)}$ is zero for
${\displaystyle i<0}$,
and
${\displaystyle H^{0}(X, \ensuremath{\mathcal{A}})}$ can be identified with
${\displaystyle \ensuremath{\mathcal{A}}}(X)$.
For any short exact sequence of sheaves
${\displaystyle 0\to {\mathcal {A}}\to {\mathcal {B}}\to {\mathcal {C}}\to 0}$,
there is a long exact sequence of
cohomology groups:
\[
{\displaystyle 0\to H^{0}(X,{\mathcal {A}})\to H^{0}(X,{\mathcal {B}})\to H^{0} (X,{\mathcal {C}})\to H^{1}(X,{\mathcal {A}})\to \cdots .}
\]
\subsection{Thickened nerve of the covering}
Here we have to provide some further topological definitions.
The nerve of an open covering is a construction of an abstract simplicial complex
from an open covering of $M$.
\begin{definition}
Let
$I$ be an index set and $\mathcal{U}$ be a family of open subsets
$U_{i}$ of $M$ indexed by $i\in I$.
The nerve of $\mathcal{U}$ is a set of finite subsets of the index-set $I$.
It contains all finite subsets $J\subseteq I$ such that the intersection of the
$U_{i}$ whose subindices are in $J$ is non-empty, i.e.,
$N(\mathcal{U})= \left\{ J\subseteq I:\bigcap _{j\in J} U_{j} \neq \varnothing \right\}$, where $J$
is a finite set.
\end{definition}
\begin{definition}
If $J\in N(\mathcal{U})$, then any subset of $J$
is also in $N(\mathcal{U})$,
making $N(\mathcal{U})$ an abstract simplicial complex,
often called the nerve complex of $C$.
\end{definition}
Next we recall the notation of a manifold made out of simplices:
A simplicial manifold is a simplicial complex for which the geometric realization
is homeomorphic to a topological manifold.
This is essentially the concept of a triangulation in topology.
This can mean simply that a neighborhood of each vertex
(i.e., the set of simplices that contain that point as a vertex) is homeomorphic to a $n$-dimensional ball.
For our further purposes we will need the following proposition from \cite{BS}:
\begin{proposition}
(Proposition (5.9) of \cite{BS}).
If $C \to C'$ is a morphism of simplicial cochain complexes such that
$C_p \to C_p'$ is a cohomology equivalence for each $p$, then $|C| \to |C'|$ is a cohomology equivalence.
\end{proposition}
\subsection{Definition of the cosimplicial sheaf complex}
In this subsection, following the general ideas of
\cite{CM, wag} we formulate definition of a cohomology $H^*(\mathcal{U}, \ensuremath{\mathcal{A}})$ of
a cosimplicial sheaf $\ensuremath{\mathcal{A}}$ complex $C_*(\mathcal{U}, \ensuremath{\mathcal{A}})$ defined
on a system $\mathcal{U}$ of domains on smooth complex manifold $M$.
\begin{definition}
Let $\Gamma (\ensuremath{\mathcal{A}}, U)$ be a local section defined on $U$ for a coherent sheaf $\ensuremath{\mathcal{A}}$.
Then one defines a chain-cochain complex
\begin{equation}
\label{sheaf-complex}
{C}_{k}(\mathcal{U}, \ensuremath{\mathcal{A}}) =
\Gamma \left(\ensuremath{\mathcal{A}}, \; \bigcap_
{
i_1 \le \ldots \le i_k, k \ge 0, \atop U_{0}\stackrel{h_1}{\longrightarrow} \ldots \stackrel{h_k}{\longrightarrow} U_k } U_{ i_k}\right).
\end{equation}
where the intersection is over all $k$-strings of $h$-
embeddings between opens $U_i\in \mathcal{U}_{i+1}$,
and the boundary operator $\delta = \sum(-1)^i\delta_i$ is given by the standard formula
\begin{equation}\label{deltas} \delta_{i}\Gamma(h_1, \ldots , h_{k+1})= \left\{ \begin{array}{lll}
h_{1}^{*}\Gamma(h_2, \ldots , h_{k+1}), \ \ \mbox{if $i=0$}\\
\Gamma(h_1, \ldots, h_{i+1}h_{i}, \ldots, h_{k+1}), \ \ \mbox{if $0<i< k+1$}\\
\Gamma(h_1, \ldots, h_k), \ \ \mbox{if $i= k+1$}
\end{array}
\right. \end{equation}
originating from application of the Chevalley-Eilenberg \cite{CE} functor.
\end{definition}
Note that according to \cite{CM}, one has
\begin{lemma}
\label{lemmaCM}
The complex \eqref{sheaf-complex} is a graded differential algebra with the usual product
\begin{equation}
\label{multiplication}
(\Gamma_1\cdot\Gamma_2)(h_1, \ldots , h_{k+k'})= (-1)^{kk'}\Gamma_1(h_1, \ldots , h_{k})\; (h_{1}^{*}
\ldots h_{k}^{*}) \;\Gamma_2(h_{k+1}, \ldots h_{k+k'}),
\end{equation}
for $\Gamma_1\in C_{k}(G, \mathcal{U})$ and $\Gamma_2\in C_{k'}(G, \mathcal{U})$.
\end{lemma}
\begin{definition}
For $\ensuremath{\mathcal{A}}=\ensuremath{\mathcal{F}}_M(\Theta(ln, kn), \mathcal{U})$ beeing local $\ensuremath{\mathcal{F}}_M$-section defined on $U$,
we introduce the cosimplicial sheaf complex of mero functions by \eqref{sheaf-complex}.
\end{definition}
\subsection{Definition of the cosimplicial sheaf cohomology $H^l_{k, \; cos}(\mathcal G(\bf z), \mathcal{U})$ for the sheaf of mero functions on $M$}
Next we define
\begin{definition}
$H_*^{*}( \ensuremath{\mathcal{G}}({\bf z)}, \mathcal{U})$ as the cohomology of the double complex ${C}_k( \ensuremath{\mathcal{S}}^{l}, \mathcal{U})$ given by
\eqref{sheaf-complex}
, where
\[
0\longrightarrow \ensuremath{\mathcal{A}}\longrightarrow \ensuremath{\mathcal{S}}^0\longrightarrow\ldots \longrightarrow \ensuremath{\mathcal{S}}^d\longrightarrow...,
\]
is a bounded resolution by $\mathcal{U}$-acyclic sheaves.
\end{definition}
By the usual arguments \cite{CM}, such resolutions always exist, and the definition does not depend on
the choice of the resolution.
Let us now recall \cite{wag} the universal notion of Stein spaces which we will use
in further discussions.
\begin{definition}
An open set $U\subset M$ of a complex manifold $M$ is called a Stein
open set, if
the coherent
sheaf cohomology vanishes on $U$, i.e.,
$H^i_k( \ensuremath{\mathcal{A}}, U) = 0$, $i \ge 1$, $k \ge 0$,
and for all coherent sheaves $\ensuremath{\mathcal{A}}$ on $M$.
\end{definition}
In the language of \cite{CM} the Stein spaces are called $\mathcal{U}$-acyclic.
\subsection{Graded differential
cohomology $H^*_{*, dg} (\Gamma(\mathcal G({\bf z})), M)$
of global sections of the sheaf $\mathcal G({\bf z})$ }
In this subsection we recall and make applications of certain facts \cite{wag}
on construction of the graded differential cohomology $H^*_{*, dg} (\Gamma(\mathcal G({\bf z})), M)$
of global sections of the sheaf $\mathcal G({\bf z})$ of an infinite-dimensional Lie algebra $\mathfrak g$ formal series.
This
cohomology is calculated
by associating to $\mathfrak g$ a graded differential
algebra $C^*(\mathfrak g) = (Hom(\Lambda^*(\mathfrak g),\mathbb{C}), d)$,
the
cohomological Chevalley-Eilenberg complex \cite{CE} described below.
\begin{definition}
Let $\mathfrak a$ be a
Lie algebra over $\mathbb{C}$.
The Chevalley--Eilenberg chain complex is a
projective resolution
$W^*(\mathfrak a) \to \mathbb{C}$
of the trivial $\mathfrak a$-module $\mathbb{C}$ in the abelian category of $\mathfrak a$-modules
(what is the same as $U({\mathfrak a})$-modules, where $U({\mathfrak a})$ is the universal
enveloping algebra of $\mathfrak a$).
Graded components of the underlying $\mathbb{C}$-module
of this resolution is given by
$W_p(\mathfrak a)= U(\mathfrak a) \otimes_k \Lambda^p \mathfrak a$,
and it has the obvious $U({\mathfrak a})$-module structure by multiplication in the first tensor factor,
because $\Lambda^p \mathfrak a$ is free as a $\mathbb{C}$-module.
For $u \in U({\mathfrak a})$, and ${\bf v}_p \in {\mathfrak a}^{\otimes p}$, the differential is given by
\begin{eqnarray}
d\left(u \otimes {\bf \bigwedge v}_p \right) =
\sum_{i = 1}^p (-1)^{i+1} u\; v_i \otimes {\bf \bigwedge v}_{(p, \widehat {i}) }
+\sum_{i < j} (-1)^{i+j} u\otimes [v_i, v_j] \wedge {\bf \bigwedge v}_{(p, \widehat {i}, \widehat {j})},
\end{eqnarray}
Then, let
$g = (\bigoplus_{i=0}^{n}g^i, {\partial})$,
be a cohomological graded differential Lie algebra (which we will denote dgla in notations).
As it was mentioned in \cite{wag},
there exist two functors,
$C_{*,dg}$ and $C^*_{dg}$,
associating to $(g, {\partial})$ graded differential coalgebras
$C_{*,dg}(g)$ and $C^*_{dg}(g)$.
$C_{*,dg}(g)$ is called the Quillen functor, \cite{Qui}.
It was explicitly constructed in \cite{HinSch}.
The cohomology version was
used in \cite{Hae} and
\cite{SchSta}.
Explicitly, it is given by
\begin{displaymath}
C_{k,dg}(g) = \bigoplus_{k=p+q}C_{dg}^p(g)^q =
\bigoplus_{k=p+q}S^{-p}(g_{(q+1)}),
\end{displaymath}
as graded vector spaces.
Here $S^{-p}(g_{(q+1)})$
is the graded symmetric algebra
$S^*$ on the shifted by one graded vector space $g_{(q+1)}$.
The differential on $C_{*, dg}(g)$ is the
direct sum of the graded homological Chevalley-Eilenberg \cite{CE} differential
in the tensor direction (with degree reversed in order to have a cohomological differential) and the differential induced on $S^*(g_{(q+1)})^*$ by $ {\partial}$.
We denote by $\ensuremath{\Gamma}(\mathfrak{F}, M)$
the graded differential Lie algebra of global sections of the sheaf $\mathfrak{F}$ on $M$.
Consider the sheaf of $G$-valued series for $\mathfrak g$. It
constitutes a sheaf of Lie algebras.
According to \cite{HinSch} (proposition .), for any sheaf of Lie algebras
$\mathfrak{h}$ there is another sheaf of differential
graded Lie algebras constituing a resolution of $\mathfrak{h}$.
It is the sheaf of cosimplicial Lie algebras given by taking $\mathfrak{h}$ on
the ${\rm \check C}$ech complex \eqref{sheaf-complex} associated to a covering ${\mathcal U}$ by Stein open
sets, suitably normalized by the Thom-Sullivan functor, see \cite{HinSch, wag}.
One can see that such sheaf of differential graded Lie algebras is given by
the sheaf $\mathcal G({\bf z})$ taken our complex \eqref{sheaf-complex}
on $\mathcal{U}$ given by Stein open sets.
\subsection{Cosimplicial cohomology
$H_{*\; cos}^*(\check C(G,
\mathcal{U}))$}
Developing ideas of \cite{wag, CE, S}, we
give the definition of the cohomology $H_{*\; cos}^*(\check C(G, \mathcal{U}))$
of cosimplicial Lie algebra of the complex
of mero functions for
an infinite-dimensional Lie algebra $\mathfrak g$ defined on
a specific covering.
\begin{definition}
The cosimplicial cohomology $H_{*\; cos}^*(\check C(G, \mathcal{U}))$ of the complex
$C^l_k(G, \mathcal{U})$, $l$, $k \ge 0$,
with $C^l_k(G, \mathcal{U})$ given by the complex
of
mero functonsi
for the Lie algebra $\mathfrak g$ of
$G$-valued formal series defined on a covering $\mathcal{U}$
is the cohomology of
the realization of simplicial cochain complex
obtained from applying the continuous (chain-cochain)
Chevalley--Eilenberg complex as a functor $C^*_{*, cont}$ to the
cosimplicial Lie algebra $\check C(G, \mathcal{U})$.
\end{definition}
\subsection{Computation of graded differential sheaf cohomology
via cohomology of cosimplicial Lie algebra of mero functions cohomology}
The main idea of this subsection is that we are able to compute
the graded differential algebra cohomology defined for the sheaf of $G$-valued series
for an infinite-dimensional Lie algebra $\mathcal G$
on an $n$-dimensional complex manifold $M$
via the mero function cosimplicial cohomology for $\mathcal G$ considered
on special type of open domains on $M$.
We then obtain the main result of this section
\begin{proposition}
On a complex manifold $M$ of dimension $n$, one has
\begin{displaymath}
H^*_{*\; dg} \left( \Gamma(\mathcal G({\bf z}), M) \right) \cong H^*_{*\; cos} \left( {\check C} (G,
{\mathcal U})\right),
\end{displaymath}
for any covering of $M$ by Stein open sets ${\mathcal U}$ (with respect to the cosimplicial sheaf cohomology of
coherent sheaves).
\end{proposition}
\begin{proof}
The idea of the proof is quite close to remarks to the proof of Theorem 4 of \cite{wag}.
Here we give an explicit realization of those ideas.
We consider the sheaf ${\mathfrak F}$ of global sections of $G$-valued formal series associated to an
infinite-dimensional Lie algebra
${\mathfrak g}$, and the sheaf $\mathcal G({\bf z})$ of $G$-valued mero functions.
According to \cite{HinSch}, for any sheaf of Lie algebras $\mathfrak h$ there is another sheaf of differential
graded Lie algebras constituting a resolution of $\mathfrak h$.
It is the sheaf of cosimplicial Lie
algebras given by taking ${\mathfrak h}$ on the ${\rm {\check C}}$ech complex associated to a covering $\mathcal{U}$
by Stein open sets, suitably normalized by the Thom-Sullivan functor \cite{HinSch}.
For ${\mathfrak F}$ the graded differential algebra $\Gamma(\mathfrak F, M)$ of global sections of ${\mathfrak F}$
is obtained by application of the Chevalley-Eilenberg
complex
on a Stein cover of $M$.
On the other hand the graded differential algebra $\check C(G, \mathcal{U})$
is given through Chevalley-Eilenberg simplicial chain complex
on a Stein open sets
with multiplication \eqref{multiplication}.
Our aim now is to construct an explicit isomorphisms of these two complexes.
We then find (following the lines of \cite{HinSch, wag})
a relation between cohomology of
the sheaf of graded differential algebras associated to $\mathfrak g$
and
the sheaf of graded differential algebra ${\check C}(G, \mathcal{U})$ of mero functions for a Lie algebra $\mathfrak g$.
Let, as in \cite{wag}, denote by
$N_{*}$ the thickened nerve of the covering
${\mathcal{U}}$, i.e., the simplicial complex manifold associated to the
covering ${\mathcal{U}}$.
On $M$, there exists an inclusion
\begin{equation}
\label{inclusion}
f :
\check C(G, N_{M,q}) \hookrightarrow {\mathfrak F}(G, N_{M,q}),
\end{equation}
of graded differential algebras $\check C(G, N_{M,q})$ and ${\mathfrak F}(G, N_{M,q})$ on $N_{M,q}$.
By applying the modification of the Quillen functor \cite{wag},
this inclusion induces
\begin{displaymath}
\widetilde{f} : C^*_{dg}({\mathfrak F}, N_{*}) \to C^*_{cont}(\mathcal G({\bf z}), N_{*}),
\end{displaymath}
a morphism of simplicial cochain complexes.
By Proposition 5.9 in \cite{BS}, the morphism $\widetilde{f}$
induces a cohomology equivalence between the realizations of the two simplicial
cochain complexes $C_k^l(\mathcal G(\bf z), M)$ and $C^l_k(G, \mathcal{U})$.
The conditions of the lemma are fulfilled because
of the isomorphism of the cohomologies on a Stein open set of
the covering and the K\"unneth theorem \cite{wag}.
Using Proposition
6.2 of
\cite{BS}, and involving partitions of unity, one shows that the cohomology of the realization of the simplicial cochain complex on the left hand side gives the graded differential cohomology of $\ensuremath{\Gamma}(X,\mathfrak{g})$.
\end{proof}
\section{Relation of cosimplicial and singular cohomology}
Gelfand and Fuks \cite{Fuk}
calculated cohomology of the Lie algebra of formal vector fields in $n$ complex variables
$W_n$.
In particular, they proved \cite{Fuk, FF1988}
\begin{theorem}
There exists a manifold $X(n)$ such that the continuous cohomology of $W_n$ is equivalent to singular cohomology of $X(n)$
\begin{displaymath}
H^*_{cont}(W_n) \cong H^*_{sing}(X(n)).
\end{displaymath}
\end{theorem}
In \cite{BS} they
showed that for $\mathbb{R}^n$ or, more generally, for a starshaped open set $U$ of an
$n$-dimensional manifold $M$, the Lie algebra of
$C^{\infty}$-vector fields $Vect(U)$ has the same cohomology as $W_n$.
In \cite{wag} it was proven that
the same is true for the Lie algebra of holomorphic vector fields on a
disk of radius $R$ in $\mathbb{C}^n$.
In this paper, we consider cohomology of mero
functions provided by bilinear pairings for an
arbitrary $n$-formal parameter Lie algebra $G$-valued series localized on a
complex $n$-dimensional manifold $M$.
For purposes of determining the cosimplicial cohomology, we use the machinery of
chequered necklaces \cite{MT} associated with $G$-valued series.
\subsection{A-matrix}
In this subsection we
discuss a number of elliptic
functions that we will need. The Weierstrass elliptic function with periods
$\sigma$, $\varsigma \in \mathbb{C}^{\ast }$ is defined by
\begin{equation}
\wp (z,\sigma ,\varsigma )=z^{-2}+ \sum_{m,n\in \mathbb{Z},(m,n)\neq
(0,0)} \left[ (z-m\sigma -n\varsigma )^{-2}- (m\sigma +n\varsigma )^{-2}\right]. \label{Weierstrass}
\end{equation}
Choosing $\varsigma =2\pi i$ and $\sigma =2\pi i\tau $ (where $\tau$
lies in the complex upper half-plane $\mathbb{H}$), we define
\begin{eqnarray} P_{2}(\tau ,z) &=&\wp (z,2\pi i\tau ,2\pi i)+E_{2}(\tau ) \notag \\ &=&\frac{1}{z^{2}}+\sum_{k=2}^{\infty }(k-1)E_{k}(\tau )z^{k-2}. \label{P2} \end{eqnarray} Here, $E_{k}(\tau )$ is equal to $0$ for $k$ odd, and for $k$ even is the
Eisenstein series \cite{Se}
\begin{equation*} E_{k}(\tau )=-\frac{B_{k}}{k!}+\frac{2}{(k-1)!}\sum_{n\geq 1}\sigma _{k-1}(n)q^{n}. \end{equation*}
Here we take $q=\exp (2\pi i\tau )$; $\sigma
_{k-1}(n)=\sum_{d\mid n}d^{k-1}$, and $B_{k}$ is the $k$th Bernoulli number defined by
\begin{eqnarray*}
\frac{t}{e^{t}-1}-1+\frac{t}{2} = \sum_{k\geq 2}B_{k}\frac{t^{k}}{k!}
= {\frac{1}{12}}{t}^{2}-{\frac{1}{720}}{t}^{4}+{\frac{1}{30240}}{t}^{6}+O({t }^{8}). \end{eqnarray*} $P_{2}$ can be alternatively expressed as \begin{equation} P_{2}(\tau ,z)=\frac{q_{z}}{(q_{z}-1)^{2}}+\sum_{n\geq 1}\frac{nq^{n}}{ 1-q^{n}}(q_{z}^{n}+q_{z}^{-n}), \label{P2exp} \end{equation} where $q_{z}=\exp (z)$.
If $k\geq 4$ the
first three Eisenstein series $E_{2}(\tau ),E_{4}(\tau ),E_{6}(\tau )$ are algebraically independent and generate a weighted polynomial algebra
$Q= \mathbb{C}[E_{2}(\tau )$, $E_{4}(\tau ),E_{6}(\tau )]$.
We define $P_{1}(\tau ,z)$ by \begin{equation} P_{1}(\tau ,z)=\frac{1}{z}-\sum_{k\geq 2}E_{k}(\tau )z^{k-1}, \label{P1} \end{equation} where $P_{2}=-\frac{d}{dz}P_{1}$ and $P_{1}+zE_{2}$ is the classical Weierstrass zeta function.
We also define $P_{0}(\tau ,z)$, up to a choice of the logarithmic branch, by \begin{equation} P_{0}(\tau ,z)=-\log (z)+\sum_{k\geq 2}\frac{1}{k}E_{k}(\tau )z^{k}, \label{P0} \end{equation} where $P_{1}=-\frac{d}{dz}P_{0}$.
Define elliptic functions $P_{k}(\tau ,z)\,$ for $k\geq 3\,$ from the analytic expansion \begin{equation} P_{1}(\tau ,z-w)=\sum_{k\geq 1}P_{k}(\tau ,z)w^{k-1} \label{P1Pnexpansion} \end{equation} where \begin{equation} P_{k}(\tau ,z)=\frac{(-1)^{k-1}}{(k-1)!}\frac{d^{k-1}}{dz^{k-1}}P_{1}(\tau ,z)=\frac{1}{z^{k}}+E_{k}+O(z). \label{Pkdef} \end{equation}
Finally, define for $k,l=1,2,\ldots $ \begin{eqnarray} C(k,l) &=&C(k,l,\tau )=(-1)^{k+1}\frac{(k+l-1)!}{(k-1)!(l-1)!}E_{k+l}(\tau ), \label{Ckldef} \\ D(k,l,z) &=&D(k,l,\tau ,z)=(-1)^{k+1}\frac{(k+l-1)!}{(k-1)!(l-1)!} P_{k+l}(\tau ,z). \label{Dkldef} \end{eqnarray} $\,$Note that $C(k,l)=C(l,k)$ and $D(k,l,z)=(-1)^{k+l}D(l,k,z)$. These naturally arise in the analytic expansions (in appropriate domains) \begin{equation} P_{2}(\tau ,z-w)=\frac{1}{(z-w)^{2}}+\sum_{k,l\geq 1}C(k,l)z^{l-1}w^{k-1}, \label{P2expansion} \end{equation} and for $k\geq 1$ \begin{eqnarray} P_{k+1}(\tau ,z) &=&\frac{1}{z^{k+1}}+\frac{1}{k}\sum_{l\geq 1}C(k,l)z^{l-1}, \label{Pkexpansion} \\ P_{k+1}(\tau ,z-w) &=&\frac{1}{k}\sum_{l\geq 1}D(k,l,w)z^{l-1}. \label{Pkzwexpansion} \end{eqnarray}
Notation here is as follows: $A(\tau,\epsilon )$ is the infinite
matrix with $(k,l)$-entry
\begin{equation}
A(k,l,\tau,\epsilon )=\frac{\epsilon ^{(k+l)/2}}{\sqrt{kl}}
C(k,l,\tau); \label{Aki1} \end{equation}
Our setup is
facilitated by an alternate description in terms of combinatorial
gadgets that we call chequered necklaces \cite{MT}.
They are certain kinds of graphs with nodes labeled by positive integers and edges labeled by quasimodular forms, and they play an important role in this section.
It is useful to introduce an interpretation for complexes
in terms of the sum of weights of certain graphs.
In particular, in this subsection, we construct a special simplicial manifold
defined
over the space of chequered necklaces.
\subsection{Explicit construction of the simplicial manifold $X(G,n)$}
Now we will show how to associate elements of $G$ to chequered necklaces.
Recall that in general each element of $G$ has a form $W_g(z_i)=\sum\limits_{m \in \mathbb{Z}} g_{i, m} z^m$,
Consider the elements
$F(x_1, x_2, (h, \zeta))= ( \theta,
W_{\overline{h}}(\zeta)\; W_{g_1}(z_1) \; W_{g_2}(z_2) \; W_{h}(\zeta) )$, that
belong to the space
$\Theta(ln, kn)$.
Here $\overline{h}$ is dual to $h \in G$ with respect to the bilinear pairing.
Performing the summation over $h\in G$,
$G(x_1, x_2, \zeta)=\sum\limits_{h \in G} (\theta,
W_{\overline{h}}(\zeta)\; W_{g_1}(z_1) \; W_{g_2}(z_2) \; W_{h}(\zeta) )$,
we obtain (with appropriate choice of the complex parameter $\zeta$ \cite{MT}),
an element $C(g_1, g_2, z_1-z_2, \tau)$ which expandes as \eqref{Aki1} in terms of $C(m, m', \tau)$
or $D(m, m', z_1-z_2, \tau)$.
Thus we see make a connection between $G$-elements and a chequered necklaces.
Let us explicitly construct the special manifold $X(G, n)$.
A simplicial manifold is a simplicial complex for which the geometric realization
is homeomorphic to a topological manifold.
This is essentially the concept of a triangulation in topology.
This can mean simply that a neighborhood of each vertex
(i.e., the set of simplices that contain that point as a vertex) is homeomorphic to an $n$-dimensional ball.
Now let us see how sets of chequerd necklaces turn into a simplicial manifold.
Chequered necklaces are one-simplexes in form of graphs.
They are associated to elements of $G$.
If ${\bf z}_n$-dependence of $G$-elements is taken into account then
a chequered necklaces is considered as an $(n+2)$-simplex (due to dependence on extra two complex parameters
$\tau$ and $\epsilon$.
The set of chequered necklaces $\mathcal{N}_m$, $m \ge 2$, is a set of special graphs.
The spaces of simplicial complex are sums of weights of certain graphs (period matrix).
Chequered necklaces $\mathcal{N}_m$ are connected graphs with $m \geq 2$ nodes, $(m-2)$ of
which have valency $2$ and two of which have valency $1$ (these latter are the end nodes),
together with an orientation,
on the edges.
These graphs represent simpleces.
Graphs have vertices
labeled by positive integers and edges
are labeled alternatively by $1$ or $2$ as one moves along the graph, e.g.,
\begin{equation*}
\ \overset{k_{1}}{\bullet }\overset{1}{\longrightarrow }\overset{k_{2}}{ \bullet }\overset{2}{\longrightarrow }\overset{k_{3}}{\bullet }\overset{1}{ \longrightarrow }\overset{k_{4}}{\bullet }\overset{2}{\longrightarrow } \overset{k_{5}}{\bullet }\overset{1}{\longrightarrow }\overset{k_{6}}{ \bullet } \end{equation*}
\end{definition}
To complete definition of neighborhoods, one defines a weight function
\begin{equation*}
\omega :\mathcal{N}_m \longrightarrow
\mathbb{C}[E_{2}(\tau), E_{4}(\tau), E_{6} (\tau),\epsilon],
\end{equation*}
as follows: if a chequered necklace $\mathcal N_n$ has edges $E$ labeled as
$\overset {k}{\bullet }\overset{a}{\longrightarrow }\overset{l}{\bullet }$, $a=1$, $2$ then we
define
\begin{eqnarray}
\omega (E) &=&A(k,l,\tau, \epsilon ),
\notag
\\
\omega (\mathcal N_m) &=&\prod \omega (E), \label{wtedge} \end{eqnarray}
where $A(k,l,\tau,\epsilon)$ is given by
(\ref{Aki1}) and the
product is taken over all edges $E$ of $\mathcal N_m$.
Here $\tau$, $\epsilon$ are complex parameters.
We further define
$\omega(\mathcal N_{0})=1$.
Recall that $\left\{E_{2}(\tau), E_{4}(\tau), E_{6} (\tau)\right\}$
form a basis for the Eisenstein series.
After a normalization
the weight functions define a map from $\mathcal{N}_m$ to an $(n+2)$-dimensional ball in $\mathbb{C}$.
The metrix on $X(G, n)$ is given by non-degenerate bilinear pairing $(., .)$.
Since $\omega(\mathcal N)$ is a map from $\mathcal N$ to
$\mathbb{C}[E_{2}(\tau_{a}), E_{4}(\tau_{a}), E_{6} (\tau), \epsilon]$,
all the transition function of $X$ can be expressed via $\omega(\mathcal N)$.
Together with functions on graphs these simpleces are homeomorphic to an $n$-dimensional ball.
Now let us associate chequered necklace to elements of $G$ and form $X(G, n)$.
According to the construction of Section \ref{functional},
non-commutative coefficients of a formal Lie-algebraic series are elements of the algebraic completion $G$ of a
$\mathcal G$-module.
For each element $g_{j, k}$, $k \in \mathbb{C}$, $j \ge \mathbb{Z}$, associate a diagram \cite{MT} representing it as action of
generators on the union element.
The properties of non-degenerate bilinear pairing $(.,.)$ allow us to find appropriate diagram for
the element $\widetilde g_{j, k}$ of $\widetilde G$ dual to $g_{j, k}$.
Recall the chequered necklace construction for elements $g_{j, k}$ used in \cite{MT}.
Chequered necklace is in one to one correspondence with the formation of an $G$-element,
Associate a knot of such diagram to a point of $X(G, n)$.
Each point of the $G$-necklace is endowed with a power of $z_j$, $j \in \mathbb{Z}$.
Let us associate to a point on the $G$-necklace the
zero power of $z_j$ the zero point of a local domain $U_j$ ob $X(G, n)$.
The union $V(G, n)$ of all chequered necklaces together with local domains $U_j$ present
in the definition of the double complex constitutes the cells of a skeleton for $G$.
Now let us define the cohomology of the simplicial manifold $X(G, n)$.
In our setup, the
cohomology is an invariant
associating a graded ring with the manifold $X(G, n)$.
Every continuous map $f: X \to Y$ determines a homomorphism from the cohomology
ring of Y to that of X; this puts strong restrictions on the possible maps from X to Y.
Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be
computable in practice for spaces of interest.
Now we are able to define the chain-cochain complex as the space
$C^m_k(\omega_k(\mathcal N_m))= \left\{ \omega_k(\mathcal N_m)\right\}$,
of $\omega$-forms
associated to a vertex on $X(G, n)$ as simplicial complex.
The boundary operator defined as
$d_k(\omega_k(\mathcal N_{m+1}) ) = P_{\mathcal N}.\omega_{k}(\mathcal N_m)$,
where the operator $P_{\mathcal N}$ extends an $m$-vertex chequered necklace by one vertex.
Note that according to properties of meromorphic functions with prescribed behavior,
and the identification of $G$-elements with necklace elements,
the extension of a chequered necklace leads to the shift $k\to (k-1)$.
The cohomology of $X(G, n)$ is defined as the cohomology of the complex $C^m_k(\omega(\mathcal N_m))$.
\subsection{A counterpart of Bott-Segal theorem}
In this subsection we prove an analogue of Bott-Segal theorem \cite{BS}.
Being equipped with the technique of chequered necklaces, we prove the following
\begin{proposition}
\label{guga}
There exists a manifold $X(G, n)$ such that the cosimplicial cohomology of mero
functions for $\mathfrak g$
on a smooth complex manifold $M$
is equivalent to the
cohomology of $X(G, n)$, i.e.,
\begin{displaymath}
\label{quota}
H^*_{* \; cos}(G, \; \mathcal{U}
) \cong H^*_{*}(X(G, n)).
\end{displaymath}
\end{proposition}
\begin{proof}
The manifold $X(G, n)$ and
its cohomology $H^*_{* \; sing}(X(G, n))$ were constructed in previous subsections.
According to the construction of Section \ref{functional},
non-commutative coefficients of a formal Lie-algebraic series are elements of a $\mathcal G$-module $G$.
For each element $g_{j, k}$, $k \in \mathbb{C}$, $j \ge \mathbb{Z}$, we associate (by using the summation over matrix elements
described in corresponding subsection)
a diagram \cite{MT} representing it as action of
generators on the union element.
The properties of non-degenerate bilinear pairing $(.,.)$ allow us to find appropriate diagram for
the element $\widetilde g_{j, k}$ of $\widetilde G$ dual to $g_{j, k}$.
Recall the chequered necklace construction for elements $g_{j, k}$ used in \cite{MT}.
Associate a knot of such diagram to a point of $X(G, n)$.
Each point of the $G$-necklace is endowed with a power of $z_j$, $j \in \mathbb{Z}$.
Let us associate to a point on the $G$-necklace the
zero power of $z_j$ the zero point of a local domain $U_j$ ob $X(G, n)$.
The union $V(G, n)$ of all chequered necklaces together with local domains $U_j$ present
in the definition of the double complex constitutes the cells of a skeleton for $G$.
Thus, we obtain an analog of a $2n$-skeleton for $W_n$ of formal vector fields.
In contrast to \cite{Fuk, FF1988} it is endowed with a power of $j$-th formal parameter $z_j$.
We define a map
$\pi: V(G,n) \to G(\infty,n)$,
from the the $G$-skeleton
to an infinite Grassmanian $G(\infty, n)$.
Since the inverse image of the union of the cells is not a manifold,
we consider
an open neighborhood of the inverse image under $\pi$ of the $G$-skeleton of the Grassmannian
$G(\infty, n)$.
The union of such open neighborhoods constitutes the manifold $X(G, p)$.
This is the cohomology of double Lie-algebraic complexes $C^l_k(\Theta(n,k) )$ which is the union of complexes
$C^l_k(\Theta(1, k) )$ for each local coordinate and $g_{jk}$-generators.
It coincides with
the cosimplicial cohomology $H^*_{*\; cos}( {\mathcal G}({\bf z}), \mathcal{U})$ of $W_n$ of $n$ complex
variables.
\end{proof}
\begin{remark}
It is also analogue of cohomology equivalence of $H^*_*(G, \mathcal{U})$ and cohomology of a groupoid
shown in \cite{CM}.
\end{remark}
Note that another way to prove Proposition \ref{guga} is to use the same technique as in
\cite{Fuk, FF1988} since
for the complex $C^l_k({\mathcal G}({\bf z}), \mathcal{U})$ there exist converging spectral sequences.
i.e., to show that
an isomorphism
of the Hochschild-Serre spectral sequence \cite{Ho} for the subalgebra $gl(n)$
with the Leray spectral of the restriction to the $2n$ skeleton of
the universal $U(n)$ principal bundle.
\section{Conclusions}
In this section we list multiple
applications of the research of this paper are in conformal field theory
\cite{Fei, wag, BZF, BPZ, DiMaSe, TUY}, in deformation theory \cite{BG, HinSch}, and in the theory of foliations \cite{Bott}.
\subsection{Applications in conformal field theory and moduli spaces}
In \cite{Fei, BeiFeiMaz}
applications of cosimplicial computations on compact Riemann surfaces in
conformal field \cite{DiMaSe, BZF, BPZ} theory were treated.
As we deal with special homology, we replace the sheaf of
holomorphic vector fields
by the sheaf of meromorhic functions associated to corresponding Lie algebra.
In \cite{Fei}, for Riemann surface $\Sigma^{(g)}$,
Feigin calculated the cosimplicial homology of
$Lie(\Sigma^{(g)})$ with values in the
representations mentioned in Introduction.
It is possible to compute cosimplicial homology of a space of meromorphic function complexes
associated to various Lie algebras.
The space of coinvariants on the right hand side
defining so-called modular functor is usually associated to
locally defined objects.
We will
obtain
its homological description in terms of
globally defined objects.
The space of coinvariants supposed to be
the continuous
dual to the local ring completion of the moduli space of compact Riemann surfaces of genus $g\geq 2$ at the point $\Sigma^{(g)}$,
provided that $\Sigma^{(g)}$ is a smooth point.
This gives an important link between
Lie algebra homology and the geometry of the moduli space.
\subsection{Applications in deformation theory}
{\it Deformations of complex manifolds.}
Cosimplician considerations above are applicable to
cohomology computations
in the
deformation theory of complex manifolds
\cite{Ma, Fei, HinSch, GerSch}.
The completion of a local ring of moduli space at a given point $M$
is isomorphic to the dual of the Lie algebra of $M$-infinitesimal automorphisms zero-th homology group.
This links Lie algebra homology and
geometry of the moduli space in a formal neighborhood of a point.
We expect results in this direction for higher dimensional complex manifolds.
In \cite{wag} we find the condition for the first cohomology in the case of
higher dimensional complex manifolds $M$.
For restricted function cohomology one can consider also related the deformation
theory following Kodaira and Spencer \cite{Kod}.
{\it Deformations of Lie algebras.}
It is well known that the Lie algebra cohomology with values in the adjoint representation $H^*(L,L)$ of a Lie algebra
$L$ answers questions about deformations of $L$ as an algebraic object.
For example, $H^2(L,L)$ can be interpreted as the space of equivalence classes of infinitesimal deformations of $L$, see \cite{Fuk, FF1988}.
There arise natural questions of this type for
bi-graded differential Lie algebras resulting from chain complex constructions.
For a disk
$D\subset\mathbb{C}^n$, holomorphic vector fields
are rigid, i.e.,
\cite{wag}
$H^*_{cont}(Hol(D),Hol(D)) = 0$.
Using cosimplicial cohomology results, we will study rigidity of
bi-graded differential Lie algebras resulting from chain complex constructions.
For a
compact Riemann surface $\Sigma^{(g)}$
of genus $g\geq 2$, we expect to find a relations for cohomologies in terms of
elements of Fr\'echet spaces
given by the polynomials on $T_{\Sigma^{(g)}}{\mathcal N}(g,0)$.
It's the space of formal power series on $T_{\Sigma^{(g)}}{\mathcal N}(g,0)^*$.
This could be interpreted as a relation between cohomology with
adjoint coefficients of $\mathfrak{g}$,
i.e., graded differential deformations of global sections of $\mathfrak{g}$, and deformations of the underlying manifold.
As it is explained in \cite{Fei, wag},
the choice of the coefficients in the Lie algebra cohomology determines a geometric object on the moduli space in a formal neighborhood of a point.
Namely, trivial coefficients correspond to the structure sheaf, adjoint coefficients correspond to vector fields, adjoint coefficients in the universal enveloping algebra correspond to differential operators.
\subsection{Applications in foliation theory.}
Applications in foliation theory are inspired by the link between cohomology of Lie algebras and characteristic classes of foliations
\cite{Fuk, FF1988}.
In \cite{wag} the author considered the
case of characteristic classes of $g$-structures.
For a complex manifold $M$, and ${\mathcal U} = \{U_i\}_{i\in I}$ a covering of $M$ by open sets such that $I$ is a countable directed index set,
consider the sheave
of meromorphic functions in cosimplicial setup given above.
Denote by $W_{2n}|_{hol}$ the Lie subalgebra of $W_{2n}$ generated by the $\frac{\partial}{\partial z_i}$ for $i=1,\ldots,n$.
Given a
structure of meromorphic functions associated to such a covering,
we obtain
that
the space of moduli of
$\{\omega_U\}_{U\in{\mathcal U}}$ is isomorphic
to the moduli space of
$W_{2n}|_{hol}$-valued differential forms $\omega$.
To such a structure
we may assign as in \cite{wag}
characteristic classes by considering
$H^*(|C^*_{cont}( {C}({\mathcal U}, \ensuremath{\mathcal{F}}
))|)$.
The cosimplicial meromorphic function
structure is defined
such that by inserting $p$-times
$\chi _{U_{i_0}\cap\ldots\cap U_{i_q}}$ into each
$c\in C^p_{cont}(\prod_{i_0<\ldots<i_q} \ensuremath{\mathcal{F}} (U_{i_0}\cap\ldots\cap U_{i_q}))$,
one associates
an element $\chi$
of the generalized ${\rm C}$ech-de Rham complex
associated to the covering ${\mathcal U}$ on $M$.
By the standard reasoning
this $\chi$
provide
a well-defined cohomology class $[ \chi ]$,
the characteristic class associated to the cosimplicial $\ensuremath{\mathcal{F}}$-structure.
Finally, a relation to factorization algebras \cite{HK} will be cleared elsewhere.
\section*{Acknowledgments}
The author would like to thank H. V. L\^e, A. Lytchak, P. Somberg, and P. Zusmanovich for related discussions.
\end{document} | arXiv |
\begin{document}
\title{Approximate Pure Nash Equilibria in Weighted Congestion Games: Existence, Efficient Computation, and Structure hanks{This work was partially supported by the grant NRF-RF2009-08 ``Algorithmic aspects of coalitional games'' and the EC-funded STREP Project FP7-ICT-258307 EULER.}
\begin{abstract} We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by potential function arguments. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria is such games is a {\sf PLS}-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. Do such equilibria exist? And if so, can we compute them efficiently?
We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an ``approximation'' of such games by a new class of potential games that we call $\Psi$-games. This allows us to show that these games have $d!$-approximate pure Nash equilibria, where $d$ is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing $O(1)$-approximate pure Nash equilibria when $d$ is a constant. For games with linear latency functions, the approximation guarantee is $\frac{3+\sqrt{5}}{2}+O(\gamma)$ for arbitrarily small $\gamma>0$; for latency functions with maximum degree $d\geq 2$, it is $d^{2d+o(d)}$. The running time is polynomial in the number of bits in the representation of the game and $1/\gamma$. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree $d\geq 2$: polynomially-long sequences of best-response moves from any initial state to a $d^{O(d^2)}$-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as $d$ is constant.
To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of $\Psi$-games. The concept of approximating non-potential games by potential ones seems to be novel and might have further applications. \end{abstract}
\section{Introduction} \label{sec:intro} Among other solution concepts, the notion of the pure Nash equilibrium plays a central role in Game Theory. Pure Nash equilibria in a game characterize situations with non-cooperative deterministic players in which no player has any incentive to unilaterally deviate from the current situation in order to achieve a higher payoff. Unfortunately, it is well known that there are games that do not have pure Nash equilibria. Furhermore, even in games where the existence of equilibria is guaranteed, their computation can be a computationally hard task. Such negative results significantly question the importance of pure Nash equilibria as solution concepts that characterize the behavior of rational players.
Approximate pure Nash equilibria, which characterize situations where no player can {\em significantly improve} her payoff by unilaterally deviating from her current strategy, could serve as alternative solution concepts\footnote{Actually, approximate pure Nash equilibria may be more desirable as solution concepts in practical decision making settings since they can accommodate small modeling inaccuracies due to uncertainty (e.g., see the arguments in \cite{CGC04}).} provided that they exist and can be computed efficiently. In this paper, we present the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our main contribution is a polynomial-time algorithm that computes $O(1)$-approximate pure Nash equilibria under mild restrictions on the game parameters; these restrictions apply to important subclasses of games in which not even the existence of such approximate equilibria was known prior to our work.
\noindent{\bf Problem statement and related work.} In a weighted congestion game, players compete over a set of resources. Each player has a positive weight. Each resource incurs a latency to all players that use it; this latency depends on the total weight of the players that use the resource according to a resource-specific, non-negative, and non-decreasing latency function. Among a given set of strategies (over sets of resources), each player aims to select one selfishly, trying to minimize her individual total cost, i.e., the sum of the latencies on the resources in her strategy. Typical examples include weighted congestion games in networks, where the network links correspond to the resources and each player has alternative paths that connect two nodes as strategies.
The case of unweighted congestion games (i.e., when all players have unit weight) has been widely studied in literature. Rosenthal \cite{R73} proved that these games admit a potential function with the following remarkable property: the difference in the potential value between two states (i.e., two snapshots of strategies) that differ in the strategy of a single player equals to the difference of the cost experienced by this player in these two states. This immediately implies the existence of a pure Nash equilibrium. Any sequence of improvement moves by the players strictly decreases the value of the potential and a state corresponding to a local minimum of the potential will eventually be reached; this corresponds to a pure Nash equilibrium. For weighted congestion games, potential functions exist only in the case where the latency functions are linear or exponential (see \cite{FKS05,HK10,PS06}). Actually, in games with polynomial latency functions (of constant maximum degree higher than $1$), pure Nash equilibria may not exist \cite{HK10}. In general, the problem of deciding whether a given weighted congestion game has a pure Nash equilibrium is {\sf NP}-hard \cite{DS08}.
Potential functions provide only inefficient proofs of existence of pure Nash equilibria. Fabrikant et al. \cite{FabrikantPT04} proved that the problem of computing a pure Nash equilibrium in a (unweighted) congestion game is {\sf PLS}-complete (informally, as hard as it could be given that there is an associated potential function; see \cite{Johnson88}). This negative result holds even in the case of linear latency functions \cite{AckermannRV08}. One consequence of {\sf PLS}-completeness results is that almost all states in some congestion games are such that any sequence of players' improvement moves that originates from these states and reaches pure Nash equilibria is exponentially long. Such phenomena have been observed even in very simple weighted congestion games (see \cite{ARV09,EKM07}). Efficient algorithms are known only for special cases. For example, Fabrikant et al. \cite{FabrikantPT04} show that the Rosenthal's potential function can be (globally) minimized efficiently by a flow computation in unweighted congestion games in networks when the strategy sets of the players are symmetric.
The above negative results have led to the study of the complexity of approximate pure Nash equilibria (or, simply, approximate equilibria). A $\rho$-approximate (pure Nash) equilibrium is a state, from which no player has an incentive to deviate so that she decreases her cost by a factor larger than $\rho$. In our recent work \cite{CFGS11}, we present an algorithm for computing $O(1)$-approximate equilibria for unweighted congestion games with polynomial latency functions of constant maximum degree. The restriction on the latency functions is necessary since, for more general latency functions, Skopalik and V\"ocking \cite{SkopalikV08} show that the problem is still {\sf PLS}-complete for any polynomially computable $\rho$ (see also the discussion in \cite{CFGS11}). Improved bounds are known for special cases. For symmetric unweighted congestion games, Chien and Sinclair \cite{ChienS07} prove that the $(1+\epsilon)$-improvement dynamics converges to a $(1 + \epsilon$)-approximate equilibrium after a polynomial number of steps; this result holds under mild assumptions on the latency functions and the participation of the players in the dynamics. Efficient algorithms for approximate equilibria have been recently obtained for other classes of games such as constraint satisfaction \cite{BhalgatCK10,NT09}, anonymous games \cite{DP07}, network formation \cite{AC09}, and facility location games \cite{CH10}.
In light of the negative results mentioned above, several authors have considered other properties of the dynamics of congestion games. The papers \cite{AwerbuchAEMS08,FM09, GMV05} consider the question of whether efficient states (in the sense that the total cost of the players, or social cost, is small compared to the optimum one) can be reached by best-response moves in linear weighted congestion games. In particular, Awerbuch et al. \cite{AwerbuchAEMS08} show that using almost unrestricted sequences of ($1+\epsilon$)-improvement best-response moves, the players rapidly converge to efficient states. Unfortunately, these states are not approximate equilibria, in general. Similar approaches have been followed in the context of other games as well, such as multicast \cite{CKM+08,CCL+07}, cut \cite{CMS06}, and valid-utility games \cite{MV04}.
\noindent{\bf Our contribution.} To the best of our knowledge, no efficient algorithm for computing approximate equilibria is known for (any broad enough subclass of) weighted congestion games. We fill this gap by presenting an algorithm for computing $O(1)$-approximate equilibria in weighted congestion games with polynomial latency functions of constant maximum degree. For games with linear latency functions, the approximation guarantee is $\frac{3+\sqrt{5}}{2}+O(\gamma)$ for arbitrarily small $\gamma>0$; for latency functions of maximum degree $d\geq 2$, it is $d^{2d+o(d)}$. The algorithm runs in time that is polynomial in the number of bits in the representation of the game and $1/\gamma$.
This result is much more surprising than it looks at first glance. In particular, weighted congestion games with superlinear latency functions do not admit potential functions, the main tool that is exploited by all known positive algorithmic results for (approximate) equilibria in congestion games. Given this, it is not even clear that $O(1)$-approximate equilibria exist. In order to bypass this obstacle, we introduce a new class of potential games (that we call $\Psi$-games), which ``approximate'' weighted congestion games with polynomial latency functions in the following sense. $\Psi$-games of degree $1$ are linear weighted congestion games. Each weighted congestion game of degree $d\geq 2$ has a corresponding $\Psi$-game of degree $d$ defined in such a way that any $\rho$-approximate equilibrium in the latter is a $d!\rho$-approximate equilibrium for the former. As an intermediate new result, we obtain that weighted congestion games with polynomial latency functions of degree $d$ have $d!$-approximate equilibria.
So, our algorithm is actually applied to $\Psi$-games. It has a simple general structure, similar to our recent algorithm for unweighted congestion games \cite{CFGS11}, but has also important differences that are due to the dependency of the cost of each player on the weights of other players. Given a $\Psi$-game of degree $d$ and an arbitrary initial state, the algorithm computes a sequence of best-response player moves of length that is bounded by a polynomial in the number of bits in the representation of the game and $1/\gamma$. The sequence consists of phases so that the players that participate in each phase experience costs that are polynomially related. This is crucial in order to obtain convergence in polynomial time. Within each phase, the algorithm coordinates the best-response moves according to two different but simple criteria; this is the main tool that guarantees that the effect of a phase to previous ones is negligible and, eventually, an approximate equilibrium is reached. The approximation guarantee is slightly higher than a quantity that characterizes the potential functions of $\Psi$-games; this quantity (which we call the {\em stretch}) is defined as the worst-case ratio of the potential value at an almost exact pure Nash equilibrium over the globally optimum potential value and is almost $\frac{3+\sqrt{5}}{2}$ for linear weighted congestion games and $d^{d+o(d)}$ for $\Psi$-games of degree $d\geq 2$. Our analysis follows the same main steps as in our recent paper \cite{CFGS11} but uses significantly more involved arguments due to the definition of $\Psi$-games.
We also present a similar but slightly inferior algorithm that is applied directly to weighted congestion games of maximum degree $d\geq 2$ and reveals a rather surprising structural property of their Nash dynamics: starting from any initial state, the algorithm identifies a polynomially-long sequence of best-response moves that lead to a $d^{O(d^2)}$-approximate equilibrium. Even though the definition of this algorithm does not make any use of properties of $\Psi$-games, the analysis is heavily based on them, similarly to the analysis of our main algorithm.
We remark that, following the classical definition of polynomial latency functions in the literature, we assume that they have non-negative coefficients. This is a necessary limitation since the problem of computing a $\rho$-approximate equilibrium in (unweighted) congestion games with linear latency functions with negative offsets is {\sf PLS}-complete for any polynomial-time computable $\rho\geq 1$ \cite{CFGS11}.
\noindent{\bf Roadmap.} We begin with preliminary general definitions in Section \ref{sec:prelim}. Section \ref{sec:psi-games} is devoted to $\Psi$-games and their properties. We present our algorithm and its analysis in Section \ref{sec:algo} and conclude with open problems in Section \ref{sec:open}. Due to lack of space, most of the proofs as well as our structural result appears in Appendix.
\section{Definitions and preliminaries}\label{sec:prelim} In general, a \emph{game} can be defined as follows. It has a set of $n$ players ${\cal N}$; each player $u\in {\cal N}$ has a set of available strategies $\Sigma_u$. A snapshot of strategies, with one strategy per player, is called a {\em state}. Each state $S\in \prod_{u\in {\cal N}}{\Sigma_u}$ incurs a positive cost $c_u(S)$ to player $u$. Players act selfishly; each of them aims to select a strategy that minimizes her cost, given the strategies of the other players. Given a state $S$ and a strategy $s'_u\in \Sigma_u$ for player $u$, we denote by $(S_{-u},s'_u)$ the state obtained from $S$ when player $u$ {\em deviates} to strategy $s'_u$. For a state $S$, an {\em improvement move} (or, simply, a {\em move}) for player $u$ is the deviation to any strategy $s'_u$ that (strictly) decreases her cost, i.e., $c_u(S_{-u},s'_u) < c_u(S)$. For $\rho\geq 1$, such a move is called a $\rho$-{\em move} if it satisfies $c_u(S_{-u},s'_u) < \frac{c_u(S)}{\rho}$. A {\em best-response move} is a move that minimizes the cost of the player (of course, given the strategies of the other players). So, from state $S$, a move of player $u$ to strategy $s_u$ is a best-response move (and is denoted by ${\mathcal{BR}}_u(S)$) when $c_u(S_{-u},s'_u) = \min_{s\in \Sigma_u}c_u(S_{-u},s)$. A state $S$ is called a {\em pure Nash equilibrium} (or, simply, an {\em equilibrium}) when $c_u(S)\leq c_u(S_{-u},s'_u)$ for every player $u\in {\cal N}$ and every strategy $s'_u\in \Sigma_u$, i.e., when no player has a move. In this case, we say that no player has (any incentive to make) a move. Similarly, a state is called a $\rho$-{\em approximate pure Nash equilibrium} (henceforth called, simply, a $\rho$-{\em approximate equilibrium}) when no player has a $\rho$-move. Also, a state is called a $\rho$-approximate equilibrium for a subset of players $A\subseteq {\cal N}$ if no player in $A$ has a $\rho$-move. We use the term {\em Nash dynamics} of a game in order to refer to the directed graph with nodes that correspond to the possible states of the game and directed edges that indicate improvement player moves; pure Nash equilibria correspond to sinks of the Nash dynamics.
A \emph{weighted congestion game} ${\cal G}$ can be represented by the tuple $\left(N, E, (w_u)_{u\in {\cal N}}, (\Sigma_u)_{u \in {\cal N}}, (f_e)_{e \in E}\right)$. There is a set of $n$ {\em players} ${\cal N}$ and a set of {\em resources} $E$. Each player $u$ has a positive weight $w_u$ and a set of available {\em strategies} $\Sigma_u$; each strategy $s_u$ in $\Sigma_u$ consists of a non-empty set of resources, i.e., $s_u\subseteq 2^E$. Each resource $e\in E$ has a non-negative and non-decreasing {\em latency function} $f_e$ defined over non-negative reals, which denotes the latency incurred to the players using resource $e$; this latency depends on the total weight of players whose strategies include the particular resource. For a state $S$, let us define $N_e(S)$ to be the multi-set of the weights of the players that use resource $e$ in $S$, i.e., $N_e(S)=\{w_u: u\in {\cal N} \mbox{ such that } e\in s_u\}$. Also, we use the notation $L(A)$ to denote the sum of the elements of a finite multi-set of reals $A$. Then, the latency incurred by resource $e$ to a player $u$ that uses it is $f_e(L(N_e(S)))$. The {\em cost} of a player $u$ at a state $S$ is the total latency she experiences at the resources in her strategy $s_u$ multiplied by her weight, i.e., $c_u(S)=w_u\sum_{e\in s_u}{f_e(L(N_e(S)))}$. We consider weighted congestion games in which the resources have polynomial latency functions with (integer) maximum degree $d\geq 1$ with non-negative coefficients. More precisely, the latency function of resource $e$ is $f_e(x) = \sum_{k=0}^d{a_{e,k}x^k}$ with $a_{e,k}\geq 0$. The special case of linear weighted congestion games (i.e., with latency functions of degree $1$) is of particular interest. In general, the size of the representation of a weighted congestion game is the number of bits required to represent the parameters $a_{e,k}$ of the latency functions, the weights of the players, and their strategy sets. In weighted congestion games in networks, the network links are the resources. Each player $u$ aims to connect a pair of nodes $(s_u,t_u)$ and her strategies are all paths connecting $s_u$ with $t_u$ in the network. Note that the representation of such games does not need to keep the whole set of strategies explicitly; it just has to represent the parameters $a_{e,k}$, the weight and the source-destination node pair of each player, and the network.
Unweighted congestion games (i.e., when $w_u=1$ for each player $u\in{\cal N}$) as well as linear weighted congestion games are potential games. They admit a {\em potential function} $\Phi:\prod_u{\Sigma_u}\mapsto {\mathbb{R}}^+$, defined over all states of the game, with the following property: for any two states $S$ and $(S_{-u},s'_u)$ that differ only in the strategy of player $u$, it holds that $\Phi(S_{-u},s'_u) - \Phi(S) = c_u(S_{-u},s'_u)-c_u(S)$. Clearly, the local minima of the potential function correspond to states that are pure Nash equilibria. The existence of a potential function also implies that the Nash dynamics of the corresponding game is acyclic. Potential functions for the two classes of games mentioned above have been presented by Rosenthal \cite{R73} and Fotakis et al. \cite{FKS05}, respectively. Unfortunately, weighted congestion games with polynomial latency functions of degree at least $2$ are not potential games and may not even have pure Nash equilibria \cite{HK10}.
\section{$\Psi$-games}\label{sec:psi-games} Our aim in this section is to define a new class of games which we call $\Psi$-games and study their properties. We will need the following interesting family of functions which have also been used in \cite{C09} in a slightly different context.
\begin{dfn} For integer $k\geq 0$, the function $\Psi_k$ mapping finite multi-sets of reals to the reals is defined as follows: $\Psi_k(\emptyset)=0$ for any integer $k\geq 1$, $\Psi_0(A)=1$ for any (possibly empty) multi-set $A$, and for any non-empty multi-set $A=\{\alpha_1, \alpha_2, ..., \alpha_\ell\}$ and integer $k\geq 1$, \[\Psi_k(A)=k! \sum_{1\leq d_1 \leq ... \leq d_k \leq \ell}{\prod_{t=1}^{k}{\alpha_{d_t}}}.\] \end{dfn} So, $\Psi_k(A)$ is essentially the sum of all monomials of total degree $k$ on the elements of $A$. Each term in the sum has coefficient $k!$. Clearly, $\Psi_1(A)=L(A)$. For $k\geq 2$, compare $\Psi_k(A)$ with $L(A)^k$ which can also be expressed as the sum of the same terms, albeit with different coefficients in $\{1, ..., k!\}$, given by the multinomial theorem.
We are ready to define $\Psi$-games. A $\Psi$-game ${\cal G}$ of (integer) degree $d\geq 1$ can be represented by the tuple $({\cal N},E,(w_u)_{u\in {\cal N}},(\Sigma_u)_{u\in {\cal N}},(a_{e,k})_{e\in E, k=0, 1, ..., d})$. Similarly to weighted congestion games, there is a set of $n$ players ${\cal N}$ and a set of resources $E$. Each player $u$ has a weight $w_u$ and a set of available strategies $\Sigma_u$; each strategy $s_u \in\Sigma_u$ consists of a non-empty set of resources, i.e., $s_u\subseteq 2^E$. Each resource $e$ is associated with $d+1$ non-negative numbers $a_{e,k}$ for $k=0, 1, ..., d$. Again, for a state $S$, we define $N_e(S)$ to be the multi-set of weights of the players that use resource $e$ at state $S$. Then, the cost of a player $u$ at a state $S$ is defined as $$\hat{c}_u(S) = w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}\Psi_k(N_e(S))}}.$$ Of course, the general definitions in the beginning of Section \ref{sec:prelim} apply also to $\Psi$-games. With some abuse in notation, we also use $\mathbf{0}$ to refer to the pseudo-state in which no player selects any strategy and ${\mathcal{BR}}_u(\mathbf{0})$ to denote the best-response of player $u$ assuming that no other player participates in the game.
Clearly, given a weighted congestion game with polynomial latency functions of maximum degree $d$, there is a corresponding $\Psi$-game with degree $d$, i.e., the one with the same sets of players, resources, and strategy sets, and parameter $a_{e,k}$ for each resource $e$ and integer $k=0, 1, ..., d$ equal to the corresponding coefficient of the latency function $f_e$. Observe that $\Psi$-games of degree $1$ are linear weighted congestion games. As we will see below, in a sense, a $\Psi$-game of degree $d\geq 2$ is an approximation of its corresponding weighted congestion game.
We remark here that a different approximation of weighted congestion games has been recently considered by Kollias and Roughgarden \cite{KR11}. Given a weighted congestion game, they define a new game by answering the following question: how should the product of the total weight of the players that use the resource times its latency be shared as cost among these players so that the resulting game is a potential game? Their games use a different sharing than the weight-proportional one used by weighted congestion games. In contrast, our approach is to define an artificial latency on each resource $e$ (by replacing the term $a_{e,k}L(N_e(S))^k$ with $a_{e,k}\Psi_k(N_e(S))$ in the latency functions) so that weight-proportional sharing yields a potential game. This guarantees the relation between approximate equilibria in weighted congestion games and $\Psi$-games presented in Lemma \ref{lem:approx} below, which is crucial for our purposes.
\noindent{\bf Properties of $\Psi$-games.} We begin with a very important property of $\Psi$-games. \begin{theorem}\label{thm:psi-potential} The function $\Phi(S) = \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e(S))}}$ is a potential function for $\Psi$-games of degree $d$. \end{theorem}
As a corollary, we conclude that the Nash dynamics of $\Psi$-games are acyclic; hence, these games admit pure Nash equilibria. Recall that $\Psi$-games of degree $1$ are linear weighted congestion games; for this specific case, Theorem \ref{thm:psi-potential} has been proved in \cite{FKS05}.
In the following, we study the relation between the approximation guarantee of a state for a $\Psi$-game and its corresponding weighted congestion game with polynomial latency functions.
\begin{claim}\label{claim:approx} Consider a weighted congestion game with polynomial latency functions of degree $d$ and its corresponding $\Psi$-game. Then, for each player $u$ and state $S$, $c_u(S)\leq \hat{c}_u(S)\leq d! c_u(S)$. \end{claim}
Using Claim \ref{claim:approx}, we can obtain a relation between approximate equilibria as well. \begin{lemma}\label{lem:approx} Any $\rho$-approximate pure Nash equilibrium for a $\Psi$-game of degree $d$ is a $d!\rho$-approximate pure Nash equilibrium for the corresponding weighted congestion game with polynomial latencies. \end{lemma}
Since pure Nash equilibria always exist in $\Psi$-games, the last statement (applied with $\rho=1$) implies the following.
\begin{theorem} Every weighted congestion game with polynomial latency functions of maximum degree $d$ has a $d!$-approximate pure Nash equilibrium. \end{theorem}
\noindent{\bf Subgames and partial potentials.} We now define restrictions of the potential function of $\Psi$-games. Given a state $S$ and a set of players $A\subseteq {\cal N}$, we denote by $N_e^A(S)$ the multiset of the weights of players in $A$ that use resource $e$ in $S$. Then, we define $$\Phi^A(S) = \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N^A_e(S))}}.$$ We can think of $\Phi^A$ as the potential of a subgame in which only the players of $A$ participate.
We also use the notion of the {\em partial} potential to account for the contribution of subsets of players to the potential function. Consider sets of players $A$ and $B$ with $B\subseteq A\subseteq {\cal N}$. Then, the $B$-partial potential of the subgame among the players in $A$ is defined as $$\Phi_B^A(S) = \Phi^A(S) - \Phi^{A\setminus B}(S).$$ When $A={\cal N}$, we remove the superscript from partial potentials, i.e., $\Phi_B(S) = \Phi_B^{\cal N}(S)$. Also, when $B$ is a singleton containing player $u$, we simplify the notation of the partial potential to $\Phi_u^A(S)$. Furthermore, observe that $\Phi^A_A(S)=\Phi^A(S)$.
The next four claims present basic properties of partial potentials. \begin{claim}\label{claim:partial-potential-bound} Let $S$ be a state of a $\Psi$-game and let $B\subseteq A\subseteq {\cal N}$. Then, $\Phi_B^A(S) \leq \Phi_B(S)$. \end{claim}
\begin{claim}\label{claim:equal-partial-potential} Let $A\subseteq {\cal N}$ be a set of players and let $S$ and $S'$ be states such that each player in $A$ uses the same strategy in $S$ and $S'$. Then, for every set of players $B\subseteq A$, $\Phi^A_B(S)=\Phi^A_B(S')$. \end{claim}
\begin{claim}\label{claim:u-partial-potential} Let $S$ be a state of a $\Psi$-game and let $u$ be a player. Then, $\Phi_u(S) = \hat{c}_u(S)$. \end{claim}
\begin{claim}\label{claim:partial-potential} Let $u$ be a player and $A\subseteq {\cal N}$ a set of players that contains $u$. Then, for any two states $S$ and $S'$ that differ only in the strategy of player $u$, it holds that $\Phi_A(S)-\Phi_A(S') = \hat{c}_u(S)-\hat{c}_u(S')$. \end{claim} In particular, Claim \ref{claim:partial-potential} implies that the $A$-partial potential can be thought of as a potential function defined over all states in which each player in ${\cal N}\setminus A$ uses the same strategy.
We proceed with the following interesting property that shows that the potential function of $\Psi$-games is cost-revealing. It also implies that the potential of a state lower-bounds the total cost of all players. \begin{lemma}\label{lem:cost-vs-potential-d} For every state $S$ of a $\Psi$-game and any set of players $A\subseteq {\cal N}$, it holds that $\Phi_A(S)\leq \sum_{u\in A}{\hat{c}_u(S)}$. \end{lemma}
\noindent{\bf The stretch of the potential function.} An important quantity for our purposes is the {\em stretch} of the potential function of $\Psi$-games; a general definition that applies to every potential game follows. \begin{dfn} Consider a potential game with a positive potential function $\Phi$ and let $S^*$ be the state of minimum potential. The $\rho$-stretch of the potential function of the game is the maximum over all $\rho$-approximate pure Nash equilibria $S$ of the ratio $\Phi(S)/\Phi(S^*)$. \end{dfn}
The next two statements provide bounds on the $\rho$-stretch of the potential function of $\Psi$-games of degree $1$ (i.e., linear weighted congestion games) and $d\geq 2$, respectively.
\begin{lemma}\label{lem:stretch-linear} For every $\rho\in [1,11/10]$, the $\rho$-stretch of the potential function of a linear weighted congestion game is at most $\frac{3+\sqrt{5}}{2} +6(\rho-1)$. \end{lemma}
\begin{lemma}\label{lem:stretch-d} The $\rho$-stretch of the potential function of a $\Psi$-game of degree $d\geq 2$ is at most $\rho(\rho+1)^d(d+1)^{d+1}$. \end{lemma}
In the rest of the paper, we denote by $\theta_d(\rho)$ the upper bounds on the $\rho$-stretch given by Lemmas \ref{lem:stretch-linear} and \ref{lem:stretch-d}, namely $\theta_1(\rho)=\frac{3+\sqrt{5}}{2}+6(\rho-1)$ and $\theta_d(\rho) = \rho(\rho+1)^d(d+1)^{d+1}$. The next lemma extends these bounds to partial potentials.
\begin{lemma}\label{lem:stretch-partial} Consider a $\Psi$-game of degree $d$ and a state $S$ which is a $\rho$-approximate pure Nash equilibrium for a set of players $R\subseteq {\cal N}$. Then, $\Phi_R(S) \leq \theta_d(\rho) \Phi_R(S^*)$ for any state $S^*$ such that each player in ${\cal N}\setminus R$ uses the same strategy in $S$ and $S^*$. \end{lemma}
\section{The algorithm}\label{sec:algo} In this section we describe our algorithm (Algorithm 1; see the table below). The algorithm takes as input a $\Psi$-game ${\cal G}$ of degree $d$ with $n$ players, an arbitrary initial state $S$ of the game, and a small positive parameter $\gamma$. It produces as output a state of ${\cal G}$. The algorithm starts by initializing its parameters, namely $\hat{c}_{\max}$, $\hat{c}_{\min}$, $m$, $g$, $q$, and $p$ (lines 1-6). It first computes the minimum possible cost $\hat{c}_{\min}$ among all players and the maximum cost $\hat{c}_{\max}$ experienced by players in the initial state $S$. Then, it sets the parameter $m$ equal to $\log{\left(\hat{c}_{\max}/\hat{c}_{\min}\right)}$; in this way, $m$ is polynomial in the number of bits in the representation of the game (i.e., polynomial in the number of bits necessary to store the parameters $a_{e,k}$ and the weights of the players). Then, the parameter $q$ is set close to $1$ (namely, $q=1+\gamma$) and parameter $p$ is set close to $\theta_d(q)$ (namely, $p=\left(\frac{1}{\theta_d(q)} - 2\gamma\right)^{-1}$). Recall that $\theta_d(q)$ is the bound on the $q$-stretch of the potential function of $\Psi$-games of degree $d$ in the statements of Lemmas \ref{lem:stretch-linear} (for $d=1$) and \ref{lem:stretch-d} (for $d\geq 2$).
\IncMargin{3em} \RestyleAlgo{boxed} \LinesNumbered \begin{algorithm} \SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up} \SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress} \SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{A $\Psi$-game ${\cal G}$ of degree $d$ with a set ${\cal N}$ of $n$ players, an arbitrary initial state $S$, and $\gamma>0$ with $\gamma\in (0,1/10]$ if $d=1$ and $\gamma\in (0,\frac{1}{8\theta_d(2)}]$, otherwise} \Output{A state of ${\cal G}$}
$\hat{c}_{\min}\leftarrow\min_{u\in {\cal N}}{\hat{c}_u(\mathbf{0}_{-u},{\mathcal{BR}}_u(\mathbf{0}))}$\;
$\hat{c}_{\max}\leftarrow \max_{u\in {\cal N}}{\hat{c}_u(S)}$\;
$m\leftarrow\log{\left(\hat{c}_{\max}/\hat{c}_{\min}\right)}$\;
$g\leftarrow2\left(1+m(1+\gamma^{-1})\right)^{d}d^{d}n\gamma^{-3}$\;
$q\leftarrow1+\gamma$\;
$p \leftarrow \left(\frac{1}{\theta_d(q)} - 2\gamma\right)^{-1}$\; \label{alg:step2}
\lFor{$i\leftarrow 0$ \KwTo $m$ \label{a}}{$b_i \leftarrow \hat{c}_{\max}g^{-i}$\;} \label{alg:step3}
\While{there exists a player $u\in {\cal N}$ such that $\hat{c}_u(S)\in [b_1,+\infty)$ and $\hat{c}_u(S_{-u},{\mathcal{BR}}_u(S))<\hat{c}_u(S)/q$}
{
$S\leftarrow (S_{-u},{\mathcal{BR}}_u(S))$\;
}
$F \leftarrow \emptyset$\; \label{alg:step1}
\For{phase $i\leftarrow 1$ \KwTo $m-1$ \label{main}}{
\While{there exists a player $u\in {\cal N}\setminus F$ such that either $\hat{c}_u(S) \in [b_{i}, +\infty)$ and $\hat{c}_u(S_{-u},{\mathcal{BR}}_u(S))<\hat{c}_u(S)/p$ or $\hat{c}_u(S) \in [b_{i+1}, b_{i})$ and $\hat{c}_u(S_{-u},{\mathcal{BR}}_u(S))<\hat{c}_u(S)/q$}
{
$S\leftarrow (S_{-u},{\mathcal{BR}}_u(S))$\;
}
$F \leftarrow F \cup \{u \in {\cal N}\setminus F:\hat{c}_u(S) \in [b_{i}, +\infty)\}$\;
} \caption{Computing approximate equilibria in $\Psi$-games.}\label{alg} \end{algorithm} \DecMargin{3em}
Then, the algorithm runs a sequence of phases; within each phase, it coordinates best-response moves of the players. This process starts (line 7) by computing a decreasing sequence of boundaries $b_0$, $b_1$, $b_2$, ..., $b_m$ that will be used to define the sets of players that are considered to move within each phase. Then, it executes phase $0$ (lines 8-10). During this phase, as long as there are players of cost at least $b_1$ that have a $q$-move, they play a best-response strategy. Hence, after the end of the phase, all players with cost higher than $b_1$ are in a $q$-approximate equilibrium. Then, the algorithm uses set $F$ to keep the players whose strategies have been irrevocably decided; $F$ is initialized to $\emptyset$ in line 11. Phases $1$ to $m-1$ (lines 12-17) constitute the heart of our algorithm. During each such phase $i$, the algorithm repeatedly checks whether, in the current state, there is a player that either has cost higher than $b_i$ that has a $p$-move or her cost is in $[b_{i+1},b_i)$ and has a $q$-move. While such a player is found, she deviates to her best-response strategy. The phase terminates when no such player exists and the algorithm irrevocably decides the strategy of the players that have cost at least $b_i$. These players are included in set $F$; at this point, they are guaranteed to be at a $p$-approximate equilibrium. Subsequent moves by other players may either increase their cost or decrease the cost they could experience by deviating to another strategy. As we will show, these changes are not significant and each player will still be at an almost $p$-approximate equilibrium at the end of all phases. The fact that plays a crucial role towards proving such a claim is that, at the end of each phase $i$, any player with cost in $[b_{i+1},b_i)$ is guaranteed to be in a $q$-approximate equilibrium. Note that $b_m\leq \hat{c}_{\min}$ and, eventually, all players will be included in set $F$.
We remark that the sequence of the phases is similar to the one in our algorithm for unweighted congestion games with polynomial latency functions of constant degree $d$ in \cite{CFGS11}. However, there is an important difference. In that context, each player is considered to move during only two consecutive phases; these phases are defined statically based only on the characteristics of the particular player. The main reason that allows this is that the cost that a player may experience by following a specific strategy may change by at most a polynomial factor (namely, at most $n^d$) during the execution of the algorithm. This is not the case in the context of $\Psi$-games since the fact that the cost of a player depends on the weights of the other players does not satisfy this polynomial relation. So, in the current algorithm, the players that are considered to move within each phase are decided {\em dynamically} based on the cost they experience during a phase. In this way, a player may (be considered to) move in many different phases.
Below, we will prove the following statement.
\begin{theorem}\label{thm:main} Algorithm 1 computes a $\hat\rho_d$-approximate equilibrium for every $\Psi$-game of constant degree $d$, where $\hat\rho_1=\frac{3+\sqrt{5}}{2}+O(\gamma)$ and $\hat\rho_d \in d^{d+o(d)}$. The running time is polynomial in $\gamma^{-1}$ and in the number of bits in the representation of the game. \end{theorem}
Combined with Lemma \ref{lem:approx}, Theorem \ref{thm:main} immediately yields the following result for weighted congestion games. \begin{theorem} When Algorithm 1 is applied to the $\Psi$-game corresponding to a weighted congestion game with polynomial latency functions of constant degree $d$, it computes a state which is a $\rho_d$-approximate equilibrium for the latter, where $\rho_1=\frac{3+\sqrt{5}}{2}+O(\gamma)$ and $\rho_d \in d^{2d+o(d)}$ for $d\geq 2$. \end{theorem}
The rest of this section is devoted to proving Theorem \ref{thm:main}. Throughout the section we consider the application of the algorithm on a $\Psi$-game of degree $d$ and denote by $S^i$ the state computed by the algorithm after the execution of phase $i$ for $i=0, 1, ..., m-1$. Also, we use $R_i$ to denote the set of players that make at least one move during phase $i$. Our arguments are split in three parts. First, we present a key property maintained by our algorithm stating that the $R_i$-partial potential is small when the phase $i\geq 1$ starts. Then, we use this fact together with the parameters of the algorithm to prove that the running time is polynomial. The proof of the approximation guarantee follows. Recall that the players whose strategies are irrevocably decided during phase $j\geq 1$ are at a $p$-approximate equilibrium at the end of the phase. The purpose of the third part of the proof is to show that for each such player, neither her cost increases significantly nor the cost she would experience by deviating to another strategy decreases significantly after phase $j$. Hence, the approximation guarantee in the final state computed by the algorithm is slightly higher than $p$.
We remark that the analysis follows the same general steps as in our recent paper on unweighted congestion games \cite{CFGS11}. However, due to the definition of $\Psi$-games and the dependency of players' cost on the weights, different and significantly more involved arguments are required, especially in the first and third step.
The key property maintained by our algorithm is the following.
\begin{lemma}\label{lem:potential-bound-per-phase} For every phase $i\geq 1$, it holds that $\Phi_{R_i}(S^{i-1}) \leq \gamma^{-1} n b_{i}$. \end{lemma}
We will now use Lemma \ref{lem:potential-bound-per-phase} and the properties of $\Psi$-games to prove that the algorithm terminates quickly.
\begin{lemma}\label{lem:complexity} The algorithm terminates after a number of steps that is polynomial in the number of bits in the representation of the game and $\gamma^{-1}$. \end{lemma}
\begin{proof} Clearly, if the number of strategies is polynomial in the number of resources, computing a best-response strategy for a player $u$ can be trivially performed in polynomial time (by the definition of $\hat{c}_u$). This is also the case for weighted congestion games in networks (where the number of strategies of a player can be exponential) using a shortest path computation. So, it remains to bound the total number of player moves.
At the initial state, the total cost of the players and, consequently (by Lemma \ref{lem:cost-vs-potential-d}), its potential is at most $n\hat{c}_{\max}$. Each of the players that move during phase $0$ decreases her cost and, consequently (by Theorem \ref{thm:psi-potential}), the potential by at least $(q-1)b_1=\gamma g^{-1} \hat{c}_{\max}$. Hence, the total number of moves in phase $0$ is at most $n \gamma^{-1} g$. For $i\geq 1$, we have $\Phi_{R_i}(S^i)\leq n b_i \gamma^{-1}$ (by Lemma \ref{lem:potential-bound-per-phase}). Each of the players in $R_i$ that move during phase $i$ decreases her cost and, consequently (by Claim \ref{claim:partial-potential}), the $R_i$-partial potential by at least $(q-1)b_{i+1} = b_i g^{-1} \gamma$. Hence, phase $i$ completes after at most $n g \gamma^{-2}$ moves. In total, we have at most $m n g \gamma^{-2}$ moves. The theorem follows by observing that $g$ depends polynomially on $m$, $n$, and $\gamma^{-1}$. \end{proof}
It remains to prove that our algorithm computes approximate equilibria. Our proofs will exploit Lemma \ref{lem:potential-bound-per-phase} as well as the following lemma which relates the cost of a player in a state to the partial potential of two different subgames. \begin{lemma}\label{lem:effect} Consider a $\Psi$-game of degree $d$, a player $u$ and a set of players $R\subseteq {\cal N}\setminus \{u\}$. Then, for every state $S$ and every $\epsilon>0$, it holds that $$\hat{c}_u(S) \leq (1+\epsilon) \Phi_u^{{\cal N}\setminus R}(S)+\xi_\epsilon \Phi_R^{{\cal N}\setminus \{u\}}(S),$$ where $\xi_\epsilon=(1+1/\epsilon)^{d}d^{d}-1$. \end{lemma}
Using Lemmas \ref{lem:potential-bound-per-phase} and \ref{lem:effect}, we will show that neither the cost of a player increases significantly after the phase at the end of which her strategy was irrevocably decided (in Lemma \ref{lem:cost-increases-only-slightly}), nor the cost she would experience by deviating to another strategy decreases significantly (in Lemma \ref{lem:deviation-cost-decreases-only-slightly}).
\begin{lemma}\label{lem:cost-increases-only-slightly} Let $u$ be a player whose strategy was irrevocably decided at phase $j$. Then, $\hat{c}_u(S^{m-1}) \leq (1+2\gamma)\hat{c}_u(S^j)$. \end{lemma}
\begin{proof} For every $i>j$ and $\epsilon>0$, we apply Lemma \ref{lem:effect} for strategy $S^i$, player $u$, and the set of players $R_i$ that move during phase $i$ to obtain \begin{eqnarray*} \hat{c}_u(S^i) &\leq & (1+\epsilon) \Phi_u^{{\cal N}\setminus R_i}(S^i)+\xi_\epsilon\Phi_{R_i}^{{\cal N}\setminus\{u\}}(S^i)\\ &=& (1+\epsilon) \Phi_u^{{\cal N}\setminus R_i}(S^{i-1})+\xi_\epsilon\Phi_{R_i}^{{\cal N}\setminus\{u\}}(S^i)\\ &\leq & (1+\epsilon) \Phi_u(S^{i-1})+\xi_\epsilon\Phi_{R_i}(S^i)\\ &\leq& (1+\epsilon) \hat{c}_u(S^{i-1})+\xi_\epsilon\Phi_{R_i}(S^{i-1}). \end{eqnarray*} The equality holds by Claim \ref{claim:equal-partial-potential} since the players in ${\cal N} \setminus R_i$ do not move during phase $i$. The second inequality follows by Claim \ref{claim:partial-potential-bound}. The last one follows by Claim \ref{claim:u-partial-potential} and since the $R_i$-partial potential decreases during phase $i$.
We now set $\epsilon=(1+\gamma)^{1/m}-1$. This implies that $(1+\epsilon)^m=1+\gamma$. Also, by Claim \ref{claim:concave} (in Appendix \ref{sec:app:two-ineq}), we get $\epsilon \geq \frac{\gamma}{m}(1+\gamma)^{1/m-1}\geq (m(1+\gamma^{-1}))^{-1}$ and, by the definition of the parameters $g$ and $\gamma$, $\xi_\epsilon= (1+m(1+\gamma^{-1}))^d d^d -1 \leq \frac{g\gamma^3}{2n}\leq \frac{g\gamma}{2(1+\gamma^{-1})n}$. Using the above inequality together with these observations, we obtain \begin{eqnarray*} \hat{c}_u(S^{m-1}) &\leq & (1+\epsilon)^{m-1-j} \hat{c}_u(S^{j})+\xi_\epsilon\sum_{i=j+1}^{m-1}{(1+\epsilon)^{m-1-i}\Phi_{R_i}(S^{i-1})}\\ &\leq & (1+\epsilon)^{m} \hat{c}_u(S^{j})+(1+\epsilon)^{m}\xi_\epsilon\sum_{i=j+1}^{m-1}{\Phi_{R_i}(S^{i-1})}\\ &\leq& (1+\gamma)\hat{c}_u(S^{j})+(1+\gamma)\xi_\epsilon\sum_{i=j+1}^{m-1}{nb_i \gamma^{-1}}\\ &= & (1+\gamma) \hat{c}_u(S^{j})+(1+\gamma^{-1})\xi_\epsilon n b_j\sum_{i=1}^{m-1-j}{g^{-i}}\\ &\leq & (1+\gamma) \hat{c}_u(S^{j})+2(1+\gamma^{-1})\xi_\epsilon n b_jg^{-1}\\ &\leq & (1+\gamma) \hat{c}_u(S^{j})+\gamma b_j\\ &\leq & (1+2\gamma)\hat{c}_u(S^j). \end{eqnarray*} The second inequality is obvious, the third one follows by Lemma \ref{lem:potential-bound-per-phase} and by the relation between $\epsilon$ and $\gamma$, the equality follows by the definition of $b_i$, the fourth inequality follows since $g\geq 2$ which implies that $\sum_{i\geq 1}{g^{-i}}\leq 2g^{-1}$, the fifth one follows by our observation about $\xi_\epsilon$ above, and the last one follows since, by the definition of the algorithm, the fact that the strategy of player $u$ is irrevocably decided at phase $j$ implies that $\hat{c}_u(S^j)\geq b_j$. \end{proof}
\begin{lemma}\label{lem:deviation-cost-decreases-only-slightly} Let $u$ be a player whose strategy was irrevocably decided at phase $j$ and let $s'_u$ be any of her strategies. Then, $\hat{c}_u(S^{m-1}_{-u},s'_u) \geq (1-2\gamma)\hat{c}_u(S^j_{-u},s'_u)$. \end{lemma}
We are now ready to use the last two lemmas in order to prove the approximation guarantee of the algorithm. This will complete the proof of Theorem \ref{thm:main}. \begin{lemma}\label{lem:apx-bound} Given a $\Psi$-game of degree $d$, the algorithm computes a $\hat\rho_d$-approximate equilibrium with $\hat\rho_1\leq \frac{3+\sqrt{5}}{2}+O(\gamma)$ and $\hat\rho_d\leq d^{d+o(d)}$. \end{lemma}
\section{Conclusions and open problems}\label{sec:open} Due to lack of space, the modification of Algorithm 1 that yields our structural result for weighted congestion games with superlinear latency functions is presented in Appendix \ref{sec:modified-alg}.
Our work reveals interesting open problems. The obvious one is whether approximate equilibria with a better approximation guarantee can be computed in polynomial time. We believe that our techniques have reached their limits for linear weighted congestion games. However, in the case of superlinear latency functions, approximations of weighted congestion games by potential games different than $\Psi$-games might yield improved (existential or algorithmic) approximation guarantees. On the conceptual level, it is interesting to further explore applications of approximations of non-potential games by potential ones like the one we have exploited in the current paper.
\begin{thebibliography}{99} \bibitem{AckermannRV08} H. Ackermann, H. R{\"o}glin, and B. V{\"o}cking. \newblock On the impact of combinatorial structure on congestion games. \newblock {\em Journal of the ACM}, 55(6), 2008.
\bibitem{ARV09} H. Ackermann, H. R{\"o}glin, and B. V{\"o}cking. \newblock Pure Nash equilibria in player-specific and weighted congestion games. \newblock {\em Theoretical Computer Science}, 410(17): 1552--1563, 2009.
\bibitem{AC09} E. Anshelevich and B. Caskurlu. Exact and approximate equilibria for optimal group network formation. In {\em Proceedings of the 17th Annual Symposium on Algorithms (ESA)}, LNCS 5757, Springer, pages 239--250, 2009.
\bibitem{AwerbuchAEMS08} B. Awerbuch, Y. Azar, A. Epstein, V.~S. Mirrokni, and A.
Skopalik. \newblock Fast convergence to nearly optimal solutions in potential games. \newblock In {\em Proceedings of the 9th ACM Conference on Electronic Commerce (EC)}, ACM, pages 264--273,
2008.
\bibitem{BhalgatCK10} A. Bhalgat, T. Chakraborty, and S. Khanna. \newblock Approximating pure Nash equilibrium in cut, party affiliation, and
satisfiability games. \newblock In {\em Proceedings of the 11th ACM Conference on Electronic Commerce (EC)}, ACM, pages 73--82.
2010.
\bibitem{C09} I. Caragiannis. Efficient coordination mechanisms for unrelated machine scheduling. In {\em Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}, pp. 815--824, 2009. Full version available at {\tt arxiv.org}.
\bibitem{CFGS11} I. Caragiannis, A. Fanelli, N. Gravin, and A. Skopalik. Efficient computation of approximate pure Nash equilibria in congestion games. In {\em Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS)}, pp. 532-541, 2011. Full version available at {\tt arxiv.org}.
\bibitem{CH10} J. Cardinal and M. Hoefer. Non-cooperative facility location and covering games. {\em Theoretical Computer Science}, 411(16-18): 1855--1876, 2010.
\bibitem{CKM+08} M. Charikar, H. J. Karloff, C. Mathieu, J. Naor, and M. E. Saks. Online multicast with egalitarian cost sharing. In {\em Proceedings of the 20th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)}, ACM, pages 70--76, 2008.
\bibitem{CCL+07} C. Chekuri, J. Chuzhoy, L. Lewin-Eytan, J. Naor, and A. Orda. Non-cooperative multicast and facility location games. {\em IEEE Journal on Selected Areas in Communications}, 25(6): 1193--1206, 2007.
\bibitem{ChienS07} S. Chien and A. Sinclair. \newblock Convergence to approximate Nash equilibria in congestion games. \newblock {\em Games and Economic Behavior}, 71(2): 315--327, 2011.
\bibitem{CGC04} N. Christin, J. Grossklags, and J. Chuang. Near rationality and competitive equilibria in networked systems. In {\em Proceedings of the ACM SIGCOMM Workshop on Practice and Theory of Incentives in Networked Systems (PINS)}, ACM, pages 213--219, 2004.
\bibitem{CMS06} G. Christodoulou, V. S. Mirrokni, and A. Sidiropoulos. Convergence and approximation in potential games. In {\em Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS)}, LNCS 3884, Springer, pages 349--360, 2006.
\bibitem{DP07} C. Daskalakis and C. H. Papadimitriou. Computing equilibria in anonymous games. In {\em Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS)}, IEEE, pages 83--93, 2007.
\bibitem{DS08} J. Dunkel and A. S. Schulz. On the complexity of pure-strategy Nash equilibria in congestion and local-effect games. {\em Mathematics of Operations Research}, 33(4): 851--868, 2008.
\bibitem{EKM07} E. Even-Dar, A. Kesselman, and Y. Mansour. Convergence time to Nash equilibrium in load balancing. {\em ACM Transactions on Algorithms}, 3(3), 2007.
\bibitem{FKS05} D. Fotakis, S. Kontogiannis, and P. G. Spirakis. Selfish unsplittable flows. {\em Theoretical Computer Science}, 340(3), pp. 514--538, 2005.
\bibitem{FabrikantPT04} A. Fabrikant, C.~H. Papadimitriou, and K. Talwar. \newblock The complexity of pure nash equilibria. \newblock In {\em Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC)}, ACM, pages 604--612, 2004.
\bibitem{FM09} A. Fanelli and L. Moscardelli. On best response dynamics in weighted congestion games with polynomial delays. In {\em Proceedings of the 5th International Workshop on Internet and Network Economics (WINE)}, LNCS 5929, Springer, pages 55--66, 2009.
\bibitem{GMV05} M. X. Goemans, V. S. Mirrokni, and A. Vetta. Sink equilibria and convergence. In {\em Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS)}, pp. 142-154, 2005.
\bibitem{HK10} T. Harks and M. Klimm. On the existence of pure Nash equilibria in weighted congestion games. \newblock In {\em Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP)}, Part 1, LNCS 6198, Springer, pages 79--89, 2010.
\bibitem{Johnson88} D.~S. Johnson, C.~H. Papadimitriou, and M. Yannakakis. \newblock How easy is local search? \newblock {\em Journal of Computer and System Sciences}, 37: 79--100, 1988.
\bibitem{KR11} K. Kollias and T. Roughgarden. Restoring pure equilibria to weighted congestion games. In {\em Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP)}, Part II, LNCS 6756, Springer, pages 539--551, 2011.
\bibitem{MV04} V. S. Mirrokni and A. Vetta. Convergence issues in competitive games. In {\em Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX)}, LNCS 3122, Springer, pages 183--192, 2004.
\bibitem{NT09} T. Nguyen and E. Tardos. Approximate pure Nash equilibria via Lov\'asz local lemma. In {\em Proceedings of the 5th International Workshop on Internet and Network Economics (WINE)}, LNCS 5929, Springer, pages 160--171, 2009.
\bibitem{PS06} P. N. Panagopoulou and P. G. Spirakis. Algorithms for pure Nash equilibria in weighted congestion games. {\em ACM Journal of Experimental Algorithmics}, 11, 2006.
\bibitem{R73} R.~W. Rosenthal. \newblock A class of games possessing pure-strategy Nash equilibria. \newblock {\em International Journal of Game Theory}, 2:65--67, 1973.
\bibitem{SkopalikV08} A. Skopalik and B. V{\"o}cking. \newblock Inapproximability of pure Nash equilibria. \newblock In {\em Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC)}, ACM, pages 355--364, 2008.
\end{thebibliography}
\appendix \section{Two technical inequalities}\label{sec:app:two-ineq} The following technical inequalities are extensively used in our proofs and are included here for easy reference.
\begin{lemma}[Minkowski inequality]\label{lem:minkowski} $\sum_{t=1}^s{(\alpha_t+\beta_t)^k}\leq \left(\left(\sum_{t=1}^s{\alpha_t^k}\right)^{1/k}+\left(\sum_{t=1}^s{\beta_t^k}\right)^{1/k}\right)^k$, for any integer $k\geq 1$ and $\alpha_t,\beta_t\geq 0$. \end{lemma}
\begin{claim}\label{claim:concave} For every $\alpha\in (0,1)$ and $z>1$, it holds that $z^{\alpha}-1 \geq \alpha (z-1) z^{\alpha-1}$. \end{claim}
\begin{proof} The function $h(x)=x^{\alpha}$ is concave in $[1,+\infty)$. This means that, for every $z>1$, the line connecting points $(1,1)$ and $(z, h(z))$ has slope higher than the derivative of $h$ at point $z$, i.e., $\frac{z^{\alpha}-1}{z-1} \geq \alpha z^{\alpha-1}$. Equivalently, $z^{\alpha}-1 \geq \alpha (z-1) z^{\alpha-1}$. \end{proof}
\section{Omitted proofs from Section \ref{sec:psi-games}} The following lemma is proved in (the full version of) \cite{C09} and is extensively used in our proofs.
\begin{lemma}\label{lem:properties} For any integer $k\geq 1$, any finite multi-set of non-negative reals $A$, and any non-negative real $b$ the following hold: \[\begin{array}{l l} \mbox{a. } L(A)^k \leq \Psi_k(A) \leq k! L(A)^k & \mbox{d. } \Psi_k(A\cup\{b\}) -\Psi_k(A) = k b\Psi_{k-1}(A\cup \{b\})\\ \mbox{b. } \Psi_{k-1}(A)^{k} \leq \Psi_{k}(A)^{k-1} & \mbox{e. } \Psi_k(A) \leq k\Psi_1(A)\Psi_{k-1}(A)\\ \mbox{c. } \Psi_k(A\cup\{b\}) = \sum_{t=0}^k{\frac{k!}{(k-t)!}b^t\Psi_{k-t}(A)} & \mbox{f. } \Psi_k(A\cup \{b\}) \leq \left(\Psi_k(\{b\})^{1/k}+\Psi_k(A)^{1/k}\right)^k \end{array}\] \end{lemma}
\subsection{Proof of Theorem \ref{thm:psi-potential}} Consider a player $u$, a state $S$ in which $u$ plays strategy $s_u$ and state $(S_{-u},s'_u)$ where $u$ has deviated to strategy $s'_u$. Using the definition of the potential function, we have \begin{eqnarray*} \Phi(S)-\Phi(S_{-u},s'_u) &=& \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e(S))}} - \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e(S_{-u},s'_u))}}\\ &=& \sum_{e\in s_u \setminus s'_u}{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\left(\Psi_{k+1}(N_e(S))-\Psi_{k+1}(N_e(S_{-u},s'_u))\right)}}\\ & & +\sum_{e\in s'_u \setminus s_u}{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\left(\Psi_{k+1}(N_e(S))-\Psi_{k+1}(N_e(S_{-u},s'_u))\right)}}\\ &=& \sum_{e\in s_u \setminus s'_u}{\sum_{k=0}^d{a_{e,k}w_u\Psi_{k}(N_e(S))}} - \sum_{e\in s'_u \setminus s_u}{\sum_{k=0}^d{a_{e,k}w_u\Psi_{k}(N_e(S_{-u},s'_u))}}\\ &=& w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}\Psi_{k}(N_e(S))}} - w_u\sum_{e\in s'_u}{\sum_{k=0}^d{a_{e,k}\Psi_{k}(N_e(S_{-u},s'_u))}}\\ &=& \hat{c}_u(S) - \hat{c}_u(S_{-u},s'_u). \end{eqnarray*} The third equality follows by Lemma \ref{lem:properties}d and the facts that $N_e(S) = N_e(S_{-u},s'_u)\cup \{w_u\}$ for every resource $e\in s_u\setminus s'_u$ and $N_e(S_{-u},s'_u) = N_e(S)\cup \{w_u\}$ for every resource $e\in s'_u\setminus s_u$. The last equality follows by the definition of $\hat{c}_u$. \hspace*{\fill}\sq
\subsection{Proof of Claim \ref{claim:approx}} We will use Lemma \ref{lem:properties}a and the definitions of $c_u(S)$ and $\hat{c}_u(S)$. Let $s_u$ be the strategy of player $u$ at state $S$. Using the first inequality of Lemma \ref{lem:properties}a, we have \begin{eqnarray*} c_u(S) &=& w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}L(N_e(S))^k}} \leq w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}\Psi_k(N_e(S))}}= \hat{c}_u(S). \end{eqnarray*} Also, using the second inequality in Lemma \ref{lem:properties}a, we have \begin{eqnarray*} \hat{c}_u(S) &=& w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}\Psi_k(N_e(S))}} \leq w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}k!L(N_e(S))^k}}\leq d! w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}L(N_e(S))^k}}\\ &=& d! c_u(S). \end{eqnarray*} \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:approx}} Let $S$ be $\rho$-approximate equilibrium for a $\Psi$-game of degree $d$, $u$ a player and $s'_u$ a strategy of $u$ different than her strategy $s_u$ in $S$. Using the $\rho$-approximate equilibrium condition for player $u$ and Claim \ref{claim:approx}, we have \begin{eqnarray*} c_u(S) &\leq & \hat{c}_u(S) \leq \rho \hat{c}_u(S_{-u},s'_u)= d! \rho \cdot c_u(S_{-u},s'_u). \end{eqnarray*} \hspace*{\fill}\sq
\subsection{Proof of Claim \ref{claim:partial-potential-bound}} Let $k\geq 1$ be an integer and consider a resource $e$ which is used by at least one player of $B$ in $S$. By the definition of $\Psi_k$, observe that $\Psi_k(N^A_e(S))-\Psi_k(N^{A\setminus B}_e(S))$ is equal to $k!$ times the sum of all monomials of degree $k$ among the elements of $N^A_e(S)$ that contain at least one element in $N^B_e(S)$. Similarly, $\Psi_k(N_e(S))-\Psi_k(N^{{\cal N}\setminus B}_e(S))$ is equal to $k!$ times the sum of all monomials of degree $k$ among the elements of $N_e(S)$ that contain at least one element in $N^B_e(S)$. Since $N^A_e(S)\subseteq N_e(S)$, we have that \begin{eqnarray*} \Psi_k(N^A_e(S))-\Psi_k(N^{A\setminus B}_e(S)) &\leq & \Psi_k(N_e(S))-\Psi_k(N^{{\cal N}\setminus B}_e(S)). \end{eqnarray*} The inequality holds trivially (with equality) if no player from $B$ uses resource $e$ in $S$. Using this inequality and the definition of the partial potential, we have \begin{eqnarray*} \Phi^A_B(S) &=& \Phi^A(S)-\Phi^{A\setminus B}(S) = \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\left(\Psi_{k+1}(N^A_e(S))-\Psi_{k+1}(N^{A\setminus B}_e(S))\right)}}\\ &\leq & \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\left(\Psi_{k+1}(N_e(S))-\Psi_{k+1}(N^{{\cal N}\setminus B}_e(S))\right)}} = \Phi(S)-\Phi^{{\cal N}\setminus B}(S)\\ &=& \Phi_B(S). \end{eqnarray*} \hspace*{\fill}\sq
\subsection{Proof of Claim \ref{claim:equal-partial-potential}} Observe that $N^{A'}_e(S)=N^{A'}_e(S')$ for each resource $e$ and any $A'\subseteq A$. By the definition of the potential of the subgame among the players of $A'$, we have $\Phi^{A'}(S)=\Phi^{A'}(S')$. Then, by the definition of the partial potential, $\Phi^A_B(S)=\Phi^A(S)-\Phi^{A\setminus B}(S) = \Phi^A(S')-\Phi^{A\setminus B}(S')=\Phi^A_B(S')$. \hspace*{\fill}\sq
\subsection{Proof of Claim \ref{claim:u-partial-potential}} Let $s_u$ be the strategy of player $u$ in $S$. We use the definition of the partial potential, the definitions of the potential for the original game and the subgame among the players in ${\cal N}\setminus \{u\}$, Lemma \ref{lem:properties}d, and the definition of $\hat{c}_u(S)$ to obtain \begin{eqnarray*} \Phi_u(S) &=& \Phi(S)-\Phi^{{\cal N}\setminus \{u\}}(S) = \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\left(\Psi_{k+1}(N_e(S))-\Psi_{k+1}(N^{{\cal N}\setminus \{u\}}_e(S))\right)}}\\ &=& \sum_{e\in s_u}{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\left(\Psi_{k+1}(N_e(S))-\Psi_{k+1}(N^{{\cal N}\setminus \{u\}}_e(S))\right)}} = w_u \sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}\Psi_{k}(N_e(S))}}\\ &=& \hat{c}_u(S). \end{eqnarray*} \hspace*{\fill}\sq
\subsection{Proof of Claim \ref{claim:partial-potential}} We have \begin{eqnarray*} \Phi_A(S)-\Phi_A(S') &=& \Phi(S)-\Phi^{{\cal N}\setminus A}(S)-\Phi(S')+\Phi^{{\cal N}\setminus A}(S') = \Phi(S)-\Phi(S') = \hat{c}_u(S)-\hat{c}_u(S'). \end{eqnarray*} The first equality follows by the definition of the $A$-partial potential, the second one follows by Claim \ref{claim:equal-partial-potential} since each player in ${\cal N}\setminus A$ uses the same strategy in $S$ and $S'$ and the last one follows by Theorem \ref{thm:psi-potential}. \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:cost-vs-potential-d}}
Let $A=\{u_1, u_2, ..., u_{|A|}\}$. Let $A_0=\emptyset$ and $A_t=\{u_1, ..., u_t\}$ for $t=1, 2, ..., |A|$. Then, using the definition of the partial potential and Claims \ref{claim:partial-potential-bound} and \ref{claim:u-partial-potential}, we have \begin{eqnarray*}
\Phi_A(S) &=& \Phi(S)-\Phi^{{\cal N}\setminus A}(S) = \sum_{t=1}^{|A|}{\left(\Phi^{{\cal N}\setminus A_{t-1}}(S) - \Phi^{{\cal N}\setminus A_{t}}(S)\right)}\\
&=& \sum_{t=1}^{|A|}{\Phi^{{\cal N}\setminus A_{t-1}}_{u_t}(S)} \leq \sum_{t=1}^{|A|}{\Phi_{u_t}(S)} = \sum_{u \in A}{\hat{c}_u(S)}. \end{eqnarray*} \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:stretch-linear}} Let $S^*$ be the state of minimum potential and $S$ be a $\rho$-approximate equilibrium. For each player $u$, we denote by $s_u$ and $s^*_u$ the strategies she plays at states $S$ and $S^*$, respectively. Using the $\rho$-approximate equilibrium condition $c_u(S)\leq \rho\cdot c_u(S_{-u},s^*_u)$, the definition of the cost of player $u$, and the definition of function $\Psi_1$, we obtain \begin{eqnarray*} \sum_u{c_u(S)} & \leq & \rho w_u\sum_{e\in s^*_u}{\left(a_{e,1}\Psi_1(N_e(S_{-u},s^*_u))+a_{e,0}\right)}\\ & \leq & \rho w_u\sum_{e\in s^*_u}{\left(a_{e,1}\Psi_1(N_e(S)\cup \{w_u\})+a_{e,0}\right)}\\ &=& \rho w_u\sum_{e\in s^*_u}{\left(a_{e,1}\Psi_1(N_e(S))+a_{e,1}w_u+a_{e,0}\right)}. \end{eqnarray*} By summing over all players, by exchanging sums, and using the definition of $N_e(S^*)$, we obtain \begin{eqnarray*} \sum_u{c_u(S)} &\leq & \rho \sum_u{w_u\sum_{e\in s^*_u}{\left(a_{e,1}\Psi_1(N_e(S))+a_{e,1}w_u+a_{e,0}\right)}}\\ &=& \rho\sum_e{\left(a_{e,1}\Psi_1(N_e(S))\sum_{u:e\in s^*_u}{w_u}+a_{e,1}\sum_{u:e\in s^*_u}{w^2_u}+a_{e,0}\sum_{u:e\in s^*_u}{w_u}\right)}\\ &=& \rho\sum_e{\left(a_{e,1}\Psi_1(N_e(S))\Psi_1(N_e(S^*))+a_{e,1}\sum_{u:e\in s^*_u}{w^2_u}+a_{e,0}\Psi_1(N_e(S^*))\right)}. \end{eqnarray*} We now apply the inequality $xy\leq \frac{\sqrt{5}-1}{2(3-\sqrt{5})}y^2+\frac{\sqrt{5}-2}{3-\sqrt{5}}x^2$ that holds for any pair of non-negative $x$ and $y$ on the rightmost part of the above derivation to obtain \begin{eqnarray*} & & \sum_u{c_u(S)} \\ &\leq & \rho\sum_e{\left(\frac{\sqrt{5}-1}{2(3-\sqrt{5})}a_{e,1}\Psi_1(N_e(S^*))^2+\frac{\sqrt{5}-2}{3-\sqrt{5}}a_{e,1}\Psi_1(N_e(S))^2+a_{e,1}\sum_{u:e\in s^*_u}{w^2_u}+a_{e,0}\Psi_1(N_e(S^*))\right)}\\ &=& \rho\sum_e{\left(\frac{5-\sqrt{5}}{4(3-\sqrt{5})}a_{e,1}\left(\Psi_1(N_e(S^*))^2+\sum_{u:e\in s^*_u}{w^2_u}\right)+a_{e,0}\Psi_1(N_e(S^*))\right)}\\ & & -\rho\sum_e{\frac{7-3\sqrt{5}}{4(3-\sqrt{5})}a_{e,1}\left(\Psi_1(N_e(S^*))^2-\sum_{u:e\in s^*_u}{w^2_u}\right)}+\frac{\sqrt{5}-2}{3-\sqrt{5}}q\sum_e{a_{e,1}\Psi_1(N_e(S))^2}. \end{eqnarray*} Now, observe that $\Psi_1(N_e(S^*))^2 \geq \sum_{u:e\in s^*_u}{w^2_u}$ for every resource $e$. Furthermore, $\Psi_1(N_e(S^*))^2+\sum_{u:e\in s^*_u}{w^2_u}=\Psi_2(N_e(S^*))$. Hence, we have \begin{eqnarray}\nonumber \sum_u{c_u(S)} &\leq & \rho\sum_e{\left(\frac{5-\sqrt{5}}{4(3-\sqrt{5})}a_{e,1}\Psi_2(N_e(S^*))+a_{e,0}\Psi_1(N_e(S^*))\right)}+\frac{\sqrt{5}-2}{3-\sqrt{5}}\rho\sum_e{a_{e,1}\Psi_1(N_e(S))^2}\\\nonumber &\leq & \frac{5-\sqrt{5}}{2(3-\sqrt{5})} \rho\sum_e{\left(\frac{a_{e,1}}{2}\Psi_2(N_e(S^*))+a_{e,0}\Psi_1(N_e(S^*))\right)}+\frac{\sqrt{5}-2}{3-\sqrt{5}}\rho\sum_e{a_{e,1}\Psi_1(N_e(S))^2}\\\label{eq:total-cost-linear} &= & \frac{5-\sqrt{5}}{2(3-\sqrt{5})}\rho \Phi(S^*)+\frac{\sqrt{5}-2}{3-\sqrt{5}}\rho\sum_e{a_{e,1}\Psi_1(N_e(S))^2}. \end{eqnarray}
We now use the definition of $\Phi(S)$, the fact that for every player $u$ and resource $e\in s_u$, it holds that $w_u\leq \Psi_1(N_e(S))$, and the definition of the cost of player $u$. We have \begin{eqnarray*} \Phi(S) &=& \sum_e{\left(\frac{a_{e,1}}{2}\Psi_2(N_e(S))+a_{e,0}\Psi_1(N_e(S))\right)}\\ &=& \sum_e{\left(\frac{a_{e,1}}{2}\sum_{u:e \in s_u}{\left(w_u \Psi_1(N_e(S))+w^2_u\right)}+a_{e,0}\sum_{u:e\in s_u}{w_u}\right)}\\ &\leq & \sum_e{\left(\frac{a_{e,1}}{2}\sum_{u:e \in s_u}{\left((6-2\sqrt{5})w_u \Psi_1(N_e(S))+(2\sqrt{5}-4)w^2_u\right)}+a_{e,0}\sum_{u:e\in s_u}{w_u}\right)}\\ &=& (3-\sqrt{5})\sum_u{w_u\sum_{e\in s_u}{\left(a_{e,1}\Psi_1(N_e(S))+a_{e,0}\right)}}+(\sqrt{5}-2)\sum_e{a_{e,1}\sum_{u:e\in s_u}{w^2_u}}\\ && +(\sqrt{5}-2)\sum_e{a_{e,0}\sum_{u:e\in s_u}{w_u}}\\ &=& (3-\sqrt{5})\sum_u{c_u(S)} +(\sqrt{5}-2)\sum_e{a_{e,1}\sum_{u:e\in s_u}{w^2_u}}+(\sqrt{5}-2)\sum_e{a_{e,0}\sum_{u:e\in s_u}{w_u}}. \end{eqnarray*} By applying inequality (\ref{eq:total-cost-linear}) to the rightmost part of this derivation, we obtain \begin{eqnarray*} \Phi(S) &\leq & \frac{5-\sqrt{5}}{2}\rho \Phi(S^*)+(\sqrt{5}-2)\rho\sum_e{a_{e,1}\Psi_1(N_e(S))^2}+(\sqrt{5}-2)\sum_e{a_{e,1}\sum_{u:e\in s_u}{w^2_u}}\\ && +(\sqrt{5}-2)\sum_e{a_{e,0}\Psi_1(N_e(S))}\\ &\leq & \frac{5-\sqrt{5}}{2}\rho\Phi(S^*)+(2\sqrt{5}-4)\rho\sum_e{\left(\frac{a_{e,1}}{2}\left(\Psi_1(N_e(S))^2+\sum_{u:e\in s_u}{w^2_u}\right)+a_{e,0}\Psi_1(N_e(S))\right)}\\ &=& \frac{5-\sqrt{5}}{2}\rho\Phi(S^*)+(2\sqrt{5}-4)\rho\sum_e{\left(\frac{a_{e,1}}{2}\Psi_2(N_e(S))+a_{e,0}\Psi_1(N_e(S))\right)}\\ &=& \frac{5-\sqrt{5}}{2}\rho\Phi(S^*)+(2\sqrt{5}-4)\rho\Phi(S). \end{eqnarray*} The last inequality implies that $\Phi(S)$ is not larger than $\frac{(5-\sqrt{5})\rho}{2(1-(2\sqrt{5}-4)\rho)} \Phi(S^*)$ which can be easily proved to be at most $\left(\frac{3+\sqrt{5}}{2}+6(\rho-1)\right) \Phi(S^*)$ when $\rho\in [1,11/10]$. \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:stretch-d}} Consider a $\rho$-approximate equilibrium $S$ of a $\Psi$-game and let $S^*$ be the state of minimum potential. We denote by $s_u$ and $s^*_u$ the strategy of player $u$ at states $S$ and $S^*$, respectively.
By Lemma \ref{lem:cost-vs-potential-d}, the $\rho$-approximate equilibrium condition $\hat{c}_u(S)\leq \rho \cdot \hat{c}_u(S_{-u},s^*_u)$, and the definition of the potential function, we have \begin{eqnarray*} \frac{1}{\rho}\Phi(S) &\leq & \frac{1}{\rho}\sum_u{\hat{c}_u(S)}\\ &\leq & \sum_u{\hat{c}_u(S_{-u},s^*_u)}\\ &=& \sum_u{w_u\sum_{e\in s^*_u}{\sum_{k=0}^d{a_{e,k}\Psi_k(N_e(S_{-u},s^*_u))}}}\\ &=& \sum_e{\sum_{k=0}^d{a_{e,k}\sum_{u:e\in s^*_u}{w_u\Psi_k(N_e(S_{-u},s^*_u))}}}. \end{eqnarray*} We now use the fact that $N_e(S_{-u},s^*_u)\subseteq N_e(S)\cup \{w_u\}$, Lemma \ref{lem:properties}c, and the fact that $\Psi_{t+1}(N_e(S^*))\geq (t+1)! \sum_{u:e\in s^*_u}{w^{t+1}_u}$ to obtain \begin{eqnarray*} \frac{1}{\rho}\Phi(S) &\leq & \sum_e{\sum_{k=0}^d{a_{e,k}\sum_{u:e\in s^*_u}{w_u\Psi_k(N_e(S) \cup \{w_u\})}}}\\ &=& \sum_e{\sum_{k=0}^d{a_{e,k}\sum_{u:e\in s^*_u}{w_u\sum_{t=0}^k{\frac{k!}{(k-t)!}\Psi_{k-t}(N_e(S)) w^t_u}}}}\\ &=& \sum_e{\sum_{k=0}^d{a_{e,k}\sum_{t=0}^k{\frac{k!}{(k-t)!}\Psi_{k-t}(N_e(S))\sum_{u:e\in s^*_u}{w^{t+1}_u}}}}\\ &\leq & \sum_e{\sum_{k=0}^d{a_{e,k}\sum_{t=0}^k{\frac{k!}{(k-t)!(t+1)!}\Psi_{k-t}(N_e(S))\Psi_{t+1}(N_e(S^*))}}}\\ &=& \sum_e{\sum_{k=0}^{d}{\frac{a_{e,k}}{k+1}\sum_{t=1}^{k+1}{\left(\begin{array}{c}k+1\\t\end{array}\right)\Psi_{k+1-t}(N_e(S))\Psi_{t}(N_e(S^*))}}}. \end{eqnarray*} Using Lemma \ref{lem:properties}b (observe that it implies that $\Psi_t(A) \leq \Psi_{k+1}(A)^{\frac{t}{k+1}}$ for any non-negative integer $t\leq k+1$ and multi-set of reals $A$), the binomial theorem, inequality $\alpha^\lambda+\beta^{\lambda} \leq (\alpha+\beta)^{\lambda}$ for every $\alpha,\beta\geq 0$ and $\lambda \geq 1$, and the definition of the potential function, we obtain \begin{eqnarray*} \frac{1}{\rho}\Phi(S) &\leq & \sum_e{\sum_{k=0}^{d}{\frac{a_{e,k}}{k+1}\sum_{t=1}^{k+1}{\left(\begin{array}{c}k+1\\t\end{array}\right)\Psi_{k+1}(N_e(S))^\frac{k+1-t}{k+1}\Psi_{k+1}(N_e(S^*))^\frac{t}{k+1}}}}\\ &=& \sum_e{\sum_{k=0}^{d}{\frac{a_{e,k}}{k+1}\left(\left(\Psi_{k+1}(N_e(S))^\frac{1}{k+1}+\Psi_{k+1}(N_e(S^*))^\frac{1}{k+1}\right)^{k+1}-\Psi_{k+1}(N_e(S))\right)}}\\ &\leq & \sum_e{\sum_{k=0}^{d}{\frac{a_{e,k}}{k+1}\left(\Psi_{k+1}(N_e(S))^\frac{1}{d+1}+\Psi_{k+1}(N_e(S^*))^\frac{1}{d+1}\right)^{d+1}}}-\Phi(S). \end{eqnarray*} We now apply Minkowski inequality twice on the double sum at the rightmost part of this last inequality and use the definition of the potential function to obtain \begin{eqnarray*} (1+1/\rho)\Phi(S) &\leq & \sum_e{\left(\left(\sum_{k=0}^{d}{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e(S))}\right)^\frac{1}{d+1}+\left(\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e(S^*))}\right)^\frac{1}{d+1}\right)^{d+1}}\\ &\leq & \left(\left(\sum_e{\sum_{k=0}^{d}{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e(S))}}\right)^\frac{1}{d+1}+\left(\sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e(S^*))}}\right)^\frac{1}{d+1}\right)^{d+1}\\ &=& \left(\left(\Phi(S)\right)^{\frac{1}{d+1}}+\left(\Phi(S^*)\right)^{\frac{1}{d+1}}\right)^{d+1}. \end{eqnarray*} The above inequality yields \begin{eqnarray}\label{eq:potential-d} \left(\Phi(S)\right)^{\frac{1}{d+1}} &\leq & \frac{1}{(1+1/\rho)^{\frac{1}{d+1}}-1} \left(\Phi(S^*)\right)^{\frac{1}{d+1}}. \end{eqnarray} By Claim \ref{claim:concave}, we have $(1+1/\rho)^{\frac{1}{d+1}}-1 \geq \left(\rho^{\frac{1}{d+1}}(\rho+1)^{\frac{d}{d+1}}(d+1)\right)^{-1}$. Using this observation, inequality (\ref{eq:potential-d}) implies that \begin{eqnarray*} \Phi(S)&\leq & \rho(\rho+1)^d(d+1)^{d+1}\Phi(S^*) \end{eqnarray*} as desired. \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:stretch-partial}} In our proof, we will use the property \begin{eqnarray}\label{eq:psi-property} \Psi_k(A\cup B) &=& \sum_{t=0}^k{\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k-t}(A)\Psi_t(B)} \end{eqnarray} for every two multi-sets of positive reals $A$ and $B$. To see why (\ref{eq:psi-property}) holds, observe that the product $\Psi_{k-t}(A)\Psi_t(B)$ equals $(k-t)! t!$ times the sum of all products of monomials of degree $k-t$ with elements of $A$ with monomials of degree $t$ with elements of $B$.
Given state $S$ in the original game, we define the $\Psi$-game $\left(R,(w_u)_{u\in R}, (\Sigma_u)_{u\in R}, (\bar{a}_{e,t})_{e\in E, t=0, ..., d}\right)$ with $$\bar{a}_{e,t} = \sum_{k=t}^d{a_{e,k}\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k-t}(N^{{\cal N}\setminus R}_e(S))}.$$ Observe that the parameters $\bar{a}_{e,k}$ depend only on the strategies of players in ${\cal N}\setminus R$ in $S$.
Now, given any state $S'$ in the original game, we denote by $\bar{S}'$ the state in the new game in which each player in $R$ uses the strategy she uses in $S'$. We also use the notation $\bar{c}_u$ for the cost of a player $u\in R$ in the new game and $\bar{\Phi}$ for its potential function.
We will first show that $\bar{c}_u(\bar{S}')=\hat{c}_u(S')$ for every state $\bar{S}'$ of the new game such that each player $u\in {\cal N}\setminus R$ uses the same strategy in $S'$ and $S$. Consequently, since state $S$ is a $\rho$-approximate equilibrium for the players in $R$ in the original game, state $\bar{S}$ is a $\rho$-approximate equilibrium in the new game. We have \begin{eqnarray*} \bar{c}_u(\bar{S}') &=& w_u\sum_{e\in s_u}{\sum_{t=0}^d{\bar{a}_{e,t}\Psi_t(N_e(\bar{S}'))}} = w_u\sum_{e\in s_u}{\sum_{t=0}^d{\Psi_t(N^R_e(S'))}\sum_{k=t}^d{a_{e,k}\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k-t}(N^{{\cal N}\setminus R}_e(S))}}\\ &=& w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}\sum_{t=0}^k{\left(\begin{array}{c}k\\t\end{array}\right)\Psi_{k-t}(N^{{\cal N}\setminus R}_e(S'))\Psi_t(N^R_e(S'))}}} = w_u\sum_{e\in s_u}{\sum_{k=0}^d{a_{e,k}\Psi_k(N_e(S'))}}\\ &=& \hat{c}_u(S'). \end{eqnarray*} The first equality follows by the definition of $\bar{c}_u(\bar{S}')$, the second one follows since $N_e(\bar{S}')=N^R_e(S')$ and by the definition of $\bar{a}_{e,k}$, the third one follows by exchanging the sums and since each player in ${\cal N}\setminus R$ use the same strategy in states $S$ and $S'$ (hence, $N_e^{{\cal N}\setminus R}(S)=N_e^{{\cal N}\setminus R}(S')$), the fourth one follows by equality (\ref{eq:psi-property}), and the last one follows by the definition of $\hat{c}_u(S')$.
We now show that $\bar{\Phi}(\bar{S}')=\Phi_R(S')$. We have \begin{eqnarray*} \bar{\Phi}(\bar{S}') &=& \sum_e{\sum_{t=0}^d{\frac{\bar{a}_{e,t}}{t+1}\Psi_{t+1}(N_e(\bar{S}'))}}\\ &=& \sum_e{\sum_{t=0}^d{\Psi_{t+1}(N_e^R(S'))\sum_{k=t}^d{a_{e,k}\frac{k!}{(t+1)!(t-k)!}\Psi_{k-t}(N^{{\cal N}\setminus R}_e(S))}}}\\ &=& \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\sum_{t=0}^k{\left(\begin{array}{c}k+1\\t+1\end{array}\right)\Psi_{k-t}(N^{{\cal N}\setminus R}_e(S'))\Psi_{t+1}(N_e^R(S'))}}}\\ &=& \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\sum_{t=1}^{k+1}{\left(\begin{array}{c}k+1\\t\end{array}\right)\Psi_{k+1-t}(N^{{\cal N}\setminus R}_e(S'))\Psi_{t}(N_e^R(S'))}}}\\ &=& \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\left(\sum_{t=0}^{k+1}{\left(\begin{array}{c}k+1\\t\end{array}\right)\Psi_{k+1-t}(N^{{\cal N}\setminus R}_e(S'))\Psi_{t}(N_e^R(S'))}-\Psi_{k+1}(N_e^{{\cal N}\setminus R}(S'))\right)}}\\ &=& \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e^R(S'))}}-\sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N_e^{{\cal N}\setminus R}(S'))}}\\ &=& \Phi(S')-\Phi^{{\cal N}\setminus R}(S')\\ &=& \Phi_R(S'). \end{eqnarray*} The first equality follows by the definition of $\bar{\Phi}(\bar{S}')$, the second one follows since $N_e(\bar{S}')=N^R_e(S')$ and by the definition of $\bar{a}_{e,k}$, the third one follows by exchanging the sums and since each player in ${\cal N}\setminus R$ use the same strategy in states $S$ and $S'$, the fourth one follows by simply changing the counter in the rightmost sum, the fifth one is obvious, the sixth one follows by property (\ref{eq:psi-property}), and the last two ones follow by the definition of the (partial) potentials.
Since the state $\bar{S}$ is a $\rho$-approximate equilibrium for the new game, the bounds on the $\rho$-stretch established in Lemmas \ref{lem:stretch-linear} and \ref{lem:stretch-d} imply that $\bar{\Phi}(\bar{S}) \leq \theta_d(\rho) \bar{\Phi}(\bar{S}^*)$. By our last equality above, we obtain that $\Phi_R(S) \leq \theta_d(\rho) \Phi_R(S^*)$ and the proof is complete. \hspace*{\fill}\sq
\section{Omitted proofs from Section \ref{sec:algo}}
\subsection{Proof of Lemma \ref{lem:potential-bound-per-phase}} In order to prove the key property maintained by our algorithm, we will need the following lemma which relates the $R_i$-partial potential to the cost they experience when they make their last move within phase $i$.
\begin{lemma}\label{lem:last-moves} Let $\hat{c}(u)$ denote the cost of player $u\in R_i$ just after making her last move within phase $i\geq 1$. Then, $$\Phi_{R_i}(S^i) \leq \sum_{u\in R_i}{\hat{c}(u)}.$$ \end{lemma}
\begin{proof}
Rename the players in $R_i$ as $u_1, u_2, ..., u_{|R_i|}$ so that $u_j$ is the $j$-th player that performed her last move within phase $i\geq 1$. Also, denote by $S^{i,j}$ the state in which player $u_j$ performed her last move. Let $R^{|R_i|}_i = \emptyset$ and $R^j_i = \{u_{j+1}, u_{j+2}..., u_{|R_i|}\}$ for $j=0, 1, 2, ..., |R_i|-1$. Then, \begin{eqnarray*}
\Phi_{R_i}(S^i) &=& \Phi(S^i)-\Phi^{{\cal N}\setminus R_i}(S^i) = \sum_{j=1}^{|R_i|}{\left(\Phi^{{\cal N}\setminus R_i^{j}}(S^i)-\Phi^{{\cal N}\setminus R_i^{j-1}}(S^i)\right)} = \sum_{j=1}^{|R_i|}{\Phi_{u_{j}}^{{\cal N}\setminus R_i^j}(S^i)}\\
&= & \sum_{j=1}^{|R_i|}{\Phi_{u_j}^{{\cal N}\setminus R_i^j}(S^{i,j})} \leq \sum_{j=1}^{|R_i|}{\Phi_{u_j}(S^{i,j})} = \sum_{u\in R_i}{\hat{c}(u)}. \end{eqnarray*} The first three inequalities follow by the definition of the partial potential functions and the definition of sets $R_i^j$. The fourth inequality follows by Claim \ref{claim:equal-partial-potential} since players in ${\cal N}\setminus R^j_i$ do not move after state $S^{i,j}$ and until the end of the phase. The inequality follows by Claim \ref{claim:partial-potential-bound} and the last equality follows by Claim \ref{claim:u-partial-potential} and the definition of $\hat{c}(u)$. \end{proof}
We now proceed to the proof of Lemma \ref{lem:potential-bound-per-phase}. For the sake of contradiction, we assume that $\Phi_{R_i}(S^{i-1}) > \gamma^{-1}n b_i$ and we denote by $P_i$ and $Q_i$ the set of players in $R_i$ whose last move was a $p$-move and $q$-move, respectively. Since each player in $P_i$ decreases her cost by at least $(p-1)\hat{c}(u)$ during her last move within phase $i$ (see Claim \ref{claim:partial-potential}), we have \begin{eqnarray*} \Phi_{R_{i}}(S^{i}) &\leq & \Phi_{R_{i}}(S^{i-1})- (p-1)\sum_{u \in P_i} \hat{c}(u). \end{eqnarray*} By Lemma \ref{lem:last-moves} and the fact that each player in $Q_i$ experiences a cost of at most $b_i$ when she makes her last move within phase $i$, we have \begin{eqnarray*} \sum_{u\in P_i}{\hat{c}(u)} &\geq & \Phi_{R_i}(S^i)-\sum_{u\in Q_i}{\hat{c}(u)} \geq \Phi_{R_i}(S^i)-nb_i. \end{eqnarray*} Using the last two inequalities and our assumption, we obtain that \begin{eqnarray*} \Phi_{R_i}(S^i) &\leq & \Phi_{R_i}(S^{i-1}) - (p-1) \Phi_{R_i}(S^i)+(p-1) nb_i\\ &< & (1+(p-1)\gamma)\Phi_{R_i}(S^{i-1}) - (p-1) \Phi_{R_i}(S^i) \end{eqnarray*} which implies that \begin{eqnarray*} \Phi_{R_i}(S^i) &< & \left(\frac{1}{p}+\gamma\right)\Phi_{R_i}(S^{i-1}). \end{eqnarray*}
Now, consider state $S^{i-1}$ and let $X_i$ and $Y_i$ be the sets of players in $R_i$ with cost at least $b_i$ and smaller than $b_i$, respectively. Notice that, by the definition of phase $i-1$, $S^{i-1}$ is a $q$-approximate equilibrium for the players in $X_i$. We construct a new $\Psi$-game of degree $d$ among the players in $\cal N$ as follows. The new game has all resources of the original game; the parameters $a_{e,k}$ for these resources are the same as in the original game. In addition, the new game has a new resource $e_u$ for each player $u\in Y_i$; the parameters for this resource are $a_{e_u,0}=b_i/w_u$ and $a_{e_u,k}=0$ for $k=1, ..., d$. Each player in ${\cal N} \setminus Y_i$ has the same set of strategies in the two games. The strategy set of player $u\in Y_i$ consists of the strategy $s_u$ she uses in $S^{i-1}$ as well as strategy $s'_u\cup \{e_u\}$ for each strategy $s'_u\not=s_u$ she has in the original game.
Let $\bar{S}^{i-1}$ be the state of the new game in which all players play their strategies in $S^{i-1}$. Clearly, state $\bar{S}^{i-1}$ is a $q$-approximate equilibrium for the players in $X_i$. Also, at state $\bar{S}^{i-1}$, each player $u\in Y_i$ experiences a cost equal to the cost she experiences at state $S^{i-1}$ of the original game, i.e., smaller than $b_i$. In the new game, any deviation of $u$ would include resource $e_u$ and would increase the cost of player $u$ to at least $w_u a_{e_u,0}=b_i$. Hence, $\bar{S}^{i-1}$ is a $q$-approximate equilibrium for the players of $Y_i$ as well. We use $\bar\Phi$ to denote the potential of the new game. Since the players use the same strategies in states $S^{i-1}$ and $\bar{S}^{i-1}$ and the parameters $a_{e,k}$ of the original resources are the same in both games, we have $\bar\Phi_{R_i}(\bar{S}^{i-1}) = \Phi_{R_i}(S^{i-1})$.
Now, let $\bar{S}^i$ be the state in which each player in ${\cal N}\setminus Y_i$ uses her strategy in $S^i$ and the strategies for the players in $Y_i$ are defined as follows. Let $u$ be a player of $Y_i$ and $s'_u$ be the strategy she uses at state $S^i$ of the original game. Her strategy in state $\bar{S}^i$ of the new game is $s'_u\cup \{e_u\}$ if $s'_u\not=s_u$ and $s_u$ otherwise. Observe that, by the definition of the partial potential, we have that the partial potential $\bar\Phi_{R_i}(\bar{S}^i)$ of the new game at state $\bar{S}^i$ is by at most $\sum_{u\in Y_i}{a_{e_u,0} \Psi_1(N_{e_u}(\bar{S}^i))} \leq nb_i$ higher than the partial potential of the original game at state $S^i$ (due to the contribution of the additional resources to the potential value). Hence, \begin{eqnarray*} \bar{\Phi}_{R_i}(\bar{S}^i) &\leq & \Phi_{R_i}(S^i)+nb_i <\left(\frac{1}{p}+2\gamma\right) \Phi_{R_i}(S^{i-1}) = \left(\frac{1}{p}+2\gamma\right) \bar{\Phi}_{R_i}(\bar{S}^{i-1}) = \frac{1}{\theta_d(q)} \bar{\Phi}_{R_i}(\bar{S}^{i-1}). \end{eqnarray*} So, we have identified a state $\bar{S}^{i-1}$ of the new game which is a $q$-approximate equilibrium for the players in $R_i$ and another state $\bar{S}^i$ such that the players in ${\cal N}\setminus R_i$ use the same strategies in $\bar{S}^{i-1}$ and $\bar{S}^i$ and $\bar\Phi_{R_i}(\bar{S}^{i-1}) > \theta_d(q) \bar\Phi_{R_i}(\bar{S}^i)$. This contradicts Lemma \ref{lem:stretch-partial} and, subsequently, it also contradicts our assumption $\Phi_{R_i}(S^{i-1})>\gamma^{-1}nb_i$. The lemma follows. \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:effect}} In order to prove the lemma, we will need the following technical claim. \begin{claim}\label{claim:technical} For any $\alpha,\beta\geq 0$ and integer $d\geq 1$, it holds that $(\alpha+\beta)^{d+1}\leq (1+\epsilon)\alpha^{d+1}+(1+1/\epsilon)^d d^d \beta^{d+1}$. \end{claim}
\begin{proof} Consider the function $h(\alpha) = (\alpha+\beta)^{d+1} - (1+\epsilon)\alpha^{d+1}$. By setting its derivative equal to $0$, we obtain that it is maximized for $\alpha=\beta\left((1+\epsilon)^{1/d}-1\right)^{-1}$ to the value $\frac{1+\epsilon}{((1+\epsilon)^{1/d}-1)^d}\beta^{d+1}$. By Claim \ref{claim:concave}, we have that $(1+\epsilon)^{1/d}-1 \geq \frac{\epsilon}{d(1+\epsilon)^{1-1/d}}$. Hence, $h(\alpha) \leq (1+1/\epsilon)^d d^{d} \beta^{d+1}$ as desired. \end{proof}
Now, let $k$ be an integer such that $1\leq k\leq d+1$, $A$ a multiset of reals, and $b\geq 0$. Using Lemma \ref{lem:properties}f, inequality $\alpha^\lambda+\beta^{\lambda} \leq (\alpha+\beta)^{\lambda}$ for every $\alpha,\beta\geq 0$ and $\lambda \geq 1$, and Claim \ref{claim:technical}, we have \begin{eqnarray}\nonumber \Psi_k(A\cup \{b\})-\Psi_k(A) &\leq & \left(\Psi_k(\{b\})^{1/k}+\Psi_k(A)^{1/k}\right)^k - \Psi_k(A)\\\nonumber &\leq & \left(\Psi_k(\{b\})^{\frac{1}{d+1}}+\Psi_k(A)^{\frac{1}{d+1}}\right)^{d+1} - \Psi_k(A)\\\label{eq:use-technical} &\leq & (1+\epsilon)\Psi_k(\{b\})+\xi_\epsilon \Psi_k(A). \end{eqnarray}
Also, let $Q={\cal N}\setminus (R\cup\{u\})$ and define $$\delta_{e,t} = \sum_{k=\max\{t-1,0\}}^d{\frac{a_{e,k}}{k+1}\left(\begin{array}{c}k+1\\t\end{array}\right)\Psi_{k+1-t}(N_e^Q(S))}$$ for each resource $e$ and $t=0, 1, ..., d+1$. Also, let $P$ be a possibly empty set such that $P\subseteq R\cup\{u\}$. By the definition of function $\Psi_{k+1}$ and by exchanging the sums, we have \begin{eqnarray}\nonumber \Phi^{P\cup Q}(S) &=& \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\Psi_{k+1}(N^{P\cup Q}_e(S))}}\\\nonumber &=& \sum_e{\sum_{k=0}^d{\frac{a_{e,k}}{k+1}\sum_{t=0}^{k+1}\left(\begin{array}{c}k+1\\t\end{array}\right)\Psi_{t}(N^{P}_e(S))\Psi_{k+1-t}(N^Q_e(S))}}\\\nonumber &=& \sum_e{\sum_{t=0}^{d+1}{\Psi_{t}(N^{P}_e(S))\sum_{k=\max\{t-1,0\}}^d{\frac{a_{e,k}}{k+1}\left(\begin{array}{c}k+1\\t\end{array}\right)\Psi_{k+1-t}(N_e^Q(S))}}}\\\label{eq:alternative} &=& \sum_e{\sum_{t=0}^{d+1}{\delta_{e,t}\Psi_t(N^P_e(S))}}. \end{eqnarray}
By Claim \ref{claim:u-partial-potential} and the definition of the partial potential we have $\hat{c}_u(S)=\Phi_u(S)=\Phi(S)-\Phi^{{\cal N}\setminus \{u\}}(S)$. Using the alternative expression for the potentials $\Phi(S)$ and $\Phi^{{\cal N}\setminus \{u\}}(S)$ (i.e., equality (\ref{eq:alternative})) as well as inequality (\ref{eq:use-technical}), we obtain \begin{eqnarray*} \hat{c}_u(S) &=& \sum_e{\sum_{k=0}^{d+1}{\delta_{e,k} \left(\Psi_k(N^{{R\cup\{u\}}}_e(S))-\Psi_k(N^{R}_e(S))\right)}}\\ &=& \sum_{e\in s_u}{\sum_{k=1}^{d+1}{\delta_{e,k} \left(\Psi_k(N^{{R\cup\{u\}}}_e(S))-\Psi_k(N^{R}_e(S))\right)}}\\ &\leq & \sum_{e\in s_u}{\sum_{k=1}^{d+1}{\delta_{e,k} \left((1+\epsilon)\Psi_k(N^{\{u\}}_e(S))+\xi_\epsilon\Psi_k(N^{{R}}_e(S))\right)}}. \end{eqnarray*} The second equality follows since $\Psi_0(A)=1$ for every (possibly empty) multiset of reals $A$. Using the fact again together with the fact $\Psi_k(\emptyset)=0$ for $k\geq 1$, as well as the definitions of the potentials, we obtain \begin{eqnarray*} \hat{c}_u(S) &\leq& (1+\epsilon)\sum_{e\in s_u}{\sum_{k=0}^{d+1}{\delta_{e,k} \left(\Psi_k(N^{\{u\}}_e(S))-\Psi_k(\emptyset)\right)}}+ \xi_\epsilon\sum_{e\in s_u}{\sum_{k=0}^{d+1}{\delta_{e,k} \left(\Psi_k(N^{R}_e(S))-\Psi_k(\emptyset)\right)}}\\ &\leq & (1+\epsilon)\sum_e{\sum_{k=0}^{d+1}{\delta_{e,k} \left(\Psi_k(N^{\{u\}}_e(S))-\Psi_k(\emptyset)\right)}}+ \xi_\epsilon\sum_{e}{\sum_{k=0}^{d+1}{\delta_{e,k} \left(\Psi_k(N^{R}_e(S))-\Psi_k(\emptyset)\right)}}\\ &=& (1+\epsilon) \left(\Phi^{{\cal N}\setminus R}(S)-\Phi^{{\cal N}\setminus (R\cup\{u\})}(S)\right)+ \xi_\epsilon\left(\Phi^{{\cal N}\setminus \{u\}}(S)-\Phi^{{\cal N}\setminus (R\cup \{u\})}(S)\right)\\ &=& (1+\epsilon) \Phi^{{\cal N}\setminus R}_u(S)+\xi_\epsilon\Phi^{{\cal N}\setminus \{u\}}_R(S) \end{eqnarray*} and the proof is complete. \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:deviation-cost-decreases-only-slightly}} For every $i>j$ and $\epsilon>0$, we apply Lemma \ref{lem:effect} for state $(S^{i-1}_{-u},s'_u)$, player $u$, and the set $R_i$ of players that move during phase $i$ to obtain \begin{eqnarray*} \hat{c}_u(S^{i-1}_{-u},s'_u) & \leq & (1+\epsilon) \Phi_u^{{\cal N}\setminus R_i}(S^{i-1}_{-u},s'_u) + \xi_\epsilon\Phi_{R_i}^{{\cal N}\setminus\{u\}}(S^{i-1}_{-u},s'_u)\\ & = & (1+\epsilon) \Phi_u^{{\cal N}\setminus R_i}(S^{i}_{-u},s'_u) + \xi_\epsilon\Phi_{R_i}^{{\cal N}\setminus\{u\}}(S^{i-1})\\ & \leq & (1+\epsilon) \Phi_u(S^{i}_{-u},s'_u) + \xi_\epsilon\Phi_{R_i}(S^{i-1})\\ & = & (1+\epsilon) \hat{c}_u(S^{i}_{-u},s'_u) + \xi_\epsilon\Phi_{R_i}(S^{i-1}) \end{eqnarray*} and, equivalently, \begin{eqnarray*} \hat{c}_u(S^{i}_{-u},s'_u) &\geq & \frac{1}{1+\epsilon}\hat{c}_u(S^{i-1}_{-u},s'_u)-\frac{\xi_\epsilon}{1+\epsilon}\Phi_{R_i}(S^{i-1}). \end{eqnarray*} The first equality in the derivation above follows by Claim \ref{claim:equal-partial-potential} since the players in ${\cal N}\setminus R_i$ use the same strategies in states $(S^{i-1}_{-u},s'_u)$ and $(S^i_{-u},s'_u)$ and since all players besides $u$ use the same strategies in states $(S^{i-1}_{-u},s'_u)$ and $S^{i-1}$. The second inequality follows by Claim \ref{claim:partial-potential-bound} and the last equality follows by Claim \ref{claim:u-partial-potential}.
We now set $\epsilon=(1+\gamma)^{1/m}-1$. This implies that $(1+\epsilon)^{-m}=(1+\gamma)^{-1}\geq 1-\gamma$. Also, by Claim \ref{claim:concave}, we get $\epsilon \geq \frac{\gamma}{m}(1+\gamma)^{1/m-1}\geq (m(1+\gamma^{-1}))^{-1}$ and, by the definition of the parameter $g$, $\xi_\epsilon= (1+m(1+\gamma^{-1})^d d^d -1 \leq \frac{g\gamma^3}{2n}$. Using the above inequality together with these observations, we obtain \begin{eqnarray*} \hat{c}_u(S^{m-1}_{-u},s'_u) &\geq & (1+\epsilon)^{j-m+1}\hat{c}_u(S^{j}_{-u},s'_u)-\xi_\epsilon\sum_{i=j+1}^{m-1}{(1+\epsilon)^{i-m-2}\Phi_{R_i}(S^{i-1})}\\ &\geq & (1+\epsilon)^{-m}\hat{c}_u(S^{j}_{-u},s'_u)-\xi_\epsilon\sum_{i=j+1}^{m-1}{\Phi_{R_i}(S^{i-1})}\\ &\geq &(1-\gamma)\hat{c}_u(S^{j}_{-u},s'_u)-\xi_\epsilon\sum_{i=j+1}^{m-1}{nb_i\gamma^{-1}}\\ &= & (1-\gamma)\hat{c}_u(S^{j}_{-u},s'_u)-\xi_\epsilon n\gamma^{-1}b_j\sum_{i=1}^{m-1-j}{g^{-i}}\\ &\geq & (1-\gamma) \hat{c}_u(S^{j}_{-u},s'_u)-2\xi_\epsilon n\gamma^{-1}b_jg^{-1}\\ &\geq & (1-\gamma) \hat{c}_u(S^{j}_{-u},s'_u)-\gamma^2 b_j\\ &\geq & (1-\gamma) \hat{c}_u(S^{j}_{-u},s'_u)-\gamma \hat{c}_u(S^j)/p\\ &\geq & (1-2\gamma) \hat{c}_u(S^{j}_{-u},s'_u). \end{eqnarray*} The second inequality is obvious, the third inequality follows by Lemma \ref{lem:potential-bound-per-phase} and by the relation between $\epsilon$ and $\gamma$, the equality follows by the definition of $b_i$, the fourth inequality follows since $g\geq 2$ which implies that $\sum_{i\geq 1}{g^{-i}}\leq 2g^{-1}$, the fifth inequality follows by our observation about $\xi_\epsilon$ above, the sixth inequality follows since $\gamma\leq 1/p$ (this can be seen by inspecting the values of $\gamma$ and $p$ in the definition of the algorithm and the bound on $\theta_d(1+\gamma)$ provided by Lemma \ref{lem:stretch-d}) and $\hat{c}_u(S^j)$ is higher than $b_j$ when the strategy of player $u$ is irrevocably decided at the end of phase $j$, and the last inequality follows since player $u$ has no incentive to make a $p$-move at state $S^j$. \hspace*{\fill}\sq
\subsection{Proof of Lemma \ref{lem:apx-bound}} Consider the application of the algorithm to a $\Psi$-game and let $u$ be any player whose strategy is irrevocably decided at the end of phase $j$ of the algorithm. Also, let $s'_u$ be any other strategy of this player. By Lemmas \ref{lem:cost-increases-only-slightly} and \ref{lem:deviation-cost-decreases-only-slightly} and since, by the definition of the algorithm, player $u$ has no incentive to make a $p$-move at state $S^j$, we have \begin{eqnarray*} \frac{\hat{c}_u(S^{m-1})}{\hat{c}_u(S^{m-1}_{-u},s'_u)} &\leq & \frac{(1+2\gamma)}{(1-2\gamma)}\cdot \frac{\hat{c}_u(S^j)}{\hat{c}_u(S^j_{-u},s'_u)} \leq \frac{1+2\gamma}{1-2\gamma} \left(\frac{1}{\theta_d(1+\gamma)}-2\gamma\right)^{-1}. \end{eqnarray*} Hence, the right-hand side of the above inequality upper-bounds the approximation guarantee of the algorithm. For $d=1$, the parameter $\gamma$ takes values in $(0,1/10]$. Since $\gamma\in (0,1/10]$ and $\theta_1(1+\gamma)=\frac{3+\sqrt{5}}{2}+6\gamma$ (see Lemma \ref{lem:stretch-linear}), by making simple calculations, we obtain that the algorithm computes a $\hat\rho_1$-approximate equilibrium with $$\hat\rho_1 \leq \frac{3+\sqrt{5}}{2}+110 \gamma.$$ For larger values of $d$, the algorithm uses $\gamma\in (0,\frac{1}{8\theta_d(2)}]$. Since $\theta_d(1+\gamma)$ is non-decreasing in $\gamma$, we have that $\left(\frac{1}{\theta_d(1+\gamma)}-2\gamma \right)^{-1} \leq \frac{4}{3}\theta_d(2)$. Also, we have that $\gamma < 1/34$ and hence $\frac{1+2\gamma}{1-\gamma}\leq \frac{9}{8}$. By using the value for $\theta_d(2)$ from Lemma \ref{lem:stretch-d}, we have that the algorithm computes a $\hat\rho_d$-approximate equilibrium with $\hat\rho_d \leq 3^{d+1} (d+1)^{d+1} \in d^{d+o(d)}$. \hspace*{\fill}\sq
\section{The structure of the Nash dynamics of weighted congestion games with superlinear latency functions}\label{sec:modified-alg} Algorithm 1 identifies a short sequence of best-response moves in the $\Psi$-game on input. When the degree of the $\Psi$-game is higher than $1$, the sequence may include non-improvement moves for the corresponding weighted congestion game. In this section, we present an algorithm that is applied directly to a weighted congestion game with polynomial latency functions of maximum degree $d\geq 2$. The algorithm (Algorithm 2, see the table below) is very similar to Algorithm 1; the main difference is that decisions are based on the cost of the players in the original weighted congestion game (so $\hat{c}_u$ in Algorithm 1 has been replaced by $c_u$ in Algorithm 2). In addition, the parameters $q$ and $p$ used by Algorithm 2 are higher than the ones used in Algorithm 1. The main reason is that the only available tool we have in order to guarantee convergence to an approximate equilibrium is the potential function of the corresponding $\Psi$-game. Hence, parameters $q$ and $p$ are sufficiently high so that the moves performed by Algorithm 2 are also improvement moves for the corresponding $\Psi$-game. Due to technical reasons, $\gamma$ is now restricted to smaller (but still constant) positive values. We remark that, in the description of Algorithm 2, ${\mathcal{BR}}_u$ denotes the best-response of player $u$ in the weighted congestion game.
The analysis of the algorithm will follow the same lines with the analysis of Algorithm 1. Again, the main idea in the analysis is to show that the algorithm computes an approximate equilibrium for the corresponding $\Psi$-game (with a slightly worse approximation guarantee) which is also an approximate equilibrium for the original weighted congestion game. Our main statement for Algorithm 2 is the following.
\begin{theorem}\label{thm:main-w} For every weighted congestion game with polynomial latency functions of constant maximum degree $d\geq 2$, Algorithm 2 identifies a sequence of best-response moves from any initial state to a $\rho_d$-approximate equilibrium, where $\rho_d \in d^{O(d^2)}$. The length of the sequence is polynomial in $\gamma^{-1}$ and in the number of bits in the representation of the game. \end{theorem}
In the following, we consider the application of the algorithm on a weighted congestion game with polynomial latency functions of degree $d$. We denote by $S^i$ the state computed by the algorithm after the execution of phase $i$ for $i=0, 1, ..., m-1$. Also, we use $R_i$ to denote the set of players that make at least one move during phase $i$. Similarly to the analysis of the algorithm for $\Psi$-games, we first aim to show that the algorithm computes a $d^{O(d^2)}$-approximate equilibrium for the corresponding $\Psi$-game. Then, the result will follow by Lemma \ref{lem:approx}.
\IncMargin{3em} \RestyleAlgo{boxed} \LinesNumbered \begin{algorithm} \SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up} \SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress} \SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{A weighted congestion game ${\cal G}$ with polynomial latency functions of maximum degree $d$ with a set ${\cal N}$ of $n$ players, an arbitrary initial state $S$, and $\gamma\in (0, \left(4\cdot d! \theta_d(2(d!)^2)\right)^{-1}]$} \Output{A state of ${\cal G}$}
$c_{\min}\leftarrow\min_{u\in {\cal N}}{c_u(\mathbf{0}_{-u},{\mathcal{BR}}_u(\mathbf{0}))}$\;
$c_{\max}\leftarrow \max_{u\in {\cal N}}{c_u(S)}$\;
$m\leftarrow\log{\left(c_{\max}/c_{\min}\right)}$\;
$g\leftarrow2\left(1+m(1+\gamma^{-1})\right)^{d}d^{d}n\gamma^{-3}$\;
$q\leftarrow d! (1+\gamma)$\;
$p \leftarrow \left(\frac{1}{d! \theta_d(d! q)} - 2\gamma\right)^{-1}$\; \label{alg-w:step2}
\lFor{$i\leftarrow 0$ \KwTo $m$ \label{a-w}}{$b_i \leftarrow c_{\max}g^{-i}$\;} \label{alg-w:step3}
\While{there exists a player $u\in {\cal N}$ such that $c_u(S)\in [b_1,+\infty)$ and $c_u(S_{-u},{\mathcal{BR}}_u(S))<c_u(S)/q$}
{
$S\leftarrow (S_{-u},{\mathcal{BR}}_u(S))$\;
}
$F \leftarrow \emptyset$\; \label{alg-w:step1}
\For{phase $i\leftarrow 1$ \KwTo $m-1$ \label{main-w}}{
\While{there exists a player $u\in {\cal N}\setminus F$ such that either $c_u(S) \in [b_{i}, +\infty)$ and $c_u(S_{-u},{\mathcal{BR}}_u(S))<c_u(S)/p$ or $c_u(S) \in [b_{i+1}, b_{i})$ and $c_u(S_{-u},{\mathcal{BR}}_u(S))<c_u(S)/q$}
{
$S\leftarrow (S_{-u},{\mathcal{BR}}_u(S))$\;
}
$F \leftarrow F \cup \{u \in {\cal N}\setminus F:c_u(S) \in [b_{i}, +\infty)\}$\;
} \caption{Computing approximate equilibria in weighted congestion games with polynomial latency functions.}\label{alg-w} \end{algorithm} \DecMargin{3em}
Again, the proof will use the same arguments as before. First, we prove the key property that the $R_i$-partial potential is small when the phase $i\geq 1$ starts. Then, we use this fact together with the parameters of the algorithm to prove that the running time is polynomial. The proof of the approximation guarantee for the corresponding $\Psi$-game follows. Again, the purpose of the third part of the proof is to show that for each player whose strategy is irrevocably decided at the end of phase $j$, neither her cost in the $\Psi$-game increases significantly nor the cost she would experience by deviating to another strategy decreases significantly after phase $j$. Hence, the approximation guarantee with respect to the $\Psi$-game in the final state computed by the algorithm is slightly higher than $p$. In our proofs, we use the terms $W$-cost and $\Psi$-cost in order to distinguish between the cost experienced by the players in the original weighted congestion game and the corresponding $\Psi$-game.
We will use the following fact that follows by Claim \ref{claim:approx}.
\begin{claim}\label{claim:modify} Let ${\cal G}$ be a weighted congestion game with polynomial latency functions of degree $d$ and ${\cal G}'$ its corresponding $\Psi$-game. A $\rho$-move in ${\cal G}$ is a $\rho/d!$-move in ${\cal G}'$. A $\rho$-approximate equilibrium in ${\cal G}$ is a $d!\rho$-approximate equilibrium in ${\cal G}'$. \end{claim}
\begin{proof} Let $S$ be a state of ${\cal G}$ and consider the deviation of player $u$ to strategy $s'_u$ which is a $\rho$-move. Then, \begin{eqnarray*} \hat{c}_u(S) &\geq & c_u(S) \geq \rho c_u(S_{-u},s'_u) \geq \frac{\rho}{d!} \hat{c}_u(S_{-u},s'_u). \end{eqnarray*} Now, assume that state $S$ is a $\rho$-approximate equilibrium for ${\cal G}$. For every player $u$ and every strategy $s'_u$, we have \begin{eqnarray*} \hat{c}_u(S) &\leq & d! c_u(S) \leq d! \rho c_u(S_{-u},s'_u) \leq d! \rho \hat{c}_u(S_{-u},s'_u), \end{eqnarray*} i.e., $S$ is a $d!\rho$-approximate equilibrium for game ${\cal G}'$. \end{proof}
Since the parameters $q$ and $p$ used by our algorithm are strictly higher than $d!$, the above claim immediately implies that the players that move in each step actually make an improvement move in the $\Psi$-game as well.
\subsection{Proving the key property} The key property maintained by Algorithm 2 is the following.
\begin{lemma}\label{lem:potential-bound-per-phase-w} For every phase $i\geq 1$ of Algorithm 2, it holds that $\Phi_{R_i}(S^{i-1}) \leq \gamma^{-1} n b_{i}$. \end{lemma}
\begin{proof} In order to prove it, we will need Lemma \ref{lem:last-moves}. Note that the proof of Lemma \ref{lem:last-moves} works for every sequence of improvement moves by players in a set $R_i$ in a $\Psi$-game and does not depend on any particular algorithm. Since, in every phase $i$ of Algorithm 2, the players of $R_i$ do follow improvement moves in the $\Psi$-game, the proof is valid in this case as well.
Now, the argument proceeds very similarly to the proof of Lemma \ref{lem:potential-bound-per-phase}. We include the full proof here since many minor modifications are required. Again, for the sake of contradiction, we assume that $\Phi_{R_i}(S^{i-1}) > \gamma^{-1}n b_i$ and we denote by $P_i$ and $Q_i$ the set of players in $R_i$ whose last move was a $p$-move and $q$-move (in the weighted congestion game), respectively. By Claim \ref{claim:modify}, each player in $P_i$ decreases her $\Psi$-cost by at least $(p/d!-1)\hat{c}(u)$ during her last move within phase $i$. Hence, we have \begin{eqnarray*} \Phi_{R_{i}}(S^{i}) &\leq & \Phi_{R_{i}}(S^{i-1})- (p/d!-1)\sum_{u \in P_i} \hat{c}(u). \end{eqnarray*} Now, observe that each player in $Q_i$ experiences a $W$-cost of at most $b_i$ when she makes her last move within phase $i$, i.e., a $\Psi$-cost at most $d!b_i$ (by Claim \ref{claim:approx}). Using this fact and Lemma \ref{lem:last-moves}, we have \begin{eqnarray*} \sum_{u\in P_i}{\hat{c}(u)} &\geq & \Phi_{R_i}(S^i)-\sum_{u\in Q_i}{\hat{c}(u)} \geq \Phi_{R_i}(S^i)-d! nb_i. \end{eqnarray*} Using the last two inequalities and our assumption, we obtain that \begin{eqnarray*} \Phi_{R_i}(S^i) &\leq & \Phi_{R_i}(S^{i-1}) - (p/d!-1) \Phi_{R_i}(S^i)+(p-d!) nb_i\\ &< & (1+(p-d!)\gamma)\Phi_{R_i}(S^{i-1}) - (p/d!-1) \Phi_{R_i}(S^i) \end{eqnarray*} which implies that \begin{eqnarray*} \Phi_{R_i}(S^i) &\leq & d! \left(\frac{1}{p}+\gamma\right)\Phi_{R_i}(S^{i-1}). \end{eqnarray*}
Now, we adapt the argument used in the proof of Lemma \ref{lem:potential-bound-per-phase} in order to reach the desired contradiction. Consider state $S^{i-1}$ and let $X_i$ and $Y_i$ be the sets of players in $R_i$ with $W$-cost at least $b_i$ and smaller than $b_i$, respectively. Notice that, by the definition of phase $i-1$ and Claim \ref{claim:modify}, $S^{i-1}$ is a $d!q$-approximate equilibrium for the players in $X_i$ (with respect to the $\Psi$-game). We construct a new $\Psi$-game of degree $d$ among the players in $\cal N$ as follows. The new game has all resources of the original game; the parameters $a_{e,k}$ for these resources are the same as in the original game. In addition, the new game has a new resource $e_u$ for each player $u\in Y_i$; the parameters for this resource are $a_{e_u,0}=d!b_i/w_u$ and $a_{e_u,k}=0$ for $k=1, ..., d$. Each player in ${\cal N} \setminus Y_i$ has the same set of strategies in the two games. The strategy set of player $u\in Y_i$ consists of the strategy $s_u$ she uses in $S^{i-1}$ as well as strategy $s'_u\cup \{e_u\}$ for each strategy $s'_u\not=s_u$ she has in the original game.
Let $\bar{S}^{i-1}$ be the state of the new game in which all players play their strategies in $S^{i-1}$. Clearly, state $\bar{S}^{i-1}$ is a $d!q$-approximate equilibrium for the players in $X_i$ (with respect to the $\Psi$-game). Also, at state $\bar{S}^{i-1}$, each player $u\in Y_i$ experiences a $\Psi$-cost equal to the $\Psi$-cost she experiences at state $S^{i-1}$ of the original game, i.e., smaller than $d!b_i$. In the new $\Psi$-game, any deviation of $u$ would include resource $e_u$ and would increase the $\Psi$-cost of player $u$ to at least $w_u a_{e_u,0}=d!b_i$. Hence, $\bar{S}^{i-1}$ is a $d!q$-approximate equilibrium for the players of $Y_i$ as well. We use $\bar\Phi$ to denote the potential of the new $\Psi$-game. Since the players use the same strategies in states $S^{i-1}$ and $\bar{S}^{i-1}$ and the parameters $a_{e,k}$ of the original resources are the same in both games, we have $\bar\Phi_{R_i}(\bar{S}^{i-1}) = \Phi_{R_i}(S^{i-1})$.
Now, let $\bar{S}^i$ be the state in which each player in ${\cal N}\setminus Y_i$ uses her strategy in $S^i$ and the strategies for the players in $Y_i$ are defined as follows. Let $u$ be a player of $Y_i$ and $s'_u$ be the strategy she uses at state $S^i$ of the original game. Her strategy in state $\bar{S}^i$ of the new $\Psi$-game is $s'_u\cup \{e_u\}$ if $s'_u\not=s_u$ and $s_u$ otherwise. Observe that, by the definition of the partial potential, we have that the partial potential $\bar\Phi_{R_i}(\bar{S}^i)$ of the new $\Psi$-game at state $\bar{S}^i$ is by at most $\sum_{u\in Y_i}{a_{e_u,0} \Psi_1(N_{e_u}(\bar{S}^i))} \leq d!nb_i$ higher than the partial potential of the original $\Psi$-game at state $S^i$ (due to the contribution of the additional resources to the potential value). Using these observations, our assumption, and the definition of parameter $p$, we have \begin{eqnarray*} \bar{\Phi}_{R_i}(\bar{S}^i) &\leq & \Phi_{R_i}(S^i)+d!nb_i < d!\left(\frac{1}{p}+2\gamma\right) \Phi_{R_i}(S^{i-1}) = d!\left(\frac{1}{p}+2\gamma\right) \bar{\Phi}_{R_i}(\bar{S}^{i-1}) = \frac{1}{\theta_d(d!q)} \bar{\Phi}_{R_i}(\bar{S}^{i-1}). \end{eqnarray*} So, we have identified a state $\bar{S}^{i-1}$ of the new $\Psi$-game which is a $d!q$-approximate equilibrium for the players in $R_i$ and another state $\bar{S}^i$ such that the players in ${\cal N}\setminus R_i$ use the same strategies in $\bar{S}^{i-1}$ and $\bar{S}^i$ and $\bar\Phi_{R_i}(\bar{S}^{i-1}) > \theta_d(d!q) \bar\Phi_{R_i}(\bar{S}^i)$. This contradicts Lemma \ref{lem:stretch-partial} and, subsequently, it also contradicts our assumption $\Phi_{R_i}(S^{i-1})>\gamma^{-1}nb_i$. The lemma follows. \end{proof}
\subsection{Bounding the running time} We will now use Lemma \ref{lem:potential-bound-per-phase-w} and the properties of $\Psi$-games to prove that the algorithm terminates quickly. Again, we assume that each player can efficiently compute her best-response strategy at any state (including the pseudo-state $\mathbf{0}$).
\begin{lemma}\label{lem:complexity-w} Algorithm 2 terminates after a number of steps that is polynomial in the number of bits in the representation of the game and $\gamma^{-1}$. \end{lemma}
\begin{proof} At the initial state, the W-cost of each player is at most $c_{\max}$. Hence, by Claim \ref{claim:approx}, the total $\Psi$-cost of the players and, consequently (by Lemma \ref{lem:cost-vs-potential-d}), the potential of the initial state is at most $d!n\hat{c}_{\max}$. By Claim \ref{claim:modify}, each one of the players that move during phase $0$ decreases her $\Psi$-cost and, consequently (by Theorem \ref{thm:psi-potential}), the potential by at least $(q/d!-1)b_1=\gamma g^{-1} \hat{c}_{\max}$. Hence, the total number of moves in phase $0$ is at most $d!n \gamma^{-1} g$. For $i\geq 1$, we have $\Phi_{R_i}(S^i)\leq n b_i \gamma^{-1}$ (by Lemma \ref{lem:potential-bound-per-phase-w}). By Claim \ref{claim:modify}, each one of the players in $R_i$ that move during phase $i$ decreases her $\Psi$-cost and, consequently (by Claim \ref{claim:partial-potential}), the $R_i$-partial potential by at least $(q/d!-1)b_{i+1} = b_i g^{-1} \gamma$. Hence, phase $i$ completes after at most $n g \gamma^{-2}$ moves. In total, we have at most $m n g \gamma^{-2}$ moves (since $\gamma^{-1}\geq d!$). The theorem follows by observing that $g$ depends polynomially on $m$, $n$, and $\gamma^{-1}$. \end{proof}
\subsection{Proving the approximation guarantee} The proof of the approximation guarantee will use the following lemma (it is analogous to Lemmas \ref{lem:cost-increases-only-slightly} and \ref{lem:deviation-cost-decreases-only-slightly} in the analysis of Algorithm 1).
\begin{lemma}\label{lem:cost-changes-only-slightly-w} Let $u$ be a player whose strategy was irrevocably decided at phase $j$ of Algorithm 2 and let $s'_u$ be any of her strategies. Then, $\hat{c}_u(S^{m-1}) \leq (1+2\gamma)\hat{c}_u(S^j)$ and $\hat{c}_u(S^{m-1}_{-u},s'_u) \geq (1-2\gamma)\hat{c}_u(S^j_{-u},s'_u)$. \end{lemma}
\begin{proof} The proofs of the two parts are identical to the proofs of Lemmas \ref{lem:cost-increases-only-slightly} and \ref{lem:deviation-cost-decreases-only-slightly}, respectively. All that needs to be changed is the justification of two inequalities. At the end of the proof of Lemma \ref{lem:cost-increases-only-slightly}, we used the inequality $b_j\leq \hat{c}_u(S^j)$. This holds in our case as well since the fact that the strategy of player $u$ was irrevocably decided at phase $j$ implies that $c_u(S)\geq b_j$ and, by Claim \ref{claim:approx}, we also have $\hat{c}_u(S)\geq c_u(S)$. Similarly, at the end of the proof of Lemma \ref{lem:deviation-cost-decreases-only-slightly}, we used the inequalities $\gamma b_j\leq \hat{c}_u(S^j)/p \leq \hat{c}_u(S^j_{-u},s'_u)$. What we need is essentially to show that inequality $\gamma b_j\leq \hat{c}_u(S^j_{-u},s'_u)$ holds. We have \begin{eqnarray*} \gamma b_j & \leq & \gamma c_u(S^j) \leq c_u(S^j)/p \leq c_u(S^j_{-u},s'_u) \leq \hat{c}_u(S^j_{-u},s'_u). \end{eqnarray*} The first inequality is due to the fact that the strategy of player $u$ was irrevocably decided at phase $j$, the second one follows since $\gamma\leq 1/p$, the third one follows since, at state $S^j$, player $u$ has no $p$-move in the original weighted congestion game, and the last one follows by Claim \ref{claim:approx}. \end{proof}
We are now ready to use the last lemma in order to prove the approximation guarantee. This will complete the proof of Theorem \ref{thm:main}. \begin{lemma} Algorithm 2 computes a $d^{O(d^2)}$-approximate equilibrium for the weighted congestion game on input. \end{lemma}
\begin{proof} Consider the application of the algorithm and let $u$ be any player whose strategy is irrevocably decided at the end of phase $j$ of the algorithm. Also, let $s'_u$ be any other strategy of this player. We will show that $c_u(S^{m-1}) \leq 6 (d!)^2 \theta_d(2(d!)^2) \cdot c_u(S^{m-1}_{-u},s'_u)$; the lemma will then follow since the bound for $\theta_d(2(d!)^2)$ given by Lemma \ref{lem:stretch-d} is $d^{O(d^2)}$. We have \begin{eqnarray*} \frac{c_u(S^{m-1})}{c_u(S^{m-1}_{-u},s'_u)} &\leq & d!\frac{\hat{c}_u(S^{m-1})}{\hat{c}_u(S^{m-1}_{-u},s'_u)}\\ &\leq & d! \frac{(1+2\gamma)}{(1-2\gamma)}\cdot \frac{\hat{c}_u(S^j)}{\hat{c}_u(S^j_{-u},s'_u)}\\ &\leq & \frac{(1+2\gamma)}{(1-2\gamma)} \cdot d! p\\ & =& d! \frac{1+2\gamma}{1-2\gamma} \left(\frac{1}{d!\theta_d((1+\gamma)(d!)^2)}-2\gamma\right)^{-1}\\ &\leq & d! \frac{1+2\gamma}{1-2\gamma} \left(\frac{1}{d!\theta_d(2(d!)^2)}-2\gamma\right)^{-1}\\ &\leq & 2 (d!)^2 \frac{1+2\gamma}{1-2\gamma} \theta_d(2(d!)^2)\\ &\leq & 6 (d!)^2 \theta_d(2(d!)^2). \end{eqnarray*} The first inequality follows by Claim \ref{claim:approx}, the second one follows by Lemma \ref{lem:cost-changes-only-slightly-w}, the third one follows since, at state $S^j$, player $u$ has no $p$-move in the original weighted congestion game and, consequently (by Claim \ref{claim:modify}), no $d!p$-move in the $\Psi$-game, the equality follows by the definition of parameter $p$, the fourth inequality follows since $\theta_d$ is non-decreasing, the fifth inequality follows by the definition of parameter $\gamma$, and the last inequality follows since $\gamma\leq 1/4$. \end{proof}
\end{document} | arXiv |
\begin{document}
\title{Communication Efficient Distributed Learning for \\Kernelized Contextual Bandits}
\author{\name Chuanhao Li \email [email protected] \\
\addr Department of Computer Science\\
University of Virginia\\
Charlottesville, VA 22903, USA\\
\AND
\name Huazheng Wang \email [email protected] \\
\addr School of Electrical Engineering and Computer Science\\
Oregon State University\\
Princeton, NJ 08544, USA\\
\AND
\name Mengdi Wang \email [email protected] \\
\addr Department of Electrical and Computer Engineering\\
Princeton University\\
Princeton, NJ 08544, USA\\
\AND
\name Hongning Wang \email [email protected] \\
\addr Department of Computer Science\\
University of Virginia\\
Charlottesville, VA 22903, USA
}
\maketitle
\begin{abstract} We tackle the communication efficiency challenge of learning kernelized contextual bandits in a distributed setting. Despite the recent advances in communication-efficient distributed bandit learning, existing solutions are restricted to simple models like multi-armed bandits and linear bandits, which hamper their practical utility. In this paper, instead of assuming the existence of a linear reward mapping from the features to the expected rewards, we consider non-linear reward mappings, by letting agents collaboratively search in a reproducing kernel Hilbert space (RKHS). This introduces significant challenges in communication efficiency as distributed kernel learning requires the transfer of raw data, leading to a communication cost that grows linearly w.r.t. time horizon $T$. We addresses this issue by equipping all agents to communicate via a common Nystr\"{o}m embedding that gets updated adaptively as more data points are collected. We rigorously proved that our algorithm can attain sub-linear rate in both regret and communication cost. \end{abstract}
\begin{keywords}
kernelized contextual bandit, distributed learning, communication efficiency \end{keywords}
\section{Introduction} \label{sec:intro}
Contextual bandit algorithms have been widely used for a variety of real-world applications, including recommender systems \citep{li2010contextual}, display advertisement \citep{li2010exploitation} and clinical trials \citep{durand2018contextual}. While most existing bandit solutions assume a centralized setting (i.e., all the data reside in and all the actions are taken by a central server),
there is
increasing research effort on distributed bandit learning lately \citep{wang2019distributed,dubey2020differentially,shi2021federated,huang2021federated,li2022asynchronous}, where $N$ clients, under the coordination of a central server, collaborate
to minimize the overall cumulative regret incurred over a finite time horizon $T$. In many distributed application scenarios, communication is the main bottleneck, e.g., communication in a network of mobile devices can be slower than local computation by several orders of magnitude \citep{huang2013depth}. Therefore, it is vital for distributed bandit learning algorithms to attain sub-linear rate (w.r.t. time horizon $T$) in both cumulative regret and communication cost.
However, prior works in this line of research are restricted to linear models \citep{wang2019distributed}, which could oversimplify the problem and thus leads to inferior performance in practice. In centralized setting, kernelized bandit algorithms, e.g., KernelUCB \citep{valko2013finite} and IGP-UCB \citep{chowdhury2017kernelized}, are proposed to address this issue by modeling the unknown reward mapping as a non-parametric function lying in a reproducing kernel Hilbert space (RKHS), i.e., the expected reward is linear w.r.t. an action feature map of possibly infinite dimensions. Despite the strong modeling capability of kernel method, collaborative exploration in the RKHS gives rise to additional challenges in designing a communication efficient bandit algorithm. Specifically, unlike distributed linear bandit where the clients can simply communicate the $d \times d$ sufficient statistics \citep{wang2019distributed}, where $d$ is the dimension of the input feature vector, the \emph{joint kernelized estimation} of the unknown reward function requires communicating either 1) the $p \times p$ sufficient statistics in the RKHS, where $p$ is the dimension of the RKHS that is possibly infinite, or 2) the set of input feature vectors that grows linearly w.r.t. $T$. Neither of them is practical due to the huge communication cost.
In this paper, we propose the first communication efficient algorithm for distributed kernel bandits, which tackles the aforementioned challenge via a low-rank approximation of the empirical kernel matrix.
In particular, we extended the Nystr\"{o}m method \citep{nystrom1930praktische} to distributed learning for kernelized contextual bandits. In this solution, all clients first project their local data to a finite RKHS spanned by a common dictionary, i.e., a small subset of the original dataset, and then they only need to communicate the embedded statistics for collaborative exploration. To ensure effective regret reduction after each communication round, as well as ensuring the dictionary remains representative for the entire distributed dataset throughout the learning process, the frequency of dictionary update and synchronization of embedded statistics is controlled by measuring the amount of new information each client has gained since last communication. We rigorously prove that the proposed algorithm incurs an $O(N^{2} \gamma_{NT}^{3})$ communication cost, where $\gamma_{NT}$ is the maximum information gain that is known to be $O\bigl(\log (NT) \bigr)$ for kernels with exponentially decaying eigenvalues, which includes the most commonly used Gaussian kernel, while attaining the optimal $O(\sqrt{NT}\gamma_{NT})$ cumulative regret.
\section{Related Works} \label{sec:related_works} To balance exploration and exploitation in stochastic linear contextual bandits, LinUCB algorithm \citep{li2010contextual,abbasi2011improved} is commonly used, which selects arm optimistically w.r.t. a constructed confidence set on the unknown linear reward function. By using kernels and Gaussian processes, studies in \cite{srinivas2009gaussian,valko2013finite,chowdhury2017kernelized} further extend UCB algorithms to non-parametric reward functions in RKHS, i.e., the feature map associated with each arm is possibly infinite.
Recent years have witnessed increasing research efforts in distributed bandit learning, i.e., multiple agents collaborate in pure exploration \cite{hillel2013distributed,tao2019collaborative,du2021collaborative}, or regret minimization \cite{shi2021federated,wang2019distributed,li2022asynchronous}.
They mainly differ in the relations of learning problems solved by the agents (i.e., homogeneous vs., heterogeneous) and the type of communication network (i.e., peer-to-peer (P2P) vs., star-shaped).
Most of these works assume linear reward functions, and the clients communicate by transferring the $O(d^{2})$ sufficient statistics. Korda et al. \cite{korda2016distributed} considered a peer-to-peer (P2P) communication network and assumed that the clients form clusters, i.e., each cluster is associated with a unique bandit problem.
Huang et al. \cite{huang2021federated} considered a star-shaped communication network as in our paper, but their proposed phase-based elimination algorithm only works in the fixed arm set setting.
The closest works to ours are \citep{wang2019distributed,dubey2020differentially,li2022asynchronous}, which proposed event-triggered communication protocols to obtain sub-linear communication cost over time for distributed linear bandits with a time-varying arm set. In comparison, distributed kernelized contextual bandits still remain under-explored. The only existing work in this direction \citep{dubey2020kernel} considered heterogeneous agents, where each agent is associated with an additional feature describing the task similarity between agents. However, they assumed a local communication setting, where the agent immediately shares the new raw data point to its neighbors after each interaction, and thus the communication cost is still linear over time.
Another closely related line of works is kernelized bandits with approximation, where Nystr\"{o}m method is adopted to improve computation efficiency in a centralized setting. Calandriello et al. \cite{calandriello2019gaussian} proposed an algorithm named BKB, which uses Ridge Leverage Score sampling (RLS) to re-sample a new dictionary from the updated dataset after each interaction with the environment. A recent work by Zenati et al. \cite{zenati2022efficient} further improved the computation efficiency of BKB by adopting an online sampling method to update the dictionary. However, both of them updated the dictionary at each time step to ensure the dictionary remains representative w.r.t. the growing dataset, and therefore are not applicable to our problem. This is because the dataset is stored cross clients in a distributed manner, and projecting the dataset to the space spanned by the new dictionary requires communication with all clients, which is prohibitively expensive in terms of communication. Calandriello et al. \cite{calandriello2020near} also proposed a variant of BKB, named BBKB, for batched Gaussian process optimization. BBKB only needs to update the dictionary occasionally according to an adaptive schedule, and thus partially addresses the issue mentioned above. However, as BBKB works in a centralized setting, their adaptive schedule can be computed based on the whole batch of data, while in our decentralized setting, each client can only make the update decision according to the data that is locally available. Moreover, in BBKB, all the interactions are based on a fixed model estimation over the whole batch, which is mentioned in their Appendix A.4 as a result of an inherent technical difficulty. In comparison, our proposed method effectively addresses this difficulty with improved analysis, and thus allows each client to utilize newly collected data to update its model estimation on the fly.
\section{Preliminaries} In this section, we first formulate the problem of distributed kernelized contextual bandits.
Then, as a starting point,
we propose and analyze a naive UCB-type algorithm for distributed kernelized contextual bandit problem, named {DisKernelUCB}{}. This demonstrates the challenges in designing a communication efficient algorithm for this problem, and also lays down the foundation for further improvement on communication efficiency in Section \ref{sec:method}.
\subsection{Distributed Kernelized Contextual Bandit Problem} \label{subsec:problem_formulation}
Consider a learning system with 1) $N$ clients that are responsible for taking actions and receiving feedback from the environment, and 2) a central server that coordinates the communication among the clients. The clients cannot directly communicate with each other, but only with the central server, i.e., a star-shaped communication network. Following prior works \citep{wang2019distributed,dubey2020differentially}, we assume the $N$ clients interact with the environment in a round-robin manner for a total number of $T$ rounds.
Specifically, at round $l \in [T]$, each client $i \in [N]$ chooses an arm $\bx_{t}$ from a candidate set $\cA_{t}$, and then receives the corresponding reward feedback $y_{t}=f(\bx_{t}) + \eta_{t} \in \bR$, where the subscript $t:=N (l-1) + i$ indicates this is the $t$-th interaction between the learning system and the environment, and we refer to it as time step $t$ \footnote{The meaning of index $t$ is slightly different from prior works, e.g. DisLinUCB in \citep{wang2019distributed}, but this is only to simplify the use of notation and does not affect the theoretical results}. Note that $\cA_{t}$ is a time-varying subset of $\cA \subseteq \bR^{d}$ that is possibly infinite, $f$ denotes the unknown reward function shared by all the clients, and $\eta_{t}$ denotes the noise.
Denote the sequence of indices corresponding to the interactions between client $i$ and the environment up to time $t$ as $\cN_{t}(i)=\left\{1 \leq s \leq t: i_{s}=i \right\}$ (if $s \;\mathrm{mod}\; N =0$, then $i_{s}=N$; otherwise $i_{s}=s \;\mathrm{mod}\; N$) for $t=1,2,\dots,NT$.
By definition, $|\cN_{Nl}(i)|=l,\forall l\in[T]$, i.e., the clients have equal number of interactions at the end of each round $l$.
\paragraph{Kernelized Reward Function} We consider an unknown reward function $f$ that lies in a RKHS, denoted as $\cH$, such that the reward can be equivalently written as $$y_{t} = \btheta_{\star}^{\top} \phi(\bx_{t}) + \eta_{t},$$ where $\btheta_{\star} \in \cH$ is an unknown parameter, and $\phi: \bR^{d} \rightarrow \cH$ is a known feature map associated with $\cH$. We assume $\eta_{t}$ is zero-mean $R$-sub-Gaussian conditioned on $\sigma\bigl( (\bx_{s},\eta_{s})_{s \in \cN_{t-1}(i_{t})}, \bx_{t} \bigr), \forall t$, which denotes the $\sigma$-algebra generated by client $i_{t}$'s previously pulled arms and the corresponding noise. In addition, there exists a positive definite kernel $k(\cdot,\cdot)$ associated with $\cH$,
and we assume $\forall \bx \in \cA$ that, $\lVert \bx \rVert_{k} \leq L$ and $\lVert f \rVert_{k} \leq S$ for some $L,S >0$.
\paragraph{Regret and Communication Cost} The goal of the learning system is to minimize the cumulative (pseudo) regret for all $N$ clients, i.e., $R_{NT}=\sum_{t=1}^{NT} r_{t}$, where $r_{t}=\max_{\bx \in \cA_{t}} \phi(\bx)^{\top} \btheta_{\star}- \phi(\bx_{t})^{\top} \btheta_{\star}$. Meanwhile, the learning system also wants to keep the communication cost $C_{NT}$ low, which is measured by the total number of scalars being transferred across the system up to time step $NT$.
\subsection{Distributed Kernel UCB} \label{subsec:diskernel_ucb}
As a starting point to studying the communication efficient algorithm in Section \ref{sec:method} and demonstrate the challenges in designing a communication efficient distributed kernelized contextual bandit algorithm, here we first introduce and analyze a naive algorithm where the $N$ clients collaborate on learning the exact parameters of kernel bandit, i.e., the mean and variance of estimated reward. We name this algorithm Distributed Kernel UCB, or {DisKernelUCB}{} for short, and its description is given in Algorithm \ref{alg:diskernelucb}.
\begin{algorithm}[t]
\caption{Distributed Kernel UCB ({DisKernelUCB}{})}
\label{alg:diskernelucb}
\begin{algorithmic}[1]
\STATE \textbf{Input} threshold $D>0$
\STATE \textbf{Initialize} $t_\text{last}=0$, $\cD_{0}(i)=\Delta \cD_{0}(i)=\emptyset, \forall i \in [N]$
\FOR{ round $l=1,2,...,T$}
\FOR{ client $i = 1,2,...,N$}
\STATE Client $i$ chooses arm $\bx_{t} \in \cA_{t}$ according to Eq~\eqref{eq:UCB_exact} and observes reward $y_{t}$, where $t=N(l-1)+i$
\STATE Client $i$ updates
$\bK_{\cD_{t}(i),\cD_{t}(i)}, \by_{\cD_{t}(i)}$, where $\cD_{t}(i)=\cD_{t-1}(i) \cup \{t\}$; and its upload buffer $\Delta \cD_{t}(i)=\Delta \cD_{t-1}(i) \cup \{t\}$\\
\textit{// Global Synchronization}
\IF{the event $\cU_{t}(D)$ defined in Eq~\eqref{eq:sync_event_exact} is true}
\STATE \textbf{Clients} $\forall j \in [N]$: send $\{(\bx_{s},y_{s})\}_{s \in \Delta \cD_{t}(j)}$ to server, and reset $\Delta \cD_{t}(j)=\emptyset$
\STATE \textbf{Server}: aggregates and sends back $\{(\bx_{s},y_{s})\}_{s \in [t]}$; sets $t_{\text{last}}=t$
\STATE \textbf{Clients} $\forall j \in [N]$: update $\bK_{\cD_{t}(j),\cD_{t}(j)}, \by_{\cD_{t}(i)}$, where $\cD_{t}(j)=[t]$
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic} \end{algorithm}
\paragraph{Arm Selection} For each round $l \in [T]$, when client $i \in [N]$ interacts with the environment, i.e., the $t$-th interaction between the learning system and the environment where $t=N(l-1)+i$,
it chooses arm $\bx_{t} \in \cA_{t}$ based on the UCB of the mean estimator (line 5): \begin{equation} \label{eq:UCB_exact}
\bx_{t}=\argmax_{\bx \in \cA_{t}} \hat{\mu}_{t-1,i}(\bx) + \alpha_{t-1,i} \hat{\sigma}_{t-1,i}(\bx) \end{equation} where $\hat{\mu}_{t,i}(\bx)$ and $\hat{\sigma}^{2}_{t,i}(\bx)$ denote client $i$'s local estimated mean reward for arm $\bx \in \cA$ and its variance, and $\alpha_{t-1,i}$ is a carefully chosen scaling factor to balance exploration and exploitation (see Lemma \ref{lem:regret_comm_diskernelucb} for proper choice).
To facilitate further discussion, for time step $t \in [NT]$, we denote the sequence of time indices for the data points that have been used to update client $i$'s local estimate as $\cD_{t}(i)$, which include both data points collected locally and those shared by the other clients. If the clients never communicate, $\cD_{t}(i)=\cN_{t}(i),\forall t,i$; otherwise, $\cN_{t}(i) \subset \cD_{t}(i) \subseteq [t]$, with $\cD_{t}(i)=[t]$ recovering the centralized setting, i.e., each new data point collected from the environment immediately becomes available to all the clients in the learning system.
The design matrix and reward vector for client $i$ at time step $t$ are denoted by $\bX_{\cD_{t}(i)}=[\bx_{s}]_{s \in \cD_{t}(i)}^{\top} \in \bR^{|\cD_{t}(i)| \times d}, \by_{t,i}=[y_{s}]_{s \in \cD_{t}(i)}^{\top} \in \bR^{|\cD_{t}(i)|}$, respectively. By applying the feature map $\phi(\cdot)$ to each row of $\bX_{\cD_{t}(i)}$, we obtain $\bPhi_{\cD_{t}(i)} \in \bR^{|\cD_{t}(i)| \times p}$, where $p$ is the dimension of $\cH$ and is possibly infinite. Since the reward function is linear in $\cH$, client $i$ can construct the Ridge regression estimator $\hat{\theta}_{t,i} = (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda I)^{-1} \bPhi_{\cD_{t}(i)}^{\top} \by_{t,i}$,
where $\lambda > 0$ is the regularization coefficient. This gives us the estimated mean reward and variance in primal form for any arm $\bx \in \cA$, i.e., $\hat{\mu}_{t,i}(\bx) = \phi(\bx)^{\top} \bA_{t,i}^{-1} \bb_{t,i}$ and $\hat{\sigma}_{t,i}(\bx) = \sqrt{\phi(\bx)^{\top} \bA_{t,i}^{-1} \phi(\bx) }$,
where $\bA_{t,i}=\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)}+ \lambda \bI$ and $\bb_{t,i}=\bPhi_{\cD_{t}(i)}^{\top} \by_{t,i}$. Then using the kernel trick, we can obtain their equivalence in the dual form that only involves entries of the kernel matrix, and avoids directly working on $\cH$ which is possibly infinite: \begin{align*}
\hat{\mu}_{t,i}(\bx) &= \bK_{\cD_{t}(i)}(\bx)^{\top} \bigl( \bK_{\cD_{t}(i), \cD_{t}(i)} + \lambda I \bigr)^{-1} \by_{\cD_{t}(i)} \\
\hat{\sigma}_{t,i}(\bx) &= \lambda^{-1/2}\sqrt{ k(\bx, \bx) - \bK_{\cD_{t}(i)}(\bx)^{\top} \bigl( \bK_{\cD_{t}(i), \cD_{t}(i)} + \lambda I \bigr)^{-1} \bK_{\cD_{t}(i)}(\bx) } \end{align*}
where $\bK_{\cD_{t}(i)}(\bx) = \bPhi_{\cD_{t}(i)} \phi(\bx) = [k(\bx_{s}, \bx)]^{\top}_{s \in \cD_{t}(i)} \in \bR^{|\cD_{t}(i)|}$, and $\bK_{\cD_{t}(i), \cD_{t}(i)} = \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} = [k(\bx_{s}, \bx_{s^{\prime}})]_{s, s^{\prime} \in \cD_{t}(i)} \in \bR^{|\cD_{t}(i)| \times |\cD_{t}(i)|}$.
\paragraph{Communication Protocol} To reduce the regret in future interactions with the environment, the $N$ clients need to collaborate via communication, and a carefully designed communication protocol is essential in ensuring the communication efficiency.
In prior works like DisLinUCB \cite{wang2019distributed}, after each round of interaction with the environment, client $i$ checks whether the event $\{(|\cD_{t}(i)|-|\cD_{t_\text{last}}(i)|) \log(\frac{\det(\bA_{t,i})}{\det(\bA_{t_\text{last},i})})>D\}$ is true, where $t_\text{last}$ denotes the time step of last global synchronization. If true, a new global synchronization is triggered, such that the server will require all clients to upload their sufficient statistics since $t_\text{last}$, aggregate them to compute $\{\bA_{t},\bb_{t}\}$, and then synchronize the aggregated sufficient statistics with all clients, i.e., set $\{\bA_{t,i},\bb_{t,i}\}=\{\bA_{t},\bb_{t}\},\forall i \in [N]$.
Using kernel trick, we can obtain an equivalent event-trigger in terms of the kernel matrix, \begin{equation} \label{eq:sync_event_exact}
\cU_{t}(D) = \left\{(|\cD_{t}(i_{t})|-|\cD_{t_\text{last}}(i_{t})|) \log\left(\frac{\det(\bI + \lambda^{-1} \bK_{\cD_{t}(i_{t}), \cD_{t}(i_{t})} )}{\det(\bI + \lambda^{-1} \bK_{\cD_{t}(i_{t})\setminus \Delta \cD_{t}(i_{t}), \cD_{t}(i_{t}) \setminus \Delta \cD_{t}(i_{t})})}\right) > D \right\}. \end{equation} where $D>0$ denotes the predefined threshold value. If event $\cU_{t}(D)$ is true (line 7), a global synchronization is triggered (line 7-10), where the local datasets of all $N$ clients are synchronized to $\{(\bx_{s},y_{s})\}_{s \in [t]}$. We should note that the transfer of raw data $(\bx_{s},y_{s})$ is necessary for the update of the kernel matrix and reward vector in line 6 and line 10, which will be used for arm selection at line 5. This is an inherent disadvantage of kernelized estimation in distributed settings, which, as we mentioned in Section \ref{sec:related_works}, is also true for the existing distributed kernelized bandit algorithm \cite{dubey2020kernel}. Lemma \ref{lem:regret_comm_diskernelucb} below shows that in order to obtain the optimal order of regret, {DisKernelUCB}{} incurs a communication cost linear in $T$ (proof given in the appendix), which is expensive for an online learning problem. \begin{lemma}[Regret and Communication Cost of {DisKernelUCB}{}] \label{lem:regret_comm_diskernelucb} With threshold $D=\frac{T}{N \gamma_{NT}}$,
$\alpha_{t,i}=\sqrt{\lambda} \lVert \btheta_{\star} \rVert + R \sqrt{4 \ln{N/\delta}+ 2\ln{\det( \bI + \bK_{\cD_{t}(i), \cD_{t}(i)} /\lambda)}}$, we have \begin{align*}
R_{NT} = O \bigl( \sqrt{NT}(\lVert \theta_{\star} \rVert \sqrt{\gamma_{NT}} + \gamma_{NT} ) \bigr), \end{align*} with probability at least $1-\delta$, and \begin{align*}
C_{NT} = O(T N^{2} d ). \end{align*}
where $\gamma_{NT}:=\max_{\cD \subset \cA:|\cD|=NT} \frac{1}{2}\log \det(\bK_{\cD,\cD}/\lambda + \bI)$ is the maximum information gain after $NT$ interactions \citep{chowdhury2017kernelized}. It is problem-dependent and can be bounded for specific arm set $\cA$ and kernel function $k(\cdot, \cdot)$. For example, $\gamma_{NT} = O(d \log(NT))$ for linear kernel and $\gamma_{NT} = O(\log(NT)^{d+1})$ for Gaussian kernel. \end{lemma} \begin{remark} \label{rmk:1} In the distributed linear bandit problem, to attain $O(d\sqrt{NT} \ln(NT))$ regret, DisLinUCB \citep{wang2019distributed} requires a total number of $O(N^{0.5} d \log(NT))$ synchronizations, and {DisKernelUCB}{} matches this result under linear kernel, as it requires $O(N^{0.5} \gamma_{NT})$ synchronizations. We should note that the communication cost for each synchronization in DisLinUCB is fixed, i.e., $O(Nd^{2})$ to synchronize the sufficient statistics with all the clients, so in total $C_{NT}=O(N^{1.5} d^{3} \ln(NT) )$. However, this is not the case for {DisKernelUCB}{} that needs to send raw data, because the communication cost for each synchronization in {DisKernelUCB}{} is not fixed, but depends on the number of unshared data points on each client. Even if the total number of synchronizations is small, {DisKernelUCB}{} could still incur $C_{NT}=O(TN^{2}d)$ in the worse case. Consider the extreme case where synchronization only happens once, but it happens near $NT$, then we still have $C_{NT}=O(T N^{2} d )$. The time when synchronization gets triggered depends on $\{\cA_{t} \}_{t \in [NT]}$, which is out of the control of the algorithm.
Therefore, in the following section, to improve the communication efficiency of {DisKernelUCB}{}, we propose to let each client communicate embedded statistics in some small subspace during each global synchronization. \end{remark}
\section{Approximated Distributed Kernel UCB} \label{sec:method} In this section, we propose and analyze a new algorithm that improves the communication efficiency of {DisKernelUCB}{} using the Nystr\"{o}m approximation, such that the clients only communicate the embedded statistics during event-triggered synchronizations. We name this algorithm Approximated Distributed Kernel UCB, or {Approx-DisKernelUCB}{} for short. Its description is given in Algorithm \ref{algo:Sync-KernelUCB-Approx}.
\begin{algorithm}[t]
\caption{Approximated Distributed Kernel UCB ({Approx-DisKernelUCB}{})}
\label{algo:Sync-KernelUCB-Approx}
\begin{algorithmic}[1]
\STATE \textbf{Input:} threshold $D>0$, regularization parameter $\lambda>0$, $\delta \in (0,1)$ and kernel function $k(\cdot, \cdot)$.
\STATE \textbf{Initialize} $\tilde{\mu}_{0,i}(\bx)= 0, \tilde{\sigma}_{0,i}(\bx)=\sqrt{k(\bx,\bx)}$, $\cN_{0}(i)=\cD_{0}(i)=\emptyset$, $\forall i\in[N]$; $\cS_{0}=\emptyset$, $t_{\text{last}}=0$
\FOR{ round $l=1,2,...,T$}
\FOR{ client $i = 1,2,...,N$}
\STATE [Client $i$] selects arm $\bx_{t} \in \cA_{t}$ according to Eq~\eqref{eq:UCB_approx}
and observes reward $y_{t}$, where $t:=N(l-1)+i$
\STATE [Client $i$]
updates $\bZ_{\cD_{t}(i);\cS_{t_{\text{last}}}}^{\top}\bZ_{\cD_{t}(i);\cS_{t_{\text{last}}}}$ and $\bZ_{\cD_{t}(i);\cS_{t_{\text{last}}}}^{\top} \by_{\cD_{t}(i)}$ using $\bigl(\bz(\bx_{t};\cS_{t_\text{last}}),y_{t}\bigr)$;
sets $\cN_{t}(i)=\cN_{t-1}(i)\cup \{t\}$, and $\cD_{t}(i)=\cD_{t-1}(i)\cup \{t\}$ \\
\textit{// Global Synchronization}
\IF{the event $\cU_{t}(D)$ defined in Eq~\eqref{eq:sync_event} is true}
\STATE [Clients $\forall i $] sample $\cS_{t,i}=\text{RLS}(\cN_{t}(i),\bar{q},\tilde{\sigma}_{t_\text{last},i}^{2})$, and send $\{(\bx_{s},y_{s})\}_{s \in \cS_{t,i}}$ to server
\STATE [Server] aggregates and sends $\{(\bx_{s},y_{s})\}_{s \in \cS_{t}}$ back to all clients, where $\cS_{t}=\cup_{i \in [N]} \cS_{t,i}$
\STATE [Clients $\forall i$] compute and send $\{\bZ_{\cN_{t}(i);\cS_{t}}^{\top}\bZ_{\cN_{t}(i);\cS_{t}}, \bZ_{\cN_{t}(i);\cS_{t}}^{\top}\by_{\cN_{t}(i)}\}$ to server
\STATE [Server] aggregates
$\sum_{i=1}^{N}\bZ_{\cN_{t}(i);\cS_{t}}^{\top}\bZ_{\cN_{t}(i);\cS_{t}}, \sum_{i=1}^{N}\bZ_{\cN_{t}(i);\cS_{t}}^{\top}\by_{\cN_{t}(i)}$
and sends it back
\STATE [Clients $\forall i$]
updates $\bZ_{\cD_{t}(i);\cS_{t}}^{\top}\bZ_{\cD_{t}(i);\cS_{t}}$ and $\bZ_{\cD_{t}(i);\cS_{t}}^{\top} \by_{\cD_{t}(i)}$; sets $\cD_{t}(i)=\cup_{i=1}^{N}\cN_{t}(i)=[t]$ and $t_\text{last}=t$
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic} \end{algorithm}
\subsection{Algorithm}
\paragraph{Arm selection} For each round $l \in [T]$, when client $i \in [N]$ interacts with the environment, i.e., the $t$-th interaction between the learning system and the environment where $t:=N(l-1)+i$,
instead of using the UCB for the exact estimator in Eq~\eqref{eq:UCB_exact}, client $i$ chooses arm $\bx_{t} \in \cA_{t}$ that maximizes the UCB for the approximated estimator (line 5): \begin{equation} \label{eq:UCB_approx}
\bx_{t}=\argmax_{\bx \in \cA_{t,i}} \tilde{\mu}_{t-1,i}(\bx) + \alpha_{t-1,i} \tilde{\sigma}_{t-1,i}(\bx) \end{equation} where $\tilde{\mu}_{t-1,i}(\bx)$ and $\tilde{\sigma}_{t-1,i}(\bx)$ are approximated using Nyestr\"{o}m method, and the statistics used to compute these approximations are much more efficient to communicate as they scale with the maximum information gain $\gamma_{NT}$ instead of $T$.
Specifically, Nystr\"{o}m method works by projecting some original dataset $\cD$ to the subspace defined by a small representative subset $\cS \subseteq \cD$, which is called the dictionary. The orthogonal projection matrix is defined as \begin{align*}
\bP_{\cS} = \bPhi_{\cS}^{\top} \bigl( \bPhi_{\cS} \bPhi_{\cS}^{\top} \bigr)^{-1} \bPhi_{\cS}=\bPhi_{\cS}^{\top} \bK_{\cS,\cS}^{-1} \bPhi_{\cS} \in \bR^{p \times p} \end{align*} We then take eigen-decomposition of $\bK_{\cS,\cS} = \bU \mathbf{\Lambda} \bU^{\top}$ to rewrite the orthogonal projection as $\bP_{\cS}=\bPhi_{\cS}^{\top} \bU \mathbf{\Lambda}^{-1/2} \mathbf{\Lambda}^{-1/2} \bU^{\top} \bPhi_{\cS}$, and define the Nystr\"{o}m embedding function \begin{align*}
z(\bx;\cS) = \bP_{\cS}^{1/2} \phi(\bx) =\mathbf{\Lambda}^{-1/2} \bU^{\top} \bPhi_{\cS} \phi(\bx) = \bK_{\cS,\cS}^{-1/2} \bK_{\cS}(\bx) \end{align*}
which maps the data point $\bx$ from $\bR^{d}$ to $\bR^{|\cS|}$.
Therefore, we can approximate the Ridge regression estimator in Section \ref{subsec:diskernel_ucb} as $\tilde{\theta}_{t,i} = \tilde{\bA}_{t,i}^{-1} \tilde{\bb}_{t,i}$,
where $\tilde{\bA}_{t,i}=\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} \bP_{\cS}+ \lambda \bI $, and $\tilde{\bb}_{t,i}=\bP_{\cS}\bPhi_{\cD_{t}(i)}^{\top} \by_{\cD_{t}(i)}$, and thus the approximated mean reward and variance in Eq~\eqref{eq:UCB_approx} can be expressed as $\tilde{\mu}_{t,i}(\bx) = \phi(\bx)^{\top} \tilde{\bA}_{t,i}^{-1} \tilde{\bb}_{t,i}$ and $\tilde{\sigma}_{t,i}(\bx) = \sqrt{ \phi(\bx)^{\top} \tilde{\bA}_{t,i}^{-1} \phi(\bx)}$,
and their kernelized representation are (see appendix for detailed derivation) \begin{align*}
\tilde{\mu}_{t,i}(\bx) & = z(\bx;\cS)^{\top} \bigl( \bZ_{\cD_{t}(i);\cS}^{\top}\bZ_{\cD_{t}(i);\cS} + \lambda \bI\bigr)^{-1} \bZ_{\cD_{t}(i);\cS}^{\top} \by_{\cD_{t}(i)} \\
\tilde{\sigma}_{t,i}(\bx) & = \lambda^{-1/2}\sqrt{ k(\bx, \bx) - z(\bx;\cS)^{\top} \bZ_{\cD_{t}(i);\cS}^{\top}\bZ_{\cD_{t}(i);\cS} [ \bZ_{\cD_{t}(i);\cS}^{\top}\bZ_{\cD_{t}(i);\cS} + \lambda \bI]^{-1} z(\bx|\cS) } \end{align*}
where $\bZ_{\cD_{t}(i);\cS} \in \bR^{|\cD_{t}(i)|\times |\cS|}$ is obtained by applying $z(\cdot;\cS)$ to each row of $\bX_{\cD_{t}(i)}$, i.e., $\bZ_{\cD_{t}(i);\cS}= \bPhi_{\cD_{t}(i)} \bP_{\cS}^{1/2}$.
We can see that
the computation of $\tilde{\mu}_{t,i}(\bx)$ and $\tilde{\sigma}_{t,i}(\bx)$ only requires the embedded statistics: matrix $\bZ_{\cD_{t}(i);\cS}^{\top}\bZ_{\cD_{t}(i);\cS} \in \bR^{|\cS| \times |\cS|}$ and vector $\bZ_{\cD_{t}(i);\cS}^{\top} \by_{\cD_{t}(i)} \in \bR^{|\cS|}$, which, as we will show later, makes joint kernelized estimation among $N$ clients much more efficient in communication.
After obtaining the new data point $(\bx_{t}, y_{t})$, client $i$ immediately updates both $\tilde{\mu}_{t-1,i}(\bx)$ and $\tilde{\sigma}_{t-1,i}(\bx)$ using the newly collected data point $(\bx_{t},y_{t})$, i.e., by projecting $\bx_{t}$ to the finite dimensional RKHS spanned by $\bPhi_{\cS_{t_\text{last}}}$ (line 6). Recall that, we use $\cN_{t}(i)$ to denote the sequence of indices for data collected by client $i$, and denote by $\cD_{t}(i)$ the sequence of indices for data that has been used to update client $i$'s model estimation $\tilde{\mu}_{t,i}$. Therefore, both of them need to be updated to include time step $t$.
\paragraph{Communication Protocol}
With the approximated estimator, the size of message being communicated across the learning system is reduced. However, a carefully designed event-trigger is still required to minimize the total number of global synchronizations up to time $NT$. Since the clients can no longer evaluate the exact kernel matrices in Eq~\eqref{eq:sync_event_exact}, we instead use the event-trigger in Eq~\eqref{eq:sync_event}, which can be computed using the approximated variance from last global synchronization as,
\begin{equation} \label{eq:sync_event}
\cU_{t}(D) = \left\{\sum_{s \in \cD_{t}(i) \setminus \cD_{t_\text{last}}(i)} \tilde{\sigma}_{t_\text{last},i}^{2}(\bx_{s}) > D\right\} \end{equation} Similar to Algorithm \ref{alg:diskernelucb}, if Eq \eqref{eq:sync_event} is true, global synchronization is triggered, where both the dictionary and the embedded statistics get updated. During synchronization, each client first samples a subset $\cS_{t}(i)$ from $\cN_{t}(i)$ (line 8) using Ridge Leverage Score sampling (RLS) \citep{calandriello2019gaussian,calandriello2020near}, which is given in Algorithm \ref{alg:rls}, and then sends $\{(\bx_{s},y_{s})\}_{s \in \cS_{t}(i)}$ to the server.
The server aggregates the received local subsets to construct a new dictionary $\{(\bx_{s},y_{s})\}_{s \in \cS_{t}}$, where $\cS_{t}=\cup_{i=1}^{N}\cS_{t}(i)$, and then sends it back to all $N$ clients (line 9). Finally, the $N$ clients use this updated dictionary to re-compute the embedded statistics of their local data, and then synchronize it with all other clients via the server (line 10-12).
\begin{algorithm}[h]
\caption{$\quad \text{Ridge Leverage Score Sampling (RLS)}$} \label{alg:rls}
\begin{algorithmic}[1]
\STATE \textbf{Input:} dataset $\cD$, scaling factor $\bar{q}$, (possibly delayed and approximated) variance function $\tilde{\sigma}^{2}(\cdot)$
\STATE \textbf{Initialize} new dictionary $\cS=\emptyset$
\FOR{$s \in \cD$}
\STATE Set $\tilde{p}_{s}=\bar{q}\tilde{\sigma}^{2}(\bx_{s})$
\STATE Draw $q_{s} \sim \text{Bernoulli}(\tilde{p}_{s})$
\STATE If $q_{s}=1$, add $s$ into $\cS$
\ENDFOR
\STATE \textbf{Output:} $\cS$
\end{algorithmic} \end{algorithm} Intuitively, in Algorithm \ref{algo:Sync-KernelUCB-Approx}, the clients first agree upon a common dictionary $\cS_{t}$ that serves as a good representation of the whole dataset at the current time $t$, and then project their local data to the subspace spanned by this dictionary before communication, in order to avoid directly sending the raw data as in Algorithm \ref{alg:diskernelucb}. Then using the event-trigger, each client monitors the amount of new knowledge it has gained through interactions with the environment from last synchronization. When there is a sufficient amount of new knowledge, it will inform all the other clients to perform a synchronization. As we will show in the following section, the size of $\cS_{t}$ scales linearly w.r.t. the maximum information gain $\gamma_{NT}$, and therefore it improves both the local computation efficiency on each client, and the communication efficiency during the global synchronization.
\subsection{Theoretical Analysis} \label{subsec:theoretical_analysis} Denote the sequence of time steps when global synchronization is performed, i.e., the event $\cU_{t}(D)$ in Eq~\eqref{eq:sync_event} is true, as $\{t_{p}\}_{p=1}^{B}$, where $B \in [NT]$ denotes the total number of global synchronizations. Note that in Algorithm \ref{algo:Sync-KernelUCB-Approx}, the dictionary is only updated during global synchronization, e.g., at time $t_{p}$, the dictionary $\{(\bx_{s}, y_{s})\}_{s \in \cS_{t_{p}}}$ is sampled from the whole dataset $\{(\bx_{s}, y_{s})\}_{s \in [t_{p}]}$ in a distributed manner, and remains fixed for all the interactions happened at $t \in [t_{p}+1,t_{p+1}]$. Moreover, at time $t_{p}$, all the clients synchronize their embedded statistics, so that $\cD_{t_{p}}(i)=[t_{p}],\forall i \in [N]$.
Since Algorithm \ref{algo:Sync-KernelUCB-Approx} enables local update on each client, for time step $t \in [t_{p}+1,t_{p}]$, new data points are collected and added into $\cD_{t}(i)$, such that $\cD_{t}(i) \supseteq [t_{p}]$. This \textit{decreases} the approximation accuracy of $\cS_{t_{p}}$, as new data points may not be well approximated by $\cS_{t_{p}}$. For example, in extreme cases, the new data could be orthogonal to the dictionary. To formally analyze the accuracy of the dictionary, we adopt the definition of $\epsilon$-accuracy from \cite{calandriello2017second}. Denote by $\bar{\bS}_{t,i} \in \bR^{|\cD_{t}(i)| \times |\cD_{t}(i)|}$ a diagonal matrix, with its $s$-th diagonal entry equal to $\frac{1}{\sqrt{\tilde{p}_{s}}}$ if $s \in \cS_{t_{p}}$ and $0$ otherwise. Then if \begin{align*}
(1-\epsilon_{t,i}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI) \preceq \bPhi_{\cD_{t}(i)}^{\top} \bar{\bS}_{t,i}^{\top} \bar{\bS}_{t,i}\bPhi_{\cD_{t}(i)} + \lambda \bI \preceq (1+\epsilon_{t,i}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI), \end{align*}
we say the dictionary $\{(\bx_{s},y_{s})\}_{s \in \cS_{t_{p}}}$ is $\epsilon_{t,i}$-accurate w.r.t. dataset $\{(\bx_{s},y_{s})\}_{s \in \cD_{t}(i)}$.
As shown below, the accuracy of the dictionary for Nystr\"{o}m approximation is essential as it affects the width of the confidence ellipsoid, and thus affects the cumulative regret. Intuitively, in order to ensure its accuracy throughout the learning process, we need to 1) make sure the RLS procedure in line 8 of Algorithm \ref{algo:Sync-KernelUCB-Approx} that happens at each global synchronization produces a representative set of data samples, and 2) monitor the extent to which the dictionary obtained in previous global synchronization has degraded over time, and when necessary, trigger a new global synchronization to update it. Compared with prior work that freezes the model in-between consecutive communications \cite{calandriello2020near}, the analysis of $\epsilon$-accuracy for {Approx-DisKernelUCB}{} is unique to our paper and the result is presented below. \begin{lemma} \label{lem:dictionary_accuracy_global}
With $\bar{q}=6\frac{1+\epsilon}{1-\epsilon} \log(4NT/\delta)/\epsilon^{2}$, for some $\epsilon \in [0,1)$, and threshold $D>0$, Algorithm \ref{algo:Sync-KernelUCB-Approx} guarantees that the dictionary is accurate with constant $\epsilon_{t,i}:=\bigl(\epsilon+1- \frac{1}{1+\frac{1+\epsilon}{1-\epsilon}D} \bigr)$, and its size $|\cS_{t}| = O(\gamma_{NT})$ for all $t \in [NT]$. \end{lemma}
Based on Lemma \ref{lem:dictionary_accuracy_global}, we can construct the following confidence ellipsoid for unknown parameter $\theta_{\star}$.
\begin{lemma}[Confidence Ellipsoid of Approximated Estimator] \label{lem:confidence_ellipsoid_approx}
Under the condition that $\bar{q}=6\frac{1+\epsilon}{1-\epsilon} \log(4NT/\delta)/\epsilon^{2}$, for some $\epsilon \in [0,1)$, and threshold $D>0$, with probability at least $1-\delta$, we have $\forall t,i$ that \small \begin{align*}
\lVert\tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}}
& \leq \Bigl( \frac{1}{\sqrt{-\epsilon + 1/(\frac{1+\epsilon}{1-\epsilon}D)}} + 1 \Bigr) \sqrt{\lambda} \lVert \btheta_{\star} \rVert + 2R \sqrt{\ln{N/\delta}+ \gamma_{NT}} := \alpha_{t,i}. \end{align*}
\end{lemma}
Using Lemma \ref{lem:confidence_ellipsoid_approx}, we obtain the regret and communication cost upper bound of {Approx-DisKernelUCB}{}, which is given in Theorem \ref{thm:regret_comm_sync} below. \begin{theorem}[Regret and Communication Cost of {Approx-DisKernelUCB}{}] \label{thm:regret_comm_sync} Under the same condition as Lemma \ref{lem:confidence_ellipsoid_approx}, and by setting $D=\frac{1}{N}, \epsilon < \frac{1}{3}$, we have \begin{align*}
R_{NT} = O \bigl( \sqrt{NT}(\lVert \theta_{\star} \rVert \sqrt{\gamma_{NT}} + \gamma_{NT} ) \bigr)
\end{align*} with probability at least $1-\delta$, and \begin{align*}
C_{NT} = O\bigl( N^{2} \gamma_{NT}^{3} \bigr)
\end{align*} \end{theorem} Here we provide a proof sketch for Theorem \ref{thm:regret_comm_sync}, and the complete proof can be found in appendix. \begin{proof}[Proof Sketch] Similar to the analysis of {DisKernelUCB}{} in Section \ref{subsec:diskernel_ucb} and DisLinUCB from \citep{wang2019distributed}, the cumulative regret incurred by {Approx-DisKernelUCB}{} can be decomposed in terms of `good' and `bad' epochs, and bounded separately. Here an epoch refers to the time period in-between two consecutive global synchronizations, e.g., the $p$-th epoch refers to $[t_{p-1}+1,t_{p}]$. Now consider an imaginary centralized agent that has immediate access to each data point in the learning system, and denote by $A_{t}=\sum_{s=1}^{t} \phi_{s} \phi_{s}^{\top}$ for $t \in [NT]$ the matrix constructed by this centralized agent. We call the $p$-th epoch a good epoch if $\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) \leq 1$,
otherwise it is a bad epoch. Note that
$\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{1}],[t_{1}]})}{\det(\bI)})+\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{2}],[t_{2}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{1}],[t_{1}]})})+\dots+\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[NT],[NT]})}{\det(\bI + \lambda^{-1}\bK_{[t_{B}],[t_{B}]})}) = \ln(\det(\bI + \lambda^{-1} \bK_{[NT],[NT]})) \leq 2 \gamma_{NT}$, where the last equality is due to the matrix determinant lemma, and the last inequality is by the definition of the maximum information gain $\gamma_{NT}$ in Lemma \ref{lem:regret_comm_diskernelucb}. Then based on the pigeonhole principle, there can be at most $2 \gamma_{NT}$ bad epochs.
By combining Lemma \ref{lem:rls} and Lemma \ref{lem:confidence_ellipsoid_approx},
we can bound the cumulative regret incurred during all good epochs, i.e., $R_{good} = O(\sqrt{NT}\gamma_{NT})$, which matches the optimal regret attained by the KernelUCB algorithm in centralized setting.
Our analysis deviates from that of {DisKernelUCB}{} in the bad epochs, because of the difference in the event-trigger. Previously, the event-trigger of {DisKernelUCB}{} directly bounds the cumulative regret each client incurs during a bad epoch, i.e., \small
$\sum_{t \in \cD_{t_{p}}(i) \setminus \cD_{t_{p-1}}(i) }\hat{\sigma}_{t-1,i}(\bx_{t}) \leq \sqrt{ (|\cD_{t_{p}}(i)|-|\cD_{t_{p-1}}(i)|) \log\left(\det(\bI + \lambda^{-1} \bK_{\cD_{t}(i_{t}), \cD_{t}(i_{t})} )/\det(\bI + \lambda^{-1} \bK_{\cD_{t}(i_{t})\setminus \Delta \cD_{t}(i_{t}), \cD_{t}(i_{t}) \setminus \Delta \cD_{t}(i_{t})})\right)} < \sqrt{D}$.
\normalsize However, the event trigger of {Approx-DisKernelUCB}{} only bounds part of it, i.e., $\sum_{t \in \cD_{t_{p}}(i) \setminus \cD_{t_{p-1}}(i) }\tilde{\sigma}_{t-1,i}(\bx_{t}) \leq \sqrt{ (|\cD_{t_{p}}(i)|-|\cD_{t_{p-1}}(i)|) D } $, which leads to $R_{bad}=O(\sqrt{T}\gamma_{NT} N \sqrt{D})$ that is slightly worse than that of {DisKernelUCB}{}, i.e., a $\sqrt{T}$ factor in place of the $\sqrt{\gamma_{NT}}$ factor. By setting $D=1/N$, we have $R_{NT}=O(\sqrt{NT}\gamma_{NT})$. Note that, to make sure $\epsilon_{t,i}=\bigl(\epsilon+1- \frac{1}{1+\frac{1+\epsilon}{1-\epsilon} \frac{1}{N}} \bigr) \in [0,1)$ is still well-defined, we can set $\epsilon < 1/3$.
For communication cost analysis, we bound the total number of epochs $B$ by upper bounding the total number of summations like $\sum_{s = t_{p-1}+1}^{t_{p}} \hat{\sigma}^{2}_{t_{p-1}}(\bx_{s})$, over the time horizon $NT$. Using Lemma \ref{lem:rls}, our event-trigger in Eq~\eqref{eq:sync_event} provides a lower bound $\sum_{s = t_{p-1}+1}^{t_{p}} \hat{\sigma}^{2}_{t_{p-1}}(\bx_{s}) \geq \frac{1-\epsilon}{1+\epsilon}D$. Then in order to apply the pigeonhole principle, we continue to upper bound the summation over all epochs, $\sum_{p=1}^{B} \sum_{s = t_{p-1}+1}^{t_{p}} \hat{\sigma}^{2}_{t_{p-1}}(\bx_{s}) = \sum_{p=1}^{B} \sum_{s = t_{p-1}+1}^{t_{p}} \hat{\sigma}^{2}_{s-1}(\bx_{s}) \frac{\hat{\sigma}^{2}_{t_{p-1}}(\bx_{s})}{\hat{\sigma}^{2}_{s-1}(\bx_{s})}$ by deriving a uniform bound for the ratio $\frac{\hat{\sigma}^{2}_{t_{p-1}}(\bx_{s})}{\hat{\sigma}^{2}_{s-1}(\bx_{s})} \leq \frac{\hat{\sigma}^{2}_{t_{p-1}}(\bx_{s})}{\hat{\sigma}^{2}_{t_{p}}(\bx_{s})} \leq 1 + \sum_{s=t_{p-1}+1}^{t_{p}} \hat{\sigma}^{2}_{t_{p-1}}(\bx_{s}) \leq 1 + \frac{1+\epsilon}{1-\epsilon} \sum_{s=t_{p-1}+1}^{t_{p}} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{s}) $ in terms of the communication threshold $D$ on each client. This leads to the following upper bound about the total number of epochs $B \leq \frac{1+\epsilon}{1-\epsilon}[ \frac{1}{D} + \frac{1+\epsilon}{1-\epsilon} (N + L^{2}/(\lambda D)) ] 2 \gamma_{NT}$, and with $D=1/N$, we have $C_{NT} \leq B \cdot N \gamma_{NT}^{2} = O(N^{2} \gamma_{NT}^{3}) $, which completes the proof.
\end{proof}
\begin{remark} \label{rmk:2} Compared with {DisKernelUCB}{}'s $O(TN^{2}d)$ communication cost, {Approx-DisKernelUCB}{} removes the linear dependence on $T$, but introduces an additional $\gamma_{NT}^{3}$ dependence due to the communication of the embedded statistics. In situations where $\gamma_{NT} \ll T^{1/3} d^{1/3}$, {DisKernelUCB}{} is preferable. As mentioned in Lemma \ref{lem:regret_comm_diskernelucb}, the value of $\gamma_{NT}$, which affects how much the data can be compressed,
depends on the specific arm set of the problem and the kernel function of the choice. By Mercer's Theorem, one can represent the kernel using its eigenvalues, and $\gamma_{NT}$ characterizes how fast its eigenvalues decay. Vakili et al. \citep{vakili2021information} showed that for kernels whose eigenvalues decay exponentially, i.e., $\lambda_{m}=O(\exp(- m^{\beta_{e}}))$, for some $\beta_{e}>0$, $\gamma_{NT}=O(\log^{1+\frac{1}{\beta_{e}}}(NT))$. In this case, {Approx-DisKernelUCB}{} is far more efficient than {DisKernelUCB}{}. This includes Gaussian kernel, which is widely used for GPs and SVMs. For kernels that have polynomially decaying eigenvalues, i.e., $\lambda_{m}=O(m^{-\beta_{p}})$, for some $\beta_{p} > 1$, $\gamma_{NT} = O(T^{\frac{1}{\beta_{p}}} \log^{1-\frac{1}{\beta_{p}}}(NT))$. Then as long as $\beta_{p} > 3$, {Approx-DisKernelUCB}{} still enjoys reduced communication cost.
\end{remark}
\section{Experiments}
In order to evaluate {Approx-DisKernelUCB}{}'s effectiveness in reducing communication cost, we performed extensive empirical evaluations on both synthetic and real-world datasets, and the results (averaged over 3 runs) are reported in Figure \ref{fig:synthetic_exp_results}, \ref{fig:uci_exp_results} and \ref{fig:recommendation_exp_results}, respectively. We included {DisKernelUCB}{},
DisLinUCB \citep{wang2019distributed}, OneKernelUCB, and NKernelUCB \citep{chowdhury2017kernelized} as baselines, where One-KernelUCB learns a shared bandit model across all clients' aggregated data where data aggregation happens immediately after each new data point becomes available, and N-KernelUCB learns a separated bandit model for each client with no communication. For all the kernelized algorithms, we used the Gaussian kernel $k(x, y) = \exp(-\gamma \lVert x-y \rVert^{2})$. We did a grid search of $\gamma \in \{0.1, 1, 4\}$ for kernelized algorithms, and set $D=20$ for DisLinUCB and {DisKernelUCB}{}, $D=5$ for {Approx-DisKernelUCB}{}. For all algorithms, instead of using their theoretically derived exploration coefficient $\alpha$, we followed the convention \cite{li2010contextual,zhou2020neural} to use grid search for $\alpha$ in $\{0.1, 1, 4\}$. Due to space limit, here we only present the experiment results and discussions. Details about the experiment setup are presented in appendix.
\begin{figure}
\caption{Experiment results on synthetic datasets with different reward function $f(\bx)$.}
\label{fig:a}
\label{fig:b}
\label{fig:synthetic_exp_results}
\end{figure}
\begin{figure}
\caption{Experiment results on UCI datasets.}
\label{fig:c}
\label{fig:d}
\label{fig:e}
\label{fig:uci_exp_results}
\end{figure}
\begin{figure}
\caption{Experiment results on MovieLens \& Yelp datasets.}
\label{fig:f}
\label{fig:g}
\label{fig:recommendation_exp_results}
\end{figure}
When examining the experiment results presented in Figure \ref{fig:synthetic_exp_results}, \ref{fig:uci_exp_results} and \ref{fig:recommendation_exp_results}, we can first look at the cumulative regret and communication cost of OneKernelUCB and NKernelUCB, which correspond to the two extreme cases where the clients communicate in every time step to learn a shared model, and each client learns its own model independently with no communication, respectively. OneKernelUCB achieves the smallest cumulative regret in all experiments, while also incurring the highest communication cost, i.e., $O(T N^{2} d)$. This demonstrates the need of efficient data aggregation across clients for reducing regret. Second, we can observe that {DisKernelUCB}{} incurs the second highest communication cost in all experiments due to the transfer of raw data, as we have discussed in Remark \ref{rmk:1}, which makes it prohibitively expensive for distributed setting. On the other extreme, we can see that DisLinUCB incurs very small communication cost thanks to its closed-form solution, but fails to capture the complicated reward mappings in most of these datasets, e.g. in Figure \ref{fig:a}, \ref{fig:d} and \ref{fig:f}, it leads to even worse regret than NKernelUCB that learns a kernelized bandit model independently for each client.
In comparison, the proposed {Approx-DisKernelUCB}{} algorithm enjoys the best of both worlds in most cases, i.e., it can take advantage of the superior modeling power of kernels to reduce regret, while only requiring a relatively low communication cost for clients to collaborate. On all the datasets, {Approx-DisKernelUCB}{} achieved comparable regret with {DisKernelUCB}{} that maintains exact kernelized estimators, and sometimes even getting very close to OneKernelUCB, e.g., in Figure \ref{fig:b} and \ref{fig:c}, but its communication cost is only slightly higher than that of DisLinUCB.
\section{Conclusion} In this paper, we proposed the first communication efficient algorithm for distributed kernel bandits using Nystr\"{o}m approximation. Clients in the learning system project their local data to a finite RKHS spanned by a shared dictionary, and then communicate the embedded statistics for collaborative exploration. To ensure communication efficiency, the frequency of dictionary update and synchronization of embedded statistics are controlled by an event-trigger.
The algorithm is proved to incur $O(N^{2} \gamma_{NT}^{3})$ communication cost, while attaining the optimal $O(\sqrt{NT}\gamma_{NT})$ cumulative regret.
We should note that the total number of synchronizations required by {Approx-DisKernelUCB}{} is $N \gamma_{NT}$, which is $\sqrt{N}$ worse than {DisKernelUCB}{}.
An important future direction of this work is to investigate whether this part can be further improved. It is also interesting to extend the proposed algorithm to
P2P setting, at the absence of a central server to coordinate the update of the shared dictionary and the exchange of embedded statistics. Due to the delay in propagating messages, it may be beneficial to utilize possible local structures in the network of clients, and approximate each block of the kernel matrix separately \citep{si2014memory}, i.e., each block corresponds to a group of clients, instead of directly approximating the complete matrix.
\appendix \section{Technical Lemmas}
\begin{lemma}[Lemma 12 of \citep{abbasi2011improved}] \label{lem:quadratic_det_inequality} Let $A$, $B$ and $C$ be positive semi-definite matrices with finite dimension, such that $A=B+C$. Then, we have that: \begin{align*}
\sup_{\bx \neq \textbf{0}} \frac{\bx^{\top} A \bx}{\bx^{\top} B \bx} \leq \frac{\det(A)}{\det(B)} \end{align*} \end{lemma} \begin{lemma}[Extension of Lemma \ref{lem:quadratic_det_inequality} to kernel matrix] \label{lem:quadratic_det_inequality_infinite} Define positive definite matrices $A=\lambda \bI + \bPhi_{1}^{\top}\bPhi_{1} + \bPhi_{2}^{\top}\bPhi_{2}$ and $B=\lambda \bI + \bPhi_{1}^{\top}\bPhi_{1}$, where $\bPhi_{1}^{\top}\bPhi_{1},\bPhi_{2}^{\top}\bPhi_{2} \in \bR^{p \times p}$ and $p$ is possibly infinite. Then, we have that: \begin{align*}
\sup_{\phi \neq \textbf{0}} \frac{\phi^{\top} A \phi}{\phi^{\top} B \phi} \leq \frac{\det(\bI+ \lambda^{-1} \bK_{A})}{\det(\bI + \lambda^{-1} \bK_{B})} \end{align*} where $\bK_{A}=\begin{bmatrix} \bPhi_{1}\\\bPhi_{2} \end{bmatrix} \begin{bmatrix} \bPhi_{1}^{\top},\bPhi_{2}^{\top} \end{bmatrix}$ and $\bK_{B}=\bPhi_{1} \bPhi_{1}^{\top}$. \end{lemma} \begin{proof}[Proof of Lemma \ref{lem:quadratic_det_inequality_infinite}] Similar to the proof of Lemma 12 of \citep{abbasi2011improved}, we start from the simple case when $ \bPhi_{2}^{\top}\bPhi_{2}=m m^{\top}$, where $m \in \bR^{p}$. Using Cauchy-Schwartz inequality, we have \begin{align*}
(\phi^{\top} m)^{2} = (\phi^{\top}B^{1/2}B^{-1/2}m)^{2} \leq \lVert B^{1/2} \phi \rVert^{2} \lVert B^{-1/2} m \rVert^{2} = \lVert \phi \rVert^{2}_{B} \lVert m \rVert^{2}_{B^{-1}}, \end{align*} and thus, \begin{align*}
\phi^{\top} (B + m m^{\top}) \phi \leq \phi^{\top} B \phi + \lVert \phi \rVert^{2}_{B} \lVert m \rVert^{2}_{B^{-1}} = (1+ \lVert m \rVert^{2}_{B^{-1}}) \lVert \phi \rVert^{2}_{B}, \end{align*} so we have \begin{align*}
\frac{\phi^{\top} A \phi}{\phi^{\top} B \phi} \leq 1 + \lVert m \rVert_{B^{-1}}^{2} \end{align*} for any $\phi$. Then using the kernel trick, e.g., see the derivation of Eq (27) in \cite{zenati2022efficient}, we have \begin{align*}
1 + \lVert m \rVert_{B^{-1}}^{2} = \frac{\det(\bI+ \lambda^{-1} \bK_{A})}{\det(\bI + \lambda^{-1} \bK_{B})}, \end{align*} which finishes the proof of this simple case. Now consider the general case where $\bPhi_{2}^{\top} \bPhi_{2} = m_{1} m_{1}^{\top} + m_{2} m_{2}^{\top} + \dots + m_{t-1} m_{t-1}^{\top}$. Let's define $V_{s}=B+m_{1} m_{1}^{\top} + m_{2} m_{2}^{\top} + \dots + m_{s-1} m_{s-1}^{\top}$ and the corresponding kernel matrix $\bK_{V_{s}}=\begin{bmatrix} \bPhi_{1}\\ m_{1}^{\top} \\ \dots \\ m_{s-1}^{\top} \end{bmatrix} \begin{bmatrix} \bPhi_{1}^{\top}, m_{1}, \dots, m_{s-1} \end{bmatrix}$, and note that $\frac{\phi^{\top} A \phi}{\phi^{\top} B \phi} = \frac{\phi^{\top} V_{t} \phi}{\phi^{\top} V_{t-1} \phi} \frac{\phi^{\top} V_{t-1} \phi}{\phi^{\top} V_{t-2} \phi} \dots \frac{\phi^{\top} V_{2} \phi}{\phi^{\top} B \phi}$.
Then we can apply the result for the simple case on each term in the product above, which gives us \begin{align*}
\frac{\phi^{\top} A \phi}{\phi^{\top} B \phi} & \leq \frac{\det(\bI+ \lambda^{-1} \bK_{V_{t}})}{\det(\bI + \lambda^{-1} \bK_{V_{t-1}})} \frac{\det(\bI+ \lambda^{-1} \bK_{V_{t-1}})}{\det(\bI + \lambda^{-1} \bK_{V_{t-2}})} \dots \frac{\det(\bI+ \lambda^{-1} \bK_{V_{2}})}{\det(\bI + \lambda^{-1} \bK_{B})} \\
& = \frac{\det(\bI+ \lambda^{-1} \bK_{V_{t}})}{\det(\bI + \lambda^{-1} \bK_{B})} = \frac{\det(\bI+ \lambda^{-1} \bK_{A})}{\det(\bI + \lambda^{-1} \bK_{B})}, \end{align*} which finishes the proof.
\end{proof}
\begin{lemma}[Eq (26) and Eq (27) of \cite{zenati2022efficient}] \label{lem:sum_sqr_log_det} Let $\{\phi_{t}\}_{t=1}^{\infty}$ be a sequence in $\bR^{p}$, $V \in \bR^{p \times p}$ a positive definite matrix, where $p$ is possibly infinite, and define $V_{t}=V + \sum_{s=1}^{t} \phi_{s} \phi_{s}^{\top}$. Then we have that
\begin{align*}
\sum_{t=1}^{n} \min\bigl(\lVert \phi_{t} \rVert^{2}_{V_{t-1}^{-1}}, 1\bigr) \leq 2 \ln\bigl( \det(\bI + \lambda^{-1} \bK_{V_{t}})\bigr), \end{align*} where $\bK_{V_{t}}$ is the kernel matrix corresponding to $V_{t}$ as defined in Lemma \ref{lem:quadratic_det_inequality_infinite}. \end{lemma}
\begin{lemma}[Lemma 4 of \citep{calandriello2020near}] \label{lem:bound_variance_ratio} For $t > t_\text{last}$, we have for any $\bx \in \bR^{d}$ \begin{align*}
\hat{\sigma}_{t}^{2}(\bx) \leq \hat{\sigma}_{t_\text{last}}^{2}(\bx) \leq \bigl( 1+\sum_{s=t_\text{last}+1}^{t} \hat{\sigma}_{t_\text{last}}^{2}(\bx_{s}) \bigr) \hat{\sigma}_{t}^{2}(\bx) \end{align*} \end{lemma}
\section{Confidence Ellipsoid for DisKernelUCB} In this section, we construct the confidence ellipsoid for DisKernelUCB as shown in Lemma \ref{lem:confidence_ellipsoid_diskernelucb}. \begin{lemma}[Confidence Ellipsoid for DisKernelUCB] \label{lem:confidence_ellipsoid_diskernelucb} Let $\delta \in (0,1)$. With probability at least $1-\delta$, for all $t \in [NT], i \in [N]$, we have \begin{align*}
\lVert \hat{\theta}_{t,i} - \theta_{\star} \rVert_{\bA_{t,i}} \leq \sqrt{\lambda} \lVert \theta_{\star} \rVert + R\sqrt{ 2 \ln(N/\delta) + \ln( \det( \bK_{ \cD_{t}(i), \cD_{t}(i) }/\lambda + \bI) ) }. \end{align*} \end{lemma}
The analysis is rooted in \citep{zenati2022efficient} for kernelized contextual bandit, but with non-trivial extensions: we adopted the stopping time argument from \citep{abbasi2011improved} to remove a logarithmic factor in $T$ (this improvement is hinted in Section 3.3 of \citep{zenati2022efficient} as well); and this stopping time argument is based on a special `batched filtration' that is different for each client, which is required to address the dependencies due to the event-triggered distributed communication. This problem also exists in prior works of distributed linear bandit, but was not addressed rigorously (see Lemma H.1. of \citep{wang2019distributed}).
Recall that the Ridge regression estimator
\begin{align*}
\hat{\theta}_{t,i} & = \bA_{t,i}^{-1} \sum_{s \in \cD_{t}(i)} \phi_{s} y_{s} = \bA_{t,i}^{-1} \sum_{s \in \cD_{t}(i)} \phi_{s} ( \phi_{s} ^{\top} \theta_{\star} + \eta_{s} ) \\
& = \theta_{\star} - \lambda \bA_{t,i}^{-1} \theta_{\star} + \bA_{t,i}^{-1} \sum_{s \in \cD_{t}(i)} \phi_{s} \eta_{s}, \end{align*} and thus, we have \begin{equation} \label{eq:ellipsoid_intermediate} \begin{split}
\lVert \bA_{t,i}^{1/2} (\hat{\theta}_{t,i} - \theta_{\star}) \rVert & = \lVert -\lambda \bA_{t,i}^{-1/2} \theta_{\star} + \bA_{t,i}^{-1/2} \sum_{s \in \cD_{t}(i)} \phi_{s} \eta_{s} \rVert \\
& \leq \lVert \lambda \bA_{t,i}^{-1/2} \theta_{\star} \rVert + \lVert \bA_{t,i}^{-1/2} \sum_{s \in \cD_{t}(i)} \phi_{s} \eta_{s} \rVert \\
& \leq \sqrt{\lambda} \lVert \theta_{\star} \rVert + \lVert \bA_{t,i}^{-1/2} \sum_{s \in \cD_{t}(i)} \phi_{s} \eta_{s} \rVert \end{split} \end{equation} where the first inequality is due to the triangle inequality, and the second is due to the property of Rayleigh quotient, i.e., $\lVert \bA_{t,i}^{-1/2} \theta_{\star} \rVert \leq \lVert \theta_{\star} \rVert \sqrt{\lambda_{max}(\bA_{t,i}^{-1}) } \leq \lVert \theta_{\star} \rVert \frac{1}{\sqrt{\lambda}} $.
\paragraph{Difference from standard argument} Note that the second term may seem similar to the ones appear in the self-normalized bound in previous works of linear and kernelized bandits \citep{abbasi2011improved,chowdhury2017kernelized,zenati2022efficient}. However, a main difference is that $\cD_{t}(i)$, i.e., the sequence of indices for the data points used to update client $i$, is constructed using the event-trigger as defined in Eq~\eqref{eq:sync_event_exact}
. The event-trigger is data-dependent, and thus it is a delayed and permuted version of the original sequence $[t]$. It is delayed in the sense that the length $|\cD_{t}(i)| < t$ unless $t$ is the global synchronization step. It is permuted in the sense that every client receives the data in a different order, i.e., before the synchronization, each client first updates using its local new data, and then receives data from other clients at the synchronization. This prevents us from directly applying Lemma 3.1 of \citep{zenati2022efficient}, and requires a careful construction of the filtration, as shown in the following paragraph.
\paragraph{Construction of filtration}
For some client $i$ at time step $t$, the sequence of time indices in $\cD_{t}(i)$ is arranged in the order that client $i$ receives the corresponding data points, which includes both data added during local update in each client, and data received from the server during global synchronization. The former adds one data point at a time, while the latter adds a batch of data points, which, as we will see, break the assumption commonly made in standard self-normalized bound \citep{abbasi2011improved,chowdhury2017kernelized,zenati2022efficient}. Specifically, we denote $\cD_{t}(i)[k]$, for $k \leq |\cD_{t}(i)|$, as the $k$-th element in this sequence, i.e., $(\bx_{\cD_{t}(i)[k]}, y_{\cD_{t}(i)[k]})$ is the $k$-th data point received by client $i$. Then we denote $\cB_{t}(i)=\{k_{1}, k_{2},\dots\}$ as the sequence of $k$'s that marks the end of each batch (a singel data point added by local update is also considered a batch). We can see that if the $k$-th element is in the middle of a batch, i.e., $k \in [k_{q-1}, k_{q}]$, it has dependency all the way to the $k_{q}$'s element, since this index can only be determined until some client triggers a global synchronization at time step $\cD_{t}(i)[k_{q}]$.
Denote by $\cF_{k,i}=\sigma \bigl( (\bx_{s}, \eta_{s})_{s \in \cD_{\infty}(i)[1:k-1]}, \bx_{\cD_{\infty}(i)[k]} \bigr)$ the $\sigma$-algebra generated by the sequence of data points up to the $k$-th element in $\cD_{\infty}(i)$. As we mentioned, because of the dependency of the index on the future data points, for some $k$-th element that is inside a batch, i.e., $k \in [k_{q-1}, k_{q}]$, $\bx_{\cD_{\infty}(i)[k]}$ is not $\cF_{k,i}$-measurable and $\eta_{\cD_{\infty}(i)[k]}$ is not $\cF_{k+1,i}$-measurable, which violate the assumption made in standard self-normalized bound \citep{abbasi2011improved,chowdhury2017kernelized,zenati2022efficient}. However, they become measurable if we condition on $\cF_{k_{q},i}$. In addition, recall that in Section \ref{subsec:problem_formulation} we assume $\eta_{\cD_{\infty}(i)[k]}$ is zero-mean $R$-sub-Gaussian conditioned on $\sigma\bigl( (\bx_{s},\eta_{s})_{s \in \cN_{\cD_{\infty}(i)[k]-1}(i_{\cD_{\infty}(i)[k]})}, \bx_{\cD_{\infty}(i)[k]} \bigr)$, which is the $\sigma$-algebra generated by the sequence of local data collected by client $i_{\cD_{\infty}(i)[k]}$. We can see that as $\sigma\bigl( (\bx_{s},\eta_{s})_{s \in \cN_{\cD_{\infty}(i)[k]-1}(i_{\cD_{\infty}(i)[k]})}, \bx_{\cD_{\infty}(i)[k]} \bigr) \subseteq \cF_{k,i} $, $\eta_{\cD_{\infty}(i)[k]}$ is zero-mean $R$-sub-Gaussian conditioned on $\cF_{k,i}$. Basically, our assumption of $R$-sub-Gaussianity conditioned on \emph{local sequence} instead of \emph{global sequence} of data points, prevents the situation where the noise depends on data points that have not been communicated to the current client yet, i.e., they are not included in $\cF_{k,i}$.
With our `batched filtration' $\{\cF_{k,i}\}_{k \in \cB_{\infty}(i)}$ for each client $i$, we have everything we need to establish a time-uniform self-normalized bound that resembles
Lemma 3.1 of \citep{zenati2022efficient}, but with improved logarithmic factor using the stopping time argument from \citep{abbasi2011improved}. Then we can take a union bound over $N$ clients to obtain the uniform bound over all clients and time steps. \paragraph{Super-martingale \& self-normalized bound} First, we need the following lemmas adopted from \citep{zenati2022efficient} to our `batched filtration'. \begin{lemma} \label{lem:super_martingale} Let $\upsilon \in \bR^{d}$ be arbitrary and consider for any $k \in \cB_{\infty}(i)$, $i \in [N]$, we have \begin{align*}
M_{k,i}^{\upsilon} = \exp\left( - \frac{1}{2} \lambda \upsilon^{\top}\upsilon + \sum_{s \in \cD_{\infty}(i)[1:k] } [ \frac{\eta_{s}<\upsilon, X_{s}>}{R} - \frac{1}{2} \upsilon^{\top} X_{s} X_{s}^{\top} \upsilon ] \right) \end{align*} is a $\cF_{k+1,i}$-super-martingale, and $\bE[ M_{k,i}^{\upsilon} ] \leq \exp(-\frac{1}{2} \lambda \upsilon^{\top}\upsilon )$. Let $\tilde{k}$ be a stopping time w.r.t. the filtration $\{\cF_{k,i}\}_{k \in \cB_{\infty}(i)}$. Then $M_{\tilde{k},i}^{\upsilon}$ is almost surely well-defined and $\bE[M_{\tilde{k},i}^{\upsilon}] \leq 1$. \end{lemma} \begin{proof}[Proof of Lemma \ref{lem:super_martingale}] To show that $\{M_{k,i}^{\upsilon}\}_{ k \in \cB_{\infty}(i) }$ is a super-martinagle, we denote \begin{align*}
D_{k,i}^{\upsilon} = \exp\left( \frac{\eta_{\cD_{\infty}(i)[k]}<\upsilon, X_{\cD_{\infty}(i)[k]}>}{R} - \frac{1}{2} <\upsilon, X_{\cD_{\infty}(i)[k]}>^{2} \right), \end{align*}
with $D_{0,i}^{\upsilon} = \exp(-\frac{1}{2} \lambda \upsilon^{\top}\upsilon )$, and as we have showed earlier, $\eta_{\cD_{\infty}(i)[k]}$ is $R$-sub-Gaussian conditioned on $\cF_{k,i}$. Therefore, $\bE[ D_{k,i}^{\upsilon} | \cF_{k,i} ] \leq 1$. Moreover, $D_{k,i}^{\upsilon}$ and $M_{k,i}^{\upsilon}$ are $\cF_{k+1,i}$-measurable. Then we have \begin{align*}
\bE[ M_{k,i}^{\upsilon} | \cF_{k,i} ] & = \bE[D_{1,i}^{\upsilon} D_{2,i}^{\upsilon} \dots D_{k-1,i}^{\upsilon} D_{k,i}^{\upsilon} | \cF_{k,i}] \\
& = D_{0,i}^{\upsilon} D_{1,i}^{\upsilon} D_{2,i}^{\upsilon} \dots D_{k-1,i}^{\upsilon} \bE[ D_{k,i}^{\upsilon} | \cF_{k,i}] \leq M_{k-1,i}^{\upsilon}, \end{align*} which shows $\{M_{k,i}^{\upsilon}\}_{ k \in \cB_{\infty}(i) }$ is a super-martinagle, with $\bE[ M_{k,i}^{\upsilon} ] \leq D_{0,i}^{\upsilon}=\exp(-\frac{1}{2} \lambda \upsilon^{\top}\upsilon )$. Then using the same argument as Lemma 8 of \citep{abbasi2011improved}, we have that $M_{\tilde{k},i}^{\upsilon}$ is almost surely well-defined, and $\bE[M_{\tilde{k},i}^{\upsilon}] \leq D_{0,i}^{\upsilon}=\exp(-\frac{1}{2} \lambda \upsilon^{\top}\upsilon )$. \end{proof}
\begin{lemma} \label{lem:self_normalized_bound} Let $\tilde{k}$ be a stopping time w.r.t. the filtration $\{\cF_{k,i}\}_{k \in \cB_{\infty}(i)}$. Then for $\delta > 0$, we have \small \begin{align*}
& P\Big( \lVert ( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:\tilde{k}] } X_{s} X_{s}^{\top} )^{-1/2} (\sum_{s \in \cD_{\infty}(i)[1:\tilde{k}] } X_{s} \eta_{s} ) \rVert > R\sqrt{2 \ln(1/\delta) + \ln( \det( \bK_{ \cD_{\infty}(i)[1:\tilde{k}], \cD_{\infty}(i)[1:\tilde{k}] }/\lambda + \bI) ) }\Big) \\
& \leq \delta. \end{align*} \normalsize \end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:self_normalized_bound}] Using $m:=( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} X_{s}^{\top} )^{-1} (\sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} \eta_{s} )$, we can rewrite $M_{\tilde{k},i}^{\upsilon}$ as \begin{align*}
M_{\tilde{k},i}^{\upsilon} & = \exp\left( -\frac{1}{2} (\upsilon-m)^{\top} ( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} X_{s}^{\top} ) (\upsilon-m) \right) \\
& \quad \times \exp\left( \frac{1}{2} \lVert ( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} X_{s}^{\top} )^{-1/2} (\sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} \eta_{s} ) \rVert^{2} \right). \end{align*} Then based on Lemma \ref{lem:super_martingale}, we have \begin{align*}
& \bE\Big[ \exp\Big( -\frac{1}{2} (\upsilon-m)^{\top} ( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} X_{s}^{\top} ) (\upsilon-m) \Big) \Big] \\
& \quad + \bE\Big[\exp\Big( \frac{1}{2} \lVert ( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} X_{s}^{\top} )^{-1/2} (\sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} \eta_{s} ) \rVert^{2} \Big) \Big] \leq \exp(-\frac{1}{2} \lambda \upsilon^{\top}\upsilon ) \end{align*} Following the Laplace's method as in proof of Lemma 3.1 of \citep{zenati2022efficient}, we have \begin{align*}
\bE\Big[\exp\Big( \frac{1}{2} \lVert ( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:\tilde{k}] } X_{s} X_{s}^{\top} )^{-1/2} (\sum_{s \in \cD_{\infty}(i)[1:\tilde{k}] } X_{s} \eta_{s} ) \rVert^{2} \Big) \Big] \leq \sqrt{ \frac{\det( \bK_{ \cD_{\infty}(i)[1:\tilde{k}], \cD_{\infty}(i)[1:\tilde{k}] }+ \lambda \bI) }{\lambda^{\tilde{k}} }} \end{align*} By applying Markov-Chernov bound, we finish the proof. \end{proof}
\paragraph{Proof of Lemma \ref{lem:confidence_ellipsoid_diskernelucb}} Now using the stopping time argument as in \citep{abbasi2011improved}, and a union bound over clients, we can bound the second term in Eq~\eqref{eq:ellipsoid_intermediate}. First, define the bad event \begin{align*}
B_{k}(\delta) = \bigl\{ \omega \in \Omega: \lVert ( \lambda \bI + &\sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} X_{s}^{\top} )^{-1/2} (\sum_{s \in \cD_{\infty}(i)[1:k] } X_{s} \eta_{s} ) \rVert > \\
\quad \quad \quad \quad & R \sqrt{ 2 \ln(1/\delta) + \ln( \det( \bK_{ \cD_{\infty}(i)[1:k], \cD_{\infty}(i)[1:k] }/\lambda + \bI) )} \bigr\}, \end{align*} and $\tilde{k}(\omega)=\min( k \geq 0: \omega \in B_{k}(\delta) )$, which is a stopping time. Moreover, $\cup_{k \in \cB_{\infty}(i)}B_{k}(\delta)=\{\omega: \tilde{k} < \infty\}$. Then using Lemma \ref{lem:self_normalized_bound}, we have \small \begin{align*}
& P\bigl( \cup_{k \in \cB_{\infty}(i)}B_{k}(\delta) \bigr) = P(\tilde{k} < \infty) \\
& \leq P\Big( \lVert ( \lambda \bI + \sum_{s \in \cD_{\infty}(i)[1:\tilde{k}] } X_{s} X_{s}^{\top} )^{-1/2} (\sum_{s \in \cD_{\infty}(i)[1:\tilde{k}] } X_{s} \eta_{s} ) \rVert > R \sqrt{ 2 \ln(1/\delta) + \ln( \det( \bK_{ \cD_{\infty}(i)[1:\tilde{k}], \cD_{\infty}(i)[1:\tilde{k}] }/\lambda + \bI) )} \Big) \\
& \leq \delta \end{align*} \normalsize Note that $\cB_{\infty}(i)$ is the sequence of indices $k$ in $\cD_{\infty}(i)$ when client $i$ gets updated. Therefore, the result above is equivalent to \begin{align*}
\lVert \bA_{t,i}^{-1/2} \sum_{s \in \cD_{t}(i)} \phi_{s} \eta_{s} \rVert \leq R\sqrt{ 2 \ln(1/\delta) + \ln( \det( \bK_{ \cD_{t}(i), \cD_{t}(i) }/\lambda + \bI) ) } \end{align*} for all $t \geq 1$, with probability at least $1-\delta$. Then by taking a union bound over $N$ clients, we finish the proof.
\section{Proof of Lemma \ref{lem:regret_comm_diskernelucb}: Regret and Communication Cost of {DisKernelUCB}{}} \label{sec:proof_regret_diskernelucb}
Based on Lemma \ref{lem:confidence_ellipsoid_diskernelucb} and the arm selection rule in Eq~\eqref{eq:UCB_exact}, we have \begin{align*}
f(\bx_{t}^{\star}) & \leq \hat{\mu}_{t-1,i_{t}}(\bx_{t}^{\star}) + \alpha_{t-1,i_{t}}\hat{\sigma}_{t-1,i_{t}}(\bx_{t}^{\star}) \leq \hat{\mu}_{t-1,i_{t}}(\bx_{t}) + \alpha_{t-1,i_{t}} \hat{\sigma}_{t-1,i_{t}}(\bx_{t}), \\
f(\bx_{t}) & \geq \hat{\mu}_{t-1,i_{t}}(\bx_{t}) - \alpha_{t-1,i_{t}} \hat{\sigma}_{t-1,i_{t}}(\bx_{t}), \end{align*} and thus $r_{t}=f(\bx_{t}^{\star})-f(\bx_{t}) \leq 2 \alpha_{t-1,i_{t}} \hat{\sigma}_{t-1,i_{t}}(\bx_{t})$, for all $t \in [NT]$, with probability at least $1-\delta$.
Then following similar steps as DisLinUCB of \citep{wang2019distributed}, we can obtain the regret and communication cost upper bound of {DisKernelUCB}{}. \subsection{Regret Upper Bound} \label{subsec:regret_proof_diskernelucb} We call the time period in-between two consecutive global synchronizations as an epoch, i.e., the $p$-th epoch refers to $[t_{p-1}+1,t_{p}]$, where $p \in [B]$ and $0 \leq B \leq NT$ denotes the total number of global synchronizations. Now consider an imaginary centralized agent that has immediate access to each data point in the learning system, and denote by $A_{t}=\sum_{s=1}^{t} \phi_{s} \phi_{s}^{\top}$ and $\bK_{[t],[t]}$ for $t \in [NT]$ the covariance matrix and kernel matrix constructed by this centralized agent.
Then similar to \citep{wang2019distributed}, we call the $p$-th epoch a good epoch if \begin{align*}
\ln\left(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}\right) \leq 1, \end{align*} otherwise it is a bad epoch.
Note that $\ln(\det(I + \lambda^{-1} \bK_{[NT],[NT]})) \leq 2 \gamma_{NT}$ by definition of $\gamma_{NT}$, i.e., the maximum information gain. Since $\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{1}],[t_{1}]})}{\det(\bI)})+\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{2}],[t_{2}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{1}],[t_{1}]})})+\dots+\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[NT],[NT]})}{\det(\bI + \lambda^{-1}\bK_{[t_{B}],[t_{B}]})}) = \ln(\det(I + \lambda^{-1} \bK_{[NT],[NT]})) \leq 2 \gamma_{NT}$, and due to the pigeonhole principle, there can be at most $2 \gamma_{NT}$ bad epochs.
If the instantaneous regret $r_{t}$ is incurred during a good epoch, we have \begin{align*}
r_{t} & \leq 2 \alpha_{t-1,i_{t}} \lVert \phi_{t} \rVert_{\bA_{t-1,i_{t}}^{-1}} \leq 2 \alpha_{t-1,i_{t}}\lVert \phi_{t} \rVert_{\bA_{t-1}^{-1}} \sqrt{\lVert \phi_{t} \rVert^{2}_{\bA_{t-1,i_{t}}^{-1}}/\lVert \phi_{t} \rVert^{2}_{\bA_{t-1}^{-1}}} \\
& = 2 \alpha_{t-1,i_{t}}\lVert \phi_{t} \rVert_{\bA_{t-1}^{-1}} \sqrt{ \frac{\det(\bI + \lambda^{-1}\bK_{[t-1],[t-1]})}{\det(\bI + \lambda^{-1}\bK_{\cD_{t-1}(i_{t}),\cD_{t-1}(i_{t})})} } \\
& \leq 2 \sqrt{e} \alpha_{t-1,i_{t}}\lVert \phi_{t} \rVert_{\bA_{t-1}^{-1}}
\end{align*} where the second inequality is due to Lemma \ref{lem:quadratic_det_inequality_infinite}, and the last inequality is due to the definition of good epoch, i.e., $\frac{\det(\bI + \lambda^{-1}\bK_{[t-1],[t-1]})}{\det(\bI + \lambda^{-1}\bK_{\cD_{t-1}(i_{t}),\cD_{t-1}(i_{t})})} \leq \frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})} \leq e$.
Define $\alpha_{NT}:=\sqrt{\lambda} \lVert \theta_{\star} \rVert + \sqrt{ 2 \ln(N/\delta) + \ln( \det( \bK_{ [NT],[NT] }/\lambda + \bI) ) }$. Then using standard arguments, the cumulative regret incurred in all good epochs can be bounded by, \begin{align*}
R_{good} & = \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) \leq 1\} \sum_{t=t_{p-1}}^{t_{p}} r_{t} \leq \sum_{t=1}^{NT} 2 \sqrt{e} \alpha_{t-1,i_{t}} \lVert \phi_{t} \rVert_{\bA_{t-1}^{-1}} \\
& \leq 2 \sqrt{e} \alpha_{NT} \sum_{t=1}^{NT} \lVert \phi_{t} \rVert_{\bA_{t-1}^{-1}} \leq 2 \sqrt{e} \alpha_{NT} \sqrt{ NT \cdot 2 \ln( \det(\bI + \lambda^{-1}\bK_{[NT],[NT]}) ) } \\
& \leq 2 \sqrt{e} \alpha_{NT} \sqrt{ NT \cdot 4 \gamma_{NT} } =O\Big(\sqrt{NT}( \lVert \theta_{\star} \rVert \sqrt{\gamma_{NT}} + \gamma_{NT})\Big) \end{align*} where the third inequality is due to Cauchy-Schwartz and Lemma \ref{lem:sum_sqr_log_det}, and the forth is due to the definition of maximum information gain $\gamma_{NT}$.
Then we look at the regret incurred during bad epochs. Consider some bad epoch $p$, and the cumulative regret incurred during this epoch can be bounded by \begin{align*}
& \sum_{t=t_{p-1}+1}^{t_{p}} r_{t} = \sum_{i=1}^{N} \sum_{t \in \cD_{t_{p}}(i) \setminus \cD_{t_{p-1}}(i) } r_{t} \leq 2 \alpha_{NT} \sum_{i=1}^{N} \sum_{t \in \cD_{t_{p}}(i) \setminus \cD_{t_{p-1}}(i) } \lVert \phi_{t} \rVert_{\bA_{t-1,i}^{-1}} \\
& \leq 2 \alpha_{NT} \sum_{i=1}^{N} \sqrt{ (| \cD_{t_{p}}(i) | - | \cD_{t_{p-1}}(i) |) \sum_{t \in \cD_{t_{p}}(i) \setminus \cD_{t_{p-1}}(i) } \lVert \phi_{t} \rVert_{\bA_{t-1,i}^{-1}}^{2} } \\
& \leq 2 \alpha_{NT} \sum_{i=1}^{N} \sqrt{ 2(| \cD_{t_{p}}(i) | - | \cD_{t_{p-1}}(i) |) \ln(\frac{\det(\bI + \lambda^{-1} \bK_{\cD_{t_{p}}(i),\cD_{t_{p}}(i)})}{\det(\bI + \lambda^{-1} K_{\cD_{t_{p-1}}(i), \cD_{t_{p-1}}(i)})}) } \\
& \leq 2\sqrt{2} \alpha_{NT} N \sqrt{D}
\end{align*} where the last inequality is due to our event-trigger in Eq~\eqref{eq:sync_event_exact}.
Since there can be at most $2 \gamma_{NT}$ bad epochs, the cumulative regret incurred in all bad epochs \begin{align*}
R_{bad} & \leq 2 \gamma_{NT} \cdot 2\sqrt{2} \alpha_{NT} N \sqrt{ D} = O\Big(N D^{0.5} ( \lVert \theta_{\star} \rVert \gamma_{NT} + \gamma_{NT}^{1.5})\Big)
\end{align*} Combining cumulative regret incurred during both good and bad epochs, we have \begin{align*}
R_{NT} = R_{good} + R_{bad} = O\bigl( (\sqrt{NT} + N \sqrt{D \gamma_{NT}}) ( \lVert \theta_{\star} \rVert \sqrt{\gamma_{NT}} + \gamma_{NT}) \bigr) \end{align*}
\subsection{Communication Upper Bound} For some $\alpha > 0$, there can be at most $\lceil \frac{NT}{\alpha} \rceil$ epochs with length larger than $\alpha$.
Based on our event-trigger design, we know that $(| \cD_{t_{p}}(i_{t_{p}}) | - | \cD_{t_{p-1}}(i_{t_{p}}) |) \ln(\frac{\det(\bI + \lambda^{-1} \bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1} K_{[t_{p-1}],[t_{p-1}]})}) \geq (| \cD_{t_{p}}(i_{t_{p}}) | - | \cD_{t_{p-1}}(i_{t_{p}}) |) \ln(\frac{\det(\bI + \lambda^{-1} \bK_{\cD_{t_{p}}(i_{t_{p}}),\cD_{t_{p}}(i_{t_{p}})})}{\det(\bI + \lambda^{-1} K_{\cD_{t_{p-1}}(i_{t_{p}}), \cD_{t_{p-1}}(i_{t_{p}})})}) \geq D$ for any epoch $p \in [B]$, where $i_{t_{p}}$ is the client who triggers the global synchronization at time step $t_{p}$. Then if the length of certain epoch $p$ is smaller than $\alpha$, i.e., $t_{p}-t_{p-1} \leq \alpha$, we have $\ln(\frac{\det(\bI + \lambda^{-1} \bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1} K_{[t_{p-1}],[t_{p-1}]})}) \geq \frac{N D}{\alpha}$. Since $\ln(\frac{\det(\bI + \lambda^{-1} \bK_{[t_{1}],[t_{1}]})}{\det(\bI)}) + \ln(\frac{\det(\bI + \lambda^{-1} \bK_{[t_{2}], [t_{2}]})}{\det(\bI + \lambda^{-1} \bK_{[t_{1}],[t_{1}]})}) + \dots + \ln(\frac{\det(\bI + \lambda^{-1} \bK_{[t_{B}],[t_{B}]})}{\det(\bI + \lambda^{-1} \bK_{[t_{B-1}], [t_{B-1}]})}) \leq \ln(\det(\bI + \lambda^{-1} \bK_{[NT], [NT]})) \leq 2 \gamma_{NT}$, the total number of such epochs is upper bounded by $\lceil \frac{2\gamma_{NT} \alpha}{ND} \rceil$. Combining the two terms, the total number of epochs can be bounded by, \begin{align*}
B \leq \lceil \frac{NT}{\alpha} \rceil + \lceil \frac{2\gamma_{NT} \alpha}{ND} \rceil \end{align*} where the LHS can be minimized using the AM-GM inequality, i.e., $B \leq \sqrt{ \frac{NT}{\alpha} \frac{2\gamma_{NT} \alpha}{ND} }= \sqrt{\frac{2 \gamma_{NT} T}{D}}$. To obtain the optimal order of regret, we set $D=O(\frac{T}{N \gamma_{NT}})$, so that $R_{NT}=O\bigl( \sqrt{NT} ( \lVert \theta_{\star} \rVert \sqrt{\gamma_{NT}} + \gamma_{NT}) \bigr)$. And the total number of epochs $B = O(\sqrt{N} \gamma_{NT})$. However, we should note that as {DisKernelUCB}{} communicates all the unshared raw data at each global synchronization, the total communication cost mainly depends on when the last global synchronization happens. Since the sequence of candidate sets $\{\cA_{t}\}_{t \in [NT]}$, which controls the growth of determinant, is an arbitrary subset of $\cA$, the time of last global synchronization could happen at the last time step $t=NT$. Therefore, $C_{T}=O(N^{2}T d)$ in such a worst case.
\section{Derivation of the Approximated Mean and Variance in Section \ref{sec:method}} \label{sec:derivation_approx} For simplicity, subscript $t$ is omitted in this section. The approximated Ridge regression estimator for dataset $\{(\bx_{s},y_{s})\}_{s \in \cD}$ is formulated as \begin{align*}
\tilde{\btheta}=\argmin_{\btheta \in \cH} \sum_{s \in \cD} \Big( (\bP_{\cS}\phi_{s})^{\top} \btheta - y_{s}\Big)^{2} + \lambda \lVert \btheta \rVert_{2}^{2} \end{align*} where $\cD$ denotes the sequence of time indices for data in the original dataset, $\cS \subseteq \cD$ denotes the time indices for data in the dictionary, and $\bP_{\cS} \in \bR^{p \times p}$ denotes the orthogonal projection matrix defined by $\cS$. Then by taking derivative and setting it to zero, we have $(\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD} \bP_{\cS} + \lambda \bI) \tilde{\btheta}=\bP_{\cS} \bPhi_{\cD}^{\top} \by_{\cD}$, and thus $\tilde{\theta}=\tilde{\bA}^{-1} \tilde{\bb}$, where $\tilde{\bA}=\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD} \bP_{\cS} + \lambda \bI$ and $\tilde{\bb}=\bP_{\cS} \bPhi_{\cD}^{\top} \by_{\cD}$.
Hence, the approximated mean reward and variance for some arm $\bx$ are \begin{align*}
\tilde{\mu}_{t,i}(\bx) &= \phi(\bx)^{\top} \tilde{\bA}^{-1} \tilde{\bb} \\
\tilde{\sigma}_{t,i}(\bx) &= \sqrt{ \phi(\bx)^{\top} \tilde{\bA}^{-1} \phi(\bx)} \end{align*} To obtain their kernelized representation, we rewrite \begin{align*}
&(\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD} \bP_{\cS} + \lambda \bI) \tilde{\btheta}=\bP_{\cS} \bPhi_{\cD}^{\top} \by_{\cD} \\ \Leftrightarrow ~ & \bP_{\cS} \bPhi_{\cD}^{\top} (\by_{\cD} - \bPhi_{\cD} \bP_{\cS} \tilde{\btheta}) = \lambda \tilde{\btheta} \\
\Leftrightarrow ~ & \tilde{\btheta} = \bP_{\cS} \bPhi_{\cD}^{\top} \rho \end{align*} where $\rho := \frac{1}{\lambda}(\by_{\cD} - \bPhi_{\cD} \bP_{\cS} \tilde{\btheta})=\frac{1}{\lambda}(\by_{\cD} - \bPhi_{\cD} \bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top} \rho)$. Solving this equation, we get $\rho=(\bPhi_{\cD} \bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top} + \lambda \bI)^{-1} \by_{\cD}$. Note that $\bP_{\cS}\bP_{\cS} = \bP_{\cS}$, since projection matrix $\bP_{\cS}$ is idempotent. Moreover, we have $( \bPhi^{\top}\bPhi + \lambda \bI)\bPhi^{\top}=\bPhi^{\top}(\bPhi \bPhi^{\top} + \lambda \bI)$, and $( \bPhi^{\top}\bPhi + \lambda \bI)^{-1}\bPhi^{\top}=\bPhi^{\top}(\bPhi \bPhi^{\top} + \lambda \bI)^{-1}$. Therefore, we can rewrite the approximated mean for some arm $\bx$ as \begin{align*}
\tilde{\mu}(\bx) & = \phi(\bx)^{\top} \bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD} \bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top} + \lambda \bI)^{-1} \by_{\cD} \\
& = (\bP_{\cS}^{1/2} \phi(\bx))^{\top} (\bPhi_{\cD}\bP_{\cS}^{1/2})^{\top}[\bPhi_{\cD} \bP_{\cS}^{1/2} (\bPhi_{\cD} \bP_{\cS}^{1/2})^{\top} + \lambda \bI]^{-1} \by_{\cD} \\
& = (\bP_{\cS}^{1/2} \phi(\bx))^{\top} (\bP_{\cS}^{1/2}\bPhi_{\cD}^{\top}\bPhi_{\cD}\bP_{\cS}^{1/2} + \lambda \bI)^{-1} (\bPhi_{\cD}\bP_{\cS}^{1/2})^{\top}\by_{\cD} \\
& = z(\bx;\cS)^{\top} \bigl( \bZ_{\cD;\cS}^{\top}\bZ_{\cD;\cS} + \lambda \bI\bigr)^{-1} \bZ_{\cD;\cS}^{\top} \by_{\cD} \end{align*} To derive the approximated variance, we start from the fact that $(\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI) \phi(\bx)=\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS}\phi(\bx) + \lambda \phi(\bx)$, so \begin{align*}
\phi(\bx)&=(\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1} \bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS}\phi(\bx) + \lambda (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1}\phi(\bx)\\
& =\bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD}\bP_{\cS} \phi(\bx) + \lambda (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1}\phi(\bx) \end{align*} Then we have \begin{align*}
& \phi(\bx)^{\top} \phi(\bx) \\
= & \bigl\{ \bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD}\bP_{\cS} \phi(\bx) + \lambda (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1}\phi(\bx) \bigr\}^{\top} \\
& \quad \quad \bigl\{ \bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD}\bP_{\cS} \phi(\bx) + \lambda (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1}\phi(\bx) \bigr\} \\
= & \phi(\bx)^{\top} \bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD} \bP_{\cS}\bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD}\bP_{\cS} \phi(\bx) \\
& + 2 \lambda \phi(\bx)^{\top} \bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD} \bP_{\cS} (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1}\phi(\bx) \\
& + \lambda \phi(\bx)^{\top} (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1} \lambda \bI (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1} \phi(\bx) \\
= & \phi(\bx)^{\top} \bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD}\bP_{\cS} \phi(\bx) + \lambda \phi(\bx)^{\top} (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1} \phi(\bx) \end{align*} By rearranging terms, we have \begin{align*}
\tilde{\sigma}^{2}(\bx) = & \phi(\bx)^{\top} (\bP_{\cS} \bPhi_{\cD}^{\top} \bPhi_{\cD}\bP_{\cS} +\lambda \bI)^{-1} \phi(\bx) \\
= & \frac{1}{\lambda} \bigl\{ \phi(\bx)^{\top} \phi(\bx) - \phi(\bx)^{\top} \bP_{\cS} \bPhi_{\cD}^{\top} (\bPhi_{\cD}\bP_{\cS} \bP_{\cS} \bPhi_{\cD}^{\top}+\lambda \bI)^{-1} \bPhi_{\cD}\bP_{\cS} \phi(\bx) \bigr\} \\
= & \frac{1}{\lambda} \{ k(\bx, \bx) - z(\bx;\cS)^{\top} \bZ_{\cD;\cS}^{\top}\bZ_{\cD;\cS} [ \bZ_{\cD;\cS}^{\top}\bZ_{\cD;\cS} + \lambda \bI]^{-1} z(\bx|\cS) \} \end{align*}
\section{Proof of Lemma \ref{lem:dictionary_accuracy_global}} In the following, we analyze the $\epsilon_{t,i}$-accuracy of the dictionary for all $t,i$.
At the time steps when global synchronization happens, i.e., $t_{p}$ for $p \in [B]$, $\cS_{t_{p}}$ is sampled from $[t_{p}]=\cD_{t_{p}}(i)$ using approximated variance $\tilde{\sigma}^{2}_{t_{p-1},i}$. In this case, the accuracy of the dictionary only depends on the RLS procedure, and
Calandriello et al. \citep{calandriello2020near} have already showed that the following guarantee on the accuracy and size of dictionary holds $\forall t \in \{t_{p}\}_{p \in [B]}$. \begin{lemma}[Lemma 2 of \citep{calandriello2020near}] \label{lem:rls}
Under the condition that $\bar{q}=6\frac{1+\epsilon}{1-\epsilon} \log(4NT/\delta)/\epsilon^{2}$, for some $\epsilon \in [0,1)$, with probability at least $1-\delta$, we have $\forall t \in \{t_{p}\}_{p \in [B]}$ that the dictionary $\{ (\bx_{s},y_{s}) \}_{s \in \cS_{t}}$ is $\epsilon$-accurate w.r.t. $\{ (\bx_{s},y_{s}) \}_{s \in \cD_{t}(i)}$, and $\frac{1-\epsilon}{1+\epsilon} \sigma^{2}_{t}(\bx) \leq \tilde{\sigma}^{2}_{t}(\bx) \leq \frac{1+\epsilon}{1-\epsilon} \sigma^{2}_{t}(\bx) , \forall \bx \in \cA$. Moreover, the size of dictionary $|\cS_{t}| \leq 3 (1+L^{2}/\lambda) \frac{1+\epsilon}{1-\epsilon} \bar{q} \tilde{d}$, where $\tilde{d}:=\text{Tr}(\bK_{[NT],[NT]} (\bK_{[NT],[NT]} + \lambda \bI)^{-1})$ denotes the effective dimension of the problem, and it is known that $\tilde{d}=O(\gamma_{NT})$ \citep{chowdhury2017kernelized}. \end{lemma} Lemma \ref{lem:rls} guarantees that for all $t \in \{t_{p}\}_{p \in [B]}$, the dictionary has a constant accuracy, i.e., $\epsilon_{t,i}=\epsilon,\forall i$. In addition, since the dictionary is fixed for $t \notin \{t_{p}\}_{p \in [B]}$, its size $\cS_{t} = O(\gamma_{NT}), \forall t \in [NT]$.
Then for time steps $t \notin \{t_{p}\}_{p \in [B]}$, due to the local update, the accuracy of the dictionary will degrade. However, thanks to our event-trigger in Eq~\eqref{eq:sync_event}, the extent of such degradation can be controlled, i.e., a new dictionary update will be triggered before the previous dictionary becomes completely irrelevant. This is shown in Lemma \ref{lem:dictionary_accuracy} below. \begin{lemma} \label{lem:dictionary_accuracy} Under the condition that $\{ (\bx_{s},y_{s}) \}_{s \in \cS_{t_{p}}}$ is $\epsilon$-accurate w.r.t. $\{ (\bx_{s},y_{s}) \}_{s \in \cD_{t_{p}}(i)}$, $\forall t \in [t_{p}+1, t_{p+1}], i\in[N]$, $\cS_{t_{p}}$ is $\bigl(\epsilon+1- \frac{1}{1+\frac{1+\epsilon}{1-\epsilon}D} \bigr)$-accurate w.r.t. $\cD_{t}(i)$. \end{lemma} Combining Lemma \ref{lem:rls} and Lemma \ref{lem:dictionary_accuracy} finishes the proof.
\begin{proof}[Proof of Lemma \ref{lem:dictionary_accuracy}] Similar to \citep{calandriello2019gaussian}, we can rewrite the $\epsilon$-accuracy condition of $\cS_{t_{p}}$ w.r.t. $\cD_{t}(i)$ for $t \in [t_{p}+1, t_{p+1}]$ as \begin{align*}
& (1-\epsilon_{t,i}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI) \preceq \bPhi_{\cD_{t}(i)}^{\top} \bar{\bS}_{t,i}^{\top} \bar{\bS}_{t,i}\bPhi_{\cD_{t}(i)} + \lambda \bI \preceq (1+\epsilon_{t,i}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI) \\
\Leftrightarrow & -\epsilon_{t,i} (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI) \preceq \bPhi_{\cD_{t}(i)}^{\top} \bar{\bS}_{t,i}^{\top} \bar{\bS}_{t,i}\bPhi_{\cD_{t}(i)} - \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} \preceq \epsilon_{t,i} (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI) \\
\Leftrightarrow & - \epsilon_{t,i} \bI \preceq (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} (\bPhi_{\cD_{t}(i)}^{\top} \bar{\bS}_{t,i}^{\top} \bar{\bS}_{t,i}\bPhi_{\cD_{t}(i)} - \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} \preceq \epsilon_{t,i} \bI \\
\Leftrightarrow & \lVert (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} (\bPhi_{\cD_{t}(i)}^{\top} \bar{\bS}_{t,i}^{\top} \bar{\bS}_{t,i}\bPhi_{\cD_{t}(i)} - \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} \rVert \leq \epsilon_{t,i} \\
\Leftrightarrow & \lVert \sum_{s \in \cD_{t_{p}} } (\frac{q_{s}}{\tilde{p}_{s}}-1) \psi_{s} \psi_{s}^{\top} + \sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}} } (0-1) \psi_{s} \psi_{s}^{\top} \rVert \leq \epsilon_{t,i} \end{align*} where $\psi_{s}=(\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} \phi_{s}$. Notice that the second term in the norm has weight $-1$ because the dictionary $\cS_{t_{p}}$ is fixed after $t_{p}$. With triangle inequality, now it suffices to bound \begin{align*}
\lVert \sum_{s \in \cD_{t_{p}} } (\frac{q_{s}}{\tilde{p}_{s}}-1) \psi_{s,j} \psi_{s}^{\top} + \sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}} } (0-1) \psi_{s} \psi_{s}^{\top} \rVert \leq \lVert \sum_{s \in \cD_{t_{p}} } (\frac{q_{s}}{\tilde{p}_{s}}-1) \psi_{s} \psi_{s}^{\top} \rVert + \lVert \sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}} } \psi_{s} \psi_{s}^{\top} \rVert. \end{align*} We should note that the first term corresponds to the approximation accuracy of $\cS_{t_{p}}$ w.r.t. the dataset $\cD_{t_{p}}$. And under the condition that it is $\epsilon$-accurate w.r.t. $\cD_{t_{p}}$, we have $\lVert \sum_{s \in \cD_{t_{p}} } (\frac{q_{s}}{\tilde{p}_{s}}-1) \psi_{s} \psi_{s}^{\top} \rVert \leq \epsilon$. The second term measures the difference between $\cD_{t}(i)$ compared with $\cD_{t_{p}}$, which is unique to our work. We can bound it as follows. \begin{align*}
& \lVert \sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}} } \psi_{s} \psi_{s}^{\top} \rVert \\
= & \lVert (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} (\sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}} } \phi_{s} \phi_{s}^{\top}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} \rVert \\
= & \max_{\phi \in \cH} \frac{\phi^{\top} (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} (\sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}} } \phi_{s} \phi_{s}^{\top}) (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1/2} \phi}{\phi^{\top} \phi} \\
= & \max_{\phi \in \cH} \frac{\phi^{\top} (\sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}} } \phi_{s} \phi_{s}^{\top}) \phi}{\phi^{\top} (\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI) \phi} \\
= & 1 - \min_{\phi \in \cH} \frac{\phi^{\top} (\bPhi_{\cD_{t_{p}}}^{\top}\bPhi_{\cD_{t_{p}}} + \lambda \bI) \phi}{\phi^{\top} (\bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} + \lambda \bI) \phi} \\
= & 1 - \frac{1}{\max_{\phi \in \cH} \frac{\phi^{\top} (\bPhi_{\cD_{t_{p}}}^{\top}\bPhi_{\cD_{t_{p}}} + \lambda \bI)^{-1} \phi}{\phi^{\top} (\bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1} \phi}} \\
= & 1- \frac{1}{\max_{\bx} \frac{\sigma^{2}_{t_{p},i}(\bx)}{\sigma^{2}_{t,i}(\bx)}}
\end{align*} We can further bound the term $\frac{\sigma^{2}_{t_{p},i}(\bx)}{\sigma^{2}_{t,i}(\bx)}$ using the threshold of the event-trigger in Eq~\eqref{eq:sync_event}. For any $\bx \in \bR^{d}$, \begin{align*}
\frac{\sigma^{2}_{t_{p},i}(\bx)}{\sigma^{2}_{t,i}(\bx)} \leq 1+\sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}}} \hat{\sigma}_{t_{p},i}^{2}(\bx_{s}) \leq 1 + \frac{1+\epsilon}{1-\epsilon} \sum_{s \in \cD_{t}(i) \setminus \cD_{t_{p}}} \tilde{\sigma}_{t_{p},i}^{2}(\bx_{s}) \leq 1 + \frac{1+\epsilon}{1-\epsilon} D \end{align*} where the first inequality is due to Lemma \ref{lem:bound_variance_ratio}, the second is due to Lemma \ref{lem:rls}, and the third is due to the event-trigger in Eq~\eqref{eq:sync_event}. Putting everything together, we have that if $\cS_{t_{p}}$ is $\epsilon$-accurate w.r.t. $\cD_{t_{p}}$, then it is $\bigl(\epsilon+1- \frac{1}{1 + \frac{1+\epsilon}{1-\epsilon} D} \bigr)$-accurate w.r.t. dataset $\cD_{t}(i)$, which finishes the proof.
\end{proof}
\section{Proof of Lemma \ref{lem:confidence_ellipsoid_approx}} \label{sec:proof_confidence_ellipsoid_approx}
To prove Lemma \ref{lem:confidence_ellipsoid_approx}, we need the following lemma. \begin{lemma} \label{lem:confidence_ellipsoid_intermediate} We have $\forall t,i$ that \begin{align*} \small
\lVert\tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}} \leq \Big( \lVert \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS}) \rVert + \sqrt{\lambda} \Big) \lVert \btheta_{\star} \rVert + R \sqrt{4 \ln{N/\delta}+ 2\ln{\det((1+\lambda) \bI + \bK_{\cD_{t}(i), \cD_{t}(i)})}} \end{align*} \normalsize with probability at least $1-\delta$. \end{lemma} \begin{proof}[Proof of Lemma \ref{lem:confidence_ellipsoid_intermediate}] Recall that the approximated kernel Ridge regression estimator for $\theta_{\star}$ is defined as \begin{align*}
\tilde{\btheta}_{t,i} = \tilde{\bA}_{t,i}^{-1} \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \by_{\cD_{t}(i)} \end{align*} where $\bP_{\cS}$ is the orthogonal projection matrix for the Nystr\"{o}m approximation, and $\tilde{\bA}_{t,i}=\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} \bP_{\cS} + \lambda \bI$. Then our goal is to bound \begin{align*}
& (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i} (\tilde{\btheta}_{t,i} - \btheta_{\star}) \\ = & (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i} (\tilde{\bA}_{t,i}^{-1}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \by_{\cD_{t}(i)} - \btheta_{\star})\\
= &(\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i} [\tilde{\bA}_{t,i}^{-1}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} (\bPhi_{\cD_{t}(i)} \btheta_{\star} + \mathbf{\eta}_{\cD_{t}(i)}) - \btheta_{\star}] \\
= &(\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i} (\tilde{\bA}_{t,i}^{-1}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} \btheta_{\star} - \btheta_{\star}) + (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i} \tilde{\bA}_{t,i}^{-1}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \end{align*} \paragraph{Bounding the first term} To bound the first term, we begin with rewriting \begin{align*}
& \tilde{\bA}_{t,i} (\tilde{\bA}_{t,i}^{-1}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} \btheta_{\star} - \btheta_{\star}) \\
= & \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} \btheta_{\star} - \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} \bP_{\cS} \btheta_{\star} - \lambda \btheta_{\star} \\
= & \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS}) \btheta_{\star} - \lambda \btheta_{\star} \end{align*} and by substituting this into the first term, we have \begin{align*}
& (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i} (\tilde{\bA}_{t,i}^{-1}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} \btheta_{\star} - \btheta_{\star}) \\
= & (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS}) \btheta_{\star} - \lambda (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \btheta_{\star} \\
=& (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i}^{1/2} \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS}) \btheta_{\star} - \lambda (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i}^{1/2} \tilde{\bA}_{t,i}^{-1/2} \btheta_{\star} \\
\leq & \lVert \tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}} \bigl( \lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS}) \btheta_{\star} \rVert + \lambda \lVert \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}^{-1}} \bigr) \\
\leq & \lVert \tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}} \bigl( \lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \rVert \lVert \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS}) \rVert \lVert \btheta_{\star} \rVert + \sqrt{\lambda} \lVert \btheta_{\star} \rVert \bigr) \\
\leq & \lVert \tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}} \bigl( \lVert \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS}) \rVert + \sqrt{\lambda} \bigr) \lVert \btheta_{\star} \rVert \end{align*} where the first inequality is due to Cauchy Schwartz, and the last inequality is because $\lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \rVert = \sqrt{\bPhi_{\cD_{t}(i)} \bP_{\cS} (\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} \bP_{\cS} + \lambda \bI)^{-1} \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} } \leq 1$.
\paragraph{Bounding the second term} By applying Cauchy-Schwartz inequality to the second term, we have \begin{align*}
& (\tilde{\btheta}_{t,i} - \btheta_{\star})^{\top} \tilde{\bA}_{t,i} \tilde{\bA}_{t,i}^{-1}\bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \\
\leq & \lVert \tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}} \lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \rVert \\
= & \lVert \tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}} \lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bA_{t,i}^{1/2} \bA_{t,i}^{-1/2} \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \rVert \\
\leq & \lVert \tilde{\btheta}_{t,i} - \btheta_{\star} \rVert_{\tilde{\bA}_{t,i}} \lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bA_{t,i}^{1/2} \rVert \lVert \bA_{t,i}^{-1/2} \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \rVert \end{align*}
Note that $\bP_{\cS} \bA_{t,i} \bP_{\cS} = \bP_{\cS} (\bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} + \lambda \bI) \bP_{\cS} = \tilde{\bA}_{t,i} + \lambda (\bP_{\cS} - \bI)$ and $\bP_{\cS} \preceq \bI$, so we have \begin{align*}
\lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bA_{t,i}^{1/2} \rVert & = \sqrt{ \lVert \tilde{\bA}_{t,i}^{-1/2} \bP_{\cS} \bA_{t,i}^{1/2} \bA_{t,i}^{1/2} \bP_{\cS} \tilde{\bA}_{t,i}^{-1/2}\rVert} \leq \sqrt{\lVert \tilde{\bA}_{t,i}^{-1/2} ( \tilde{\bA}_{t,i} + \lambda (\bP_{\cS} - \bI) ) \tilde{\bA}_{t,i}^{-1/2} \rVert} \\
& = \sqrt{\lVert \bI + \lambda \tilde{\bA}_{t,i}^{-1/2} (\bP_{\cS} - \bI) ) \tilde{\bA}_{t,i}^{-1/2} \rVert} \leq \sqrt{1+ \lambda \lVert \tilde{\bA}_{t,i}^{-1} \rVert \lVert \bP_{\cS} - \bI) \rVert} \\
& \leq \sqrt{1+ \lambda \cdot \lambda^{-1} \cdot 1 } = \sqrt{2} \end{align*} Then using the self-normalized bound derived for Lemma \ref{lem:self_normalized_bound}, the term $\lVert \bA_{t,i}^{-1/2} \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \rVert = \lVert \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \rVert_{\bA_{t,i}^{-1}}$ can be bounded by \begin{align*}
\lVert \bPhi_{\cD_{t}(i)}^{\top} \mathbf{\eta}_{\cD_{t}(i)} \rVert_{\bA_{t,i}^{-1}} & \leq R \sqrt{ 2 \ln(N/\delta) + \ln( \det( \bK_{ \cD_{t}(i), \cD_{t}(i) }/\lambda + \bI) ) } \\
& \leq R \sqrt{ 2 \ln(N/\delta) + 2 \gamma_{NT} } \end{align*} for $\forall t,i$, with probability at least $1-\delta$. Combining everything finishes the proof. \end{proof}
Now we are ready to prove Lemma \ref{lem:confidence_ellipsoid_approx} by further bounding the term $\lVert \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS_{t_{p}}}) \rVert$. \begin{proof}[Proof of Lemma \ref{lem:confidence_ellipsoid_approx}]
Recall that $\bar{\bS}_{t,i} \in \bR^{|\cD_{t}(i)| \times |\cD_{t}(i)|}$ denotes the diagonal matrix, whose $s$-th diagonal entry equals to $\frac{q_{s}}{\sqrt{\tilde{p}_{s}}}$, where $q_{s}=1$ if $s \in \cS_{t_{p}}$ and $0$ otherwise (note that for $s \notin \cS_{t_{p}}$, we set $\tilde{p}_{s}=1$, so $q_{s}/\tilde{p}_{s}=0$).
Therefore, $\forall s \in \cD_{t}(i) \setminus \cD_{t_{p}}$, $q_{s}=0$, as the dictionary is fixed after $t_{p}$. We can rewrite $\bPhi_{\cD_{t}(i)}^{\top} \bar{\bS}_{t,i}^{\top} \bar{\bS}_{t,i}\bPhi_{\cD_{t}(i)}=\sum_{s \in \cD_{t}(i)} \frac{q_{s}}{\tilde{p}_{s}}\phi_{s} \phi_{s}^{\top}$, where $\phi_{s}:=\phi(\bx_{s})$.
Then by definition of the spectral norm $\lVert \cdot \rVert$, and the properties of the projection matrix $\bP_{\cS_{t_{p}}}$, we have \begin{align} \label{eq:approx_error}
& \lVert \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS_{t_{p}}}) \rVert = \sqrt{\lambda_{\max}\bigl(\bPhi_{\cD_{t}(i)}(\bI - \bP_{\cS_{t_{p}}})^{2}\bPhi_{\cD_{t}(i)}^{\top} \bigr)} = \sqrt{\lambda_{\max}\bigl(\bPhi_{\cD_{t}(i)}(\bI - \bP_{\cS_{t_{p}}})\bPhi_{\cD_{t}(i)}^{\top} \bigr)}.
\end{align}
Moreover, due to Lemma \ref{lem:dictionary_accuracy}, we know $\cS_{t_{p}}$ is $\epsilon_{t,i}$-accurate w.r.t. $\cD_{t}(i)$ for $t \in [t_{p}+1,t_{p+1}]$, where $\epsilon_{t,i}=\bigl(\epsilon+1- \frac{1}{1 + \frac{1+\epsilon}{1-\epsilon} D} \bigr)$, so we have $\bI - \bP_{\cS_{t_{p}}} \preceq \frac{\lambda}{1-\epsilon_{t,i}} (\bPhi_{\cD_{t}(i)}^{\top}\bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1}$ by the property of $\epsilon$-accuracy (Proposition 10 of \cite{calandriello2019gaussian}).
Therefore,
by substituting this back to Eq~\eqref{eq:approx_error}, we have \begin{align*}
\lVert \bPhi_{\cD_{t}(i)} (\bI - \bP_{\cS_{t_{p}}}) \rVert
\leq \sqrt{\lambda_{\max}\bigl( \frac{\lambda}{1-\epsilon_{t,i}} \bPhi_{\cD_{t}(i)}(\bPhi_{\cD_{t}(i)}^{\top} \bPhi_{\cD_{t}(i)} + \lambda \bI)^{-1}\bPhi_{\cD_{t}(i)}^{\top} \bigr)} \leq \sqrt{\frac{\lambda}{-\epsilon+ \frac{1}{1 + \frac{1+\epsilon}{1-\epsilon} D}}} \end{align*} which finishes the proof. \end{proof}
\section{Proof of Theorem \ref{thm:regret_comm_sync}: Regret and Communication Cost of {Approx-DisKernelUCB}{}} \label{sec:proof_appprox_diskernelucb}
\subsection{Regret Analysis}
Consider some time step $t \in [t_{p-1}+1,t_{p}]$, where $p \in [B]$.
Due to Lemma \ref{lem:confidence_ellipsoid_approx}, i.e., the confidence ellipsoid for approximated estimator,
and the fact that $\bx_{t}=\argmax_{\bx \in \cA_{t,i}} \tilde{\mu}_{t-1,i}(\bx) + \alpha_{t-1,i} \tilde{\sigma}_{t-1,i}(\bx)$, we have \begin{align*}
f(\bx_{t}^{\star}) & \leq \tilde{\mu}_{t-1,i}(\bx_{t}^{\star}) + \alpha_{t-1,i}\tilde{\sigma}_{t-1,i}(\bx_{t}^{\star}) \leq \tilde{\mu}_{t-1,i}(\bx_{t}) + \alpha_{t-1,i} \tilde{\sigma}_{t-1,i}(\bx_{t}), \\
f(\bx_{t}) & \geq \tilde{\mu}_{t-1,i}(\bx_{t}) - \alpha_{t-1,i} \tilde{\sigma}_{t-1,i}(\bx_{t}), \end{align*} and thus $r_{t}=f(\bx_{t}^{\star})-f(\bx_{t}) \leq 2 \alpha_{t-1,i} \tilde{\sigma}_{t-1,i}(\bx_{t})$, where \begin{align*}
\alpha_{t-1,i}=\Bigg( \frac{1}{\sqrt{-\epsilon + \frac{1}{1+\frac{1+\epsilon}{1-\epsilon}D}}} + 1 \Bigg) \sqrt{\lambda} \lVert \btheta_{\star} \rVert + R \sqrt{4 \ln{N/\delta}+ 2\ln{\det((1+\lambda) \bI + \bK_{\cD_{t-1}(i), \cD_{t-1}(i)})}}. \end{align*} Note that, different from Appendix \ref{sec:proof_regret_diskernelucb}, the $\alpha_{t-1,i}$ term now depends on the threshold $D$ and accuracy constant $\epsilon$, as a result of the approximation error. As we will see in the following paragraphs, their values need to be set properly in order to bound $\alpha_{t-1,i}$.
We begin the regret analysis of {Approx-DisKernelUCB}{} with the same decomposition of good and bad epochs as in Appendix \ref{subsec:regret_proof_diskernelucb}, i.e., we call the $p$-th epoch a good epoch if $\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) \leq 1$, otherwise it is a bad epoch. Moreover, due to the pigeon-hold principle, there can be at most $2 \gamma_{NT}$ bad epochs.
As we will show in the following paragraphs, using Lemma \ref{lem:rls}, we can obtain a similar bound for the cumulative regret in good epochs as that in Appendix \ref{subsec:regret_proof_diskernelucb}, but with additional dependence on $D$ and $\epsilon$. The proof mainly differs in the bad epochs, where we need to use the event-trigger in Eq~\eqref{eq:sync_event} to bound the cumulative regret in each bad epoch. Compared with Eq~\eqref{eq:sync_event_exact}, Eq~\eqref{eq:sync_event} does not contain the number of local updates on each client since last synchronization., and as mentioned in Section \ref{subsec:theoretical_analysis}, this introduces a $\sqrt{T}$ factor in the regret bound for bad epochs in place of the $\sqrt{\gamma_{NT}}$ term in Appendix \ref{subsec:regret_proof_diskernelucb}.
\paragraph{Cumulative Regret in Good Epochs} Let's first consider some time step $t$ in a good epoch $p$, i.e., $t \in [t_{p-1}+1, t_{p}]$, and we have the following bound on the instantaneous regret \begin{align*}
r_{t} & \leq 2 \alpha_{t-1,i} \tilde{\sigma}_{t-1,i}(\bx_{t}) \leq 2 \alpha_{t-1,i} \tilde{\sigma}_{t_{p-1},i}(\bx_{t}) \leq 2 \alpha_{t-1,i} \frac{1+\epsilon}{1-\epsilon} \sigma_{t_{p-1},i}(\bx_{t}) \\
& = 2 \alpha_{t-1,i} \frac{1+\epsilon}{1-\epsilon} \sqrt{ \phi_{t}^{\top} A_{t_{p-1}}^{-1} \phi_{t}} \leq 2 \alpha_{t-1,i} \frac{1+\epsilon}{1-\epsilon} \sqrt{ \phi_{t}^{\top} A_{t-1}^{-1} \phi_{t}} \sqrt{\frac{\det(\bI + \lambda^{-1}\bK_{[t-1],[t-1]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}} \\
& \leq 2 \sqrt{e} \frac{1+\epsilon}{1-\epsilon} \alpha_{t-1,i} \sqrt{ \phi_{t}^{\top} A_{t-1}^{-1} \phi_{t}}
\end{align*} where the second inequality is because the (approximated) variance is non-decreasing, the third inequality is due to Lemma \ref{lem:rls}, the forth is due to Lemma \ref{lem:quadratic_det_inequality_infinite}, and the last is because in a good epoch, we have $\frac{\det(\bI + \lambda^{-1}\bK_{[t-1],[t-1]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})} \leq \frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})} \leq e$ for $t \in [t_{p-1}+1, t_{p}]$.
Therefore, the cumulative regret incurred in all good epochs, denoted by $R_{good}$, is upper bounded by \begin{align*}
R_{good} & \leq 2 \sqrt{e} \frac{1+\epsilon}{1-\epsilon} \sum_{t=1}^{NT} \alpha_{t-1,i} \sqrt{ \phi_{t}^{\top} A_{t-1}^{-1} \phi_{t}} \leq 2 \sqrt{e} \frac{1+\epsilon}{1-\epsilon} \alpha_{NT} \sqrt{NT \cdot \sum_{t=1}^{NT} \phi_{t}^{\top} A_{t-1}^{-1} \phi_{t} } \\
& \leq 2 \sqrt{e} \frac{1+\epsilon}{1-\epsilon} \alpha_{NT} \sqrt{NT \cdot 2\gamma_{NT} }
\end{align*} where $\alpha_{NT}:=\Bigg( \frac{1}{\sqrt{-\epsilon + \frac{1}{1+\frac{1+\epsilon}{1-\epsilon}D}}} + 1 \Bigg) \sqrt{\lambda} \lVert \btheta_{\star} \rVert + R \sqrt{4 \ln{N/\delta}+ 2\ln{\det((1+\lambda) \bI + \bK_{[NT],[NT]})}}$. \paragraph{Cumulative Regret in Bad Epochs}
The cumulative regret incurred in this bad epoch is \scriptsize \begin{align*}
& \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} \sum_{t=t_{p-1}+1}^{t_{p}} r_{t} \\
& \leq 2 \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} \sum_{t=t_{p-1}+1}^{t_{p}} \alpha_{t-1,i} \tilde{\sigma}_{t-1,i}(\bx_{t}) \\
& \leq 2 \alpha_{NT} \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} \sum_{i=1}^{N} \sum_{t \in \cN_{t_{p}}(i) \setminus \cN_{t_{p-1}}(i) }\tilde{\sigma}_{t-1,i}(\bx_{t}) \\
& \leq 2 \alpha_{NT} \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} \sum_{i=1}^{N} \sqrt{ (|\cN_{t_{p}}(i)|- |\cN_{t_{p-1}}(i)| ) \sum_{t \in \cN_{t_{p}}(i) \setminus \cN_{t_{p-1}}(i) }\tilde{\sigma}^{2}_{t-1,i}(\bx_{t}) } \\
& \leq 2 \alpha_{NT} \sqrt{D} \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} \sum_{i=1}^{N} \sqrt{ (|\cN_{t_{p}}(i)|- |\cN_{t_{p-1}}(i)| ) } \\
& \leq 2 \alpha_{NT} \sqrt{D} \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} \sum_{i=1}^{N} \sqrt{ \frac{t_{p}-t_{p-1}}{N} } \\
& \leq 2 \alpha_{NT} \sqrt{DN} \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} \sqrt{t_{p}-t_{p-1}} \\
& \leq 2 \alpha_{NT} \sqrt{DN} \sqrt{ \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} (t_{p}-t_{p-1}) \cdot \sum_{p=1}^{B} \mathbbm{1}\{\ln(\frac{\det(\bI + \lambda^{-1}\bK_{[t_{p}],[t_{p}]})}{\det(\bI + \lambda^{-1}\bK_{[t_{p-1}],[t_{p-1}]})}) > 1\} } \\
& \leq 2 \alpha_{NT} \sqrt{DN} \sqrt{2 N T \gamma_{NT}} \end{align*} \normalsize where the third inequality is due to the Cauchy-Schwartz inequality, the forth is due to our event-trigger in Eq~\eqref{eq:sync_event}, the fifth is due to our assumption that clients interact with the environment in a round-robin manner, the sixth is due to the Cauchy-Schwartz inequality again, and the last is due to the fact that there can be at most $2\gamma_{NT}$ bad epochs.
Combining cumulative regret incurred during both good and bad epochs, we have \begin{align*}
R_{NT} \leq R_{good} + R_{bad} \leq 2 \sqrt{e} \frac{1+\epsilon}{1-\epsilon} \alpha_{NT} \sqrt{NT \cdot 2\gamma_{NT} } + 2 \alpha_{NT} \sqrt{DN} \sqrt{2 N T \gamma_{NT}} \end{align*}
\subsection{Communication Cost Analysis} Consider some epoch $p$. We know that for the client $i$ who triggers the global synchronization, we have \begin{align*}
\frac{1+\epsilon}{1-\epsilon}\sum_{s = t_{p-1}+1}^{t_{p}} \sigma^{2}_{t_{p-1}}(\bx_{s}) \geq
\sum_{s = t_{p-1}+1}^{t_{p}} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{s}) \geq \sum_{s \in \cD_{t_{p}(i)} \setminus \cD_{t_{p-1}(i)}} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{s}) \geq D \end{align*} Then by summing over $B$ epochs, we have \begin{align*}
B D < \frac{1+\epsilon}{1-\epsilon} \sum_{p=1}^{B} \sum_{s = t_{p-1}+1}^{t_{p}} \sigma^{2}_{t_{p-1}}(\bx_{s}) \leq \frac{1+\epsilon}{1-\epsilon} \sum_{p=1}^{B} \sum_{s = t_{p-1}+1}^{t_{p}} \sigma^{2}_{s-1}(\bx_{s}) \frac{\sigma^{2}_{t_{p-1}}(\bx_{s})}{\sigma^{2}_{s-1}(\bx_{s})}. \end{align*} Now we need to bound the ratio $\frac{\sigma^{2}_{t_{p-1}}(\bx_{s})}{\sigma^{2}_{s-1}(\bx_{s})}$ for $s \in [t_{p-1}+1,t_{p}]$. \begin{align*}
& \frac{\sigma^{2}_{t_{p-1}}(\bx_{s})}{\sigma^{2}_{s-1}(\bx_{s})} \leq \Big[ 1 + \sum_{\tau = t_{p-1}+1}^{s} \sigma^{2}_{t_{p-1}}(\bx_{\tau}) \Big] \leq \Big[ 1 + \frac{1+\epsilon}{1-\epsilon} \sum_{\tau = t_{p-1}+1}^{s} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{\tau}) \Big] \end{align*} Note that for the client who triggers the global synchronization, we have $\sum_{s \in \cD_{t_{p}-1}(i) \setminus \cD_{t_{p-1}}(i)} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{s}) < D$, i.e., one time step before it triggers the synchronization at time $t_{p}$.
Due to the fact that the (approximated) posterior variance cannot exceed $L^{2}/\lambda$, we have $\sum_{s \in \cD_{t_{p}}(i) \setminus \cD_{t_{p-1}}(i)} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{s}) < D+ L^{2}/\lambda$. For the other $N-1$ clients, we have $\sum_{s \in \cD_{t_{p}}(i) \setminus \cD_{t_{p-1}}(i)} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{s}) < D$.
Summing them together, we have \begin{align*}
\sum_{s = t_{p-1}+1}^{t_{p}} \tilde{\sigma}^{2}_{t_{p-1}}(\bx_{s}) < (N D + L^{2}/\lambda) \end{align*} for the $p$-th epoch. By substituting this back, we have \begin{align*}
\frac{\sigma^{2}_{t_{p-1}}(\bx_{s})}{\sigma^{2}_{s-1}(\bx_{s})} \leq \Big[ 1 + \frac{1+\epsilon}{1-\epsilon} (N D + L^{2}/\lambda) \Big] \end{align*} Therefore, \begin{align*}
B D & < \frac{1+\epsilon}{1-\epsilon}\Big[ 1 + \frac{1+\epsilon}{1-\epsilon} (N D + L^{2}/\lambda) \Big] \sum_{p=1}^{B} \sum_{s = t_{p-1}+1}^{t_{p}} \sigma^{2}_{s-1}(\bx_{s}) \\
& \leq \frac{1+\epsilon}{1-\epsilon}\Big[ 1 + \frac{1+\epsilon}{1-\epsilon} (N D + L^{2}/\lambda) \Big] 2 \gamma_{NT} \end{align*} and thus the total number of epochs $B < \frac{1+\epsilon}{1-\epsilon}[ \frac{1}{D} + \frac{1+\epsilon}{1-\epsilon} (N + L^{2}/(\lambda D)) ] 2 \gamma_{NT}$.
By setting $D=\frac{1}{N}$, we have \begin{align*}
\alpha_{NT} & =\Bigg( \frac{1}{\sqrt{-\epsilon + \frac{1}{1+\frac{1+\epsilon}{1-\epsilon} \frac{1}{N}}}} + 1 \Bigg) \sqrt{\lambda} \lVert \btheta_{\star} \rVert + R \sqrt{4 \ln{N/\delta}+ 2\ln{\det((1+\lambda) \bI + \bK_{[NT],[NT]})}} \\
& \leq \Bigg( \frac{1}{\sqrt{-\epsilon + \frac{1}{1+\frac{1+\epsilon}{1-\epsilon} }}} + 1 \Bigg) \sqrt{\lambda} \lVert \btheta_{\star} \rVert + R \sqrt{4 \ln{N/\delta}+ 2\ln{\det((1+\lambda) \bI + \bK_{[NT],[NT]})}} \end{align*} because $N \geq 1$. Moreover, to ensure $-\epsilon + \frac{1}{1+\frac{1+\epsilon}{1-\epsilon} } > 0$, we need to set the constant $\epsilon < 1/3$. Therefore, \begin{align*}
R_{NT}=O\Big( \sqrt{NT} ( \lVert \theta_{\star} \rVert \sqrt{\gamma_{NT}} + \gamma_{NT}) \Big) \end{align*} and the total number of global synchronizations $B=O(N\gamma_{NT})$. Since for each global synchronization, the communication cost is $O(N\gamma_{NT}^{2})$, we have \begin{align*}
C_{NT} = O \Big( N^{2} \gamma_{NT}^{3} \Big) \end{align*}
\section{Experiment Setup}
\paragraph{Synthetic dataset} We simulated the distributed bandit setting defined in Section \ref{subsec:problem_formulation}, with $d=20,T=100, N=100$ ($NT=10^{4}$ interactions in total). In each round $l \in [T]$, each client $i \in [N]$ (denote $t=N(l-1)+i$) selects an arm from candidate set $\cA_{t}$,
where $\cA_{t}$ is uniformly sampled from a $\ell_2$ unit ball, with $|\cA_{t}|=20$.
Then the corresponding reward is generated using one of the following reward functions:
\begin{align*}
& f_{1}(\bx) = \cos(3 \bx^{\top} \btheta_{\star}) \\ & f_{2}(\bx) = (\bx^{\top} \btheta_{\star})^{3} - 3(\bx^{\top} \btheta_{\star})^{2} - (\bx^{\top} \btheta_{\star}) + 3
\end{align*} where the parameter $\btheta_{\star}$ is uniformly sampled from a $\ell_2$ unit ball.
\paragraph{UCI Datasets}
To evaluate {Approx-DisKernelUCB}{}'s performance in a more challenging and practical scenario, we performed experiments using real-world datasets: MagicTelescope, Mushroom and Shuttle from the UCI Machine Learning Repository \citep{Dua:2019}. To convert them to contextual bandit problems, we pre-processed these datasets following the steps in \citep{filippi2010parametric}. In particular, we partitioned the dataset in to $20$ clusters using k-means, and used the centroid of each cluster as the context vector for the arm and the averaged response variable as mean reward (the response variable is binarized by associating one class as $1$, and all the others as $0$). Then we simulated the distributed bandit learning problem in Section \ref{subsec:problem_formulation} with $|\cA_{t}|=20$, $T=100$ and $N=100$ ($NT=10^{4}$ interactions in total).
\paragraph{MovieLens and Yelp dataset} Yelp dataset, which is released by the Yelp dataset challenge, consists of 4.7 million rating entries for 157 thousand restaurants by 1.18 million users. MovieLens is a dataset consisting of 25 million ratings between 160 thousand users and 60 thousand movies \citep{harper2015movielens}. Following the pre-processing steps in \citep{ban2021ee}, we built the rating matrix by choosing the top 2000 users and top 10000 restaurants/movies and used singular-value decomposition (SVD) to extract a 10-dimension feature vector for each user and restaurant/movie. We treated rating greater than $2$ as positive.
We simulated the distributed bandit learning problem in Section \ref{subsec:problem_formulation} with $T=100$ and $N=100$ ($NT=10^{4}$ interactions in total). In each time step, the candidate set $\cA_{t}$ (with $|\cA_{t}|=20$) is constructed by sampling an arm with reward $1$ and nineteen arms with reward $0$ from the arm pool, and the concatenation of user and restaurant/movie feature vector is used as the context vector for the arm (thus $d=20$).
\section{Lower Bound for Distributed Kernelized Contextual Bandits}
First, we need the following two lemmas
\begin{lemma}[Theorem 1 of \cite{valko2013finite}] \label{lem:KernelUCB_upperbound} There exists a constant $C > 0$, such that for any instance of kernelized bandit with $L=S=R= 1$, the expected cumulative regret for KernelUCB algorithm is upper bounded by $\bbE[R_{T}] \leq C\sqrt{T\gamma_{T}}$, where the maximum information gain $\gamma_{T}=O\bigl((\ln(T))^{d+1}\bigr)$ for Squared Exponential kernels.
\end{lemma}
\begin{lemma}[Theorem 2 of \cite{scarlett2017lower}] \label{lem:squaredexponential_lowerbound} There always exists a set of hard-to-learn instances of kernelized bandit with $L=S=R= 1$, such that for any algorithm, for a uniformly random instance in the set, the expected cumulative regret $\bbE[R_{T}] \geq c \sqrt{T (\ln(T))^{d/2}}$ for Squared Exponential kernels,
with some constant $c$. \end{lemma}
Then we follow a similar procedure as the proof for Theorem 2 of \cite{wang2019distributed} and Theorem 5.3 of \cite{he2022simple}, to prove the following lower bound results for distributed kernelized bandit with Squared Exponential kernels.
\begin{theorem}\label{thm:diskernel_lowerbound} For any distributed kernelized bandit algorithm with expected communication cost less than $O(\frac{N}{(\ln(T))^{0.25 d + 0.5}})$, there exists a kernelized bandit instance with Squared Exponential kernel, and $L=S=R=1$, such that the expected cumulative regret for this algorithm is at least $\Omega(N \sqrt{T(\ln(T))^{d/2}})$. \end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:diskernel_lowerbound}] Here we consider kernelized bandit with Squared Exponential kernels. The proof relies on the construction of a auxiliary algorithm, denoted by \textbf{AuxAlg}, based on the original distributed kernelized bandit algorithm, denoted by \textbf{DisKernelAlg}, as shown below. For each agent $i \in [N]$, \textbf{AuxAlg} performs \textbf{DisKernelAlg}, until any communication happens between client $i$ and the server, in which case, \textbf{AuxAlg} switches to the single-agent optimal algorithm, i.e., the KernelUCB algorithm that attains the rate in Lemma \ref{lem:KernelUCB_upperbound}. Therefore, \textbf{AuxAlg} is a single-agent bandit algorithm, and the lower bound in Lemma \ref{lem:squaredexponential_lowerbound} applies: the cumulative regret that \textbf{AuxAlg} incurs for some agent $i\in[N]$ is lower bounded by \begin{align*}
\bbE[R_{\textbf{AuxAlg},i}] \geq c \sqrt{T (\ln(T))^{d/2}}, \end{align*} and by summing over all $N$ clients, we have \begin{align*}
\bbE[R_{\textbf{AuxAlg}}] = \sum_{i=1}^{N}\bbE[R_{\textbf{AuxAlg},i}] \geq c N\sqrt{T (\ln(T))^{d/2}}. \end{align*}
For each client $i \in [N]$, denote the probability that client $i$ will communicate with the server as $p_{i}$, and $p:=\sum_{i=1}^{N} p_{i}$. Note that before the communication, the cumulative regret incurred by \textbf{AuxAlg} is the same as \textbf{DisKernelAlg}, and after the communication happens, the regret incurred by \textbf{AuxAlg} is the same as KernelUCB, whose upper bound is given in Lemma \ref{lem:KernelUCB_upperbound}. Therefore, the cumulative regret that \textbf{AuxAlg} incurs for client $i$ can be upper bounded by \begin{align*}
\bbE[R_{\textbf{AuxAlg},i}] \leq \bbE[R_{\textbf{DisKernelAlg},i}] + p_{i} C\sqrt{T(\ln(T))^{d+1}}, \end{align*} and by summing over $N$ clients, we have \begin{align*}
\bbE[R_{\textbf{AuxAlg}}] & = \sum_{i=1}^{N}\bbE[R_{\textbf{AuxAlg},i}] \\
& \leq \sum_{i=1}^{N} \bbE[R_{\textbf{DisKernelAlg},i}] + (\sum_{i=1}^{N} p_{i}) C\sqrt{T (\ln(T))^{d+1}} \\
& = \bbE[R_{\textbf{DisKernelAlg}}] + p C\sqrt{T (\ln(T))^{d+1}}. \end{align*} Combining the upper and lower bounds for $\bbE[R_{\textbf{AuxAlg}}]$, we have \begin{align*}
\bbE[R_{\textbf{DisKernelAlg}}] \geq c N\sqrt{T (\ln(T))^{d/2}} - p C\sqrt{T (\ln(T))^{d+1}}. \end{align*} Therefore, for any \textbf{DisKernelAlg} with number of communications $p\leq N \frac{c}{2C (\ln(T))^{0.25 d + 0.5}}=O(\frac{N}{(\ln(T))^{0.25 d + 0.5}})$, we have \begin{align*}
\bbE[R_{\textbf{DisKernelAlg}}] \geq \frac{c}{2} N\sqrt{T (\ln(T))^{d/2}} = \Omega(N \sqrt{T(\ln(T))^{d/2}}). \end{align*}
\end{proof}
\end{document} | arXiv |
Journal of the Mathematical Society of Japan
J. Math. Soc. Japan
Volume 69, Number 3 (2017), 913-943.
Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom
Teruyuki YORIOKA
More by Teruyuki YORIOKA
Recently, David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions. The side condition method with models was introduced by Stevo Todorčević in the 1980s. The Asperó–Mota iteration enables us to force some $\Pi_2$-statements over $H(\aleph_2)$ with the continuum greater than $\aleph_2$. In this article, by using the Asperó–Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak{p}= {\mathrm{add}}(\mathcal{N}) = 2^{\aleph_0}>\aleph_2$ and ${MA}_{\aleph_1}$ fails.
J. Math. Soc. Japan, Volume 69, Number 3 (2017), 913-943.
First available in Project Euclid: 12 July 2017
https://projecteuclid.org/euclid.jmsj/1499846513
doi:10.2969/jmsj/06930913
Primary: 03E35: Consistency and independence results
Secondary: 03E05: Other combinatorial set theory 03E17: Cardinal characteristics of the continuum
side condition method $\mho$ weak club guessing sequences cardinal invariants gaps Martin's Axiom
YORIOKA, Teruyuki. Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom. J. Math. Soc. Japan 69 (2017), no. 3, 913--943. doi:10.2969/jmsj/06930913. https://projecteuclid.org/euclid.jmsj/1499846513
U. Avraham and S. Shelah, Martin's axiom does not imply that every two $\aleph_1$-dense sets of reals are isomorphic, Israel J. Math., 38 (1981), 161–176.
Mathematical Reviews (MathSciNet): MR599485
U. Abraham and S. Todorčević, Partition properties of $\omega_1$ compatible with CH, Fund. Math., 152 (1997), 165–180.
D. Asperó and M. A. Mota, Forcing consequences of PFA together with the continuum large, Trans. Amer. Math. Soc., 367 (2015), 6103–6129.
Digital Object Identifier: doi:10.1090/S0002-9947-2015-06205-9
D. Asperó and M. A. Mota, A Generalization of Martin's Axiom, Israel J. Math., 210 (2015), 193–231.
T. Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc., 281 (1984), 209–213.
Digital Object Identifier: doi:10.1090/S0002-9947-1984-0719666-7
T. Bartoszyński, Invariants of Measure and Category, Handbook of set theory, Vols. 1–3, Springer, Dordrecht, 2010, 491–555.
T. Bartoszyński and H. Judah, Set theory, On the structure of the real line, A K Peters, Ltd., Wellesley, MA, 1995.
J. Baumgartner, Applications of the proper forcing axiom, Handbook of Set-Theoretic Topology, Chapter 21, pp.,913–959.
M. Bell, On the combinatorial principle $P(\mathfrak{c})$, Fund. Math., 114 (1981), 149–157.
A. Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory, Vols. 1–3, Springer, Dordrecht, 2010, 395–489.
T. Eisworth, J. T. Moore and D. Milovich, Iterated forcing and the Continuum Hypothesis, In: Appalachian set theory 2006–2012, (eds. J. Cummings and E. Schimmerling), London Math. Society, Lecture Notes Series, Cambridge University Press, 2013.
T. Jech, Set theory, The third millennium edition, revised and expanded. Springer Monographs in Math., Springer-Verlag, Berlin, 2003.
K. Kunen, Set theory, An introduction to independence proofs, Studies in Logic and the Foundations of Math., 102, North-Holland Publishing Co., Amsterdam-New York, 1980.
K. Kunen, Set theory, Studies in Logic (London), 34, College Publications, London, 2011.
J. T. Moore, A five element basis for the uncountable linear orders, Ann. of Math. (2), 163 (2006), 669–688.
Digital Object Identifier: doi:10.4007/annals.2006.163.669
J. T. Moore, Aronszajn lines and the club filter, J. Symbolic Logic, 73 (2008), 1029–1035.
Digital Object Identifier: doi:10.2178/jsl/1230396763
Project Euclid: euclid.jsl/1230396763
J. T. Moore and D. Milovich, A tutorial on Set Mapping Reflection, In: Appalachian set theory 2006–2012, (eds. J. Cummings and E. Schimmerling), London Math. Society, Lecture Notes Series, Cambridge University Press, 2013.
S. Shelah, Proper and improper forcing, Second edition, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.
M. Scheepers, Gaps in $\omega^\omega$, In: Set Theory of the Reals, Israel Math. Conference Proceedings, 6 1993, 439–561.
R. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. (2), 94 (1971), 201–245.
Digital Object Identifier: doi:10.2307/1970860
S. Todorčević, A note on the proper forcing axiom, Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984, 209–218.
S. Todorčević, Directed sets and cofinal types, Transactions of American Math. Society, 290, no. 2, 1985, pp.,711–723.
S. Todorčević, Partition Problems in Topology, Contemporary mathematics, 84, American Math. Society, Providence, Rhode Island, 1989.
S. Todorčević, S-gaps and T-gaps, Handwritten note, August 6, 2005.
S. Todorčević and I. Farah, Some applications of the method of forcing, Yenisei Series in Pure and Applied Math., Yenisei, Moscow; Lycée, Troitsk, 1995.
S. Todorčević and B. Veličković, Martin's axiom and partitions, Compositio Math., 63 (1987), 391–408.
T. Yorioka, The diamond principle for the uniformity of the meager ideal implies the existence of a destructible gap, Arch. Math. Logic, 44 (2005), 677–683.
T. Yorioka, Combinatorial principles on $\omega_1$, cardinal invariants of the meager ideal and destructible gaps, J. Math. Soc. Japan, 57 (2005), 1217–1228.
Digital Object Identifier: doi:10.2969/jmsj/1150287311
Project Euclid: euclid.jmsj/1150287311
T. Yorioka, Some weak fragments of Martin's axiom related to the rectangle refining property, Arch. Math. Logic, 47 (2008), 79–90.
T. Yorioka, A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees, Ann. Pure Appl. Logic, 161 (2010), 469–487.
Digital Object Identifier: doi:10.1016/j.apal.2009.02.006
Mathematical Society of Japan
Forcing axioms and the continuum hypothesis
Asperό, David, Larson, Paul, and Moore, Justin Tatch, Acta Mathematica, 2013
Martin's Axioms, Measurability and Equiconsistency Results
Ihoda, Jaime I. and Shelah, Saharon, Journal of Symbolic Logic, 1989
Gregory trees, the continuum, and Martin's axiom
Kunen, Kenneth and Raghavan, Dilip, Journal of Symbolic Logic, 2009
Independence Results
Shelah, Saharon, Journal of Symbolic Logic, 1980
Souslin Forcing
A Characterization of Martin's Axiom in Terms of Absoluteness
Bagaria, Joan, Journal of Symbolic Logic, 1997
Definable well-orders of $H(\omega _2)$ and $GCH$
Asperó, David and Friedman, Sy-David, Journal of Symbolic Logic, 2012
On the number of normal measures $\aleph_1$ and $\aleph_2$ can carry
Apter, Arthur W., Tbilisi Mathematical Journal, 2008
Forcing with Sequences of Models of Two Types
Neeman, Itay, Notre Dame Journal of Formal Logic, 2014
ℛmax variations for separating club guessing principles
Ishiu, Tetsuya and Larson, Paul B., Journal of Symbolic Logic, 2012
euclid.jmsj/1499846513 | CommonCrawl |
Space Exploration Meta
Space Exploration Beta
Equation for orbital period around oblate bodies, based on J2?
In this answer I point out that the period of items (ring particles, moons, spacecraft, etc.) around an oblate body will not scale exactly as $a^{3/2}$ because the closer you are to the planet, the stronger the perturbing effects are as a result of being much closer to the near-side of the oblate "ring" than the far side of it. Mathematically that turns out to be $1/r^4$ vs $1/r^2$.
I can mindlessly calculate orbits including the $J_2$ term as shown in this answer using these radial acceleration terms assuming an equatorial orbit:
$$a_0 = -\frac{GM}{r^2},$$
$$a_2 = -\frac{3}{2} J_2 \frac{GM R^2}{r^4},$$
where $a_0$ is the radial acceleration due to the monopole term and $a_2$ is the radial acceleration due to the quadrupole term — that part of the oblateness captured within the $J_2$ coefficient, and $R$ is the normalizing radius of the body used to keep $J_2$ dimensionless.
I can rewrite this as
$$a_{tot} = -\frac{GM}{r^2} \left( 1+\frac{3}{2} J_2 \frac{R^2}{r^2} \right)$$
and just decide that for the circular equatorial case I can set $r$ equal to the semi-major axis and the "effective mass" of the central body is increased by the factor in the parenthesis, but I am not sure if I've done this right, and certainly don't know what to do if the orbit is elliptical and/or inclined.
Question: What would an equation for the period of a circular orbit taking into account $J_2$ look like? Is there something that would include either elliptical and/or inclined orbits as well?
I'm also a bit confused about the mass and its distribution. I'd like to double check that the standard gravitational parameter $GM$ represents all of the mass including that in the equatorial bulge, and that we're not somehow double-counting that by using $J_2$.
A related and (still) unanswered question is For the mathematical relationship between J2 (km^5/s^2) and dimensionless J2 - which one is derived from the other?.
orbital-mechanics mathematics
uhohuhoh
$\begingroup$ @ChrisB.Behrens thanks for your several title rewrites, but after asking over 1,000 Stack Exchange questions, I've sort-of developed a sense for the way I'd like to write titles. $\endgroup$ – uhoh Mar 7 '18 at 17:08
If you consider the orbital period to be defined as successive node crossings, that's known as the nodal period. For an orbit with semimajor axis $a$ around a spherical body with gravitational parameter $\mu$, the nodal period is equal to the Keplerian period: $T_0=2\pi \sqrt\frac{a^3}{\mu}$, however, as you point out, this changes when oblateness is taken into account. Wikipedia has one form for the expression taking the $J_2$ budge into account: $$T = T_0\left[1 - \frac{3J_2(4-5\sin^2 i)}{4\left(\frac{a}{R}\right)^2\sqrt{1-e^2}(1+e\cos\omega)^2} - \frac{3J_2(1-e\cos\omega)^3}{2\left(\frac{a}{R}\right)^2(1-e^2)^3}\right]$$
As you can see, it depends on the eccentricity $e$, argument of perigee $\omega$, and inclination $i$ of the orbit, as opposed to $T_0$ which is only a function of semimajor axis. $R$ is the equatorial radius of the body.
As an example, using this equation, an orbit around Earth with $a=6778~\textrm{km}$, $e=1\times10^{-3}$, $i=20^\circ$, and $\omega=0^\circ$ has a Keplerian period of about 92.56 minutes vs. a nodal period incorporating $J_2$ of about 92.20 minutes, the latter being a little under 22 seconds shorter.
$\begingroup$ Yikes! That's more complicated than I expected, but then again I didn't really know what to expect. I'll take it for a spin. I'm never sure what qualifies as a period for an orbit that doesn't exactly repeat itself, but it looks like the nodal period is a pretty easy one to understand, and test for. Thank you for both the equation and the explanation! $\endgroup$ – uhoh Mar 8 '18 at 4:13
Thanks for contributing an answer to Space Exploration Stack Exchange!
Not the answer you're looking for? Browse other questions tagged orbital-mechanics mathematics or ask your own question.
Returning to moderation
Why are Tiangong-1's Apogee and Perigee Graphs Wobbly?
Determine orbit type from TLE
Physical meaning of perigee advance
For the mathematical relationship between J2 (km^5/s^2) and dimensionless J2 - which one is derived from the other?
What delta-v per orbit would a spacecraft need to hover next to Saturn's rings?
Mechanics of Sun-Synchronous Orbit & North Korea's KMS-4
Secular variation of orbital elements and influence of force components
How to derive the polar form of the equations of motion?
How can I verify my reconstructed gravity field of Ceres from spherical harmonics?
How to calculate the flight path angle, γ, from a state vector?
Why are JPL using this expression to emulate Schwarzschild orbits? | CommonCrawl |
Geodesics as Hamiltonian flows
In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form of Hamilton's equations. This latter formulation is developed in this article.
Overview
It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton–Jacobi approach to the geodesic equation, this statement can be given a very intuitive meaning: geodesics describe the motions of particles that are not experiencing any forces. In flat space, it is well known that a particle moving in a straight line will continue to move in a straight line if it experiences no external forces; this is Newton's first law. The Hamiltonian describing such motion is well known to be $H=p^{2}/2m$ with p being the momentum. It is the conservation of momentum that leads to the straight motion of a particle. On a curved surface, exactly the same ideas are at play, except that, in order to measure distances correctly, one must use the Riemannian metric. To measure momenta correctly, one must use the inverse of the metric. The motion of a free particle on a curved surface still has exactly the same form as above, i.e. consisting entirely of a kinetic term. The resulting motion is still, in a sense, a "straight line", which is why it is sometimes said that geodesics are "straight lines in curved space". This idea is developed in greater detail below.
Geodesics as an application of the principle of least action
Given a (pseudo-)Riemannian manifold M, a geodesic may be defined as the curve that results from the application of the principle of least action. A differential equation describing their shape may be derived, using variational principles, by minimizing (or finding the extremum) of the energy of a curve. Given a smooth curve
$\gamma :I\to M$
that maps an interval I of the real number line to the manifold M, one writes the energy
$E(\gamma )={\frac {1}{2}}\int _{I}g({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt,$
where ${\dot {\gamma }}(t)$ is the tangent vector to the curve $\gamma $ at point $t\in I$. Here, $g(\cdot ,\cdot )$ is the metric tensor on the manifold M.
Using the energy given above as the action, one may choose to solve either the Euler–Lagrange equations or the Hamilton–Jacobi equations. Both methods give the geodesic equation as the solution; however, the Hamilton–Jacobi equations provide greater insight into the structure of the manifold, as shown below. In terms of the local coordinates on M, the (Euler–Lagrange) geodesic equation is
${\frac {d^{2}x^{a}}{dt^{2}}}+\Gamma _{bc}^{a}{\frac {dx^{b}}{dt}}{\frac {dx^{c}}{dt}}=0$
where the xa(t) are the coordinates of the curve γ(t), $\Gamma _{bc}^{a}$ are the Christoffel symbols, and repeated indices imply the use of the summation convention.
Hamiltonian approach to the geodesic equations
Geodesics can be understood to be the Hamiltonian flows of a special Hamiltonian vector field defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term.
The geodesic equations are second-order differential equations; they can be re-expressed as first-order equations by introducing additional independent variables, as shown below. Note that a coordinate neighborhood U with coordinates xa induces a local trivialization of
$T^{*}M|_{U}\simeq U\times \mathbb {R} ^{n}$
by the map which sends a point
$\eta \in T_{x}^{*}M|_{U}$
of the form $\eta =p_{a}dx^{a}$ to the point $(x,p_{a})\in U\times \mathbb {R} ^{n}$. Then introduce the Hamiltonian as
$H(x,p)={\frac {1}{2}}g^{ab}(x)p_{a}p_{b}.$
Here, gab(x) is the inverse of the metric tensor: gab(x)gbc(x) = $\delta _{c}^{a}$. The behavior of the metric tensor under coordinate transformations implies that H is invariant under a change of variable. The geodesic equations can then be written as
${\dot {x}}^{a}={\frac {\partial H}{\partial p_{a}}}=g^{ab}(x)p_{b}$
and
${\dot {p}}_{a}=-{\frac {\partial H}{\partial x^{a}}}=-{\frac {1}{2}}{\frac {\partial g^{bc}(x)}{\partial x^{a}}}p_{b}p_{c}.$
The flow determined by these equations is called the cogeodesic flow; a simple substitution of one into the other obtains the Euler–Lagrange equations, which give the geodesic flow on the tangent bundle TM. The geodesic lines are the projections of integral curves of the geodesic flow onto the manifold M. This is a Hamiltonian flow, and the Hamiltonian is constant along the geodesics:
${\frac {dH}{dt}}={\frac {\partial H}{\partial x^{a}}}{\dot {x}}^{a}+{\frac {\partial H}{\partial p_{a}}}{\dot {p}}_{a}=-{\dot {p}}_{a}{\dot {x}}^{a}+{\dot {x}}^{a}{\dot {p}}_{a}=0.$
Thus, the geodesic flow splits the cotangent bundle into level sets of constant energy
$M_{E}=\{(x,p)\in T^{*}M:H(x,p)=E\}$
for each energy E ≥ 0, so that
$T^{*}M=\bigcup _{E\geq 0}M_{E}$.
References
• Terence Tao, The Euler-Arnold Equation, 2010: http://terrytao.wordpress.com/2010/06/07/the-euler-arnold-equation/ See the discussion at the beginning
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 2.7.
• B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov, Modern Geometry: Methods and Applications, Part I, (1984) Springer-Verlag, Berlin ISBN 0-387-90872-2 See chapter 5, in particular section 33.
| Wikipedia |
Login | Create
Sort by: Relevance Date Users's collections Twitter
Group by: Day Week Month Year All time
Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
PICOSEC: Charged particle timing at sub-25 picosecond precision with a Micromegas based detector (1712.05256)
J. Bortfeldt, F. Brunbauer, C. David, D. Desforge, G. Fanourakis, J. Franchi, M. Gallinaro, I. Giomataris, D. González-Díaz, T. Gustavsson, C. Guyot, F.J. Iguaz, M. Kebbiri, P. Legou, J. Liu, M. Lupberger, O. Maillard, I. Manthos, H. Müller, V. Niaouris, E. Oliveri, T. Papaevangelou, K. Paraschou, M. Pomorski, B. Qi, F. Resnati, L. Ropelewski, D. Sampsonidis, T. Schneider, P. Schwemling, L. Sohl, M. van Stenis, P. Thuiner, Y. Tsipolitis, S.E. Tzamarias, R. Veenhof, X. Wang, S. White, Z. Zhang, Y. Zhou
March 14, 2018 physics.ins-det
The prospect of pileup induced backgrounds at the High Luminosity LHC (HL-LHC) has stimulated intense interest in developing technologies for charged particle detection with accurate timing at high rates. The required accuracy follows directly from the nominal interaction distribution within a bunch crossing ($\sigma_z\sim5$ cm, $\sigma_t\sim170$ ps). A time resolution of the order of 20-30 ps would lead to significant reduction of these backgrounds. With this goal, we present a new detection concept called PICOSEC, which is based on a "two-stage" Micromegas detector coupled to a Cherenkov radiator and equipped with a photocathode. First results obtained with this new detector yield a time resolution of 24 ps for 150 GeV muons, and 76 ps for single photoelectrons.
Measurement of Elastic pp Scattering at $\sqrt{s}$ = 8 TeV in the Coulomb-Nuclear Interference Region - Determination of the $\rho$-Parameter and the Total Cross-Section (1610.00603)
TOTEM Collaboration: G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, U. Bottigli, M. Bozzo, P. Broulím, H. Burkhardt, A. Buzzo, F. S. Cafagna, C. E. Campanella, M. G. Catanesi, M. Csanád, T. Csörgő, M. Deile, F. De Leonardis, A. D'Orazio, M. Doubek, K. Eggert, V. Eremin, F. Ferro, A. Fiergolski, F. Garcia, V. Georgiev, S. Giani, L. Grzanka, C. Guaragnella, J. Hammerbauer, J. Heino, A. Karev, J. Kašpar, J. Kopal, V. Kundrát, S. Lami, G. Latino, R. Lauhakangas, R. Linhart, E. Lippmaa, J. Lippmaa, M. V. Lokajíček, L. Losurdo, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, A. Mercadante, N. Minafra, S. Minutoli, T. Naaranoja, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, L. Paločko, V. Passaro, Z. Peroutka, V. Petruzzelli, T. Politi, J. Procházka, F. Prudenzano, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, S. Redaelli, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, B. Salvachua, A. Scribano, J. Smajek, W. Snoeys, J. Sziklai, C. Taylor, N. Turini, V. Vacek, G. Valentino, J. Welti, J. Wenninger, P. Wyszkowski, K. Zielinski
Oct. 3, 2016 hep-ph, hep-ex, nucl-ex
The TOTEM experiment at the CERN LHC has measured elastic proton-proton scattering at the centre-of-mass energy $\sqrt{s}$ = 8 TeV and four-momentum transfers squared, |t|, from 6 x $10^{-4}$ GeV$^2$ to 0.2 GeV$^2$. Near the lower end of the |t|-interval the differential cross-section is sensitive to the interference between the hadronic and the electromagnetic scattering amplitudes. This article presents the elastic cross-section measurement and the constraints it imposes on the functional forms of the modulus and phase of the hadronic elastic amplitude. The data exclude the traditional Simplified West and Yennie interference formula that requires a constant phase and a purely exponential modulus of the hadronic amplitude. For parametrisations of the hadronic modulus with second- or third-order polynomials in the exponent, the data are compatible with hadronic phase functions giving either central or peripheral behaviour in the impact parameter picture of elastic scattering. In both cases, the $\rho$-parameter is found to be 0.12 $\pm$ 0.03. The results for the total hadronic cross-section are $\sigma_{tot}$ = (102.9 $\pm$ 2.3) mb and (103.0 $\pm$ 2.3) mb for central and peripheral phase formulations, respectively. Both are consistent with previous TOTEM measurements.
Fiber Bragg Grating (FBG) sensors as flatness and mechanical stretching sensors (1512.08481)
D. Abbaneo, M. Abbas, M. Abbrescia, A.A. Abdelalim, M. Abi Akl, O. Aboamer, D. Acosta, A. Ahmad, W. Ahmed, W. Ahmed, A. Aleksandrov, R. Aly, P. Altieri, C. Asawatangtrakuldee, P. Aspell, Y. Assran, I. Awan, S. Bally, Y. Ban, S. Banerjee, V. Barashko, P. Barria, G. Bencze, N. Beni, L. Benussi, V. Bhopatkar, S. Bianco, J. Bos, O. Bouhali, A. Braghieri, S. Braibant, S. Buontempo, C. Calabria, M. Caponero, C. Caputo, F. Cassese, A. Castaneda, S. Cauwenbergh, F.R. Cavallo, A. Celik, M. Choi, S. Choi, J. Christiansen, A. Cimmino, S. Colafranceschi, A. Colaleo, A. Conde Garcia, S. Czellar, M.M. Dabrowski, G. De Lentdecker, R. De Oliveira, G. de Robertis, S. Dildick, B. Dorney, W. Elmetenawee, G. Endroczi, F. Errico, A. Fenyvesi, S. Ferry, I. Furic, P. Giacomelli, V. Golovtsov, L. Guiducci, F. Guilloux, A. Gutierrez, R.M. Hadjiiska, A. Hassan, J. Hauser, K. Hoepfner, M. Hohlmann, H. Hoorani, P. Iaydjiev, Y.G. Jeng, T. Kamon, P. Karchin, A. Korytov, S. Krutelyov, A. Kumar, H. Kim, J. Lee, T. Lenzi, L. Litov, F. Loddo, A. Madorsky, T. Maerschalk, M. Maggi, A. Magnani, P.K. Mal, K. Mandal, A. Marchioro, A. Marinov, R. Masod, N. Majumdar, J.A. Merlin, G. Mitselmakher, A.K. Mohanty, S. Muhammad, A. Mohapatra, J. Molnar, S. Mukhopadhyay, M. Naimuddin, S. Nuzzo, E. Oliveri, L.M. Pant, P. Paolucci, I. Park, G. Passeggio, L. Passamonti, B. Pavlov, B. Philipps, D. Piccolo, D. Pierluigi, H. Postema, A. Puig Baranac, A. Radi, R. Radogna, G. Raffone, A. Ranieri, G. Rashevski, C. Riccardi, M. Rodozov, A. Rodrigues, L. Ropelewski, S. RoyChowdhury, A. Russo, G. Ryu, M.S. Ryu, A. Safonov, S. Salva, G. Saviano, A. Sharma, A. Sharma, R. Sharma, A.H. Shah, M. Shopova, J. Sturdy, G. Sultanov, S.K. Swain, Z. Szillasi, A. Tatarinov, T. Tuuva, M. Tytgat, I. Vai, M. Van Stenis, R. Venditti, E. Verhagen, P. Verwilligen, P. Vitulo, S. Volkov, A. Vorobyev, D. Wang, M. Wang, U. Yang, Y. Yang, R. Yonamine, N. Zaganidis, F. Zenoni, A. Zhang
Dec. 28, 2015 physics.ins-det
A novel approach which uses Fibre Bragg Grating (FBG) sensors has been utilised to assess and monitor the flatness of Gaseous Electron Multipliers (GEM) foils. The setup layout and preliminary results are presented.
A novel application of Fiber Bragg Grating (FBG) sensors in MPGD (1512.08529)
We present a novel application of Fiber Bragg Grating (FBG) sensors in the construction and characterisation of Micro Pattern Gaseous Detector (MPGD), with particular attention to the realisation of the largest triple (Gas electron Multiplier) GEM chambers so far operated, the GE1/1 chambers of the CMS experiment at LHC. The GE1/1 CMS project consists of 144 GEM chambers of about 0.5 m2 active area each, employing three GEM foils per chamber, to be installed in the forward region of the CMS endcap during the long shutdown of LHC in 2108-2019. The large active area of each GE1/1 chamber consists of GEM foils that are mechanically stretched in order to secure their flatness and the consequent uniform performance of the GE1/1 chamber across its whole active surface. So far FBGs have been used in high energy physics mainly as high precision positioning and re-positioning sensors and as low cost, easy to mount, low space consuming temperature sensors. FBGs are also commonly used for very precise strain measurements in material studies. In this work we present a novel use of FBGs as flatness and mechanical tensioning sensors applied to the wide GEM foils of the GE1/1 chambers. A network of FBG sensors have been used to determine the optimal mechanical tension applied and to characterise the mechanical tension that should be applied to the foils. We discuss the results of the test done on a full-sized GE1/1 final prototype, the studies done to fully characterise the GEM material, how this information was used to define a standard assembly procedure and possible future developments.
Charge Transfer Properties Through Graphene for Applications in Gaseous Detectors (1512.05409)
S. Franchino, D. Gonzalez-Diaz, R. Hall-Wilton, R. B. Jackman, H. Muller, T. T. Nguyen, R. de Oliveira, E. Oliveri, D. Pfeiffer, F. Resnati, L. Ropelewski, J. Smith, M. van Stenis, C. Streli, P. Thuiner, R. Veenhof
Dec. 16, 2015 hep-ex, physics.ins-det
Graphene is a single layer of carbon atoms arranged in a honeycomb lattice with remarkable mechanical and electrical properties. Regarded as the thinnest and narrowest conductive mesh, it has drastically different transmission behaviours when bombarded with electrons and ions in vacuum. This property, if confirmed in gas, may be a definitive solution for the ion back-flow problem in gaseous detectors. In order to ascertain this aspect, graphene layers of dimensions of about 2x2cm$^2$, grown on a copper substrate, are transferred onto a flat metal surface with holes, so that the graphene layer is freely suspended. The graphene and the support are installed into a gaseous detector equipped with a triple Gaseous Electron Multiplier (GEM), and the transparency properties to electrons and ions are studied in gas as a function of the electric fields. The techniques to produce the graphene samples are described, and we report on preliminary tests of graphene-coated GEMs.
Effects of High Charge Densities in Multi-GEM Detectors (1512.04968)
S. Franchino, D. Gonzalez Diaz, R. Hall-Wilton, H. Muller, E. Oliveri, D. Pfeiffer, F. Resnati, L. Ropelewski, M. Van Stenis, C. Streli, P. Thuiner, R. Veenhof
A comprehensive study, supported by systematic measurements and numerical computations, of the intrinsic limits of multi-GEM detectors when exposed to very high particle fluxes or operated at very large gains is presented. The observed variations of the gain, of the ion back-flow, and of the pulse height spectra are explained in terms of the effects of the spatial distribution of positive ions and their movement throughout the amplification structure. The intrinsic dynamic character of the processes involved imposes the use of a non-standard simulation tool for the interpretation of the measurements. Computations done with a Finite Element Analysis software reproduce the observed behaviour of the detector. The impact of this detailed description of the detector in extreme conditions is multiple: it clarifies some detector behaviours already observed, it helps in defining intrinsic limits of the GEM technology, and it suggests ways to extend them.
Evidence for non-exponential elastic proton-proton differential cross-section at low |t| and sqrt(s) = 8 TeV by TOTEM (1503.08111)
TOTEM Collaboration: G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, U. Bottigli, M. Bozzo, A. Buzzo, F. S. Cafagna, C. E. Campanella, M. G. Catanesi, M. Csanád, T. Csörgő, M. Deile, F. De Leonardis, A. D'Orazio, M. Doubek, K. Eggert, V. Eremin, F. Ferro, A. Fiergolski, F. Garcia, V. Georgiev, S. Giani, L. Grzanka, C. Guaragnella, J. Hammerbauer, J. Heino, A. Karev, J. Kašpar, J. Kopal, V. Kundrát, S. Lami, G. Latino, R. Lauhakangas, E. Lippmaa, J. Lippmaa, M. V. Lokajíček, L. Losurdo, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, A. Mercadante, N. Minafra, S. Minutoli, T. Naaranoja, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, V. Passaro, Z. Peroutka, V. Petruzzelli, T. Politi, J. Procházka, F. Prudenzano, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, A. Scribano, J. Smajek, W. Snoeys, T. Sodzawiczny, J. Sziklai, C. Taylor, N. Turini, V. Vacek, J. Welti, P. Wyszkowski, K. Zielinski
Sept. 12, 2015 hep-ex
The TOTEM experiment has made a precise measurement of the elastic proton-proton differential cross-section at the centre-of-mass energy sqrt(s) = 8 TeV based on a high-statistics data sample obtained with the beta* = 90 optics. Both the statistical and systematic uncertainties remain below 1%, except for the t-independent contribution from the overall normalisation. This unprecedented precision allows to exclude a purely exponential differential cross-section in the range of four-momentum transfer squared 0.027 < |t| < 0.2 GeV^2 with a significance greater than 7 sigma. Two extended parametrisations, with quadratic and cubic polynomials in the exponent, are shown to be well compatible with the data. Using them for the differential cross-section extrapolation to t = 0, and further applying the optical theorem, yields total cross-section estimates of (101.5 +- 2.1) mb and (101.9 +- 2.1) mb, respectively, in agreement with previous TOTEM measurements.
Charge Transfer Properties Through Graphene Layers in Gas Detectors (1503.06596)
P. Thuiner, R. Hall-Wilton, R. B. Jackman, H. Müller, T. T. Nguyen, E. Oliveri, D. Pfeiffer, F. Resnati, L. Ropelewski, J. A. Smith, M. van Stenis, R. Veenhof
March 23, 2015 hep-ex, physics.ins-det
Graphene is a single layer of carbon atoms arranged in a honeycomb lattice with remarkable mechanical, electrical and optical properties. For the first time graphene layers suspended on copper meshes were installed into a gas detector equipped with a gaseous electron multiplier. Measurements of low energy electron and ion transfer through graphene were conducted. In this paper we describe the sample preparation for suspended graphene layers, the testing procedures and we discuss the preliminary results followed by a prospect of further applications.
Performance of a Large-Area GEM Detector Prototype for the Upgrade of the CMS Muon Endcap System (1412.0228)
D.Abbaneo, M. Abbas, M. Abbrescia, A.A. Abdelalim, M. Abi Akl, W. Ahmed, W. Ahmed, P. Altieri, R. Aly, C. Asawatangtrakuldee, A. Ashfaq, P. Aspell, Y. Assran, I. Awan, S. Bally, Y. Ban, S. Banerjee, P. Barria, L.Benussi, V. Bhopatkar, S. Bianco, J. Bos, O. Bouhali, S. Braibant, S. Buontempo, C. Calabria, M. Caponero, C. Caputo, F. Cassese, A. Castaneda, S. Cauwenbergh, F.R. Cavallo, A. Celik, M. Choi, K. Choi, S. Choi, J. Christiansen, A. Cimmino, S. Colafranceschi, A. Colaleo, A. Conde Garcia, M.M.Dabrowski, G. De Lentdecker, R. De Oliveira, G.de Robertis, S. Dildick, B.Dorney, W. Elmetenawee, G. Fabrice, M. Ferrini, S. Ferry, P. Giacomelli, J. Gilmore, L. Guiducci, A. Gutierrez, R.M. Hadjiiska, A. Hassan, J. Hauser, K. Hoepfner, M. Hohlmann, H. Hoorani, Y.G. Jeng, T. Kamon, P.E. Karchin, H.S. Kim, S. Krutelyov, A. Kumar, J. Lee, T. Lenzi, L. Litov, F. Loddo, T. Maerschalk, G. Magazzu, M. Maggi, Y. Maghrbi, A. Magnani, N. Majumdar, P.K. Mal, K. Mandal, A. Marchioro, A. Marinov, J.A. Merlin, A.K. Mohanty, A. Mohapatra, S. Muhammad, S. Mukhopadhyay, M. Naimuddin, S. Nuzzo, E. Oliveri, L.M. Pant, P. Paolucci, I. Park, G. Passeggio, B. Pavlov, B. Philipps, M. Phipps, D. Piccolo, H. Postema, G. Pugliese, A. Puig Baranac, A. Radi, R. Radogna, G. Raffone, S. Ramkrishna, A. Ranieri, C. Riccardi, A. Rodrigues, L. Ropelewski, S. RoyChowdhury, M.S. Ryu, G. Ryu, A. Safonov, A. Sakharov, S. Salva, G. Saviano, A. Sharma, S.K. Swain, J.P. Talvitie, C. Tamma, A. Tatarinov, N. Turini, T. Tuuva, J. Twigger, M. Tytgat, I. Vai, M. van Stenis, R. Venditi, E. Verhagen, P. Verwilligen, P. Vitulo, D. Wang, M. Wang, U. Yang, Y. Yang, R. Yonamine, N. Zaganidis, F. Zenoni, A. Zhang
Dec. 8, 2014 physics.ins-det
Gas Electron Multiplier (GEM) technology is being considered for the forward muon upgrade of the CMS experiment in Phase 2 of the CERN LHC. Its first implementation is planned for the GE1/1 system in the $1.5 < \mid\eta\mid < 2.2$ region of the muon endcap mainly to control muon level-1 trigger rates after the second long LHC shutdown. A GE1/1 triple-GEM detector is read out by 3,072 radial strips with 455 $\mu$rad pitch arranged in eight $\eta$-sectors. We assembled a full-size GE1/1 prototype of 1m length at Florida Tech and tested it in 20-120 GeV hadron beams at Fermilab using Ar/CO$_{2}$ 70:30 and the RD51 scalable readout system. Four small GEM detectors with 2-D readout and an average measured azimuthal resolution of 36 $\mu$rad provided precise reference tracks. Construction of this largest GEM detector built to-date is described. Strip cluster parameters, detection efficiency, and spatial resolution are studied with position and high voltage scans. The plateau detection efficiency is [97.1 $\pm$ 0.2 (stat)]\%. The azimuthal resolution is found to be [123.5 $\pm$ 1.6 (stat)] $\mu$rad when operating in the center of the efficiency plateau and using full pulse height information. The resolution can be slightly improved by $\sim$ 10 $\mu$rad when correcting for the bias due to discrete readout strips. The CMS upgrade design calls for readout electronics with binary hit output. When strip clusters are formed correspondingly without charge-weighting and with fixed hit thresholds, a position resolution of [136.8 $\pm$ 2.5 stat] $\mu$rad is measured, consistent with the expected resolution of strip-pitch/$\sqrt{12}$ = 131.3 $\mu$rad. Other $\eta$-sectors of the detector show similar response and performance.
Measurement of the forward charged particle pseudorapidity density in pp collisions at sqrt(s) = 8 TeV using a displaced interaction point (1411.4963)
The TOTEM Collaboration: G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, U. Bottigli, M. Bozzo, E. Brücken, A. Buzzo, F. S. Cafagna, M. G. Catanesi, C. Covault, M. Csanád, T. Csörgő, M. Deile, M. Doubek, K. Eggert, V. Eremin, F. Ferro, A. Fiergolski, F. Garcia, V. Georgiev, S. Giani, L. Grzanka, J. Hammerbauer, J. Heino, T. Hilden, A. Karev, J. Kašpar, J. Kopal, V. Kundrát, S. Lami, G. Latino, R. Lauhakangas, T. Leszko, E. Lippmaa, J. Lippmaa, M. V. Lokajíček, L. Losurdo, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, T. Mäki, A. Mercadante, N. Minafra, S. Minutoli, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, Z. Peroutka, J. Procházka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, A. Scribano, J. Smajek, W. Snoeys, J. Sziklai, C. Taylor, N. Turini, V. Vacek, J. Welti, J. Whitmore, P. Wyszkowski, K. Zielinski
Nov. 18, 2014 hep-ex
The pseudorapidity density of charged particles dN(ch)/deta is measured by the TOTEM experiment in pp collisions at sqrt(s) = 8 TeV within the range 3.9 < eta < 4.7 and -6.95 < eta < -6.9. Data were collected in a low intensity LHC run with collisions occurring at a distance of 11.25 m from the nominal interaction point. The data sample is expected to include 96-97\% of the inelastic proton-proton interactions. The measurement reported here considers charged particles with p_T > 0 MeV/c, produced in inelastic interactions with at least one charged particle in -7 < eta < -6 or 3.7 < eta <4.8 . The dN(ch)/deta has been found to decrease with |eta|, from 5.11 +- 0.73 at eta = 3.95 to 1.81 +- 0.56 at eta= - 6.925. Several MC generators are compared to the data and are found to be within the systematic uncertainty of the measurement.
LHC Optics Measurement with Proton Tracks Detected by the Roman Pots of the TOTEM Experiment (1406.0546)
June 2, 2014 hep-ex, physics.acc-ph
Precise knowledge of the beam optics at the LHC is crucial to fulfil the physics goals of the TOTEM experiment, where the kinematics of the scattered protons is reconstructed with the near-beam telescopes -- so-called Roman Pots (RP). Before being detected, the protons' trajectories are influenced by the magnetic fields of the accelerator lattice. Thus precise understanding of the proton transport is of key importance for the experiment. A novel method of optics evaluation is proposed which exploits kinematical distributions of elastically scattered protons observed in the RPs. Theoretical predictions, as well as Monte Carlo studies, show that the residual uncertainty of this optics estimation method is smaller than 0.25 percent.
Performance of the TOTEM Detectors at the LHC (1310.2908)
TOTEM Collaboration: G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, M. G. Bagliesi, V. Berardi, M. Berretti, E. Bossini, U. Bottigli, M. Bozzo, E. Brücken, A. Buzzo, F. S. Cafagna, M. G. Catanesi, R. Cecchi, C. Covault, M. Csanád, T. Csörgő, M. Deile, M. Doubek, K. Eggert, V. Eremin, F. Ferro, A. Fiergolski, F. Garcia, S. Giani, V. Greco, L. Grzanka, J. Heino, T. Hilden, A. Karev, J. Kašpar, J. Kopal, V. Kundrát, S. Lami, G. Latino, R. Lauhakangas, T. Leszko, E. Lippmaa, J. Lippmaa, M. Lokajíček, L. Losurdo, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, T. Mäki, A. Mercadante, N. Minafra, S. Minutoli, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, E. Pedreschi, J. Procházka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, A. Scribano, J. Smajek, W. Snoeys, F. Spinella, J. Sziklai, C. Taylor, A. Thys, N. Turini, V. Vacek, M. Vítek, J. Welti, J. Whitmore, P. Wyszkowski
Oct. 10, 2013 hep-ex, physics.ins-det
The TOTEM Experiment is designed to measure the total proton-proton cross-section with the luminosity-independent method and to study elastic and diffractive pp scattering at the LHC. To achieve optimum forward coverage for charged particles emitted by the pp collisions in the interaction point IP5, two tracking telescopes, T1 and T2, are installed on each side of the IP in the pseudorapidity region 3.1 < = |eta | < = 6.5, and special movable beam-pipe insertions - called Roman Pots (RP) - are placed at distances of +- 147 m and +- 220 m from IP5. This article describes in detail the working of the TOTEM detector to produce physics results in the first three years of operation and data taking at the LHC.
Double diffractive cross-section measurement in the forward region at LHC (1308.6722)
TOTEM Collaboration: G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, U. Bottigli, M. Bozzo, E. Brücken, A. Buzzo, F. S. Cafagna, M. G. Catanesi, M. Csanád, T. Csörgő, M. Deile, K. Eggert, V. Eremin, F. Ferro, A. Fiergolski, F. Garcia, S. Giani, V. Greco, L. Grzanka, J. Heino, T. Hilden, A. Karev, J. Kašpar, J. Kopal, V. Kundrát, K. Kurvinen, S. Lami, G. Latino, R. Lauhakangas, T. Leszko, E. Lippmaa, J. Lippmaa, M. Lokajíček, L. Losurdo, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, T. Mäki, A. Mercadante, N. Minafra, S. Minutoli, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, J. Procházka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, A. Scribano, J. Smajek, W. Snoeys, J. Sziklai, C. Taylor, N. Turini, V. Vacek, M. Vítek, J. Welti, J. Whitmore, P. Wyszkowski
Aug. 30, 2013 hep-ex
The first double diffractive cross-section measurement in the very forward region has been carried out by the TOTEM experiment at the LHC with center-of-mass energy of sqrt(s)=7 TeV. By utilizing the very forward TOTEM tracking detectors T1 and T2, which extend up to |eta|=6.5, a clean sample of double diffractive pp events was extracted. From these events, we measured the cross-section sigma_DD =(116 +- 25) mub for events where both diffractive systems have 4.7 <|eta|_min < 6.5 .
Beam Test Results for New Full-scale GEM Prototypes for a Future Upgrade of the CMS High-eta Muon System (1211.3939)
D. Abbaneo, M. Abbrescia, C. Armagnaud, P. Aspell, Y. Assran, Y. Ban, S. Bally, L. Benussi, U. Berzano, S. Bianco, J. Bos, K. Bunkowski, J. Cai, J. P. Chatelain, J. Christiansen, S. Colafranceschi, A. Colaleo, A. Conde Garcia, E. David, G. de Robertis, R. De Oliveira, S. Duarte Pinto, S. Ferry, F. Formenti, L. Franconi, T. Fruboes, A. Gutierrez, M. Hohlmann, A. E. Kamel, P. E. Karchin, F. Loddo, G. Magazzu, M. Maggi, A. Marchioro, A. Marinov, K. Mehta, J. Merlin, A. Mohapatra, T. Moulik, M. V. Nemallapudi, S. Nuzzo, E. Oliveri, D. Piccolo, H. Postema, A. Radi, G. Raffone, A. Rodrigues, L. Ropelewski, G. Saviano, A. Sharma, M. J. Staib, H. Teng, M. Tytgat, S. A. Tupputi, N. Turini, N. Smilkjovic, M. Villa, N. Zaganidis, M. Zientek
Nov. 16, 2012 physics.ins-det
The CMS GEM collaboration is considering Gas Electron Multipliers (GEMs) for upgrading the CMS forward muon system in the 1.5<|eta|<2.4 endcap region. GEM detectors can provide precision tracking and fast trigger information. They would improve the CMS muon trigger and muon momentum resolution and provide missing redundancy in the high-eta region. Employing a new faster construction and assembly technique, we built four full-scale Triple-GEM muon detectors for the inner ring of the first muon endcap station. We plan to install these or further improved versions in CMS during the first long LHC shutdown in 2013/14 for continued testing. These detectors are designed for the stringent rate and resolution requirements in the increasingly hostile environments expected at CMS after the second long LHC shutdown in 2018/19. The new prototypes were studied in muon/pion beams at the CERN SPS. We discuss our experience with constructing the new full-scale production prototypes and present preliminary performance results from the beam test. We also tested smaller Triple-GEM prototypes with zigzag readout strips with 2 mm pitch in these beams and measured a spatial resolution of 73 microns. This readout offers a potential reduction of channel count and consequently electronics cost for this system while maintaining high spatial resolution.
Measurement of the forward charged particle pseudorapidity density in pp collisions at sqrt{s} = 7 TeV with the TOTEM experiment (1205.4105)
G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, M. Bozzo, P. Brogi, E. Brücken, A. Buzzo, F. S. Cafagna, M. Calicchio, M. G. Catanesi, C. Covault, M. Csanád, T. Csörgő, M. Deile, K Eggert, V. Eremin, R. Ferretti, F. Ferro, A. Fiergolski, F. Garcia, S. Giani, V. Greco, L. Grzanka, J. Heino, T. Hilden, M. R. Intonti, J. Kašpar, J. Kopal, V. Kundrát, K. Kurvinen, S. Lami, G. Latino, R. Lauhakangas, T. Leszko, E. Lippmaa, M. Lokajíček, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, L. Magaletti, T. Mäki, A. Mercadante, N. Minafra, S. Minutoli, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, J. Procházka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, A. Santroni, A. Scribano, W. Snoeys, J. Sziklai, C. Taylor, N. Turini, V. Vacek, M. Vitek, J. Welti, J. Whitmore
May 18, 2012 hep-ex
The TOTEM experiment has measured the charged particle pseudorapidity density dN_{ch}/deta in pp collisions at sqrt{s} = 7 TeV for 5.3<|eta|<6.4 in events with at least one charged particle with transverse momentum above 40 MeV/c in this pseudorapidity range. This extends the analogous measurement performed by the other LHC experiments to the previously unexplored forward eta region. The measurement refers to more than 99% of non-diffractive processes and to single and double diffractive processes with diffractive masses above ~3.4 GeV/c^2, corresponding to about 95% of the total inelastic cross-section. The dN_{ch}/deta has been found to decrease with |eta|, from 3.84 pm 0.01(stat) pm 0.37(syst) at |eta| = 5.375 to 2.38 pm 0.01(stat) pm 0.21(syst) at |eta| = 6.375. Several MC generators have been compared to data; none of them has been found to fully describe the measurement.
Elastic Scattering and Total Cross-Section in p+p reactions measured by the LHC Experiment TOTEM at sqrt(s) = 7 TeV (1204.5689)
T. Csörgő, G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, M. Bozzo, P. Brogi, E. Brücken, A. Buzzo, F. S. Cafagna, M. Calicchio, M. G. Catanesi, C. Covault, M. Csanád, M. Deile, E. Dimovasili, M. Doubek, K. Eggert, V.Eremin, R. Ferretti, F. Ferro, A. Fiergolski, F. Garcia, S. Giani, V. Greco, L. Grzanka, J. Heino, T. Hilden, M. R. Intonti, M. Janda, J. Kašpar, J. Kopal, V. Kundrát, K. Kurvinen, S. Lami, G. Latino, R. Lauhakangas, T. Leszko E. Lippmaa, M. Lokajíček, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, L. Magaletti, G. Magazzù, A. Mercadante, M. Meucci, S. Minutoli, F. Nemes, H. Niewiadomski, E. Noschis, T. Novák, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, A.-L. Perrot, E. Pedreschi, J. Petäjäjärvi, J. Procházka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, G. Sanguinetti, A. Santroni, A. Scribano, G. Sette, W. Snoeys, F. Spinella, J. Sziklai, C. Taylor, N. Turini, V. Vacek, M. Vítek, J. Welti, J. Whitmore
April 25, 2012 hep-ex
Proton-proton elastic scattering has been measured by the TOTEM experiment at the CERN Large Hadron Collider at $\sqrt{s} = 7 $ TeV in special runs with the Roman Pot detectors placed as close to the outgoing beam as seven times the transverse beam size. The differential cross-section measurements are reported in the |t|-range of 0.36 to 2.5 GeV^2. Extending the range of data to low t values from 0.02 to 0.33 GeV^2,and utilizing the luminosity measurements of CMS, the total proton-proton cross section at sqrt(s) = 7 TeV is measured to be (98.3 +- 0.2(stat) +- 2.8(syst)) mb.
Construction and Performance of Large-Area Triple-GEM Prototypes for Future Upgrades of the CMS Forward Muon System (1111.7249)
M. Tytgat, A. Marinov, N. Zaganidis, Y. Ban, J. Cai, H. Teng, A. Mohapatra, T. Moulik, M. Abbrescia, A. Colaleo, G. de Robertis, F. Loddo, M. Maggi, S. Nuzzo, S. A. Tupputi, L. Benussi, S. Bianco, S. Colafranceschi, D. Piccolo, G. Raffone, G. Saviano, M.G. Bagliesi, R. Cecchi, G. Magazzu, E. Oliveri, N. Turini, T. Fruboes, D. Abbaneo, C. Armagnaud, P. Aspell, S. Bally, U. Berzano, J. Bos, K. Bunkowski, J. P. Chatelain, J. Christiansen, A. Conde Garcia, E. David, R. De Oliveira, S. Duarte Pinto, S. Ferry, F. Formenti, L. Franconi, A. Marchioro, K. Mehta, J. Merlin, M. V. Nemallapudi, H. Postema, A. Rodrigues, L. Ropelewski, A. Sharma, N. Smilkjovic, M. Villa, M. Zientek, A. Gutierrez, P. E. Karchin, K. Gnanvo, M. Hohlmann, M. J. Staib
Nov. 30, 2011 hep-ex, physics.ins-det
At present, part of the forward RPC muon system of the CMS detector at the CERN LHC remains uninstrumented in the high-\eta region. An international collaboration is investigating the possibility of covering the 1.6 < |\eta| < 2.4 region of the muon endcaps with large-area triple-GEM detectors. Given their good spatial resolution, high rate capability, and radiation hardness, these micro-pattern gas detectors are an appealing option for simultaneously enhancing muon tracking and triggering capabilities in a future upgrade of the CMS detector. A general overview of this feasibility study will be presented. The design and construction of small (10\times10 cm2) and full-size trapezoidal (1\times0.5 m2) triple-GEM prototypes will be described. During detector assembly, different techniques for stretching the GEM foils were tested. Results from measurements with x-rays and from test beam campaigns at the CERN SPS will be shown for the small and large prototypes. Preliminary simulation studies on the expected muon reconstruction and trigger performances of this proposed upgraded muon system will be reported.
Test beam results of the GE1/1 prototype for a future upgrade of the CMS high-$\eta$ muon system (1111.4883)
D. Abbaneo, M. Abbrescia, C. Armagnaud, P. Aspell, M. G. Bagliesi, Y. Ban, S. Bally, L. Benussi, U. Berzano, S. Bianco, J. Bos, K. Bunkowski, J. Cai, R. Cecchi, J. P. Chatelain, J. Christiansen, S. Colafranceschi, A. Colaleo, A. Conde Garcia, E. David, G. de Robertis, R. De Oliveira, S. Duarte Pinto, S. Ferry, F. Formenti, L. Franconi, K. Gnanvo, A. Gutierrez, M. Hohlmann, P. E. Karchin, F. Loddo, G. Magazzú, M. Maggi, A. Marchioro, A. Marinov, K. Mehta, J. Merlin, A. Mohapatra, T. Moulik, M. V. Nemallapudi, S. Nuzzo, E. Oliveri, D. Piccolo, H. Postema, G. Raffone, A. Rodrigues, L. Ropelewski, G. Saviano, A. Sharma, M. J. Staib, H. Teng, M. Tytgat, S. A. Tupputi, N. Turini, N. Smilkjovic, M. Villa, N. Zaganidis, M. Zientek
Gas Electron Multipliers (GEM) are an interesting technology under consideration for the future upgrade of the forward region of the CMS muon system, specifically in the $1.6<| \eta |<2.4$ endcap region. With a sufficiently fine segmentation GEMs can provide precision tracking as well as fast trigger information. The main objective is to contribute to the improvement of the CMS muon trigger. The construction of large-area GEM detectors is challenging both from the technological and production aspects. In view of the CMS upgrade we have designed and built the largest full-size Triple-GEM muon detector, which is able to meet the stringent requirements given the hostile environment at the high-luminosity LHC. Measurements were performed during several test beam campaigns at the CERN SPS in 2010 and 2011. The main issues under study are efficiency, spatial resolution and timing performance with different inter-electrode gap configurations and gas mixtures. In this paper results of the performance of the prototypes at the beam tests will be discussed.
Proton-proton elastic scattering at the LHC energy of {\surd} = 7 TeV (1110.1385)
The TOTEM Collaboration: G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, M. Bozzo, E. Brücken, A. Buzzo, F. Cafagna, M. Calicchio, M. G. Catanesi, C. Covault, M. Csanád, T. Csörgö, M. Deile, E. Dimovasili, M. Doubek, K. Eggert, V. Eremin, F. Ferro, A. Fiergolski, F. Garcia, S. Giani, V. Greco, L. Grzanka, J. Heino, T. Hilden, M. Janda, J. Kašpar, J. Kopal, V. Kundrát, K. Kurvinen, S. Lami, G. Latino, R. Lauhakangas, T. Leszko, E. Lippmaa, M. Lokajíček, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, L. Magaletti, G. Magazzú, A. Mercadante, M. Meucci, S. Minutoli, F. Nemes, H. Niewiadomski, E. Noschis, T. Novak, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, A.-L. Perrot, P. Palazzi, E. Pedreschi, J. Petäjäjärvi, J. Procházka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, A. Santroni, A. Scribano, G. Sette, W. Snoeys, F. Spinella, J. Sziklai, C. Taylor, N. Turini, V. Vacek, M. Vítek, J. Welti, J. Whitmore
Oct. 6, 2011 hep-ex
Proton-proton elastic scattering has been measured by the TOTEM experiment at the CERN Large Hadron Collider at {\surd}s = 7 TeV in dedicated runs with the Roman Pot detectors placed as close as seven times the transverse beam size (sbeam) from the outgoing beams. After careful study of the accelerator optics and the detector alignment, |t|, the square of four-momentum transferred in the elastic scattering process, has been determined with an uncertainty of d t = 0.1GeV p|t|. In this letter, first results of the differential cross section are presented covering a |t|-range from 0.36 to 2.5GeV2. The differential cross-section in the range 0.36 < |t| < 0.47 GeV2 is described by an exponential with a slope parameter B = (23.6{\pm}0.5stat {\pm}0.4syst)GeV-2, followed by a significant diffractive minimum at |t| = (0.53{\pm}0.01stat{\pm}0.01syst)GeV2. For |t|-values larger than ~ 1.5GeV2, the cross-section exhibits a power law behaviour with an exponent of -7.8_\pm} 0.3stat{\pm}0.1syst. When compared to predictions based on the different available models, the data show a strong discriminative power despite the small t-range covered.
First measurement of the total proton-proton cross section at the LHC energy of {\surd} s =7 TeV (1110.1395)
G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, M. Bozzo, P. Brogi, E. Brücken, A. Buzzo, F. Cafagna, M. Calicchio, M. G. Catanesi, C. Covault, T. Csörgö, M. Deile, K. Eggert, V. Eremin, R. Ferretti, F. Ferro, A. Fiergolski, F. Garcia, S. Giani, V. Greco, L. Grzanka, J. Heino, T. Hilden, M.R. Intonti, J. Kašpar, J. Kopal, V. Kundrát, K. Kurvinen, S. Lami, G. Latino, R. Lauhakangas, T. Leszko, E. Lippmaa, M. Lokajíček, M. Lo Vetere, F. Lucas Rodríguez, M. Macrí, L. Magaletti, A. Mercadante, S. Minutoli, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Österberg, P. Palazzi, J. Procházka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, G. Sanguinetti, A. Santroni, A. Scribano, W. Snoeys, J. Sziklai, C. Taylor, N. Turini, V. Vacek, M. Vítek, J. Welti, J. Whitmore
TOTEM has measured the differential cross-section for elastic proton-proton scattering at the LHC energy of {\srud}s = 7TeV analysing data from a short run with dedicated large {\beta} * optics. A single exponential fit with a slope B = (20:1{\pm}0:2stat {\pm}0:3syst)GeV-2 describes the range of the four-momentum transfer squared |t| from 0.02 to 0.33 GeV2. After the extrapolation to |t| = 0, a total elastic scattering cross-section of (24:8{\pm}0:2stat {\pm}1:2syst) mb was obtained. Applying the optical theorem and using the luminosity measurement from CMS, a total proton-proton cross-section of (98:3{\pm}0:2stat {\pm}2:8syst) mb was deduced which is in good agreement with the expectation from the overall fit of previously measured data over a large range of center-of-mass energies. From the total and elastic pp cross-section measurements, an inelastic pp cross-section of (73:5{\pm}0:6stat +1:8 -1:3 syst) mb was inferred. PACS 13.60.Hb: Total and inclusive cross sections
First Results from the TOTEM Experiment (1110.1008)
G. Latino, G. Antchev, P. Aspell, I. Atanassov, V. Avati, J. Baechler, V. Berardi, M. Berretti, E. Bossini, M. Bozzo, P. Brogi, E. Brucken, A. Buzzo, F. Cafagna, M. Calicchio, M.G. Catanesi, C. Covault, T. Csorgo, M. Deile, K. Eggert, V. Eremin, R. Ferretti, F. Ferro, A. Fiergolski, F. Garcia, S. Giani, V. Greco, L. Grzanka, J. Heino, T. Hilden, M.R. Intonti, J. Kaspar, J. Kopal, V. Kundrat, K. Kurvinen, S. Lami, R. Lauhakangas, T. Leszko, E. Lippmaa, M. Lokajicek, M. Lo Vetere, F. Lucas Rodriguez, M. Macri, L. Magaletti, A. Mercadante, S. Minutoli, F. Nemes, H. Niewiadomski, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Osterberg, P. Palazzi, J. Prochazka, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, E. Robutti, L. Ropelewski, G. Ruggiero, H. Saarikko, G. Sanguinetti, A. Santroni, A. Scribano, W. Snoeys, J. Sziklai, C. Taylor, N. Turini, V. Vacek, M. Vitek, J. Welti, J. Whitmore
The first physics results from the TOTEM experiment are here reported, concerning the measurements of the total, differential elastic, elastic and inelastic pp cross-section at the LHC energy of $\sqrt{s}$ = 7 TeV, obtained using the luminosity measurement from CMS. A preliminary measurement of the forward charged particle $\eta$ distribution is also shown.
Characterization of GEM Detectors for Application in the CMS Muon Detection System (1012.3675)
D. Abbaneo, S. Bally, H. Postema, A. Conde Garcia, J. P. Chatelain, G. Faber, L. Ropelewski, E. David, S. Duarte Pinto, G. Croci, M. Alfonsi, M. van Stenis, A. Sharma, L. Benussi, S. Bianco, S. Colafranceschi, D. Piccolo, G. Saviano, N. Turini, E. Oliveri, G. Magazzu', A. Marinov, M. Tytgat, N. Zaganidis, M. Hohlmann, K. Gnanvo, Y. Ban, H. Teng, J. Cai
The muon detection system of the Compact Muon Solenoid experiment at the CERN Large Hadron Collider is based on different technologies for muon tracking and triggering. In particular, the muon system in the endcap disks of the detector consists of Resistive Plate Chambers for triggering and Cathode Strip Chambers for tracking. At present, the endcap muon system is only partially instrumented with the very forward detector region remaining uncovered. In view of a possible future extension of the muon endcap system, we report on a feasibility study on the use of Micro-Pattern Gas Detectors, in particular Gas Electron Multipliers, for both muon triggering and tracking. Results on the construction and characterization of small tripleGas Electron Multiplier prototype detectors are presented.
Construction of the first full-size GEM-based prototype for the CMS high-$\eta$ muon system (1012.1524)
D. Abbaneo, S. Bally, H. Postema, A. Conde Garcia, J. P. Chatelain, G. Faber, L. Ropelewski, S. Duarte Pinto, G. Croci, M. Alfonsi, M. Van Stenis, A. Sharma, L. Benussi, S. Bianco, S. Colafranceschi, F. Fabbri, L. Passamonti, D. Piccolo, D. Pierluigi, G. Raffone, A. Russo, G. Saviano, A. Marinov, M. Tytgat, N. Zaganidis, M. Hohlmann, K. Gnanvo, M.G. Bagliesi, R. Cecchi, N. Turini, E. Oliveri, G. Magazzù, Y. Ban, H. Teng, J. Cai
Dec. 9, 2010 hep-ex, physics.ins-det
In view of a possible extension of the forward CMS muon detector system and future LHC luminosity upgrades, Micro-Pattern Gas Detectors (MPGDs) are an appealing technology. They can simultaneously provide precision tracking and fast trigger information, as well as sufficiently fine segmentation to cope with high particle rates in the high-eta region at LHC and its future upgrades. We report on the design and construction of a full-size prototype for the CMS endcap system, the largest Triple-GEM detector built to-date. We present details on the 3D modeling of the detector geometry, the implementation of the readout strips and electronics, and the detector assembly procedure.
Proceedings of the workshop: HERA and the LHC workshop series on the implications of HERA for LHC physics (0903.3861)
H. Jung, Z. J. Ajaltouni, S. Albino, G. Altarelli, F. Ambroglini, J. Anderson, G. Antchev, M. Arneodo, P. Aspell, V. Avati, M. Bahr, A. Bacchetta, M. G. Bagliesi, R. D. Ball, A. Banfi, S. Baranov, P. Bartalini, J. Bartels, F. Bechtel, V. Berardi, M. Berretti, G. Beuf, M. Biasini, I. Bierenbaum, J. Blumlein, R. E. Blair, C. Bombonati, M. Boonekamp, U. Bottigli, S. Boutle, M. Bozzo, E. Brucken, J. Bracinik, A. Bruni, G. E. Bruno, A. Buckley, A. Bunyatyan, H. Burkhardt, P. Bussey, A. Buzzo, M. Cacciari, F. Cafagna, M. Calicchio, F. Caola, M. G. Catanesi, P. L. Catastini, R. Cecchi, F. A. Ceccopieri, S. Cerci, S. Chekanov, R. Chierici, M. Ciafaloni, M. A. Ciocci, V. Coco, D. Colferai, A. Cooper-Sarkar, G. Corcella, M. Czakon, A. Dainese, M. Dasgupta, M. Deak, M. Deile, P. A. Delsart, L. Del Debbio, A. de Roeck, C. Diaconu, M. Diehl, E. Dimovasili, M. Dittmar, I. M. Dremin, K. Eggert, R. Engel, V. Eremin, S. Erhan, C. Ewerz, L. Fano, J. Feltesse, G. Ferrera, F. Ferro, R. Field, S. Forte, F. Garcia, A. Geiser, F. Gelis, S. Giani, S. Gieseke, M. A. Gigg, A. Glazov, K. Golec-Biernat, K. Goulianos, J. Grebenyuk, V. Greco, D. Grellscheid, G. Grindhammer, M. Grothe, A. Guffanti, C. Gwenlan, V. Halyo, K. Hamilton, F. Hautmann, J. Heino, G. Heinrich, T. Hilden, K. Hiller, J. Hollar, X. Janssen, S. Joseph, A. W. Jung, H. Jung, V. Juranek, J. Kaspar, O. Kepka, V. A. Khoze, Ch. Kiesling, M. Klasen, S. Klein, B. A. Kniehl, A. Knutsson, J. Kopal, G. Kramer, F. Krauss, V. Kundrat, K. Kurvinen, K. Kutak, L. Lonnblad, S. Lami, G. Latino, J. I. Latorre, O. Latunde-Dada, R. Lauhakangas, V. Lendermann, P. Lenzi, G. Li, A. Likhoded, A. Lipatov, E. Lippmaa, M. Lokajicek, M. Lo Vetere, F. Lucas Rodriguez, G. Luisoni, E. Lytken, K. Muller, M. Macri, G. Magazzu, A. Majhi, S. Majhi, P. Marage, L. Marti, A. D. Martin, M. Meucci, D. A. Milstead, S. Minutoli, A. Nischke, A. Moares, S. Moch, L. Motyka, T. Namsoo, P. Newman, H. Niewiadomski, C. Nockles, E. Noschis, G. Notarnicola, J. Nystrand, E. Oliveri, F. Oljemark, K. Osterberg, R. Orava, M. Oriunno, S. Osman, S. Ostapchenko, P. Palazzi, E. Pedreschi, A. V. Pereira, H. Perrey, J. Petajajarvi, T. Petersen, A. Piccione, T. Pierog, J. L. Pinfold, O. I. Piskounova, S. Platzer, M. Quinto, Z. Rurikova, E. Radermacher, V. Radescu, E. Radicioni, F. Ravotti, G. Rella, P. Richardson, E. Robutti, G. Rodrigo, E. Rodrigues, M. Rogal, T. C. Rogers, J. Rojo, P. Roloff, L. Ropelewski, C. Rosemann, Ch. Royon, G. Ruggiero, A. Rummel, M. Ruspa, M. G. Ryskin, D. Salek, W. Slominski, H. Saarikko, A. Sabio Vera, T. Sako, G. P. Salam, V. A. Saleev, C. Sander, G. Sanguinetti, A. Santroni, Th. Schorner-Sadenius, R. Schicker, I. Schienbein, W. B. Schmidke, F. Schwennsen, A. Scribano, G. Sette, M. H. Seymour, A. Sherstnev, T. Sjostrand, W. Snoeys, G. Somogyi, L. Sonnenschein, G. Soyez, H. Spiesberger, F. Spinella, P. Squillacioti, A. M. Stasto, A. Starodumov, H. Stenzel, Ph. Stephens, A. Ster, D. Stocco, M. Strikman, C. Taylor, T. Teubner, R. S. Thorne, Z. Trocsanyi, M. Treccani, D. Treleani, L. Trentadue, A. Trummal, J. Tully, W. K. Tung, M. Turcato, N. Turini, M. Ubiali, A. Valkarova, A. van Hameren, P. Van Mechelen, J. A. M. Vermaseren, A. Vogt, B. F. L. Ward, G. Watt, B. R. Webber, Ch. Weiss, Ch. White, J. Whitmore, R. Wolf, J. Wu, A. Yagues-Molina, S. A. Yost, G. Zanderighi, N. Zotov, M. zur Nedden
March 30, 2009 hep-ph
2nd workshop on the implications of HERA for LHC physics. Working groups: Parton Density Functions Multi-jet final states and energy flows Heavy quarks (charm and beauty) Diffraction Cosmic Rays Monte Carlos and Tools
Diffraction at TOTEM (0812.3338)
G. Antchev, P. Aspell, V. Avati, M.G. Bagliesi, V. Berardi, M. Berretti, U. Bottigli, M. Bozzo, E. Brucken, A. Buzzo, F. Cafagna, M. Calicchio, M.G. Catanesi, P.L. Catastini, R. Cecchi, M.A. Ciocci, M. Deile, E. Dimovasili, K. Eggert, V. Eremin, F. Ferro, F. Garcia, S. Giani, V. Greco, J. Heino, T. Hilden, J. Kaspar, J. Kopal, V. Kundrat, K. Kurvinen, S. Lami, G. Latino, R. Lauhakangas, E. Lippmaa, M. Lokajicek, M. Lo Vetere, F. Lucas Rodriguez, M. Macri, G. Magazzu, M. Meucci, S. Minutoli, H. Niewiadomski, E. Noschis, G. Notarnicola, E. Oliveri, F. Oljemark, R. Orava, M. Oriunno, K. Osterberg, P. Palazzi, E. Pedreschi, J. Petajajarvi, M. Quinto, E. Radermacher, E. Radicioni, F. Ravotti, G. Rella, E. Robutti, L. Ropelewski, G. Ruggiero, A. Rummel, H. Saarikko, G. Sanguinetti, A. Santroni, A. Scribano, G. Sette, W. Snoeys, F. Spinella, P. Squillacioti, A. Ster, C. Taylor, A. Trummal, N. Turini, J. Whitmore, J. Wu
Dec. 17, 2008 hep-ex
The TOTEM experiment at the LHC measures the total proton-proton cross section with the luminosity-independent method and the elastic proton-proton cross-section over a wide |t|-range. It also performs a comprehensive study of diffraction, spanning from cross-section measurements of individual diffractive processes to the analysis of their event topologies. Hard diffraction will be studied in collaboration with CMS taking advantage of the large common rapidity coverage for charged and neutral particle detection and the large variety of trigger possibilities even at large luminosities. TOTEM will take data under all LHC beam conditions including standard high luminosity runs to maximize its physics reach. This contribution describes the main features of the TOTEM physics programme including measurements to be made in the early LHC runs. In addition, a novel scheme to extend the diffractive proton acceptance for high luminosity runs by installing proton detectors at IP3 is described. | CommonCrawl |
\begin{document}
\title{\LARGE \bf Behavior and Management of Stochastic Multiple-Origin-Destination Traffic Flows Sharing a Common Link}
\author{Li Jin and Yining Wen \thanks{This work was supported in part by NYU Tandon School of Engineering and C2SMART Department of Transportation Center. The authors appreciate discussion with Profs. Saurabh Amin and Dengfeng Sun.} \thanks{L. Jin is with the Department of Civil and Urban Engineering and Y. Wen is with the Department of Mechanical and Aerospace Engineering, New York University Tandon School of Engineering, Brooklyn, NY, USA, emails: {[email protected], [email protected]}.} } \newcommand*{\QEDA}{
\ensuremath{\blacksquare}}
\maketitle
\begin{abstract} In transportation systems (e.g. highways, railways, airports), traffic flows with distinct origin-destination pairs usually share common facilities and interact extensively. Such interaction is typically stochastic due to natural fluctuations in the traffic flows. In this paper, we study the interaction between multiple traffic flows and propose intuitive but provably efficient control algorithms based on modern sensing and actuating capabilities. We decompose the problem into two sub-problems: the impact of a merging junction and the impact of a diverging junction. We use a fluid model to show that (i) appropriate choice of priority at the merging junction is decisive for stability of the upstream queues and (ii) discharging priority at the diverging junction does not affect stability. We also illustrate the insights of our analysis via an example of management of multi-class traffic flows with platooning. \end{abstract}
\textbf{Index terms}: Stochastic fluid model, Traffic flow management, Piecewise-deterministic Markov processes.
\section{Introduction}
In transportation systems such as roads \cite{kurzhanskiy2010active,osorio2013simulation,coogan2015compartmental,jin2018throughput} and airspace \cite{bertsimas1998air,sun2008multicommodity,chen2017stochastic}, traffic flows with distinct origin-destination pairs usually share common facilities (e.g. road section and airspace sector) to optimize system-wide efficiency and utilization of infrastructure. Consequently, multiple traffic flows interact extensively in the common link, and such interaction can propagate to upstream links.
Consider the typical setting in Fig.~\ref{road}. Two classes of traffic ``compete'' for getting discharged from the common link 3, which can lead congestion in link 3. Limited capacities of links 4 and 5 can also contribute to this congestion. Congestion in link 3 may further block traffic from the upstream links. \begin{figure}
\caption{Two traffic flows with distinct origin-destination pairs sharing a common highway section.}
\label{road}
\end{figure}
If both inflows at the source nodes and capacities of the links are constant, then no congestion should arise as long as the inflows are less than the capacities. However, in reality, congestion is prone to occur due to fluctuations in inflows. For example, inflows to a highway depends on traffic condition on upstream arterial roads as well as demand-disrupting events (concert, sports, etc.) Air traffic flow is heavily influenced by weather.
Furthermore, such fluctuation is typically stochastic and is best modeled probabilistically. However, how to manage traffic flows in such scenarios, especially under stochastic inflows, has not been well understood.
In this paper, we study the behavior of multiple traffic flows sharing a common link and propose intuitive but provably efficient management strategies that ensure bounded queuing delay and maximal throughput. We consider the setting where both links 1 and 2 are subject to Markovian inflows: the inflow to each link switches between two values according to a Markov chain. We assume that the inflows to these two links are statistically independent. The traffic flows have their respective fixed routes, which overlap on link 3. Link 3, the common link, has a finite storage space; once the traffic queue in link 3 attains the storage space, the flows out of links 1 and 2 will be reduced due to spillback. The limited storage space of link 3 is shared by traffic from links 1 and 2 according to pre-specified priorities. The multiclass traffic flow is discharged from link 3 according to a proportional rule: the discharge rate of a traffic class is proportional to the fraction of traffic of this class in the current queue. Such discharging rule may also cause spillback from links 4 or 5 to link 3.
In this setting, the major decision variable for traffic flow management is the priorities according to which the limited capacity of the common link is shared between two traffic classes. In road traffic, this involves signal control (urban streets \cite{osorio2013simulation}) and ramp metering (highways \cite{kurzhanskiy2010active}). In air transportation, this involves ground and/or airborne holding \cite{sun2008multicommodity,zhou2019resilient}.
Such control actions are typically costly and must be designed based on rigorous and systematic justification.
The main contributions of this paper are a set of results that help a system operator determine the priorities at the merging junction based on operational parameters (demands and capacities) to ensure guarantees of key performance metrics, viz. queuing delay and throughput. Specifically, we argue via Theorem~\ref{thm_1} that there exist a non-empty set of priorities ensuring bounded traffic queues at the merging junction and maximal throughput if and only if the average inflow of each traffic class is less than the capacity of each link on its route. We further argue via Theorem~\ref{thm_2} that the discharging rule and the possible spillback at the diverging junction does not affect stability of the system. In addition, we explicitly provide a set of priorities that stabilize the system. We expect our results to be directly relevant for road traffic management \cite{jin2018stability} and the discrete-state extension thereof to be relevant for air traffic management \cite{chen2017stochastic}.
Our study is based on a fluid model. Fluid models are commonly used for highway bottlenecks \cite[Ch. 2]{newell13}. Its discretized version is also common for air traffic management \cite{bertsimas1998air}. We are aware that queuing models (e.g. M/M/1) are also widely used in transportation studies \cite{osorio2013simulation,baykal2009modeling}. However, queuing models focus on the delay due to random headways between vehicles rather than the congestion due to demand fluctuation; therefore, fluid model fits our purpose better. In fact, we view our fluid model as a reduced-order abstraction for queuing model: it is well known that stability of queuing models is closely related to their fluid counterparts \cite{dai1995stability,bertsimas1996stability}.
Hence, our analysis in itself contributes to the literature on stochastic fluid models, which has mainly focused on controlling single/parallel links \cite{jin2018stability,kulkarni97,cassandras02} or characterizing steady-state distribution of queue sizes \cite{mitra1988stochastic,kroese2001joint}, rather than quantifying spillback-induced delay and throughput loss.
The rest of this paper is organized as follows. In Section~\ref{sec_merge}, we isolate the merging junction from the network and study its behavior. In Section~\ref{sec_system}, we add the diverging junction into our analysis and obtain results for the merge-diverge system. In Section~\ref{sec_simulate}, we present a numerical example illustrating the main results that we derived. In Section~\ref{sec_conclusion}, we summarize the main conclusions and propose several directions for future work. \section{Analysis of merging junction} \label{sec_merge}
In this section, we study the behavior of a single merging junction (see Fig.~\ref{y}). This is an important component in the merge-diverge system and turns out to play a decisive role in terms of stability and throughput analysis.
\begin{figure}
\caption{Merging junction (left); inflow to link $k\in\{1,2\}$ evolves according to a Markov chain (right).}
\label{y}
\end{figure}
\subsection{Model and main result} The merging junction consists of three links. Traffic flows out of the upstream links 1 and 2 join and enter the downstream link 3. The \emph{inflows} to links 1 and 2 are specified as follows. Let $A_k(t)$ be the inflow to link $k$ at time $t$. Then, $A_k(t)$ is a two-state Markov process with state space $\{0,a_k^+\}$ and transition rates $\lambda_k$ and $\mu_k$, as illustrated in Fig.~\ref{y}. Thus, the mean inflows are given by \begin{align*} \bar a_k=\frac{\lambda_k}{\lambda_k+\mu_k}a_k^+ \quad k=1,2. \end{align*} Note that our results can be extended to the case where $A_k(t)$ is a two-state Markov process with state space $\{a_k^-,a_k^+\}$ for some $a_k^->0$. We assume that $\{A_1(t);t>0\}$ and $\{A_2(t);t>0\}$ are independent processes. Consequently, we can use a four-state Markov chain to describe the evolution of the inflows. The state of the Markov chain is $a\in\mathcal A=\{0,a_1^+\}\times\{0,a_2^+\}$. Fig.~\ref{chain} uses a shorthand notation where ``00'' means $a=[0\ 0]^T$ and ``10'' means $a=[a_1^+\ 0]^T$. We also use the unified notation $\{\nu_{ij};i,j\in\mathcal A\}$ to denote the transition rates; e.g. $\nu_{00,10}=\lambda_1$.
\begin{figure}
\caption{Markov chain for $\{A(t)=[A_1(t)\ A_2(t)]^T;t>0\}$.}
\label{chain}
\end{figure}
The \emph{flows} $f_{13},f_{23}$ between the links are determined by the \emph{sending flows} offered by links 1 and 2 as well as the \emph{receiving flow} allowed by link 3. Specifically, let $q_k\in[0,\infty)$ be the \emph{queue length} in link $k$ and $\mathcal Q=[0,\infty)^2$ be the set of queue lengths. Then, the sending flow out of link $k$ is given by \begin{align*} s_k(q_k,a_k)=\begin{cases} a_k & q_k=0,\\ F_k & q_k>0, \end{cases} \quad k=1,2, \end{align*} where $F_k$ is the \emph{capacity} of link $k$. The receiving flow of link 3 is given by \begin{align*} r_3(q_3)=\begin{cases} R_3 & q_3<\theta,\\ F_3 & q_3=\theta, \end{cases} \end{align*} where $R_3$ is the maximal receiving flow of link 3. In this section, we focus on the merging junction and assume that link 3 is not constrained downstream and $Q_3(t)=0$ for all $t\ge0$; we will relax this assumption in the next section. Thus, the between-link flows are given by \begin{subequations} \begin{align} &f_{13}(a,q)=\min\{s_1(a_1,q_1),R_3\mathbb I_{q_2=0}+\phi_1 R_3\mathbb I_{q_2>0}\},\label{eq_f1}\\ &f_{23}(a,q)=\min\{s_2(a_2,q_2),R_3\mathbb I_{q_1=0}+\phi_2 R_3\mathbb I_{q_1>0}\}\label{eq_f2} \end{align} \end{subequations} where $\phi_k\in[0,1]$ is the \emph{priority} of link $k=1,2$ and $R_3=r_3(0)$ is the receiving flow of link 3 for $q_3=0$; the \emph{priority vector} $\phi=[\phi_1\ \phi_2]^T$ must satisfy \begin{align}
\phi\ge0,\quad|\phi|=1. \label{eq_phi} \end{align} In practice, the priority vector specify how the limited capacity of the shared link is distributed over the upstream links. A typical mechanism for implementing such capacity allocation is traffic signal control, i.e. intersection control on urban streets and ramp metering on highways. In air transportation, this is done by air traffic management instructions.
The \emph{state} of the merging junction is $(a,q)\in\mathcal A\times\mathcal Q$. The evolution of $A(t)$ is fully specified by the Markov chain in Fig.~\ref{chain}. For a given initial condition $Q(0)\in\mathcal Q$, the evolution of $Q(t)$ is governed by \begin{align*} \frac{d}{dt}Q_k(t)=A_k(t)-f_{k3}(A(t),Q(t)) \quad k=1,2,\ t>0. \end{align*} The process $\{(A(t),Q(t));t>0\}$ is actually a piecewise-deterministic Markov process \cite{davis84}.
We say that the the merging junction is \emph{stable} if the total queue size is bounded, i.e. if there exists $Z<\infty$ such that for each initial condition \begin{align} \limsup_{t\to\infty}\frac1t\int_{s=0}^t\mathsf E[Q_1(s)+Q_2(s)]ds\le Z. \label{eq_stable} \end{align} This definition of stability is motivated by \cite{dai95}. Note that stability of the merging junction depends on the inflow, the capacities, the maximal receiving flow, and the priority vector.
The main result of this section is a criterion for existence of priority vectors that stabilize the queues:
\begin{Theorem} \label{thm_1} Consider a merging junction and let $[\phi_1\ \phi_2]^T\in[0,1]^2$ be the priority vector satisfying \eqref{eq_phi}. Then, there exists a non-empty set of priority vectors that stabilize the merging junction if and only if \begin{align} \bar a_1<F_1,\ \bar a_2<F_2,\ \bar a_1+\bar a_2<R_3. \label{eq_nominal} \end{align} Furthermore, when \eqref{eq_nominal} holds, if furthermore \begin{align} \frac{\bar a_1}{F_1}+\frac{\bar a_2}{F_2}<1 \label{eq_sufficient1} \end{align} holds, then every $\phi\in\Phi$ is stabilizing; otherwise, a set of stabilizing priority vectors is given by \begin{align} \Phi_1=\{\phi\in\Phi:\phi_1>\bar a_1/R_3,\ \phi_2>\bar a_2/R_3\} \label{eq_Phi1} \end{align} and a set of destabilizing priority vectors is given by the complement of the set \begin{align*} \Phi_0=\Bigg\{&\phi\in\Phi:\frac{\bar a_1}{F_1}+\frac{\bar a_2}{F_2}+\left(1-\frac{\phi_1 R_3}{F_1}-\frac{\phi_2R_3}{F_2}\right)\\ &\times\min\left\{\frac{\bar a_1}{\phi_1 R_3},\frac{\bar a_2}{\phi_2R_3}\right\}\le1\Bigg\}. \end{align*} \end{Theorem}
The above theorem essentially states that there exist stabilizing priority vectors if and only if the average inflows are less than the respective capacities. Furthermore, we provide criteria for the stability of particular priority vectors. Note that the set $\Phi_1$ (resp. $\Phi_0$) is derived from a sufficient (resp. necessary) condition for stability; there may exist a gap between $\Phi_1$ and $\Phi_0$. For priority vectors in the gap, our results do not provide a conclusive answer regarding stability; see Section~\ref{sec_simulate} for a numerical example with a visualization of this gap.
\subsection{Proof of Theorem~\ref{thm_1}} This subsection is devoted to a series of results leading to Theorem~\ref{thm_1}. First, we derive a necessary condition for stability of the merge:
\begin{Proposition} \label{prp_necessary} Consider the merging junction and let $\phi_1\in[0,1]$ be the priority of link 1. If the traffic queues upstream to the merging junction are stable, then either \begin{align} \frac{\bar a_1}{F_1}+\frac{\bar a_2}{F_2}\le1 \label{eq_necessary1} \end{align} or \begin{align} &\frac{\bar a_1}{F_1}+\frac{\bar a_2}{F_2}+\left(1-\frac{\phi_1 R}{F_1}-\frac{\phi_2R}{F_2}\right)\min\left\{\frac{\bar a_1}{\phi_1 R},\frac{\bar a_2}{\phi_2R}\right\}\le1 \label{eq_necessary2} \end{align} holds, where $1/\phi_k:=\infty$ for $\phi_k=0$, $k=1,2$. \end{Proposition}
\emph{Proof:} Apparently, for each initial condition, the fluid queuing process $\{Q(t);t>0\}$ can always visit the state $q_1=0,q_2=0$ within finite time and with positive probability. Hence, the fluid queuing process is ergodic \cite[Theorem 2.11]{cloez2015exponential}. Hence, there exist constants $\mathsf p_{01}$, $\mathsf p_{10}$, and $\mathsf p_{11}$ such that for any initial condition \begin{align*} &\mathsf p_{00}=\lim_{t\to\infty}\frac1t\int_{s=0}^t\mathbb I_{Q_1(s)=0,Q_2(s)=0}ds\quad a.s.\\ &\mathsf p_{01}=\lim_{t\to\infty}\frac1t\int_{s=0}^t\mathbb I_{Q_1(s)=0,Q_2(s)>0}ds\quad a.s.\\ &\mathsf p_{10}=\lim_{t\to\infty}\frac1t\int_{s=0}^t\mathbb I_{Q_1(s)>0,Q_2(s)=0}ds\quad a.s.\\ &\mathsf p_{11}=\lim_{t\to\infty}\frac1t\int_{s=0}^t\mathbb I_{Q_1(s)>0,Q_2(s)>0}ds\quad a.s. \end{align*} where $\mathbb I$ is the indicator variable. If the upstream queues are stable, then \begin{subequations} \begin{align} &\bar a_1=\mathsf p_{10}F_1+\mathsf p_{11}\phi_1 R\label{eq_a1}\\ &\bar a_2=\mathsf p_{01}F_2+\mathsf p_{11}\phi_2 R \end{align} \end{subequations} Also note that \begin{subequations} \begin{align} &\mathsf p_{01}\ge0,\ \mathsf p_{10}\ge0,\ \mathsf p_{11}\ge0,\\ &\mathsf p_{01}+\mathsf p_{10}+\mathsf p_{11}\le1.\label{eq_ppp} \end{align} \end{subequations} Then, one can obtain from \eqref{eq_a1}--\eqref{eq_ppp} that either \eqref{eq_necessary1} or \eqref{eq_necessary2} holds.
$\blacksquare$
Second, we derive a sufficient condition for stability of the merge:
\begin{Proposition} \label{prp_sufficient} Consider a merging junction and let $\phi_1\in[0,1]$ be the priority of link 1. The traffic queues upstream to the merging junction are stable if either (i) \eqref{eq_nominal}--\eqref{eq_sufficient1} hold or (ii) the following inequalities \begin{subequations} \begin{align} &\bar a_1-\min\{F_1,\phi_1 R\}<0 \label{eq_sufficient2a}\\ &\bar a_2-\min\{F_2,\phi_2 R\}<0. \label{eq_sufficient2b} \end{align} \end{subequations} hold. \end{Proposition}
\emph{Proof:} We only prove the case where \eqref{eq_sufficient1} holds; the case where \eqref{eq_sufficient2a}--\eqref{eq_sufficient2b} hold can be proved analogously. Suppose that \eqref{eq_sufficient1} holds. Consider the quadratic Lyapunov function \begin{align*} V_1(a,q):=&q^T \left[\begin{array}{cc} 1 & \alpha\\ \alpha & \alpha^2 \end{array}\right] q +[\beta_i\ \alpha\beta_i]q, \\ &\quad q\in[0,\infty)^2,\ i\in\{00,10,01,11\} \end{align*} where the parameters are given by \begin{subequations} \begin{align} &\alpha:=\frac12\left(\frac{\bar a_1}{F_2-\bar a_2}+\frac{F_1-\bar a_1}{\bar a_2}\right),\\ &\beta_{00}: =1,\ \beta_{10}:=\frac{\bar a_1}{\lambda_1}+1,\label{eq_beta00}\\ &\beta_{01}:=\alpha\frac{\bar a_2}{\lambda_2}+1,\ \beta_{11}:=\frac{\bar a_1}{\lambda_1}+\alpha\frac{\bar a_2}{\lambda_2}+1.\label{eq_beta11} \end{align} \end{subequations} The above parameters are guaranteed to be positive by \eqref{eq_sufficient1}. Let $\mathcal L_1$ be the infinitesimal generator (see \cite{davis84} for definition) of the merge system. With $\alpha$ and $\beta_i$ as given above, one can show that for each $i$ and $q$ \begin{align*} \mathcal L_1V_1(i,q)=\Big((\bar a_1-f_1(q))+\alpha(\bar a_2-f_2(q))\Big)(q_1+\alpha q_2). \end{align*} Then, there exist \begin{align*} &c:=\min\{F_1-\bar a_1-\alpha\bar a_2,\alpha F_2-\bar a_1-\alpha\bar a_2\}\min\{1,\alpha\}>0\\ &d:=\max_{q_1\le\tilde q_1,q_2\le\tilde q_2}\mathcal L_1V_1(i,q)<\infty \end{align*} such that \begin{align*}
\mathcal L_1V_1(i,q)\le-c|q|+d \quad\forall i,q \end{align*} which implies \eqref{eq_stable} according to the Foster-Lyapunov drift criteria \cite[Theorem 4.3]{meyn93}.
$\blacksquare$
Finally, we optimize the priority under various scenarios. The following result characterizes the priorities that leads to maximal throughput under the stability constraint given by Proposition~\ref{prp_sufficient}:
\emph{Proof of Theorem~\ref{thm_1}:} The necessity of \eqref{eq_nominal} is apparent: if the queues are stable, then the average inflows must be less than the corresponding capacities.
To show the sufficiency of \eqref{eq_nominal}, note that \eqref{eq_nominal} implies that every $\phi\in\Phi_1$ verifies \eqref{eq_sufficient2a}--\eqref{eq_sufficient2b}. The sets $\Phi_1$ and $\Phi_1'$ naturally result from Propositions~\ref{prp_necessary} and \ref{prp_sufficient}.
$\blacksquare$ \section{Analysis of merge-diverge network} \label{sec_system}
In this section, we extend the results for a merging junction to the merge-diverge network in Fig.~\ref{yy}.
\begin{figure}
\caption{Merge and diverge in series; two classes of traffic flows share a common link.}
\label{yy}
\end{figure}
\subsection{Model and main result} The dynamics is essentially the same as that of the merging junction; the main difference results from the interaction between two traffic classes at the diverge according to the \emph{discharging rule}, which is specified by $\psi$, the fraction of class 1 traffic in the sending flow $s_3$. For the flow out of link 3, we consider a \emph{proportional} discharging rule $\psi:[0,\infty)^2\times\prod_{k=1}^2[0,F_k]\to[0,1]^2$ defined as \begin{align} &\psi(q_3^1,q_3^2,f_{13},f_{23})\nonumber\\ &=\begin{cases} \left[\frac{q_3^1}{q_3^1+q_3^2}\ \frac{q_3^2}{q_3^1+q_3^2}\right]^T & q_3^1+q_3^2>0\\ \left[\frac{f_{13}}{f_{13}+f_{23}}\ \frac{f_{23}}{f_{13}+f_{23}}\right]^T & q_3^1+q_3^2=0,f_{13}+f_{23}>0\\ [1/2\ 1/2]^T & o.w. \end{cases} \label{eq_psi} \end{align}
The sending and receiving flows are $s_k(q_k)$ and $r_k(q_k)$ respectively, $k=1,2,\ldots,5$. With a slight abuse of notation, we denote $q=[q_1\ q_2\ \cdots q_5]^T$. The inter-link flows are given by \begin{subequations} \begin{align} &f_{13}(q)=\min\left\{s_1(q_1),\Big(r_3(q_3)-(1-\phi)s_2(q_2)\Big)_+\right\} \label{eq_f13}\\ &f_{23}(q)=\min\left\{s_2(q_2),\Big(r_3(q_3)-\phi s_1(q_1)\Big)_+\right\} \label{eq_f23}\\ &f_{34}(q)=\min\left\{\psi s_3(q_3),r_4(q_4),\frac{\psi}{1-\psi}r_5(q_5)\right\} \label{eq_f34}\\ &f_{35}(q)=\min\left\{\psi s_3(q_3),r_5(q_5),\frac{1-\psi}{\psi}r_4(q_4)\right\} \label{eq_f35} \end{align} \end{subequations} In the above, \eqref{eq_f13}--\eqref{eq_f23} are direct extensions of \eqref{eq_f1}--\eqref{eq_f2} from $q_3=0$ to $q_3\ge0$. \eqref{eq_f34}--\eqref{eq_f35} essentially follow the same logic; the only difference is the impact of the discharging ratio $\psi$. That is, congestion in one traffic class (e.g. $q_5$) may undermine the flow in the other (e.g. $f_{34}$). For ease of presentation, we assume that $$R_4<F_3,\ R_5<F_3,\ F_3<R_4+R_5.$$ The other cases can be analogously studied but are less interesting.
Since the queue in link 3 is upper-bounded and since links 4 and 5 are not constrainted downstream, the merge-diverge network is stable if \eqref{eq_stable} holds. However, in this setting the upstream queues $Q_1(t)$ and $Q_2(t)$ are also affected by links 4 and 5. The main result of this section is a criterion for exisitence priority vectors that stabilize the network:
\begin{Theorem} \label{thm_2} Consider a merge-diverge system. The merging junction has a priority vector $[\phi_1\ \phi_2]^T\in[0,1]^2$ satisfying \eqref{eq_phi} . The diverging junction has a discharging rule $\psi\in[0,1]$ satisfying \eqref{eq_psi}. Then, there exists a non-empty set of static priorities $\phi\in[0,1]^2$ that stabilize the system if and only if \begin{align} \bar a_1<\min\{F_1,R_4\},\ \bar a_2<\min\{F_2,R_5\},\ \bar a_1+\bar a_2<F_3. \label{eq_nominal2} \end{align} Furthermore, when \eqref{eq_nominal} holds, a set of stabilizing priority vectors is given by \begin{align} \Phi_2=\Big\{&\phi\in\Phi:\bar a_1-\min\{F_1,\phi_1F_3,R_4,(\phi_1/\phi_2)R_5\}<0,\nonumber\\ &\bar a_2-\min\{F_2,\phi_2F_3,R_5,(\phi_2/\phi_1)R_4\}<0\Big\}. \label{eq_Phi2} \end{align} \end{Theorem}
Comparing the above result with Theorem~\ref{thm_1}, we can see that the diverge junction does not affect the existence of stabilizing priority vectors; however, it does affect the set of stabilizing priority vectors. Apparently, $\Phi_2\subset\Phi_1$, where $\Phi_1$ is given by \eqref{eq_Phi1}.
\subsection{Proof of Theorem~\ref{thm_2}}
The necessity of \eqref{eq_nominal2} is apparent: if the fluid queuing process is stable, then the average inflow has to be less than the capacity for both traffic classes.
We next prove the sufficiency of \eqref{eq_nominal2}. We only consider the case where $a_1^+>F_1$ and $a_2^+>F_2$; the other cases can be covered following analogous steps. The proof is based on the following intermediate result: \begin{Lemma} \label{lmm_qtilde} For the fluid queuing process over the merge-diverge system, suppose that $\phi\in\Phi_2$ and $a_k^+>F_k$, $k=1,2$. Then, for any $t>0$ and any $\tilde q_k\ge0$, $k=1,2$, if $Q_k(t)\ge\tilde q_k$, then \begin{align} \psi_k(A(t),Q(t))\ge\tilde\psi_k^{\tilde q_k}:=\frac{\theta_k^{\tilde q_k}}{\Theta} \label{eq_psik} \end{align} where $\theta_k^{\tilde q_k}$ is given by \begin{align} &\theta_k^{\tilde q_k} =\nonumber\\ &\int_{s=0}^{\tilde q_k/(a_k^+-\min\{F_k,\phi_kF_3\})}\Big(\min\{F_k,\phi_kF_3\}-(\theta_s^k/\Theta)F_3\Big)ds. \label{eq_theta_s} \end{align} \end{Lemma}
This result essentially states that under priority vector $\phi\in\Phi_2$, if there is a long queue in link 1 (resp. 2), then the fraction of class 1 (resp. 2) traffic in link 3 is subject to a lower bound.
\emph{Proof of Lemma~\ref{lmm_qtilde}:} Suppose that $Q_1(t)=\theta_1\ge0$ and $Q_3^1(t)=\theta_3^1\ge0$ for a given $t>0$. Since $\dot Q_1(s)\le a_1^+-\min\{F_1,\phi_1F_3\}$ for any $s$, we have \begin{align*} Q_1(s)\ge (a_1^+-\min\{F_1,\phi_1F_3\})(s-t_0) \end{align*} for each $s\in(t-\theta_1/(a_1^+-\min\{F_1,\phi_1F_3\}),t]$.
Thus, for each $s\in(t-\theta_1/(a_1^+-\min\{F_1,\phi_1F_3\}),t]$, $Q_1(s)>0$ and consequently $ f_{12}(A(s),Q(s))=\min\{F_1,\phi_1F_3\}. $ In addition, $\theta_s^1$ as specified in \eqref{eq_theta_s} is the solution to $Q_3^1(s)$ with the initial condition $Q_3^1(0)=0$, $Q_3^2(0)=\Theta$ and the constraints $Q_1(s)>0$ and $Q_2(s)$ for $t\in(0,\theta_1/(a_1^+-\min\{F_1,\phi_1F_3\})]$. The above scenario is the one where $f_{34}$ would be minimized. Hence, $f_{34}(A(t),Q(t))\ge f_{34}(\cdot,q)$, where $q=[q_1\ q_2\ q_3^1\ q_3^2]^T$ satisfies $q_1=\theta_1$ and $q_3^1=\theta^k_{\theta_1/(a_k^+-\min\{F_k,\phi_kF_3\})}$.
If $Q_3^1(t)=Q_3^2(t)=0$, then $\psi_1\ge\phi_1$, which naturally satisfies \eqref{eq_psik}. Otherwise, $ \psi_1\ge\theta^k_{\theta_1/(a_k^+-\min\{F_k,\phi_kF_3\})}/\Theta $. This completes the proof.
$\blacksquare$
Lemma~\ref{lmm_qtilde} implies that if \eqref{eq_nominal2} holds, then the fluid queuing process over the merge-diverge system admits an invariant set $\mathcal M$ such that \begin{align*} \mathcal M=\cup_{\tilde q_1\ge0,\tilde q_2\ge0}\{q\in\mathcal Q:q_k\ge\tilde q_k,\psi_k(q)\ge\tilde\psi_k^{\tilde q_k},\ k=1,2\}. \end{align*} Lemma~\ref{lmm_qtilde} also implies that there exist $\hat q_1$ and $\hat q_2$ such that \begin{align*} &\forall q:q_1\ge\hat q_1,\ \psi_1(q)\ge 1-R_5/F_3,\\ &\forall q:q_2\ge\hat q_2,\ \psi_2(q)\ge 1-R_4/F_3, \end{align*} which enable us to derive Theorem~\ref{thm_2}:
\emph{Proof of Theorem~\ref{thm_2}:} Let $$ x=\left[\begin{array}{c} (q_1-\hat q_1)_++q_3^1\\ (q_2-\hat q_2)_++q_3^2 \end{array}\right] $$ and consider the Lyapunov function \begin{align*} V_2(i,q):=&x^T \left[\begin{array}{cc} 1 & \tilde\alpha\\ \tilde\alpha & \tilde\alpha^2 \end{array}\right] x +[\beta_i\ \tilde\alpha\beta_i]x, \\ &\quad q\in\mathcal Q,\ i\in\{00,10,01,11\} \end{align*} where{ \begin{align*} \tilde\alpha:=\frac12\Big(&\frac{\bar a_1}{\min\{F_2,\phi_2F_3,R_5,(\phi_2/\phi_1) R_4\}-\bar a_2}\\ &+\frac{\min\{F_1,\phi_1F_3,R_4,(\phi_1/\phi_2)R_5\}-\bar a_1}{\bar a_2}\Big) \end{align*}} and $\beta_i$ are specified by \eqref{eq_beta00}--\eqref{eq_beta11}.
Then, following procedures analogous to the proof of Proposition~\ref{prp_sufficient}, we can show that there exist $c>0$ and $d<\infty$ such that
$$\mathcal L_2V_2(a,q)\le-c|q|+d\quad\forall a\in\mathcal A,\ \forall q\in\mathcal M$$ which implies stability.
$\blacksquare$ \section{Numerical example} \label{sec_simulate}
Now we use a numerical example to illustrate the results that we obtained in the previous sections. Consider a merge-diverge network with the parameters in Table~\ref{tab_parameters}; the parameters make practical sense for the highway traffic setting.
\begin{table}[!ht]
\caption{Model parameters}\label{tab_parameters}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Parameter & Notation & Value \\
\hline
Average inflow & $\bar a_1,\bar a_2$ & $1200$ veh/hr \\
Merging link capacity & $F_1,F_2$ & 1500 veh/hr \\
\begin{tabular}{c}Diverging link capacity \\(receiving flow)\end{tabular}
& $R_4,R_5$ & 1400 veh/hr \\
\hline
\end{tabular}
\end{center} \end{table}
In modern intelligent transportation systems, the Markovian inflow can be interpreted in a particular scenario, i.e. traffic flow with connected and autonomous vehicles traveling in platoons, or ``platooning'' \cite{al2010experimental}. \begin{figure}
\caption{Randomly arriving platoons cause Markovian switches in traffic flow.}
\label{fig_arrival}
\end{figure} In this scenario, if we assume that (i) the headways between platoons are independent and identically distributed (IID) random variables $X$ with the cumulative distribution function (CDF) $$\mathsf F_X(x)=1-e^{-\lambda x},\quad x\ge0,$$ and (ii) the lengths of platoons are IID with the CDF $$\mathsf F_Y(y)=1-e^{-\lambda y},\quad y\ge0;$$ see Fig.~\ref{fig_arrival}. Furthermore, we assume that the background traffic flow is constant. Then, the inflow is a two-state Markov process \cite{jin2018modeling}.
We study the range of stabilizing priority vectors for various values of the capacity of the common link, $F_3$. That is, for every given value of $F_3$ and given value of $\phi_1$, we check the stability conditions, i.e. verifying whether $\phi=[\phi_1\ \phi_2]^T$ is in the sets $\Phi_1'$, $\Phi_1$, and $\Phi_2$.
\begin{figure}
\caption{Stability of the merge-diverge network under various priorities and various capacities of the common link.}
\label{stable}
\end{figure}
Fig.~\ref{stable} illustrates the results; the nomenclature is explained in Table~\ref{tab_stable}.
\begin{table}[!ht]
\caption{Nomenclature for various regions in Fig.~\ref{stable}.}\label{tab_stable}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Region & $\phi\in\Phi_0$? & $\phi\in\Phi_1$? & $\phi\in\Phi_2$? \\
\hline
unstable & no & no & no \\
unknown & yes & no & no \\
merge stable & yes & yes & no \\
merge-diverge stable & yes & yes & yes \\
\hline
\end{tabular}
\end{center} \end{table}
The following observations are noteworthy. First, there exists stabilizing priority vectors if and only if the common link has sufficient capacity to discharge both traffic classes, i.e. $F_3>2400=\bar a_1+\bar a_2$. Second, there exist gaps (``unknown'') between the ``stable'' regions and the ``unstable'' region due to the gap between the necessary condition (characterized by $\Phi_0$) and the sufficient condition (characterized by $\Phi_1$) for stability. Third, merge-diverge stability requires more restrictions on $\phi$ than merge stability alone. Fourth, as long as $F_3$ is larger than a certain threshold (2600 in this example), the set of priority vectors stabilizing the merge-diverge network is insensitive to $F_3$; the reason is that in that range the downstream receiving flows $R_4$ and $R_5$ are the decisive quantities for stability. \section{Concluding remarks} \label{sec_conclusion}
In this article, we studied the behavior of two traffic flows of distinct origins and destinations sharing a common link on their routes. Both flows are generated by a Markov process, and the delay is estimated using a fluid model. We found that the way in which the limited space in the shared link is allocated to either traffic flow (characterized by the priority vector at the merging junction) plays a decisive role in the network's behavior. In general, the fractional priority of a traffic flow should be in a neighborhood (which we quantitatively specify) of the inflow-to-capacity ratio of that flow. Furthermore, although spillback may also happen at the diverging junction, it does not affect the stability or throughput or the network.
This work can serve as the basis for several directions of future work. First, modern traffic networks are equipped with real-time sensing and actuating capabilities. Therefore, the priority vector at the merging junction can be made responsive to real-time traffic condition. The advantage of a dynamic feedback priority vector is that it does not necessarily require accurate prediction of inflow or capacity. Second, our analysis can be extended to more general networks, and approximated models can be developed for scalability. Third, route choice model (road traffic) or routing algorithm (air traffic) can be added in the network extension as a second dimension of control capabilities.
\end{document} | arXiv |
OSA Publishing > Optics Express > Volume 27 > Issue 26 > Page 38359
Cold-atom clock based on a diffractive optic
R. Elvin, G. W. Hoth, M. Wright, B. Lewis, J. P. McGilligan, A. S. Arnold, P. F. Griffin, and E. Riis
R. Elvin,1 G. W. Hoth,1 M. Wright,1 B. Lewis,1 J. P. McGilligan,2,3 A. S. Arnold,1,* P. F. Griffin,1 and E. Riis1
1Department of Physics, SUPA, University of Strathclyde, Glasgow, G4 0NG, UK
2National Institute of Standards and Technology, Boulder, Colorado 80305, USA
3University of Colorado, Department of Physics, Boulder, Colorado 80309, USA
*Corresponding author: [email protected]
A. S. Arnold https://orcid.org/0000-0001-7084-6958
P. F. Griffin https://orcid.org/0000-0002-0134-7554
R Elvin
G Hoth
M Wright
B Lewis
J McGilligan
A Arnold
P Griffin
E Riis
pp. 38359-38366
•https://doi.org/10.1364/OE.378632
R. Elvin, G. W. Hoth, M. Wright, B. Lewis, J. P. McGilligan, A. S. Arnold, P. F. Griffin, and E. Riis, "Cold-atom clock based on a diffractive optic," Opt. Express 27, 38359-38366 (2019)
Cold atoms
Optical molasses
Saturation spectroscopy
Tunable diode lasers
Original Manuscript: September 23, 2019
Revised Manuscript: October 18, 2019
Manuscript Accepted: December 5, 2019
Clocks based on cold atoms offer unbeatable accuracy and long-term stability, but their use in portable quantum technologies is hampered by a large physical footprint. Here, we use the compact optical layout of a grating magneto-optical trap (gMOT) for a precise frequency reference. The gMOT collects 107 87Rb atoms, which are subsequently cooled to 20 µK in optical molasses. We optically probe the microwave atomic ground-state splitting using lin⊥lin polarised coherent population trapping and a Raman-Ramsey sequence. With ballistic drop distances of only 0.5 mm, the measured short-term fractional frequency stability is $2 \times 10 ^{-11} /\sqrt {\tau }$.
Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Performance of a prototype atomic clock based on lin‖lin coherent population trapping resonances in Rb atomic vapor
Eugeniy E. Mikhailov, Travis Horrom, Nathan Belcher, and Irina Novikova
Electromagnetically induced absorption scheme for vapor-cell atomic clock
Denis Brazhnikov, Stepan Ignatovich, Vladislav Vishnyakov, Rodolphe Boudot, and Mikhail Skvortsov
Free expanding cloud of cold atoms as an atomic standard: Ramsey fringes contrast
Stella Torres Müller, Daniel Varela Magalhães, Aida Bebeachibuli, Tiago Almeida Ortega, Mushtaq Ahmed, and Vanderlei Salvador Bagnato
Optimization of FM spectroscopy parameters for a frequency locking loop in small scale CPT based atomic clocks
I. Ben-Aroya, M. Kahanov, and G. Eisenstein
Highly reliable optical system for a rubidium space cold atom clock
Wei Ren, Yanguang Sun, Bin Wang, Wenbing Xia, Qiuzhi Qu, Jingfeng Xiang, Zuoren Dong, Desheng Lü, and Liang Liu
A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, "Optical atomic clocks," Rev. Mod. Phys. 87(2), 637–701 (2015).
S. M. Dickerson, J. M. Hogan, A. Sugarbaker, D. M. S. Johnson, and M. A. Kasevich, "Multiaxis inertial sensing with long-time point source atom interferometry," Phys. Rev. Lett. 111(8), 083001 (2013).
I. Dutta, D. Savoie, B. Fang, B. Venon, C. L. Garrido Alzar, R. Geiger, and A. Landragin, "Continuous cold-atom inertial sensor with 1nrad/sec rotation stability," Phys. Rev. Lett. 116(18), 183003 (2016).
M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-resolution magnetometry with a spinor Bose-Einstein condensate," Phys. Rev. Lett. 98(20), 200801 (2007).
N. Huntemann, C. Sanner, B. Lipphardt, Chr. Tamm, and E. Peik, "Single-ion atomic clock with 3 × 10−18 systematic uncertainty," Phys. Rev. Lett. 116(6), 063001 (2016).
S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye, "A Fermi-degenerate three-dimensional optical lattice clock," Science 358(6359), 90–94 (2017).
K. J. Arnold, R. Kaewuam, A. Roy, T. R. Tan, and M. D. Barrett, "Blackbody radiation shift assessment for a lutetium ion clock," Nat. Commun. 9(1), 1650 (2018).
S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, "27Al+ quantum-logic clock with a systematic uncertainty below 10−18," Phys. Rev. Lett. 123(3), 033201 (2019).
W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow, "Atomic clock performance enabling geodesy below the centimetre level," Nature 564(7734), 87–90 (2018).
E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr, and J. Ye, "Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks," Nat. Photonics 13(10), 714–719 (2019).
S. B. Koller, J. Grotti, St. Vogt, A. Al-Masoudi, S. Dörscher, S. Häfner, U. Sterr, and Ch. Lisdat, "Transportable optical lattice clock with 7 × 10−17 uncertainty," Phys. Rev. Lett. 118(7), 073601 (2017).
J. Grotti, S. Koller, S. Vogt, S. Häfner, U. Sterr, C. Lisdat, H. Denker, C. Voigt, L. Timmen, A. Rolland, F. N. Baynes, H. S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro, B. Rauf, F. Bregolin, A. Tampellini, P. Barbieri, M. Zucco, G. A. Costanzo, C. Clivati, F. Levi, and D. Calonico, "Geodesy and metrology with a transportable optical clock," Nat. Phys. 14(5), 437–441 (2018).
SpectraDynamics cRb-Clock .
MuQuans MuClock .
S. Knappe, V. Shah, P. D. D. Schwindt, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "A microfabricated atomic clock," Appl. Phys. Lett. 85(9), 1460–1462 (2004).
CSAC specifications at www.microsemi.com (2019).
P. Cash, W. Krzewick, P. Machado, K. R. Overstreet, M. Silveira, M. Stanczyk, D. Taylor, and X. Zhang, "Microsemi chip scale atomic clock (CSAC) technical status, applications, and future plans," IEEE Proc. EFTF (2018).
K. Bongs, V. Boyer, M. A. Cruise, A. Freise, M. Holynski, J. Hughes, A. Kaushik, Y.-H. Lien, A. Niggebaum, M. Perea-Ortiz, P. Petrov, S. Plant, Y. Singh, A. Stabrawa, D. J. Paul, M. Sorel, D. R. S. Cumming, J. H. Marsh, R. W. Bowtell, M. G. Bason, R. P. Beardsley, R. P. Campion, M. J. Brookes, T. Fernholz, T. M. Fromhold, L. Hackermuller, P. Krüger, X. Li, J. O. Maclean, C. J. Mellor, S. V. Novikov, F. Orucevic, A. W. Rushforth, N. Welch, T. M. Benson, R. D. Wildman, T. Freegarde, M. Himsworth, J. Ruostekoski, P. Smith, A. Tropper, P. F. Griffin, A. S. Arnold, E. Riis, J. E. Hastie, D. Paboeuf, D. C. Parrotta, B. M. Garraway, A. Pasquazi, M. Peccianti, W. Hensinger, E. Potter, A. H. Nizamani, H. Bostock, A. Rodriguez Blanco, G. Sinuco-Leon, I. R. Hill, R. A. Williams, P. Gill, N. Hempler, G. P. A. Malcolm, T. Cross, B. O. Kock, S. Maddox, and P. John, "The UK National Quantum Technologies Hub in sensors and metrology (Keynote Paper)," Proc. SPIE 9900, 990009 (2016).
C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, "A surface-patterned chip as a strong source of ultracold atoms for quantum technologies," Nat. Nanotechnol. 8(5), 321–324 (2013).
J. P. McGilligan, P. F. Griffin, R. Elvin, S. J. Ingleby, E. Riis, and A. S. Arnold, "Grating chips for quantum technologies," Sci. Rep. 7(1), 384 (2017).
R. Legere and K. Gibble, "Quantum scattering in a juggling atomic fountain," Phys. Rev. Lett. 81(26), 5780–5783 (1998).
K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, "Single-beam atom trap in a pyramidal and conical hollow mirror," Opt. Lett. 21(15), 1177–1179 (1996).
Q. Bodart, S. Merlet, N. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, "A cold atom pyramidal gravimeter with a single laser beam," Appl. Phys. Lett. 96(13), 134101 (2010).
X. Wu, F. Zi, J. Dudley, R. J. Bilotta, P. Canoza, and H. Müller, "Multiaxis atom interferometry with a single-diode laser and a pyramidal magneto-optical trap," Optica 4(12), 1545–1551 (2017).
M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, "Single-laser, one beam, tetrahedral magneto-optical trap," Opt. Express 17(16), 13601–13608 (2009).
M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, "Laser cooling with a single laser beam and a planar diffractor," Opt. Lett. 35(20), 3453–3455 (2010).
D. W. Sesko and C. E. Wieman, "Observation of the cesium clock transition in laser-cooled atoms," Opt. Lett. 14(5), 269–271 (1989).
J. Vanier, "Atomic clocks based on coherent population trapping: a review," Appl. Phys. B 81(4), 421–442 (2005).
F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, "Cold-atom double-Λ coherent population trapping clock," Phys. Rev. A 88(4), 042120 (2013).
X. Liu, E. Ivanov, V. I. Yudin, J. Kitching, and E. A. Donley, "Low-drift coherent population trapping clock based on laser-cooled atoms and high-coherence excitation fields," Phys. Rev. Appl. 8(5), 054001 (2017).
T. Zanon, S. Guérandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, "High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system," Phys. Rev. Lett. 94(19), 193002 (2005).
M. Abdel Hafiz, G. Coget, P. Yun, S. Guérandel, E. de Clercq, and R. Boudot, "A high-performance Raman-Ramsey Cs vapor cell atomic clock," J. Appl. Phys. 121(10), 104903 (2017).
S. V. Kargapoltsev, J. Kitching, L. Hollberg, A. V. Taichenachev, V. L. Velichansky, and V. I. Yudin, "High-contrast dark resonance in σ+ − σ− optical field," Laser Phys. Lett. 1(10), 495–499 (2004).
X. Liu, V. I. Yudin, A. V. Taichenachev, J. Kitching, and E. A. Donley, "High contrast dark resonances in a cold-atom clock probed with counterpropagating circularly polarized beams," Appl. Phys. Lett. 111(22), 224102 (2017).
J. D. Elgin, T. P. Heavner, J. Kitching, E. A. Donley, J. Denney, and E. A. Salim, "A cold-atom beam clock based on coherent population trapping," Appl. Phys. Lett. 115(3), 033503 (2019).
J.-M. Danet, M. Lours, S. Guérandel, and E. De Clercq, "Dick effect in a pulsed atomic clock using coherent population trapping," IEEE Trans. UFFC 61(4), 567–574 (2014).
S. Peil, T. B. Swanson, J. Hanssen, and J. Taylor, "Microwave-clock timescale with instability on order of 10−17," Metrologia 54(3), 247–252 (2017).
E. Blanshan, S. M. Rochester, E. A. Donley, and J. Kitching, "Light shifts in a pulsed cold-atom coherent population trapping clock," Phys. Rev. A 91(4), 041401 (2015).
J. W. Pollock, V. I. Yudin, M. Shuker, M. Y. Basalaev, A. V. Taichenachev, X. Liu, J. Kitching, and E. A. Donley, "ac Stark shifts of dark resonances probed with Ramsey spectroscopy," Phys. Rev. A 98(5), 053424 (2018).
J. P. McGilligan, P. F. Griffin, E. Riis, and A. S. Arnold, "Phase-space properties of magneto-optical traps utilising micro-fabricated gratings," Opt. Express 23(7), 8948–8959 (2015).
A. S. Arnold, J. S. Wilson, and M. G. Boshier, "A simple extended-cavity diode laser," Rev. Sci. Instrum. 69(3), 1236–1239 (1997).
A. V. Rakholia, H. J. McGuinness, and G. W. Biedermann, "Dual-axis high-data-rate atom interferometer via cold ensemble exchange," Phys. Rev. Appl. 2(5), 054012 (2014).
G. W. Hoth, R. Elvin, M. Wright, B. Lewis, A. S. Arnold, P. F. Griffin, and E. Riis, "Towards a compact atomic clock based on coherent population trapping and the grating magneto-optical trap," Proc. SPIE 10934, 109342E (2019).
Commercially available at www.kntnano.com .
Z. Warren, M. S. Shahriar, R. Tripathi, and G. S. Pati, "Experimental and theoretical comparison of different optical excitation schemes for a compact coherent population trapping Rb vapor clock," Metrologia 54(4), 418–431 (2017).
R. Boudot, S. Guerandel, E. de Clercq, N. Dimarcq, and A. Clairon, "Current status of a pulsed CPT Cs cell clock," IEEE Trans. Instrum. Meas. 58(4), 1217–1222 (2009).
R. Elvin, G. W. Hoth, M. W. Wright, J. P. McGilligan, A. S. Arnold, P. F. Griffin, and E. Riis, "Raman-Ramsey CPT with a grating magneto-optical trap," IEEE Proc. EFTF (2018).
W. Li, X. Pan, N. Song, X. Xu, and X. Lu, "A phase-locked laser system based on double direct modulation technique for atom interferometry," Appl. Phys. B 123(2), 54 (2017).
E. A. Donley, F.-X. Esnault, E. Blanshan, and J. Kitching, "Cancellation of Doppler shifts in a cold-atom CPT clock," IEEE Proc. EFTF (2013).
C. Affolderbach, S. Knappe, R. Wynands, A. V. Taĭchenachev, and V. I. Yudin, "Electromagnetically induced transparency and absorption in a standing wave," Phys. Rev. A 65(4), 043810 (2002).
J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, "Observation of Ramsey fringes using a stimulated, resonance Raman transition in a sodium atomic beam," Phys. Rev. Lett. 48(13), 867–870 (1982).
G. Santarelli, Ph. Laurent, P. Lemonde, A. Clairon, A. G. Mann, S. Chang, A. N. Luiten, and C. Salomon, "Quantum projection noise in an atomic fountain: a high stability cesium frequency standard," Phys. Rev. Lett. 82(23), 4619–4622 (1999).
C. Xi, Y. Guo-Qing, W. Jin, and Z. Ming-Sheng, "Coherent population trapping - Ramsey interference in cold atoms," Chin. Phys. Lett. 27(11), 113201 (2010).
V. I. Yudin, A. V. Taichenachev, M. Y. Basalaev, T. Zanon-Willette, T. E. Mehlstäubler, R. Boudot, J. W. Pollock, M. Shuker, E. A. Donley, and J. Kitching, "Combined error signal in Ramsey spectroscopy of clock Transitions," New J. Phys. 20(12), 123016 (2018).
V. I. Yudin, A. V. Taichenachev, M. Y. Basalaev, T. Zanon-Willette, J. W. Pollock, M. Shuker, E. A. Donley, and J. Kitching, "Generalized autobalanced Ramsey spectroscopy of clock transitions," Phys. Rev. Appl. 9(5), 054034 (2018).
Dataset DOI: 10.15129/4b93bf4c-84fe-4a0b-8e85-871884aefa6e.
Abdel Hafiz, M.
Affolderbach, C.
Al-Masoudi, A.
Arnold, A. S.
Arnold, K. J.
Barbieri, P.
Barrett, M. D.
Basalaev, M. Y.
Bason, M. G.
Baynes, F. N.
Beardsley, R. P.
Beloy, K.
Benson, T. M.
Biedermann, G. W.
Bilotta, R. J.
Blanshan, E.
Bloom, B. J.
Bodart, Q.
Bongs, K.
Boshier, M. G.
Bostock, H.
Bothwell, T.
Boudot, R.
Bouyer, P.
Bowtell, R. W.
Boyd, M. M.
Boyer, V.
Bregolin, F.
Brewer, S. M.
Brookes, M. J.
Brown, R. C.
Calonico, D.
Campbell, S. L.
Campion, R. P.
Canoza, P.
Cash, P.
Chang, S.
Chen, J.-S.
Chou, C. W.
Clairon, A.
Clements, E. R.
Clivati, C.
Coget, G.
Costanzo, G. A.
Cotter, J. P.
Cross, T.
Cruise, M. A.
Cumming, D. R. S.
Danet, J.-M.
Darkwah Oppong, N.
de Clercq, E.
Denker, H.
Denney, J.
Dickerson, S. M.
Dimarcq, N.
Donley, E. A.
Dörscher, S.
Dudley, J.
Dutta, I.
Elgin, J. D.
Elvin, R.
Esnault, F.-X.
Ezekiel, S.
Fang, B.
Fasano, R. J.
Fernholz, T.
Freegarde, T.
Freise, A.
Fromhold, T. M.
Garraway, B. M.
Garrido Alzar, C. L.
Geiger, R.
Gibble, K.
Gill, P.
Giunta, M.
Goban, A.
Griffin, P. F.
Grotti, J.
Guerandel, S.
Guérandel, S.
Guo-Qing, Y.
Guzman, J.
Hackermuller, L.
Häfner, S.
Hankin, A. M.
Hanssen, J.
Hastie, J. E.
Heavner, T. P.
Hemmer, P. R.
Hempler, N.
Hensinger, W.
Higbie, J. M.
Hill, I. R.
Himsworth, M.
Hinds, E. A.
Hinkley, N.
Hogan, J. M.
Hollberg, L.
Holleville, D.
Holynski, M.
Holzwarth, R.
Hoth, G. W.
Hughes, J.
Hume, D. B.
Huntemann, N.
Hutson, R. B.
Ingleby, S. J.
Ironside, C. N.
Ivanov, E.
Ivanov, E. N.
Jhe, W.
Jin, W.
John, P.
Johnson, D. M. S.
Kaewuam, R.
Kargapoltsev, S. V.
Kasevich, M. A.
Kaushik, A.
Kedar, D.
Kennedy, C. J.
Kim, J. A.
Kitching, J.
Knappe, S.
Kock, B. O.
Koller, S.
Koller, S. B.
Krüger, P.
Krzewick, W.
Landragin, A.
Laurent, Ph.
Lee, K. I.
Legere, R.
Legero, T.
Leibrandt, D. R.
Leiby, C. C.
Lemonde, P.
Leslie, S. R.
Levi, F.
Lewis, B.
Li, W.
Li, X.
Lien, Y.-H.
Liew, L.-A.
Lipphardt, B.
Lisdat, C.
Lisdat, Ch.
Lours, M.
Lu, X.
Ludlow, A. D.
Luiten, A. N.
Machado, P.
Maclean, J. O.
Maddox, S.
Malcolm, G. P. A.
Malossi, N.
Mann, A. G.
Margolis, H. S.
Marsh, J. H.
Marti, G. E.
Matei, D. G.
McGilligan, J. P.
McGrew, W. F.
McGuinness, H. J.
McNally, R. L.
Mehlstäubler, T. E.
Mellor, C. J.
Merlet, S.
Milani, G.
Ming-Sheng, Z.
Moreland, J.
Müller, H.
Nicolodi, D.
Niggebaum, A.
Nizamani, A. H.
Noh, H. R.
Novikov, S. V.
Nshii, C. C.
Oelker, E.
Orucevic, F.
Overstreet, K. R.
Paboeuf, D.
Pan, X.
Parrotta, D. C.
Pasquazi, A.
Pati, G. S.
Paul, D. J.
Peccianti, M.
Peik, E.
Peil, S.
Perea-Ortiz, M.
Pereira Dos Santos, F.
Petrov, P.
Picard, R. H.
Pizzocaro, M.
Plant, S.
Pollock, J. W.
Potter, E.
Rakholia, A. V.
Rauf, B.
Riehle, F.
Riis, E.
Robinson, J. M.
Rochester, S. M.
Rodriguez Blanco, A.
Rolland, A.
Roy, A.
Ruostekoski, J.
Rushforth, A. W.
Sadler, L. E.
Salim, E. A.
Salomon, C.
Sanner, C.
Santarelli, G.
Savoie, D.
Schäffer, S. A.
Schioppo, M.
Schmidt, P. O.
Scholten, R. E.
Schwindt, P. D. D.
See, P.
Sesko, D. W.
Shah, V.
Shahriar, M. S.
Shuker, M.
Silveira, M.
Sinclair, A. G.
Singh, Y.
Sinuco-Leon, G.
Smith, P.
Sonderhouse, L.
Song, N.
Sorel, M.
Stabrawa, A.
Stamper-Kurn, D. M.
Stanczyk, M.
Sterr, U.
Sugarbaker, A.
Swanson, T. B.
Taichenachev, A. V.
Tamm, Chr.
Tampellini, A.
Tan, T. R.
Taylor, D.
Taylor, J.
Thomas, J. E.
Thoumany, P.
Timmen, L.
Tripathi, R.
Tropper, A.
Vangeleyn, M.
Vanier, J.
Velichansky, V. L.
Vengalattore, M.
Venon, B.
Vogt, S.
Vogt, St.
Voigt, C.
Warren, Z.
Welch, N.
Wieman, C. E.
Wildman, R. D.
Williams, R. A.
Willis, C. R.
Wilson, J. S.
Wineland, D. J.
Wright, M.
Wright, M. W.
Wu, X.
Wynands, R.
Xi, C.
Xu, X.
Ye, J.
Yoon, T. H.
Yudin, V. I.
Yun, P.
Zampaolo, M.
Zanon, T.
Zanon-Willette, T.
Zhang, W.
Zi, F.
Zucco, M.
Appl. Phys. B (2)
Appl. Phys. Lett. (4)
Chin. Phys. Lett. (1)
IEEE Trans. Instrum. Meas. (1)
IEEE Trans. UFFC (1)
J. Appl. Phys. (1)
Laser Phys. Lett. (1)
Metrologia (2)
Nat. Commun. (1)
Nat. Nanotechnol. (1)
Nat. Photonics (1)
Nat. Phys. (1)
New J. Phys. (1)
Phys. Rev. A (4)
Phys. Rev. Appl. (3)
Phys. Rev. Lett. (10)
Proc. SPIE (2)
Rev. Mod. Phys. (1)
Rev. Sci. Instrum. (1)
Sci. Rep. (1)
Fig. 1. Schematic of the gMOT with CPT detection. The single beam required for the gMOT (red downward arrow) is produced by collimating the output of an optical fiber with a lens (blue disk). It is circularly polarised by a quarter-wave plate (green disk), and directed to a $2\times 2\, \textrm{cm}^2$ microfabricated diffractive optic (gold grating), which creates the remaining three beams. All four beams intersect to make up the trapping volume within a vacuum chamber represented by the transparent cuboid. A set of anti-Helmholtz coils (grey) co-axial with the incident trapping beam generate the required 3D magnetic field gradient. The clock signal is obtained by measuring the transmission of the probe D$_1$ light (red arrow, left) through the cold atoms (purple ball) using the two photodiodes (PD$_{\textrm {R/S}}$ for reference/signal). The beamsplitter cube splits 90/10 (reflecting/transmitting – a convention we use throughout this paper) and is non-polarising.
Fig. 2. (a) The level diagram for lin$\perp$lin CPT on the D$_{1}$ line of $^{87}$Rb. The two orthogonally polarised CPT fields are represented by the red and blue solid lines, and the levels used in the CPT state are labelled $m_{F}=0-0$. (b) Schematic optical bench for the CPT experiment including: external cavity diode laser (ECDL); acousto-optical modulator (AOM); non-polarising beam-splitter (NPBS); photodiode for reference/signal (PD$_{\textrm {R/S}}$); local oscillator (LO); loop filter (LF). The dashed region indicates the gMOT and CPT signal zone shown in Fig. 1.
Fig. 3. Experimental Ramsey-CPT fringes with a $T=1\,$ms free evolution time. We use the double-ratio technique, described in the text, to cancel common-mode laser intensity noise. Each data point represents a single run of the experiment sequence as the LO frequency is varied. The signal is normalised to the off-resonant wings.
Fig. 4. Measured Allan deviation curves for our apparatus. Experimental data, with a $T=10\,$ms free evolution time and PLL engaged, are plotted for two CPT bias field configurations: $280\,$mG (red diamonds) and $140\,$mG (blue circles). The dashed line represents $2 \times 10 ^{-11}/ \sqrt {\tau }$, which is a fit to the first six points. Inset: an example $T=10\,$ms fringe, where each point is a single experimental run. | CommonCrawl |
Grouped Dirichlet distribution
In statistics, the grouped Dirichlet distribution (GDD) is a multivariate generalization of the Dirichlet distribution It was first described by Ng et al. 2008.[1] The Grouped Dirichlet distribution arises in the analysis of categorical data where some observations could fall into any of a set of other 'crisp' category. For example, one may have a data set consisting of cases and controls under two different conditions. With complete data, the cross-classification of disease status forms a 2(case/control)-x-(condition/no-condition) table with cell probabilities
TreatmentNo Treatment
Controlsθ1θ2
Casesθ3θ4
If, however, the data includes, say, non-respondents which are known to be controls or cases, then the cross-classification of disease status forms a 2-x-3 table. The probability of the last column is the sum of the probabilities of the first two columns in each row, e.g.
TreatmentNo TreatmentMissing
Controlsθ1θ2θ1+θ2
Casesθ3θ4θ3+θ4
The GDD allows the full estimation of the cell probabilities under such aggregation conditions.[1]
Probability Distribution
Consider the closed simplex set ${\mathcal {T}}_{n}=\left\{\left(x_{1},\ldots x_{n}\right)\left|x_{i}\geq 0,i=1,\cdots ,n,\sum _{i=1}^{n}x_{n}=1\right.\right\}$ and $\mathbf {x} \in {\mathcal {T}}_{n}$. Writing $\mathbf {x} _{-n}=\left(x_{1},\ldots ,x_{n-1}\right)$ for the first $n-1$ elements of a member of ${\mathcal {T}}_{n}$, the distribution of $\mathbf {x} $ for two partitions has a density function given by
$\operatorname {GD} _{n,2,s}\left(\left.\mathbf {x} _{-n}\right|\mathbf {a} ,\mathbf {b} \right)={\frac {\left(\prod _{i=1}^{n}x_{i}^{a_{i}-1}\right)\cdot \left(\sum _{i=1}^{s}x_{i}\right)^{b_{1}}\cdot \left(\sum _{i=s+1}^{n}x_{i}\right)^{b_{2}}}{\operatorname {\mathrm {B} } \left(a_{1},\ldots ,a_{s}\right)\cdot \operatorname {\mathrm {B} } \left(a_{s+1},\ldots ,a_{n}\right)\cdot \operatorname {\mathrm {B} } \left(b_{1}+\sum _{i=1}^{s}a_{i},b_{2}+\sum _{i=s+1}^{n}a_{i}\right)}}$
where $\operatorname {\mathrm {B} } \left(\mathbf {a} \right)$ is the multivariate beta function.
Ng et al.[1] went on to define an m partition grouped Dirichlet distribution with density of $\mathbf {x} _{-n}$ given by
$\operatorname {GD} _{n,m,\mathbf {s} }\left(\left.\mathbf {x} _{-n}\right|\mathbf {a} ,\mathbf {b} \right)=c_{m}^{-1}\cdot \left(\prod _{i=1}^{n}x_{i}^{a_{i}-1}\right)\cdot \prod _{j=1}^{m}\left(\sum _{k=s_{j-1}+1}^{s_{j}}x_{k}\right)^{b_{j}}$
where $\mathbf {s} =\left(s_{1},\ldots ,s_{m}\right)$ is a vector of integers with $0=s_{0}<s_{1}\leqslant \cdots \leqslant s_{m}=n$. The normalizing constant given by
$c_{m}=\left\{\prod _{j=1}^{m}\operatorname {\mathrm {B} } \left(a_{s_{j-1}+1},\ldots ,a_{s_{j}}\right)\right\}\cdot \operatorname {\mathrm {B} } \left(b_{1}+\sum _{k=1}^{s_{1}}a_{k},\ldots ,b_{m}+\sum _{k=s_{m-1}+1}^{s_{m}}a_{k}\right)$
The authors went on to use these distributions in the context of three different applications in medical science.
References
1. Ng, Kai Wang (2008). "Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis". Journal of Multivariate Analysis. 99: 490–509.
| Wikipedia |
Sample records for riemann function
Exploring the Riemann zeta function 190 years from Riemann's birth
Nikeghbali, Ashkan; Rassias, Michael
This book is concerned with the Riemann Zeta Function, its generalizations, and various applications to several scientific disciplines, including Analytic Number Theory, Harmonic Analysis, Complex Analysis and Probability Theory. Eminent experts in the field illustrate both old and new results towards the solution of long-standing problems and include key historical remarks. Offering a unified, self-contained treatment of broad and deep areas of research, this book will be an excellent tool for researchers and graduate students working in Mathematics, Mathematical Physics, Engineering and Cryptography.
Functionals of finite Riemann surfaces
Schiffer, Menahem
This advanced monograph on finite Riemann surfaces, based on the authors' 1949-50 lectures at Princeton University, remains a fundamental book for graduate students. The Bulletin of the American Mathematical Society hailed the self-contained treatment as the source of ""a plethora of ideas, each interesting in its own right,"" noting that ""the patient reader will be richly rewarded."" Suitable for graduate-level courses, the text begins with three chapters that offer a development of the classical theory along historical lines, examining geometrical and physical considerations, existence theo
Lectures on the Riemann zeta function
Iwaniec, H
The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. Th...
Meromorphic functions and cohomology on a Riemann surface
Gomez-Mont, X.
The objective of this set of notes is to introduce a series of concepts of Complex Analytic Geometry on a Riemann Surface. We motivate the introduction of cohomology groups through the analysis of meromorphic functions. We finish by showing that the set of infinitesimal deformations of a Riemann surface (the tangent space to Teichmueller space) may be computed as a Cohomology group. (author). 6 refs
The sewing technique and correlation functions on arbitrary Riemann surfaces
Di Vecchia, P.
We describe in the case of free bosonic and fermionic theories the sewing procedure, that is a very convenient way for constructing correlation functions of these theories on an arbitrary Riemann surface from their knowledge on the sphere. The fundamental object that results from this construction is the N-point g-loop vertex. It summarizes the information of all correlation functions of the theory on an arbitrary Riemann surface. We then check explicitly the bosonization rules and derive some useful formulas. (orig.)
Riemann zeta function from wave-packet dynamics
Mack, R.; Dahl, Jens Peder; Moya-Cessa, H.
We show that the time evolution of a thermal phase state of an anharmonic oscillator with logarithmic energy spectrum is intimately connected to the generalized Riemann zeta function zeta(s, a). Indeed, the autocorrelation function at a time t is determined by zeta (sigma + i tau, a), where sigma...... index of JWKB. We compare and contrast exact and approximate eigenvalues of purely logarithmic potentials. Moreover, we use a numerical method to find a potential which leads to exact logarithmic eigenvalues. We discuss possible realizations of Riemann zeta wave-packet dynamics using cold atoms...
The Riemann zeta-function theory and applications
Ivic, Aleksandar
""A thorough and easily accessible account.""-MathSciNet, Mathematical Reviews on the Web, American Mathematical Society. This extensive survey presents a comprehensive and coherent account of Riemann zeta-function theory and applications. Starting with elementary theory, it examines exponential integrals and exponential sums, the Voronoi summation formula, the approximate functional equation, the fourth power moment, the zero-free region, mean value estimates over short intervals, higher power moments, and omega results. Additional topics include zeros on the critical line, zero-density estim
Method of construction of the Riemann function for a second-order hyperbolic equation
Aksenov, A. V.
A linear hyperbolic equation of the second order in two independent variables is considered. The Riemann function of the adjoint equation is shown to be invariant with respect to the fundamental solutions transformation group. Symmetries and symmetries of fundamental solutions of the Euler-Poisson-Darboux equation are found. The Riemann function is constructed with the aid of fundamental solutions symmetries. Examples of the application of the algorithm for constructing Riemann function are given.
On the $a$-points of the derivatives of the Riemann zeta function
Onozuka, Tomokazu
We prove three results on the $a$-points of the derivatives of the Riemann zeta function. The first result is a formula of the Riemann-von Mangoldt type; we estimate the number of the $a$-points of the derivatives of the Riemann zeta function. The second result is on certain exponential sum involving $a$-points. The third result is an analogue of the zero density theorem. We count the $a$-points of the derivatives of the Riemann zeta function in $1/2-(\\log\\log T)^2/\\log T
Minimal models on Riemann surfaces: The partition functions
Foda, O.
The Coulomb gas representation of the A n series of c=1-6/[m(m+1)], m≥3, minimal models is extended to compact Riemann surfaces of genus g>1. An integral representation of the partition functions, for any m and g is obtained as the difference of two gaussian correlation functions of a background charge, (background charge on sphere) x (1-g), and screening charges integrated over the surface. The coupling constant x (compacitification radius) 2 of the gaussian expressions are, as on the torus, m(m+1), and m/(m+1). The partition functions obtained are modular invariant, have the correct conformal anomaly and - restricting the propagation of states to a single handle - one can verify explicitly the decoupling of the null states. On the other hand, they are given in terms of coupled surface integrals, and it remains to show how they degenerate consistently to those on lower-genus surfaces. In this work, this is clear only at the lattice level, where no screening charges appear. (orig.)
Foda, O. (Katholieke Univ. Nijmegen (Netherlands). Inst. voor Theoretische Fysica)
The Coulomb gas representation of the A{sub n} series of c=1-6/(m(m+1)), m{ge}3, minimal models is extended to compact Riemann surfaces of genus g>1. An integral representation of the partition functions, for any m and g is obtained as the difference of two gaussian correlation functions of a background charge, (background charge on sphere) x (1-g), and screening charges integrated over the surface. The coupling constant x (compacitification radius){sup 2} of the gaussian expressions are, as on the torus, m(m+1), and m/(m+1). The partition functions obtained are modular invariant, have the correct conformal anomaly and - restricting the propagation of states to a single handle - one can verify explicitly the decoupling of the null states. On the other hand, they are given in terms of coupled surface integrals, and it remains to show how they degenerate consistently to those on lower-genus surfaces. In this work, this is clear only at the lattice level, where no screening charges appear. (orig.).
Orbifold Riemann surfaces: Teichmueller spaces and algebras of geodesic functions
Mazzocco, Marta [Loughborough University, Loughborough (United Kingdom); Chekhov, Leonid O [Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow (Russian Federation)
A fat graph description is given for Teichmueller spaces of Riemann surfaces with holes and with Z{sub 2}- and Z{sub 3}-orbifold points (conical singularities) in the Poincare uniformization. The corresponding mapping class group transformations are presented, geodesic functions are constructed, and the Poisson structure is introduced. The resulting Poisson algebras are then quantized. In the particular cases of surfaces with n Z{sub 2}-orbifold points and with one and two holes, the respective algebras A{sub n} and D{sub n} of geodesic functions (classical and quantum) are obtained. The infinite-dimensional Poisson algebra D{sub n}, which is the semiclassical limit of the twisted q-Yangian algebra Y'{sub q}(o{sub n}) for the orthogonal Lie algebra o{sub n}, is associated with the algebra of geodesic functions on an annulus with n Z{sub 2}-orbifold points, and the braid group action on this algebra is found. From this result the braid group actions are constructed on the finite-dimensional reductions of this algebra: the p-level reduction and the algebra D{sub n}. The central elements for these reductions are found. Also, the algebra D{sub n} is interpreted as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple point. Bibliography: 36 titles.
Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions
Biane, P.; Pitman, J.; Yor, M.
This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.
Functional models for commutative systems of linear operators and de Branges spaces on a Riemann surface
Zolotarev, Vladimir A
Functional models are constructed for commutative systems {A 1 ,A 2 } of bounded linear non-self-adjoint operators which do not contain dissipative operators (which means that ξ 1 A 1 +ξ 2 A 2 is not a dissipative operator for any ξ 1 , ξ 2 element of R). A significant role is played here by the de Branges transform and the function classes occurring in this context. Classes of commutative systems of operators {A 1 ,A 2 } for which such a construction is possible are distinguished. Realizations of functional models in special spaces of meromorphic functions on Riemann surfaces are found, which lead to reasonable analogues of de Branges spaces on these Riemann surfaces. It turns out that the functions E(p) and E-tilde(p) determining the order of growth in de Branges spaces on Riemann surfaces coincide with the well-known Baker-Akhiezer functions. Bibliography: 11 titles.
Local Extrema of the $\\Xi(t)$ Function and The Riemann Hypothesis
Kobayashi, Hisashi
In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function $\\Xi(t)=\\xi(\\tfrac{1}{2}+it)$. First, we prove that positivity of all local maxima and negativity of all local minima of $\\Xi(t)$ form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of $\\Xi(t)$ is a saddle point of the function $\\Re\\{\\xi(s)\\}$, we prove that the above properties o...
Quantized Dirac field in curved Riemann--Cartan background. I. Symmetry properties, Green's function
Nieh, H.T.; Yan, M.L.
In the present series of papers, we study the properties of quantized Dirac field in curved Riemann--Cartan space, with particular attention on the role played by torsion. In this paper, we give, in the spirit of the original work of Weyl, a systematic presentation of Dirac's theory in curved Riemann--Cartan space. We discuss symmetry properties of the system, and derive conservation laws as direct consequences of these symmetries. Also discussed is conformal gauge symmetry, with torsion effectively playing the role of a conformal gauge field. To obtain short-distance behavior, we calculate the spinor Green's function, in curved Riemann--Cartan background, using the Schwinger--DeWitt method of proper-time expansion. The calculation corresponds to a generalization of DeWitt's calculation for a Riemannian background
Averages of ratios of the Riemann zeta-function and correlations of divisor sums
Conrey, Brian; Keating, Jonathan P.
Nonlinearity has published articles containing a significant number-theoretic component since the journal was first established. We examine one thread, concerning the statistics of the zeros of the Riemann zeta function. We extend this by establishing a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.
Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function
Saker SamirH
Full Text Available On the hypothesis that the th moments of the Hardy -function are correctly predicted by random matrix theory and the moments of the derivative of are correctly predicted by the derivative of the characteristic polynomials of unitary matrices, we establish new large spaces between the zeros of the Riemann zeta-function by employing some Wirtinger-type inequalities. In particular, it is obtained that which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing.
Fractional parts and their relations to the values of the Riemann zeta function
Alabdulmohsin, Ibrahim
A well-known result, due to Dirichlet and later generalized by de la Vallée–Poussin, expresses a relationship between the sum of fractional parts and the Euler–Mascheroni constant. In this paper, we prove an asymptotic relationship between the summation of the products of fractional parts with powers of integers on the one hand, and the values of the Riemann zeta function, on the other hand. Dirichlet's classical result falls as a particular case of this more general theorem.
Bernhard Riemann
the basis for various fields of mathematics and the general relativity theory of Einstein. In 1857 ... This idea explained the work on algebraic ... theory, Riemann found the key to the problem of the distribution of primes, in that he associated it ...
Deformations of super Riemann surfaces
Ninnemann, H.
Two different approaches to (Konstant-Leites-) super Riemann surfaces are investigated. In the local approach, i.e. glueing open superdomains by superconformal transition functions, deformations of the superconformal structure are discussed. On the other hand, the representation of compact super Riemann surfaces of genus greater than one as a fundamental domain in the Poincare upper half-plane provides a simple description of super Laplace operators acting on automorphic p-forms. Considering purely odd deformations of super Riemann surfaces, the number of linear independent holomorphic sections of arbitrary holomorphic line bundles will be shown to be independent of the odd moduli, leading to a simple proof of the Riemann-Roch theorem for compact super Riemann surfaces. As a further consequence, the explicit connections between determinants of super Laplacians and Selberg's super zeta functions can be determined, allowing to calculate at least the 2-loop contribution to the fermionic string partition function. (orig.)
Ninnemann, H [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik
Two different approaches to (Konstant-Leites-) super Riemann surfaces are investigated. In the local approach, i.e. glueing open superdomains by superconformal transition functions, deformations of the superconformal structure are discussed. On the other hand, the representation of compact super Riemann surfaces of genus greater than one as a fundamental domain in the Poincare upper half-plane provides a simple description of super Laplace operators acting on automorphic p-forms. Considering purely odd deformations of super Riemann surfaces, the number of linear independent holomorphic sections of arbitrary holomorphic line bundles will be shown to be independent of the odd moduli, leading to a simple proof of the Riemann-Roch theorem for compact super Riemann surfaces. As a further consequence, the explicit connections between determinants of super Laplacians and Selberg's super zeta functions can be determined, allowing to calculate at least the 2-loop contribution to the fermionic string partition function. (orig.).
Conformal mapping on Riemann surfaces
Cohn, Harvey
The subject matter loosely called ""Riemann surface theory"" has been the starting point for the development of topology, functional analysis, modern algebra, and any one of a dozen recent branches of mathematics; it is one of the most valuable bodies of knowledge within mathematics for a student to learn.Professor Cohn's lucid and insightful book presents an ideal coverage of the subject in five pans. Part I is a review of complex analysis analytic behavior, the Riemann sphere, geometric constructions, and presents (as a review) a microcosm of the course. The Riemann manifold is introduced in
Fuzzy Riemann surfaces
Arnlind, Joakim; Hofer, Laurent; Hoppe, Jens; Bordemann, Martin; Shimada, Hidehiko
We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C(onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.
Riemann surfaces with boundaries and string theory
Morozov, A.Yu.; Roslyj, A.A.
A consideration of the cutting and joining operations for Riemann surfaces permits one to express the functional integral on a Riemann surface in terms of integrals over its pieces which are suarfaces with boundaries. This yields an expression for the determinant of the Laplacian on a Riemann surface in terms of Krichever maps for its pieces. Possible applications of the methods proposed to a study of the string perturbation theory in terms of an universal moduli space are mentioned
Super Riemann surfaces
Rogers, Alice
A super Riemann surface is a particular kind of (1,1)-dimensional complex analytic supermanifold. From the point of view of super-manifold theory, super Riemann surfaces are interesting because they furnish the simplest examples of what have become known as non-split supermanifolds, that is, supermanifolds where the odd and even parts are genuinely intertwined, as opposed to split supermanifolds which are essentially the exterior bundles of a vector bundle over a conventional manifold. However undoubtedly the main motivation for the study of super Riemann surfaces has been their relevance to the Polyakov quantisation of the spinning string. Some of the papers on super Riemann surfaces are reviewed. Although recent work has shown all super Riemann surfaces are algebraic, some areas of difficulty remain. (author)
Statistical properties of the zeros of zeta functions - beyond the Riemann case
Bogomolny, E.; Leboeuf, P.
The statistical distribution of the zeros of Dirichlet L-functions is investigated both analytically and numerically. Using the Hardy-Littlewood conjecture about the distribution of primes it is shown that the two-point correlation function of these zeros coincides with that for eigenvalues of the Gaussian unitary ensemble of random matrices, and that the distributions of zeros of different L-functions are statistically independent. Applications of these results to Epstein's zeta functions are shortly discussed. (authors) 30 refs., 3 figs., 1 tab
Generalized Riemann zeta-function regularization and Casimir energy for a piecewise uniform string
Li Xinzhou; Shi Xin; Zhang Jianzu.
The generalized zeta-function techniques will be utilized to investigate the Casimir energy for the transverse oscillations of a piecewise uniform closed string. We find that zeta-function regularization method can lead straightforwardly to a correct result. (author). 6 refs
Operator bosonization on Riemann surfaces: new vertex operators
Semikhatov, A.M.
A new formalism is proposed for the construction of an operator theory of generalized ghost systems (bc theories of spin J) on Riemann surfaces (loop diagrams of the theory of closed strings). The operators of the bc system are expressed in terms of operators of the bosonic conformal theory on a Riemann surface. In contrast to the standard bosonization formulas, which have meaning only locally, operator Baker-Akhiezer functions, which are well defined globally on a Riemann surface of arbitrary genus, are introduced. The operator algebra of the Baker-Akhiezer functions generates explicitly the algebraic-geometric Ï" function and correlation functions of bc systems on Riemann surfaces
Riemann, topology, and physics
Monastyrsky, Michael I
This significantly expanded second edition of Riemann, Topology, and Physics combines a fascinating account of the life and work of Bernhard Riemann with a lucid discussion of current interaction between topology and physics. The author, a distinguished mathematical physicist, takes into account his own research at the Riemann archives of Göttingen University and developments over the last decade that connect Riemann with numerous significant ideas and methods reflected throughout contemporary mathematics and physics. Special attention is paid in part one to results on the Riemann–Hilbert problem and, in part two, to discoveries in field theory and condensed matter such as the quantum Hall effect, quasicrystals, membranes with nontrivial topology, "fake" differential structures on 4-dimensional Euclidean space, new invariants of knots and more. In his relatively short lifetime, this great mathematician made outstanding contributions to nearly all branches of mathematics; today Riemann's name appears prom...
Riemann quasi-invariants
Pokhozhaev, Stanislav I
The notion of Riemann quasi-invariants is introduced and their applications to several conservation laws are considered. The case of nonisentropic flow of an ideal polytropic gas is analysed in detail. Sufficient conditions for gradient catastrophes are obtained. Bibliography: 16 titles.
Chiral bosonization on a Riemann surface
Eguchi, Tohru; Ooguri, Hirosi
We point out that the basic addition theorem of θ-functions, Fay's identity, implies an equivalence between bosons and chiral fermions on Riemann surfaces with arbitrary genus. We present a rule for a bosonized calculation of correlation functions. We also discuss ghost systems of n and (1-n) tensors and derive formulas for their chiral determinants. (orig.)
Interpolating and sampling sequences in finite Riemann surfaces
Ortega-Cerda, Joaquim
We provide a description of the interpolating and sampling sequences on a space of holomorphic functions on a finite Riemann surface, where a uniform growth restriction is imposed on the holomorphic functions.
Computational approach to Riemann surfaces
Klein, Christian
This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first...
The concept of a Riemann surface
Weyl, Hermann
This classic on the general history of functions was written by one of the twentieth century's best-known mathematicians. Hermann Weyl, who worked with Einstein at Princeton, combined function theory and geometry in this high-level landmark work, forming a new branch of mathematics and the basis of the modern approach to analysis, geometry, and topology.The author intended this book not only to develop the basic ideas of Riemann's theory of algebraic functions and their integrals but also to examine the related ideas and theorems with an unprecedented degree of rigor. Weyl's two-part treatment
Supermanifolds and super Riemann surfaces
Rabin, J.M.
The theory of super Riemann surfaces is rigorously developed using Rogers' theory of supermanifolds. The global structures of super Teichmueller space and super moduli space are determined. The super modular group is shown to be precisely the ordinary modular group. Super moduli space is shown to be the gauge-fixing slice for the fermionic string path integral
Riemann surfaces, Clifford algebras and infinite dimensional groups
Carey, A.L.; Eastwood, M.G.; Hannabuss, K.C.
We introduce of class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a 'gauge' group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces. (orig.)
On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral
Ostrovsky, Dmitry
Rescaled Mellin-type transforms of the exponential functional of the Bourgade–Kuan–Rodgers statistic of Riemann zeroes are conjecturally related to the distribution of the total mass of the limit lognormal stochastic measure of Mandelbrot–Bacry–Muzy. The conjecture implies that a non-trivial, log-infinitely divisible probability distribution is associated with Riemann zeroes. For application, integral moments, covariance structure, multiscaling spectrum, and asymptotics associated with the exponential functional are computed in closed form using the known meromorphic extension of the Selberg integral. (paper)
Extended Riemann-Liouville type fractional derivative operator with applications
Agarwal P.
Full Text Available The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox's H-function are also presented.
Getting superstring amplitudes by degenerating Riemann surfaces
Matone, Marco; Volpato, Roberto
We explicitly show how the chiral superstring amplitudes can be obtained through factorisation of the higher genus chiral measure induced by suitable degenerations of Riemann surfaces. This powerful tool also allows to derive, at any genera, consistency relations involving the amplitudes and the measure. A key point concerns the choice of the local coordinate at the node on degenerate Riemann surfaces that greatly simplifies the computations. As a first application, starting from recent ansaetze for the chiral measure up to genus five, we compute the chiral two-point function for massless Neveu-Schwarz states at genus two, three and four. For genus higher than three, these computations include some new corrections to the conjectural formulae appeared so far in the literature. After GSO projection, the two-point function vanishes at genus two and three, as expected from space-time supersymmetry arguments, but not at genus four. This suggests that the ansatz for the superstring measure should be corrected for genus higher than four.
The Riemann-Lovelock Curvature Tensor
Kastor, David
In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth-order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k \\le D
A Polyakov action on Riemann surfaces
Zucchini, R.
A calculation of the effective action for induced conformal gravity on higher genus Riemann surfaces is presented. Our expression, generalizing Polyakov's formula, depends holomorphically on the Beltrami and integrates the diffeomorphism anomaly. A solution of the conformal Ward identity on an arbitrary compact Riemann surfaces without boundary is presented, and its remarkable properties are studied. (K.A.) 16 refs., 2 figs
Riemann-Theta Boltzmann Machine arXiv
Krefl, Daniel; Haghighat, Babak; Kahlen, Jens
A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural network, thereby increasing the modelling capacity of the network. Both the Boltzmann machine and the derived feedforward neural network can be successfully trained via standard gradient- and non-gradient-based optimization techniques.
Ice cream and orbifold Riemann-Roch
Buckley, Anita; Reid, Miles; Zhou Shengtian
We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X,D), under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [1] and [2]), although the precise statements are considerably trickier. We expect to return to this in future publications.
Buckley, Anita; Reid, Miles; Zhou, Shengtian
We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X,D), under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of {K3} surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [1] and [2]), although the precise statements are considerably trickier. We expect to return to this in future publications.
In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k ≤ D < 4k. In D = 2k + 1 this identity implies that all solutions of pure kth-order Lovelock gravity are 'Riemann-Lovelock' flat. It is verified that the static, spherically symmetric solutions of these theories, which are missing solid angle spacetimes, indeed satisfy this flatness property. This generalizes results from Einstein gravity in D = 3, which corresponds to the k = 1 case. We speculate about some possible further consequences of Riemann-Lovelock curvature. (paper)
Riemann y los Números Primos
José Manuel Sánchez Muñoz
Full Text Available En el mes de noviembre de 1859, durante la presentación mensual de losinformes de la Academia de Berlín, el alemán Bernhard Riemann presentóun trabajo que cambiaría los designios futuros de la ciencia matemática. El tema central de su informe se centraba en los números primos, presentando el que hoy día, una vez demostrada la Conjetura de Poincaré, puede ser considerado el problema matemático abierto más importante. El presente artículo muestra en su tercera sección una traducción al castellano de dicho trabajo.
Conformal algebra of Riemann surfaces
Vafa, C.
It has become clear over the last few years that 2-dimensional conformal field theories are a crucial ingredient of string theory. Conformal field theories correspond to vacuum solutions of strings; or more precisely we know how to compute string spectrum and scattering amplitudes by starting from a formal theory (with a proper value of central charge of the Virasoro algebra). Certain non-linear sigma models do give rise to conformal theories. A lot of progress has been made in the understanding of conformal theories. The author discusses a different view of conformal theories which was motivated by the development of operator formalism on Riemann surfaces. The author discusses an interesting recent work from this point of view
Conformal scalar fields and chiral splitting on super Riemann surfaces
D'Hoker, E.; Phong, D.H.
We provide a complete description of correlation functions of scalar superfields on a super Riemann surface, taking into account zero modes and non-trivial topology. They are built out of chirally split correlation functions, or conformal blocks at fixed internal momenta. We formulate effective rules which determine these completely in terms of geometric invariants of the super Riemann surface. The chirally split correlation functions have non-trivial monodromy and produce single-valued amplitudes only upon integration over loop momenta. Our discussion covers the even spin structure as well as the odd spin structure case which had been the source of many difficulties in the past. Super analogues of Green's functions, holomorphic spinors, and prime forms emerge which should pave the way to function theory on super Riemann surfaces. In superstring theories, chirally split amplitudes for scalar superfields are crucial in enforcing the GSO projection required for consistency. However one really knew how to carry this out only in the operator formalism to one-loop order. Our results provide a way of enforcing the GSO projection to any loop. (orig.)
Super differential forms on super Riemann surfaces
Konisi, Gaku; Takahasi, Wataru; Saito, Takesi.
Line integral on the super Riemann surface is discussed. A 'super differential operator' which possesses both properties of differential and of differential operator is proposed. With this 'super differential operator' a new theory of differential form on the super Riemann surface is constructed. We call 'the new differentials on the super Riemann surface' 'the super differentials'. As the applications of our theory, the existency theorems of singular 'super differentials' such as 'super abelian differentials of the 3rd kind' and of a super projective connection are examined. (author)
Bosonization in a two-dimensional Riemann Cartan geometry
Denardo, G.; Spallucci, E.
We study the vacuum functional for a Dirac field in a two dimensional Riemann-Cartan geometry. Torsion is treated as a quantum variable while the metric is considered as a classical background field. Decoupling spinors from the non-Riemannian part of the geometry introduces a chiral Jacobian into the vacuum generating functional. We compute this functional Jacobian determinant by means of the Alvarez method. Finally, we show that the effective action for the background geometry is of the Liouville type and does not preserve any memory of the initial torsion field. (author)
Quantum Hall effect on Riemann surfaces
Tejero Prieto, Carlos
We study the family of Landau Hamiltonians compatible with a magnetic field on a Riemann surface S by means of Fourier-Mukai and Nahm transforms. Starting from the geometric formulation of adiabatic charge transport on Riemann surfaces, we prove that Hall conductivity is proportional to the intersection product on the first homology group of S and therefore it is quantized. Finally, by using the theory of determinant bundles developed by Bismut, Gillet and Soul, we compute the adiabatic curvature of the spectral bundles defined by the holomorphic Landau levels. We prove that it is given by the polarization of the jacobian variety of the Riemann surface, plus a term depending on the relative analytic torsion.
Structural stability of solutions to the Riemann problem for a non-strictly hyperbolic system with flux approximation
Meina Sun
Full Text Available We study the Riemann problem for a non-strictly hyperbolic system of conservation laws under the linear approximations of flux functions with three parameters. The approximated system also belongs to the type of triangular systems of conservation laws and this approximation does not change the structure of Riemann solutions to the original system. Furthermore, it is proven that the Riemann solutions to the approximated system converge to the corresponding ones to the original system as the perturbation parameter tends to zero.
Gaussian curvature on hyperelliptic Riemann surfaces
Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 2, May 2014, pp. 155–167. c Indian Academy of Sciences. Gaussian curvature on hyperelliptic Riemann surfaces. ABEL CASTORENA. Centro de Ciencias Matemáticas (Universidad Nacional Autónoma de México,. Campus Morelia) Apdo. Postal 61-3 Xangari, C.P. 58089 Morelia,.
Conformal deformation of Riemann space and torsion
Pyzh, V.M.
Method for investigating conformal deformations of Riemann spaces using torsion tensor, which permits to reduce the second ' order equations for Killing vectors to the system of the first order equations, is presented. The method is illustrated using conformal deformations of dimer sphere as an example. A possibility of its use when studying more complex deformations is discussed [ru
Study Paths, Riemann Surfaces, and Strebel Differentials
Buser, Peter; Semmler, Klaus-Dieter
These pages aim to explain and interpret why the late Mika Seppälä, a conformal geometer, proposed to model student study behaviour using concepts from conformal geometry, such as Riemann surfaces and Strebel differentials. Over many years Mika Seppälä taught online calculus courses to students at Florida State University in the United States, as…
Hysteresis rarefaction in the Riemann problem
Krej�í, Pavel
Ro�. 138, - (2008), s. 1-10 ISSN 1742-6588. [International Workshop on Multi-Rate Processes and Hysteresis. Cork , 31.03.2008-05.04.2008] Institutional research plan: CEZ:AV0Z10190503 Keywords : Preisach hysteresis * Riemann problem Subject RIV: BA - General Mathematics http://iopscience.iop.org/1742-6596/138/1/012010
Generalized Riemann problem for reactive flows
Ben-Artzi, M.
A generalized Riemann problem is introduced for the equations of reactive non-viscous compressible flow in one space dimension. Initial data are assumed to be linearly distributed on both sides of a jump discontinuity. The resolution of the singularity is studied and the first-order variation (in time) of flow variables is given in exact form. copyright 1989 Academic Press, Inc
Explicit solution of Riemann-Hilbert problems for the Ernst equation
Klein, C.; Richter, O.
Riemann-Hilbert problems are an important solution technique for completely integrable differential equations. They are used to introduce a free function in the solutions which can be used at least in principle to solve initial or boundary value problems. But even if the initial or boundary data can be translated into a Riemann-Hilbert problem, it is in general impossible to obtain explicit solutions. In the case of the Ernst equation, however, this is possible for a large class because the matrix problem can be shown to be gauge equivalent to a scalar one on a hyperelliptic Riemann surface that can be solved in terms of theta functions. As an example we discuss the rigidly rotating dust disk.
SO(N) WZNW models on higher-genus Riemann surfaces
Alimohammadi, M.; Arfaei, H.; Bonn Univ.
With the help of the string functions and fusion rules of SO(2N) 1 , we show that the results on SU(N) 1 correlators on higher-genus Riemann surfaces (HGRS) can be extended to the SO(2N) 1 and other level-one simply-laced WZNW models. Using modular invariance and factorization properties of Green functions we find multipoint correlators of primary and descendant fields of SO(2N+1) 1 WZNW models on higher genus Riemann surfaces. (orig.)
Reassessing Riemann's paper on the number of primes less than a given magnitude
Dittrich, Walter
In this book, the author pays tribute to Bernhard Riemann (1826–1866), mathematician with revolutionary ideas, whose work on the theory of integration, the Fourier transform, the hypergeometric differential equation, etc. contributed immensely to mathematical physics. This book concentrates in particular on Riemann's only work on prime numbers, including such then new ideas as analytical continuation in the complex plane and the product formula for entire functions. A detailed analysis of the zeros of the Riemann zeta function is presented. The impact of Riemann's ideas on regularizing infinite values in field theory is also emphasized.
On Lovelock analogs of the Riemann tensor
Camanho, Xián O.; Dadhich, Naresh
It is possible to define an analog of the Riemann tensor for Nth order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analog of the Einstein tensor. Interestingly there exist two parallel but distinct such analogs and the main purpose of this note is to reconcile both formulations. In addition we will introduce a simple tensor identity and use it to show that any pure Lovelock vacuum in odd d=2N+1 dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock-Riemann tensor. Further, in the presence of cosmological constant it is the Lovelock-Weyl tensor that vanishes.
Poisson sigma model with branes and hyperelliptic Riemann surfaces
Ferrario, Andrea
We derive the explicit form of the superpropagators in the presence of general boundary conditions (coisotropic branes) for the Poisson sigma model. This generalizes the results presented by Cattaneo and Felder [''A path integral approach to the Kontsevich quantization formula,'' Commun. Math. Phys. 212, 591 (2000)] and Cattaneo and Felder ['Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model', Lett. Math. Phys. 69, 157 (2004)] for Kontsevich's angle function [Kontsevich, M., 'Deformation quantization of Poisson manifolds I', e-print arXiv:hep.th/0101170] used in the deformation quantization program of Poisson manifolds. The relevant superpropagators for n branes are defined as gauge fixed homotopy operators of a complex of differential forms on n sided polygons P n with particular ''alternating'' boundary conditions. In the presence of more than three branes we use first order Riemann theta functions with odd singular characteristics on the Jacobian variety of a hyperelliptic Riemann surface (canonical setting). In genus g the superpropagators present g zero mode contributions
Fractal supersymmetric QM, Geometric Probability and the Riemann Hypothesis
Castro, C
The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $ s_n =1/2+i\\lambda_n $. Earlier work on the RH based on supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric probability theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this geometric probabilistic framework to compute the probability if the RH is true, we apply it directly to the the hyperbolic sine function $ \\sinh (s) $ case which obeys a trivial analog of the RH (the HSRH). Its zeros are equally spaced in the imaginary axis $ s_n = 0 + i n \\pi $. The geometric probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymme...
Large chiral diffeomorphisms on Riemann surfaces and W-algebras
Bandelloni, G.; Lazzarini, S.
The diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a Becchi-Rouet-Stora (BRS) formulation (for a given order of truncation) leading to a more algebraic setup. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the so-called W-algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between Korteweg-de Vries (KdV) flows and W-diffeomorphims
The KZB equations on Riemann surfaces
Felder, Giovanni
In this paper, based on the author's lectures at the 1995 les Houches Summer school, explicit expressions for the Friedan--Shenker connection on the vector bundle of WZW conformal blocks on the moduli space of curves with tangent vectors at $n$ marked points are given. The covariant derivatives are expressed in terms of ``dynamical $r$-matrices'', a notion borrowed from integrable systems. The case of marked points moving on a fixed Riemann surface is studied more closely. We prove a universa...
Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian
Bender, Carl M.; Brody, Dorje C.
The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction.
Jacob's ladders, Riemann's oscillators, quotient of two oscillating multiforms and set of metamorphoses of this system
Moser, Jan
In this paper we introduce complicated oscillating system, namely quotient of two multiforms based on Riemann-Siegel formula. We prove that there is an infinite set of metamorphoses of this system (=chrysalis) on critical line $\\sigma=\\frac 12$ into a butterfly (=infinite series of M\\" obius functions in the region of absolute convergence $\\sigma>1$).
Post-Quantum Cryptography: Riemann Primitives and Chrysalis
Malloy, Ian; Hollenbeck, Dennis
The Chrysalis project is a proposed method for post-quantum cryptography using the Riemann sphere. To this end, Riemann primitives are introduced in addition to a novel implementation of this new method. Chrysalis itself is the first cryptographic scheme to rely on Holomorphic Learning with Errors, which is a complex form of Learning with Errors relying on the Gauss Circle Problem within the Riemann sphere. The principle security reduction proposed by this novel cryptographic scheme applies c...
Inverse Scattering, the Coupling Constant Spectrum, and the Riemann Hypothesis
Khuri, N. N.
It is well known that the s-wave Jost function for a potential, λV, is an entire function of λ with an infinite number of zeros extending to infinity. For a repulsive V, and at zero energy, these zeros of the 'coupling constant', λ, will all be real and negative, λ n (0) n n =1/2+iγ n . Thus, finding a repulsive V whose coupling constant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but this will be a very difficult and unguided search.In this paper we make a significant enlargement of the class of potentials needed for a generalization of the above idea. We also make this new class amenable to construction via inverse scattering methods. We show that all one needs is a one parameter class of potentials, U(s;x), which are analytic in the strip, 0≤Res≤1, Ims>T 0 , and in addition have an asymptotic expansion in powers of [s(s-1)] -1 , i.e. U(s;x)=V 0 (x)+gV 1 (x)+g 2 V 2 (x)+...+O(g N ), with g=[s(s-1)] -1 . The potentials V n (x) are real and summable. Under suitable conditions on the V n 's and the O(g N ) term we show that the condition, ∫ 0 ∞ vertical bar f 0 (x) vertical bar 2 V 1 (x) dx≠0, where f 0 is the zero energy and g=0 Jost function for U, is sufficient to guarantee that the zeros g n are real and, hence, s n =1/2+iγ n , for γ n ≥T 0 .Starting with a judiciously chosen Jost function, M(s,k), which is constructed such that M(s,0) is Riemann's ξ(s) function, we have used inverse scattering methods to actually construct a U(s;x) with the above properties. By necessity, we had to generalize inverse methods to deal with complex potentials and a nonunitary S-matrix. This we have done at least for the special cases under consideration.For our specific example, ∫ 0 ∞ vertical bar f 0 (x) vertical bar 2 V 1 (x) dx=0 and, hence, we get no restriction on Img n or Res n . The reasons for the vanishing of the above integral are given, and they give us hints on what one needs to proceed further. The problem
Riemann monodromy problem and conformal field theories
Blok, B.
A systematic analysis of the use of the Riemann monodromy problem for determining correlators (conformal blocks) on the sphere is presented. The monodromy data is constructed in terms of the braid matrices and gives a constraint on the noninteger part of the conformal dimensions of the primary fields. To determine the conformal blocks we need to know the order of singularities. We establish a criterion which tells us when the knowledge of the conformal dimensions of primary fields suffice to determine the blocks. When zero modes of the extended algebra are present the analysis is more difficult. In this case we give a conjecture that works for the SU(2) WZW case. (orig.)
The Euler–Riemann gases, and partition identities
Chair, Noureddine
The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice (s-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order s of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions, the Euler generating function is identified with the inverse of the graded parafermionic partition function of order 2. As a result we obtain new sequences of partitions of integers with given restrictions. If the parity of the order s is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of s is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function θ 4 , and generate sequences in partition theory. Our partition functions coincide with the overpartitions of Corteel and Lovejoy, and jagged partitions in conformal field theory. Also, the partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore–Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi theta function is (ζ(2t))/(ζ(t) 2 ) . Finally, we propose two formulas which brings
Quantum field theory on higher-genus Riemann surfaces
Kubo, Reijiro; Yoshii, Hisahiro; Ojima, Shuichi; Paul, S.K.
Quantum field theory for b-c systems is formulated on Riemann surfaces with arbitrary genus. We make use of the formalism recently developed by Krichever and Novikov. Hamiltonian is defined properly, and the Ward-Takahashi identities are derived on higher-genus Riemann surfaces. (author)
Non-abelian bosonization in higher genus Riemann surfaces
Koh, I.G.; Yu, M.
We propose a generalization of the character formulas of the SU(2) Kac-Moody algebra to higher genus Riemann surfaces. With this construction, we show that the modular invariant partition funciton of the SO(4) k = 1 Wess-Zumino model is equivalent, in arbitrary genus Riemann surfaces, to that of free fermion theory. (orig.)
Collisionless analogs of Riemann S ellipsoids with halo
Abramyan, M.G.
A spheroidal halo ensures equilibrium of the collisionless analogs of the Riemann S ellipsoids with oscillations of the particles along the direction of their rotation. Sequences of collisionless triaxial ellipsoids begin and end with dynamically stable members of collisionless embedded spheroids. Both liquid and collisionless Riemann S ellipsoids with weak halo have properties that resemble those of bars of SB galaxies
A Riemann problem with small viscosity and dispersion
Kayyunnapara Thomas Joseph
Full Text Available In this paper we prove existence of global solutions to a hyperbolic system in elastodynamics, with small viscosity and dispersion terms and derive estimates uniform in the viscosity-dispersion parameters. By passing to the limit, we prove the existence of solution the Riemann problem for the hyperbolic system with arbitrary Riemann data.
E-string theory on Riemann surfaces
Kim, Hee-Cheol; Vafa, Cumrun [Jefferson Physical Laboratory, Harvard University, Cambridge, MA (United States); Razamat, Shlomo S. [Physics Department, Technion, Haifa (Israel); Zafrir, Gabi [Kavli IPMU (WPI), UTIAS, the University of Tokyo, Kashiwa, Chiba (Japan)
We study compactifications of the 6d E-string theory, the theory of a small E{sub 8} instanton, to four dimensions. In particular we identify N = 1 field theories in four dimensions corresponding to compactifications on arbitrary Riemann surfaces with punctures and with arbitrary non-abelian flat connections as well as fluxes for the abelian sub-groups of the E{sub 8} flavor symmetry. This sheds light on emergent symmetries in a number of 4d N = 1 SCFTs (including the 'E7 surprise' theory) as well as leads to new predictions for a large number of 4-dimensional exceptional dualities and symmetries. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Moduli of Riemann surfaces, transcendental aspects
Hain, R.
These notes are an informal introduction to moduli spaces of compact Riemann surfaces via complex analysis, topology and Hodge Theory. The prerequisites for the first lecture are just basic complex variables, basic Riemann surface theory up to at least the Riemann-Roch formula, and some algebraic topology, especially covering space theory. The first lecture covers moduli in genus 0 and genus 1 as these can be understood using relatively elementary methods, but illustrate many of the points which arise in higher genus. The notes cover more material than was covered in the lectures, and sometimes the order of topics in the notes differs from that in the lectures. We have seen in genus 1 case that M 1 is the quotient Γ 1 /X 1 of a contractible complex manifold X 1 = H by a discrete group Γ 1 = SL 2 (Z). The action of Γ 1 on X 1 is said to be virtually free - that is, Γ 1 has a finite index subgroup which acts (fixed point) freely on X 1 . In this section we will generalize this to all g >= 1 - we will sketch a proof that there is a contractible complex manifold Xg, called Teichmueller space, and a group Γ g , called the mapping class group, which acts virtually freely on X g . The moduli space of genus g compact Riemann surfaces is the quotient: M g = Γ g /X g . This will imply that M g has the structure of a complex analytic variety with finite quotient singularities. Teichmueller theory is a difficult and technical subject. Because of this, it is only possible to give an overview. In this lecture, we compute the orbifold Picard group of M g for all g >= 1. Recall that an orbifold line bundle over M g is a holomorphic line bundle L over Teichmueller space X g together with an action of the mapping class group Γ g on it such that the projection L → X g is Γ g -equivariant. An orbifold section of this line bundle is a holomorphic Γ g -equivariant section X g → L of L. This is easily seen to be equivalent to fixing a level l>= 3 and considering holomorphic
Transformation optics with artificial Riemann sheets
Xu, Lin; Chen, Huanyang
The two original versions of 'invisibility' cloaks (Leonhardt 2006 Science 312 1777-80 and Pendry et al 2006 Science 312 1780-2) show perfect cloaking but require unphysical singularities in material properties. A non-Euclidean version of cloaking (Leonhardt 2009 Science 323 110-12) was later presented to address these problems, using a very complicated non-Euclidean geometry. In this work, we combine the two original approaches to transformation optics into a more general concept: transformation optics with artificial Riemann sheets. Our method is straightforward and can be utilized to design new kinds of cloaks that can work not only in the realm of geometric optics but also using wave optics. The physics behind this design is similar to that of the conformal cloak for waves. The resonances in the interior region make the phase delay disappear and induce the cloaking effect. Numerical simulations confirm our theoretical results.
The two original versions of 'invisibility' cloaks (Leonhardt 2006 Science 312 1777–80 and Pendry et al 2006 Science 312 1780–2) show perfect cloaking but require unphysical singularities in material properties. A non-Euclidean version of cloaking (Leonhardt 2009 Science 323 110–12) was later presented to address these problems, using a very complicated non-Euclidean geometry. In this work, we combine the two original approaches to transformation optics into a more general concept: transformation optics with artificial Riemann sheets. Our method is straightforward and can be utilized to design new kinds of cloaks that can work not only in the realm of geometric optics but also using wave optics. The physics behind this design is similar to that of the conformal cloak for waves. The resonances in the interior region make the phase delay disappear and induce the cloaking effect. Numerical simulations confirm our theoretical results. (paper)
From Riemann to differential geometry and relativity
Papadopoulos, Athanase; Yamada, Sumio
This book explores the work of Bernhard Riemann and its impact on mathematics, philosophy and physics. It features contributions from a range of fields, historical expositions, and selected research articles that were motivated by Riemann's ideas and demonstrate their timelessness. The editors are convinced of the tremendous value of going into Riemann's work in depth, investigating his original ideas, integrating them into a broader perspective, and establishing ties with modern science and philosophy. Accordingly, the contributors to this volume are mathematicians, physicists, philosophers and historians of science. The book offers a unique resource for students and researchers in the fields of mathematics, physics and philosophy, historians of science, and more generally to a wide range of readers interested in the history of ideas.
From Euclidean to Minkowski space with the Cauchy-Riemann equations
Gimeno-Segovia, Mercedes; Llanes-Estrada, Felipe J.
We present an elementary method to obtain Green's functions in non-perturbative quantum field theory in Minkowski space from Green's functions calculated in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes often is unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore, we suggest to use the Cauchy-Riemann equations, which perform the analytical continuation without assuming global information on the function in the entire complex plane, but only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge quantum chromodynamics, which is known from lattice and Dyson-Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy-Riemann equations against high-frequency noise,which makes it difficult to achieve good accuracy. We also point out a few curious details related to the Wick rotation. (orig.)
Riemann zeros and phase transitions via the spectral operator on fractal strings
Herichi, Hafedh; Lapidus, Michel L
The spectral operator was introduced by Lapidus and van Frankenhuijsen (2006 Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings) in their reinterpretation of the earlier work of Lapidus and Maier (1995 J. Lond. Math. Soc. 52 15–34) on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this review, we present the rigorous functional analytic framework given by Herichi and Lapidus (2012) and within which to study the spectral operator. Furthermore, we give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is quasi-invertible (or equivalently, that its truncations are invertible) if and only if the Riemann zeta function ζ(s) does not have any zeros on the vertical line Re(s) = c. Hence, it is not invertible in the mid-fractal case when c= 1/2 , and it is quasi-invertible everywhere else (i.e. for all c ∈ (0, 1) with c≠1/2 ) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c= 1/2 and c = 1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. (review)
Riemann-Hilbert treatment of Liouville theory on the torus: the general case
Menotti, Pietro
We extend the previous treatment of Liouville theory on the torus to the general case in which the distribution of charges is not necessarily symmetric. This requires the concept of Fuchsian differential equation on Riemann surfaces. We show through a group theoretic argument that the Heun parameter and a weight constant are sufficient to satisfy all monodromy conditions. We then apply the technique of differential equations on a Riemann surface to the two-point function on the torus in which one source is arbitrary and the other small. As a byproduct, we give in terms of quadratures the exact Green function on the square and on the rhombus with opening angle 2Ï€/6 in the background of the field generated by an arbitrary charge.
A Modified Groundwater Flow Model Using the Space Time Riemann-Liouville Fractional Derivatives Approximation
Abdon Atangana
Full Text Available The notion of uncertainty in groundwater hydrology is of great importance as it is known to result in misleading output when neglected or not properly accounted for. In this paper we examine this effect in groundwater flow models. To achieve this, we first introduce the uncertainties functions u as function of time and space. The function u accounts for the lack of knowledge or variability of the geological formations in which flow occur (aquifer in time and space. We next make use of Riemann-Liouville fractional derivatives that were introduced by Kobelev and Romano in 2000 and its approximation to modify the standard version of groundwater flow equation. Some properties of the modified Riemann-Liouville fractional derivative approximation are presented. The classical model for groundwater flow, in the case of density-independent flow in a uniform homogeneous aquifer is reformulated by replacing the classical derivative by the Riemann-Liouville fractional derivatives approximations. The modified equation is solved via the technique of green function and the variational iteration method.
The Differential-Algebraic Analysis of Symplectic and Lax Structures Related with New Riemann-Type Hydrodynamic Systems
Prykarpatsky, Yarema A.; Artemovych, Orest D.; Pavlov, Maxim V.; Prykarpatski, Anatolij K.
A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic hierarchy, proposed recently by O. D. Artemovych, M. V. Pavlov, Z. Popowicz and A. K. Prykarpatski, is developed. In addition to the Lax-type representation, found before by Z. Popowicz, a closely related representation is constructed in exact form by means of a new differential-functional technique. The bi-Hamiltonian integrability and compatible Poisson structures of the generalized Riemann type hierarchy are analyzed by means of the symplectic and gradient-holonomic methods. An application of the devised differential-algebraic approach to other Riemann and Vakhnenko type hydrodynamic systems is presented.
On Riemann boundary value problems for null solutions of the two dimensional Helmholtz equation
Bory Reyes, Juan; Abreu Blaya, Ricardo; Rodríguez Dagnino, Ramón Martin; Kats, Boris Aleksandrovich
The Riemann boundary value problem (RBVP to shorten notation) in the complex plane, for different classes of functions and curves, is still widely used in mathematical physics and engineering. For instance, in elasticity theory, hydro and aerodynamics, shell theory, quantum mechanics, theory of orthogonal polynomials, and so on. In this paper, we present an appropriate hyperholomorphic approach to the RBVP associated to the two dimensional Helmholtz equation in R^2 . Our analysis is based on a suitable operator calculus.
Fermions on a Riemann surface and the Kadomtsev-Petviashvili equation
Zabrodin, A.V.
It is shown that the S matrix of free massless fermions on a Riemann surface of finite genus generates quasiperiodic solutions of the Kadomtsev-Petviashvili equation. An operator that changes the genus of a solution is constructed, and the law of composition of such operators is discussed. The construction is a generalization of the well-known operator approach in the case of soliton solutions to the general case of quasiperiodic Ï" functions
Two-dimensional time dependent Riemann solvers for neutron transport
Brunner, Thomas A.; Holloway, James Paul
A two-dimensional Riemann solver is developed for the spherical harmonics approximation to the time dependent neutron transport equation. The eigenstructure of the resulting equations is explored, giving insight into both the spherical harmonics approximation and the Riemann solver. The classic Roe-type Riemann solver used here was developed for one-dimensional problems, but can be used in multidimensional problems by treating each face of a two-dimensional computation cell in a locally one-dimensional way. Several test problems are used to explore the capabilities of both the Riemann solver and the spherical harmonics approximation. The numerical solution for a simple line source problem is compared to the analytic solution to both the P 1 equation and the full transport solution. A lattice problem is used to test the method on a more challenging problem
The exchange algebra for Liouville theory on punctured Riemann sphere
Shen Jianmin; Sheng Zhengmao
We consider in this paper the classical Liouville field theory on the Riemann sphere with n punctures. In terms of the uniformization theorem of Riemann surface, we show explicitly the classical exchange algebra (CEA) for the chiral components of the Liouville fields. We find that the matrice which dominate the CEA is related to the symmetry of the Lie group SL(n) in a nontrivial manner with n>3. (author). 10 refs
Bernhard Riemann 1826-1866 Turning Points in the Conception of Mathematics
Laugwitz, Detlef
The name of Bernard Riemann is well known to mathematicians and physicists around the world. College students encounter the Riemann integral early in their studies. Real and complex function theories are founded on Riemann's work. Einstein's theory of gravitation would be unthinkable without Riemannian geometry. In number theory, Riemann's famous conjecture stands as one of the classic challenges to the best mathematical minds and continues to stimulate deep mathematical research. The name is indelibly stamped on the literature of mathematics and physics. This book, originally written in German and presented here in an English-language translation, examines Riemann's scientific work from a single unifying perspective. Laugwitz describes Riemann's development of a conceptual approach to mathematics at a time when conventional algorithmic thinking dictated that formulas and figures, rigid constructs, and transformations of terms were the only legitimate means of studying mathematical objects. David Hi...
Robinson manifolds and Cauchy-Riemann spaces
Trautman, A
A Robinson manifold is defined as a Lorentz manifold (M, g) of dimension 2n >= 4 with a bundle N subset of C centre dot TM such that the fibres of N are maximal totally null and there holds the integrability condition [Sec N, Sec N] subset of Sec N. The real part of N intersection N-bar is a bundle of null directions tangent to a congruence of null geodesics. This generalizes the notion of a shear-free congruence of null geodesics (SNG) in dimension 4. Under a natural regularity assumption, the set M of all these geodesics has the structure of a Cauchy-Riemann manifold of dimension 2n - 1. Conversely, every such CR manifold lifts to many Robinson manifolds. Three definitions of a CR manifold are described here in considerable detail; they are equivalent under the assumption of real analyticity, but not in the smooth category. The distinctions between these definitions have a bearing on the validity of the Robinson theorem on the existence of null Maxwell fields associated with SNGs. This paper is largely a re...
Multidimensional Riemann problem with self-similar internal structure - part III - a multidimensional analogue of the HLLI Riemann solver for conservative hyperbolic systems
Balsara, Dinshaw S.; Nkonga, Boniface
Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The fastest way of endowing such sub-structure consists of making a multidimensional extension of the HLLI Riemann solver for hyperbolic conservation laws. Presenting such a multidimensional analogue of the HLLI Riemann solver with linear sub-structure for use on structured meshes is the goal of this work. The multidimensional MuSIC Riemann solver documented here is universal in the sense that it can be applied to any hyperbolic conservation law. The multidimensional Riemann solver is made to be consistent with constraints that emerge naturally from the Galerkin projection of the self-similar states within the wave model. When the full eigenstructure in both directions is used in the present Riemann solver, it becomes a complete Riemann solver in a multidimensional sense. I.e., all the intermediate waves are represented in the multidimensional wave model. The work also presents, for the very first time, an important analysis of the dissipation characteristics of multidimensional Riemann solvers. The present Riemann solver results in the most efficient implementation of a multidimensional Riemann solver with sub-structure. Because it preserves stationary linearly degenerate waves, it might also help with well-balancing. Implementation-related details are presented in pointwise fashion for the one-dimensional HLLI Riemann solver as well as the multidimensional MuSIC Riemann solver.
Solution of Riemann problem for ideal polytropic dusty gas
Nath, Triloki; Gupta, R.K.; Singh, L.P.
Highlights : • A direct approach is used to solve the Riemann problem for dusty ideal polytropic gas. • An analytical solution to the Riemann problem for dusty gas flow is obtained. • The existence and uniqueness of the solution in dusty gas is discussed. • Properties of elementary wave solutions of Riemann problem are discussed. • Effect of mass fraction of solid particles on the solution is presented. - Abstract: The Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady flow of an ideal polytropic gas with dust particles is solved analytically without any restriction on magnitude of the initial states. The elementary wave solutions of the Riemann problem, that is shock waves, rarefaction waves and contact discontinuities are derived explicitly and their properties are discussed, for a dusty gas. The existence and uniqueness of the solution for Riemann problem in dusty gas is discussed. Also the conditions leading to the existence of shock waves or simple waves for a 1-family and 3-family curves in the solution of the Riemann problem are discussed. It is observed that the presence of dust particles in an ideal polytropic gas leads to more complex expression as compared to the corresponding ideal case; however all the parallel results remain same. Also, the effect of variation of mass fraction of dust particles with fixed volume fraction (Z) and the ratio of specific heat of the solid particles and the specific heat of the gas at constant pressure on the variation of velocity and density across the shock wave, rarefaction wave and contact discontinuities are discussed.
Conformal fields. From Riemann surfaces to integrable hierarchies
I discuss the idea of translating ingredients of conformal field theory into the language of hierarchies of integrable differential equations. Primary conformal fields are mapped into (differential or matrix) operators living on the phase space of the hierarchy, whereas operator insertions of, e.g., a current or the energy-momentum tensor, become certain vector fields on the phase space and thus acquire a meaning independent of a given Riemann surface. A number of similarities are observed between the structures arising on the hierarchy and those of the theory on the world-sheet. In particular, there is an analogue of the operator product algebra with the Cauchy kernel replaced by its 'off-shell' hierarchy version. Also, hierarchy analogues of certain operator insertions admit two (equivalent, but distinct) forms, resembling the 'bosonized' and 'fermionized' versions respectively. As an application, I obtain a useful reformulation of the Virasoro constraints of the type that arise in matrix models, as a system of equations on dressing (or Lax) operators (rather than correlation functions, i.e., residues or traces). This also suggests an interpretation in terms of a 2D topological field theory, which might be extended to a correspondence between Virasoro-constrained hierarchies and topological theories. (orig.)
Riemann solvers and undercompressive shocks of convex FPU chains
Herrmann, Michael; Rademacher, Jens D M
We consider FPU-type atomic chains with general convex potentials. The naive continuum limit in the hyperbolic space–time scaling is the p-system of mass and momentum conservation. We systematically compare Riemann solutions to the p-system with numerical solutions to discrete Riemann problems in FPU chains, and argue that the latter can be described by modified p-system Riemann solvers. We allow the flux to have a turning point, and observe a third type of elementary wave (conservative shocks) in the atomistic simulations. These waves are heteroclinic travelling waves and correspond to non-classical, undercompressive shocks of the p-system. We analyse such shocks for fluxes with one or more turning points. Depending on the convexity properties of the flux we propose FPU-Riemann solvers. Our numerical simulations confirm that Lax shocks are replaced by so-called dispersive shocks. For convex–concave flux we provide numerical evidence that convex FPU chains follow the p-system in generating conservative shocks that are supersonic. For concave–convex flux, however, the conservative shocks of the p-system are subsonic and do not appear in FPU-Riemann solutions
The Riemann problem for the relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation
Shao, Zhiqiang
The relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation is studied. The Riemann problem is solved constructively. The delta shock wave arises in the Riemann solutions, provided that the initial data satisfy some certain conditions, although the system is strictly hyperbolic and the first and third characteristic fields are genuinely nonlinear, while the second one is linearly degenerate. There are five kinds of Riemann solutions, in which four only consist of a shock wave and a centered rarefaction wave or two shock waves or two centered rarefaction waves, and a contact discontinuity between the constant states (precisely speaking, the solutions consist in general of three waves), and the other involves delta shocks on which both the rest mass density and the proper energy density simultaneously contain the Dirac delta function. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine-Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of solutions involving delta shocks for the Riemann problem.
Ernst Equation and Riemann Surfaces: Analytical and Numerical Methods
Ernst, Frederick J
metric tensor components. The first two chapters of this book are devoted to some basic ideas: in the introductory chapter 1 the authors discuss the concept of integrability, comparing the integrability of the vacuum Ernst equation with the integrability of nonlinear equations of Korteweg-de Vries (KdV) type, while in chapter 2 they describe various circumstances in which the vacuum Ernst equation has been determined to be relevant, not only in connection with gravitation but also, for example, in the construction of solutions of the self-dual Yang-Mills equations. It is also in this chapter that one of several equivalent linear systems for the Ernst equation is described. The next two chapters are devoted to Dmitry Korotkin's concept of algebro-geometric solutions of a linear system: in chapter 3 the structure of such solutions of the vacuum Ernst equation, which involve Riemann theta functions of hyperelliptic algebraic curves of any genus, is contrasted with the periodic structure of such solutions of the KdV equation. How such solutions can be obtained, for example, by solving a matrix Riemann-Hilbert problem and how the metric tensor of the associated spacetime can be evaluated is described in detail. In chapter 4 the asymptotic behaviour and the similarity structure of the general algebro-geometric solutions of the Ernst equation are described, and the relationship of such solutions to the perhaps more familiar multi-soliton solutions is discussed. The next three chapters are based upon the authors' own published research: in chapter 5 it is shown that a problem involving counter-rotating infinitely thin disks of matter can be solved in terms of genus two Riemann theta functions, while in chapter 6 the authors describe numerical methods that facilitate the construction of such solutions, and in chapter 7 three-dimensional graphs are displayed that depict all metrical fields of the associated spacetime. Finally, in chapter 8, the difficulties associated with
BRST quantization of superconformal theories on higher genus Riemann surfaces
Leman Kuang
A complex contour integral method is constructed and applied to the Becchi-Rouet-Stora-Tyutin (BRST) quantization procedure of string theories on higher genus Riemann surfaces with N=0 and 1 Krichever-Novikov (KN) algebras. This method makes calculations very simple. It is shown that the critical spacetime dimension of the string theories on a genus-g Riemann surface equals that of the string theories on a genus-zero Riemann surface, and that the 'Regge intercepts' in the genus-g case are α(g)=1-3/4g-9/8g 2 and 1/2-3/4g-17/16g 2 for bosonic strings and superstrings, respectively. (orig.)
Compact Riemann surfaces an introduction to contemporary mathematics
Jost, Jürgen
Although Riemann surfaces are a time-honoured field, this book is novel in its broad perspective that systematically explores the connection with other fields of mathematics. It can serve as an introduction to contemporary mathematics as a whole as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. It is unique among textbooks on Riemann surfaces in including an introduction to Teichmüller theory. For this new edition, the author has expanded and rewritten several sections to include additional material and to improve the presentation.
Line operators from M-branes on compact Riemann surfaces
Amariti, Antonio [Physics Department, The City College of the CUNY, 160 Convent Avenue, New York, NY 10031 (United States); Orlando, Domenico [Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern (Switzerland); Reffert, Susanne, E-mail: [email protected] [Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern (Switzerland)
In this paper, we determine the charge lattice of mutually local Wilson and 't Hooft line operators for class S theories living on M5-branes wrapped on compact Riemann surfaces. The main ingredients of our analysis are the fundamental group of the N-cover of the Riemann surface, and a quantum constraint on the six-dimensional theory. The latter plays a central role in excluding some of the possible lattices and imposing consistency conditions on the charges. This construction gives a geometric explanation for the mutual locality among the lines, fixing their charge lattice and the structure of the four-dimensional gauge group.
Riemann-Hilbert approach to the time-dependent generalized sine kernel
Kozlowski, K.K.
We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance dependent correlation functions of integrable models described by a six-vertex R-matrix. This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point. Our method builds on a Riemann-Hilbert based analysis. (orig.)
Integrability of Liouville system on high genus Riemann surface: Pt. 1
Chen Yixin; Gao Hongbo
By using the theory of uniformization of Riemann-surfaces, we study properties of the Liouville equation and its general solution on a Riemann surface of genus g>1. After obtaining Hamiltonian formalism in terms of free fields and calculating classical exchange matrices, we prove the classical integrability of Liouville system on high genus Riemann surface
On an isospectrality question over compact Riemann surfaces
Srinivas Rau, S.
It is proved that for a generic compact Riemann surface X of genus g>1,(i) there are at most 2 2g unitary characters of π 1 (X) whose associated line bundles have laplacians of identical spectrum, (ii) generating cycles for π 1 (X) can be chosen to be closed geodesics whose length multiplicity is 1. (author). 5 refs
Quantum Riemann surfaces. Pt. 2; The discrete series
Klimek, S. (Dept. of Mathematics, IUPUI, Indianapolis, IN (United States)); Lesniewski, A. (Dept. of Physics, Harvard Univ., Cambridge, MA (United States))
We continue our study of noncommutative deformations of two-dimensional hyperbolic manifolds which we initiated in Part I. We construct a sequence of C{sup *}-algebras which are quantizations of a compact Riemann surface of genus g corresponding to special values of the Planck constant. These algebras are direct integrals of finite-dimensional C{sup *}-algebras. (orig.).
Colliding holes in Riemann surfaces and quantum cluster algebras
Chekhov, Leonid; Mazzocco, Marta
In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appears when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesic arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as λ-lengths in Thurston-Penner terminology. We compute the Goldman bracket explicitly in terms of these λ-lengths and show that the groupoid of flip morphisms acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.
Weyl transforms associated with the Riemann-Liouville operator
N. B. Hamadi
Full Text Available For the Riemann-Liouville transform ℛα, α∈�+, associated with singular partial differential operators, we define and study the Weyl transforms Wσ connected with ℛα, where σ is a symbol in Sm, m∈�. We give criteria in terms of σ for boundedness and compactness of the transform Wσ.
Weyl and Riemann-Liouville multifractional Ornstein-Uhlenbeck processes
Lim, S C; Teo, L P
This paper considers two new multifractional stochastic processes, namely the Weyl multifractional Ornstein-Uhlenbeck process and the Riemann-Liouville multifractional Ornstein-Uhlenbeck process. Basic properties of these processes such as locally self-similar property and Hausdorff dimension are studied. The relationship between the multifractional Ornstein-Uhlenbeck processes and the corresponding multifractional Brownian motions is established
Toeplitz operators on higher Cauchy-Riemann spaces
Engliš, Miroslav; Zhang, G.
Ro�. 22, �. 22 (2017), s. 1081-1116 ISSN 1431-0643 Institutional support: RVO:67985840 Keywords : Toeplitz operator * Hankel operator * Cauchy-Riemann operators Subject RIV: BA - General Math ematics OBOR OECD: Pure math ematics Impact factor: 0.800, year: 2016 https://www. math .uni-bielefeld.de/documenta/vol-22/32.html
The beauty of the Riemann-Silberstein vector
Bialynicki-Birula, I.
Beams of light carrying angular momentum have recently been widely studied theoretically and experimentally. In my talk I will show that the description of these beams in terms of the Riemann-Silberstein vector offers many advantages. In particular, it provides a natural bridge between the classical and the quantum description. (author)
Riemann surfaces of complex classical trajectories and tunnelling splitting in one-dimensional systems
Harada, Hiromitsu; Mouchet, Amaury; Shudo, Akira
The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann surface provides complete information on the topology of complex paths, which allows us to enumerate all the possible candidates contributing to the semiclassical sum formula for tunnelling splittings. This naturally leads to action relations among classically disjoined regions, revealing entirely non-local nature in the quantization condition. The importance of the proper treatment of Stokes phenomena is also discussed in Hamiltonians in the normal form.
Approximate Riemann solver for the two-fluid plasma model
Shumlak, U.; Loverich, J.
An algorithm is presented for the simulation of plasma dynamics using the two-fluid plasma model. The two-fluid plasma model is more general than the magnetohydrodynamic (MHD) model often used for plasma dynamic simulations. The two-fluid equations are derived in divergence form and an approximate Riemann solver is developed to compute the fluxes of the electron and ion fluids at the computational cell interfaces and an upwind characteristic-based solver to compute the electromagnetic fields. The source terms that couple the fluids and fields are treated implicitly to relax the stiffness. The algorithm is validated with the coplanar Riemann problem, Langmuir plasma oscillations, and the electromagnetic shock problem that has been simulated with the MHD plasma model. A numerical dispersion relation is also presented that demonstrates agreement with analytical plasma waves
Exploration and extension of an improved Riemann track fitting algorithm
Strandlie, A.; Frühwirth, R.
Recently, a new Riemann track fit which operates on translated and scaled measurements has been proposed. This study shows that the new Riemann fit is virtually as precise as popular approaches such as the Kalman filter or an iterative non-linear track fitting procedure, and significantly more precise than other, non-iterative circular track fitting approaches over a large range of measurement uncertainties. The fit is then extended in two directions: first, the measurements are allowed to lie on plane sensors of arbitrary orientation; second, the full error propagation from the measurements to the estimated circle parameters is computed. The covariance matrix of the estimated track parameters can therefore be computed without recourse to asymptotic properties, and is consequently valid for any number of observation. It does, however, assume normally distributed measurement errors. The calculations are validated on a simulated track sample and show excellent agreement with the theoretical expectations.
Pseudo-periodic maps and degeneration of Riemann surfaces
Matsumoto, Yukio
The first part of the book studies pseudo-periodic maps of a closed surface of genus greater than or equal to two. This class of homeomorphisms was originally introduced by J. Nielsen in 1944 as an extension of periodic maps. In this book, the conjugacy classes of the (chiral) pseudo-periodic mapping classes are completely classified, and Nielsen's incomplete classification is corrected. The second part applies the results of the first part to the topology of degeneration of Riemann surfaces. It is shown that the set of topological types of all the singular fibers appearing in one-parameter holomorphic families of Riemann surfaces is in a bijective correspondence with the set of conjugacy classes of the pseudo-periodic maps of negative twists. The correspondence is given by the topological monodromy.
Towards a theory of chaos explained as travel on Riemann surfaces
Calogero, F; Santini, P M; Gomez-Ullate, D; Sommacal, M
We investigate the dynamics defined by a set of three coupled first-order ODEs. It is shown that the system can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable), and we investigate the position and nature of its branch points. In the semi-symmetric case (g 1 = g 2 ≠g 3 ), for rational values of the coupling constants the system is isochronous and explicit formulae for the period of the solutions can be given. For irrational values, the motions are confined but feature aperiodic motion with sensitive dependence on initial conditions. The system shows a rich dynamical behaviour that can be understood in quantitative detail since a global description of the Riemann surface associated with the solutions can be achieved. The details of the description of the Riemann surface are postponed to a forthcoming publication. This toy model is meant to provide a paradigmatic first step towards understanding a certain novel kind of chaotic behaviour
Non-supersymmetric matrix strings from generalized Yang-Mills theory on arbitrary Riemann surfaces
Billo, M.; D'Adda, A.; Provero, P.
We quantize pure 2d Yang-Mills theory on an arbitrary Riemann surface in the gauge where the field strength is diagonal. Twisted sectors originate, as in Matrix string theory, from permutations of the eigenvalues around homotopically non-trivial loops. These sectors, that must be discarded in the usual quantization due to divergences occurring when two eigenvalues coincide, can be consistently kept if one modifies the action by introducing a coupling of the field strength to the space-time curvature. This leads to a generalized Yang-Mills theory whose action reduces to the usual one in the limit of zero curvature. After integrating over the non-diagonal components of the gauge fields, the theory becomes a free string theory (sum over unbranched coverings) with a U(1) gauge theory on the world-sheet. This is shown to be equivalent to a lattice theory with a gauge group which is the semi-direct product of S N and U(1) N . By using well known results on the statistics of coverings, the partition function on arbitrary Riemann surfaces and the kernel functions on surfaces with boundaries are calculated. Extensions to include branch points and non-abelian groups on the world-sheet are briefly commented upon
Quantum field theory on higher-genus Riemann surfaces, 2
Kubo, Reijiro; Ojima, Shuichi.
Quantum field theory for closed bosonic string systems is formulated on arbitrary higher-genus Riemann surfaces in global operator formalism. Canonical commutation relations between bosonic string field X μ and their conjugate momenta P ν are derived in the framework of conventional quantum field theory. Problems arising in quantizing bosonic systems are considered in detail. Applying the method exploited in the preceding paper we calculate Ward-Takahashi identities. (author)
A contribution to the great Riemann solver debate
Quirk, James J.
The aims of this paper are threefold: to increase the level of awareness within the shock capturing community to the fact that many Godunov-type methods contain subtle flaws that can cause spurious solutions to be computed; to identify one mechanism that might thwart attempts to produce very high resolution simulations; and to proffer a simple strategy for overcoming the specific failings of individual Riemann solvers.
Submaximal Riemann-Roch expected curves and symplectic packing.
Wioletta Syzdek
Full Text Available We study Riemann-Roch expected curves on $mathbb{P}^1 imes mathbb{P}^1$ in the context of the Nagata-Biran conjecture. This conjecture predicts that for sufficiently large number of points multiple points Seshadri constants of an ample line bundle on algebraic surface are maximal. Biran gives an effective lower bound $N_0$. We construct examples verifying to the effect that the assertions of the Nagata-Biran conjecture can not hold for small number of points. We discuss cases where our construction fails. We observe also that there exists a strong relation between Riemann-Roch expected curves on $mathbb{P}^1 imes mathbb{P}^1$ and the symplectic packing problem. Biran relates the packing problem to the existence of solutions of certain Diophantine equations. We construct such solutions for any ample line bundle on $mathbb{P}^1 imes mathbb{P}^1$ and a relatively smallnumber of points. The solutions geometrically correspond to Riemann-Roch expected curves. Finally we discuss in how far the Biran number $N_0$ is optimal in the case of mathbb{P}^1 imes mathbb{P}^1. In fact we conjecture that it can be replaced by a lower number and we provide evidence justifying this conjecture.
Derivative-Based Trapezoid Rule for the Riemann-Stieltjes Integral
Weijing Zhao
Full Text Available The derivative-based trapezoid rule for the Riemann-Stieltjes integral is presented which uses 2 derivative values at the endpoints. This kind of quadrature rule obtains an increase of two orders of precision over the trapezoid rule for the Riemann-Stieltjes integral and the error term is investigated. At last, the rationality of the generalization of derivative-based trapezoid rule for Riemann-Stieltjes integral is demonstrated.
Integrable systems twistors, loop groups, and Riemann surfaces
Hitchin, NJ; Ward, RS
This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do werecognize an integrable system? His own contribution then develops connections with algebraic geometry, and inclu
Riemann-Roch Spaces and Linear Network Codes
Hansen, Johan P.
We construct linear network codes utilizing algebraic curves over finite fields and certain associated Riemann-Roch spaces and present methods to obtain their parameters. In particular we treat the Hermitian curve and the curves associated with the Suzuki and Ree groups all having the maximal...... number of points for curves of their respective genera. Linear network coding transmits information in terms of a basis of a vector space and the information is received as a basis of a possibly altered vector space. Ralf Koetter and Frank R. Kschischang %\\cite{DBLP:journals/tit/KoetterK08} introduced...... in the above metric making them suitable for linear network coding....
Extended KN algebras and extended conformal field theories over higher genus Riemann surfaces
Ceresole, A.; Huang Chaoshang
A global operator formalism for extended conformal field theories over higher genus Riemann surfaces is introduced and extended KN algebra are obtained by means of the KN bases. The BBSS construction of the spin-3 operator is carried out for Kac-Moody algebra A 2 over a Riemann surface of arbitrary genus. (orig.)
Superconformal structures and holomorphic 1/2-superdifferentials on N=1 super Riemann surfaces
Kachkachi, H.; Kachkachi, M.
Using the Super Riemann-Roch theorem we give a local expression for a holomorphic 1/2-superdifferential in a superconformal structure parametrized by special isothermal coordinates on an N=1 Super Riemann Surface (SRS). This construction is done by choosing a suitable origin for these coordinates. The holomorphy of the latter with respect to super Beltrami differentials is proven. (author). 26 refs
Superconformal algebra on meromorphic vector fields with three poles on super-Riemann sphere
Wang Shikun; Xu Kaiwen.
Based upon the Riemann-Roch theorem, we construct superconformal algebra of meromorphic vector fields with three poles and the relevant abelian differential of the third kind on super Riemann sphere. The algebra includes two Ramond sectors as subalgebra, and implies a picture of interaction of three superstrings. (author). 14 refs
Two-Loop Scattering Amplitudes from the Riemann Sphere
Geyer, Yvonne; Monteiro, Ricardo; Tourkine, Piotr
The scattering equations give striking formulae for massless scattering amplitudes at tree level and, as shown recently, at one loop. The progress at loop level was based on ambitwistor string theory, which naturally yields the scattering equations. We proposed that, for ambitwistor strings, the standard loop expansion in terms of the genus of the worldsheet is equivalent to an expansion in terms of nodes of a Riemann sphere, with the nodes carrying the loop momenta. In this paper, we show how to obtain two-loop scattering equations with the correct factorization properties. We adapt genus-two integrands from the ambitwistor string to the nodal Riemann sphere and show that these yield correct answers, by matching standard results for the four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills theory. In the Yang-Mills case, this requires the loop analogue of the Parke-Taylor factor carrying the colour dependence, which includes non-planar contributions.
Zeros da função zeta de Riemann e o teorema dos números primos
Oliveira, Willian Diego [UNESP
We studied various properties of the Riemann's zeta function. Three proofs of the Prime Number Theorem were provides. Classical results on zero-free region of the zeta function, as well as their relation to the error term in the Prime Number Theorem, were studied in details Estudamos várias propriedades da função zeta de Riemann. Três provas do Teorema dos Números Primos foram fornecidas. Resultados clássicos sobre regiões livres de zeros da função zeta, bem como sua relação com o termo do...
A Riemann-Hilbert formulation for the finite temperature Hubbard model
Cavaglià , Andrea [Dipartimento di Fisica and INFN, Università di Torino,Via P. Giuria 1, 10125 Torino (Italy); Cornagliotto, Martina [Dipartimento di Fisica and INFN, Università di Torino,Via P. Giuria 1, 10125 Torino (Italy); DESY Hamburg, Theory Group,Notkestrasse 85, D-22607 Hamburg (Germany); Mattelliano, Massimo; Tateo, Roberto [Dipartimento di Fisica and INFN, Università di Torino,Via P. Giuria 1, 10125 Torino (Italy)
Inspired by recent results in the context of AdS/CFT integrability, we reconsider the Thermodynamic Bethe Ansatz equations describing the 1D fermionic Hubbard model at finite temperature. We prove that the infinite set of TBA equations are equivalent to a simple nonlinear Riemann-Hilbert problem for a finite number of unknown functions. The latter can be transformed into a set of three coupled nonlinear integral equations defined over a finite support, which can be easily solved numerically. We discuss the emergence of an exact Bethe Ansatz and the link between the TBA approach and the results by Jüttner, Klümper and Suzuki based on the Quantum Transfer Matrix method. We also comment on the analytic continuation mechanism leading to excited states and on the mirror equations describing the finite-size Hubbard model with twisted boundary conditions.
A new numerical approach for uniquely solvable exterior Riemann-Hilbert problem on region with corners
Zamzamir, Zamzana; Murid, Ali H. M.; Ismail, Munira
Numerical solution for uniquely solvable exterior Riemann-Hilbert problem on region with corners at offcorner points has been explored by discretizing the related integral equation using Picard iteration method without any modifications to the left-hand side (LHS) and right-hand side (RHS) of the integral equation. Numerical errors for all iterations are converge to the required solution. However, for certain problems, it gives lower accuracy. Hence, this paper presents a new numerical approach for the problem by treating the generalized Neumann kernel at LHS and the function at RHS of the integral equation. Due to the existence of the corner points, Gaussian quadrature is employed which avoids the corner points during numerical integration. Numerical example on a test region is presented to demonstrate the effectiveness of this formulation.
Riemann surfaces and algebraic curves a first course in Hurwitz theory
Cavalieri, Renzo
Hurwitz theory, the study of analytic functions among Riemann surfaces, is a classical field and active research area in algebraic geometry. The subject's interplay between algebra, geometry, topology and analysis is a beautiful example of the interconnectedness of mathematics. This book introduces students to this increasingly important field, covering key topics such as manifolds, monodromy representations and the Hurwitz potential. Designed for undergraduate study, this classroom-tested text includes over 100 exercises to provide motivation for the reader. Also included are short essays by guest writers on how they use Hurwitz theory in their work, which ranges from string theory to non-Archimedean geometry. Whether used in a course or as a self-contained reference for graduate students, this book will provide an exciting glimpse at mathematics beyond the standard university classes.
Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis.
Lapidus, Michel L
This research expository article not only contains a survey of earlier work but also contains a main new result, which we first describe. Given c≥0, the spectral operator [Formula: see text] can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c. Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function ζ=ζ(s): a=ζ(∂), where ∂=∂(c) is the infinitesimal shift of the real line acting on the weighted Hilbert space [Formula: see text]. In this paper, we establish a new asymmetric criterion for the Riemann hypothesis (RH), expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter [Formula: see text] (i.e. for all c in the left half of the critical interval (0,1)). This corresponds (conditionally) to a mathematical (and perhaps also, physical) 'phase transition' occurring in the midfractal case when [Formula: see text]. Both the universality and the non-universality of ζ=ζ(s) in the right (resp., left) critical strip [Formula: see text] (resp., [Formula: see text]) play a key role in this context. These new results are presented here. We also briefly discuss earlier joint work on the complex dimensions of fractal strings, and we survey earlier related work of the author with Maier and with Herichi, respectively, in which were established symmetric criteria for the RH, expressed, respectively, in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension D∈(0,1), with [Formula: see text], and of the quasi-invertibility of the family of spectral operators [Formula: see text] (with [Formula: see text]). © 2015 The Author(s) Published by the Royal Society. All rights reserved.
Contribution of non integer integro-differential operators (NIDO) to the geometrical understanding of Riemann's conjecture-(II)
Le Mehaute, Alain; El Kaabouchi, Abdelaziz; Nivanen, Laurent
Advances in fractional analysis suggest a new way for the physics understanding of Riemann's conjecture. It asserts that, if s is a complex number, the non trivial zeros of zeta function 1/(ζ(s)) =Σ n=1 ∞ (μ(n))/(n s ) in the gap [0, 1], is characterized by s=1/2 (1+2iθ). This conjecture can be understood as a consequence of 1/2-order fractional differential characteristics of automorph dynamics upon opened punctuated torus with an angle at infinity equal to π/4. This physical interpretation suggests new opportunities for revisiting the cryptographic methodologies
Polynomials, Riemann surfaces, and reconstructing missing-energy events
Gripaios, Ben; Webber, Bryan
We consider the problem of reconstructing energies, momenta, and masses in collider events with missing energy, along with the complications introduced by combinatorial ambiguities and measurement errors. Typically, one reconstructs more than one value and we show how the wrong values may be correlated with the right ones. The problem has a natural formulation in terms of the theory of Riemann surfaces. We discuss examples including top quark decays in the Standard Model (relevant for top quark mass measurements and tests of spin correlation), cascade decays in models of new physics containing dark matter candidates, decays of third-generation leptoquarks in composite models of electroweak symmetry breaking, and Higgs boson decay into two tau leptons.
Supersymmetric Dirac particles in Riemann-Cartan space-time
Rumpf, H.
A natural extension of the supersymmetric model of Di Vecchia and Ravndal yields a nontrivial coupling of classical spinning particles to torsion in a Riemann-Cartan geometry. The equations of motion implied by this model coincide with a consistent classical limit of the Heisenberg equations derived from the minimally coupled Dirac equation. Conversely, the latter equation is shown to arise from canonical quantization of the classical system. The Heisenberg equations are obtained exact in all powers of h/2Ï€ and thus complete the partial results of previous WKB calculations. The author also considers such matters of principle as the mathematical realization of anticommuting variables, the physical interpretation of supersymmetry transformations, and the effective variability of rest mass. (Auth.)
A New Riemann Type Hydrodynamical Hierarchy and its Integrability Analysis
Golenia, Jolanta Jolanta; Bogolubov, Nikolai N. Jr.; Popowicz, Ziemowit; Pavlov, Maxim V.; Prykarpatsky, Anatoliy K.
Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The well defined regularization of the model is constructed and its Lax type integrability is discussed. A generalized hydrodynamical Riemann type system is considered, infinite hierarchies of conservation laws, related compatible co-symplectic structures and Lax type representations for the special cases N = 2, 3 and N = 4 are constructed. (author)
The Picard group of the moduli space of r-Spin Riemann surfaces
Randal-Williams, Oscar
An r-Spin Riemann surface is a Riemann surface equipped with a choice of rth root of the (co)tangent bundle. We give a careful construction of the moduli space (orbifold) of r-Spin Riemann surfaces, and explain how to establish a Madsen–Weiss theorem for it. This allows us to prove the "Mumford...... conjecture� for these moduli spaces, but more interestingly allows us to compute their algebraic Picard groups (for g≥10, or g≥9 in the 2-Spin case). We give a complete description of these Picard groups, in terms of explicitly constructed line bundles....
Structural stability of Riemann solutions for strictly hyperbolic systems with three piecewise constant states
Xuefeng Wei
Full Text Available This article concerns the wave interaction problem for a strictly hyperbolic system of conservation laws whose Riemann solutions involve delta shock waves. To cover all situations, the global solutions are constructed when the initial data are taken as three piecewise constant states. It is shown that the Riemann solutions are stable with respect to a specific small perturbation of the Riemann initial data. In addition, some interesting nonlinear phenomena are captured during the process of constructing the solutions, such as the generation and decomposition of delta shock waves.
The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime
Sierra, Germán
We construct a Hamiltonian H R whose discrete spectrum contains, in a certain limit, the Riemann zeros. H R is derived from the action of a massless Dirac fermion living in a domain of Rindler spacetime, in 1 + 1 dimensions, which has a boundary given by the world line of a uniformly accelerated observer. The action contains a sum of delta function potentials that can be viewed as partially reflecting moving mirrors. An appropriate choice of the accelerations of the mirrors, provide primitive periodic orbits that are associated with the prime numbers p, whose periods, as measured by the observer's clock, are logp. Acting on the chiral components of the fermion χ ∓ , H R becomes the Berry–Keating Hamiltonian ±(x p-hat + p-hat x)/2, where x is identified with the Rindler spatial coordinate and p-hat with the conjugate momentum. The delta function potentials give the matching conditions of the fermion wave functions on both sides of the mirrors. There is also a phase shift e iϑ for the reflection of the fermions at the boundary where the observer sits. The eigenvalue problem is solved by transfer matrix methods in the limit where the reflection amplitudes become infinitesimally small. We find that, for generic values of ϑ, the spectrum is a continuum where the Riemann zeros are missing, as in the adelic Connes model. However, for some values of ϑ, related to the phase of the zeta function, the Riemann zeros appear as discrete eigenvalues that are immersed in the continuum. We generalize this result to the zeros of Dirichlet L-functions, which are associated to primitive characters, that are encoded in the reflection coefficients of the mirrors. Finally, we show that the Hamiltonian associated to the Riemann zeros belongs to class AIII, or chiral GUE, of the Random Matrix Theory. (paper)
Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor, and Scalar Curvature
Loveridge, Lee C.
Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also, the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally a derivation of Newtonian Gravity from Einstein's Equations is given.
Existence and Nonexistence of Positive Solutions for Coupled Riemann-Liouville Fractional Boundary Value Problems
Johnny Henderson
Full Text Available We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with two parameters, subject to coupled integral boundary conditions.
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
National Research Council Canada - National Science Library
Derbyshire, John
.... Is the hypothesis true or false?Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic defining a precise formula to track and identify the occurrence...
Nontrivial Solution of Fractional Differential System Involving Riemann-Stieltjes Integral Condition
Ge-Feng Yang
differential system involving the Riemann-Stieltjes integral condition, by using the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle, some sufficient conditions of the existence and uniqueness of a nontrivial solution of a system are obtained.
A variational approach to closed bosonic strings on bordered Riemann surfaces
Ohrndorf, T.
Polyakov's path integral for bosonic closed strings defined on a bordered Riemann surface is investigated by variational methods. It is demonstrated that boundary variations are generated by the Virasoro operators. The investigation is performed for both, simply connected Riemann surfaces as well as ringlike domains. It is shown that the form of the variational operator is the same on both kinds of surfaces. The Virasoro algebra arises as a consistency condition for the variation. (orig.)
Essay on Fractional Riemann-Liouville Integral Operator versus Mikusinski's
Full Text Available This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.
Riemann's and Helmholtz-Lie's problems of space from Weyl's relativistic perspective
Bernard, Julien
I reconstruct Riemann's and Helmholtz-Lie's problems of space, from some perspectives that allow for a fruitful comparison with Weyl. In Part II. of his inaugural lecture, Riemann justifies that the infinitesimal metric is the square root of a quadratic form. Thanks to Finsler geometry, I clarify both the implicit and explicit hypotheses used for this justification. I explain that Riemann-Finsler's kind of method is also appropriate to deal with indefinite metrics. Nevertheless, Weyl shares with Helmholtz a strong commitment to the idea that the notion of group should be at the center of the foundations of geometry. Riemann missed this point, and that is why, according to Weyl, he dealt with the problem of space in a "too formal" way. As a consequence, to solve the problem of space, Weyl abandoned Riemann-Finsler's methods for group-theoretical ones. However, from a philosophical point of view, I show that Weyl and Helmholtz are in strong opposition. The meditation on Riemann's inaugural lecture, and its clear methodological separation between the infinitesimal and the finite parts of the problem of space, must have been crucial for Weyl, while searching for strong epistemological foundations for the group-theoretical methods, avoiding Helmholtz's unjustified transition from the finite to the infinitesimal.
RELATIVISTIC MAGNETOHYDRODYNAMICS: RENORMALIZED EIGENVECTORS AND FULL WAVE DECOMPOSITION RIEMANN SOLVER
Anton, Luis; MartI, Jose M; Ibanez, Jose M; Aloy, Miguel A.; Mimica, Petar; Miralles, Juan A.
We obtain renormalized sets of right and left eigenvectors of the flux vector Jacobians of the relativistic MHD equations, which are regular and span a complete basis in any physical state including degenerate ones. The renormalization procedure relies on the characterization of the degeneracy types in terms of the normal and tangential components of the magnetic field to the wave front in the fluid rest frame. Proper expressions of the renormalized eigenvectors in conserved variables are obtained through the corresponding matrix transformations. Our work completes previous analysis that present different sets of right eigenvectors for non-degenerate and degenerate states, and can be seen as a relativistic generalization of earlier work performed in classical MHD. Based on the full wave decomposition (FWD) provided by the renormalized set of eigenvectors in conserved variables, we have also developed a linearized (Roe-type) Riemann solver. Extensive testing against one- and two-dimensional standard numerical problems allows us to conclude that our solver is very robust. When compared with a family of simpler solvers that avoid the knowledge of the full characteristic structure of the equations in the computation of the numerical fluxes, our solver turns out to be less diffusive than HLL and HLLC, and comparable in accuracy to the HLLD solver. The amount of operations needed by the FWD solver makes it less efficient computationally than those of the HLL family in one-dimensional problems. However, its relative efficiency increases in multidimensional simulations.
Numerical implication of Riemann problem theory for fluid dynamics
Menikoff, R.
The Riemann problem plays an important role in understanding the wave structure of fluid flow. It is also crucial step in some numerical algorithms for accurately and efficiently computing fluid flow; Godunov method, random choice method, and from tracking method. The standard wave structure consists of shock and rarefaction waves. Due to physical effects such as phase transitions, which often are indistinguishable from numerical errors in an equation of state, anomalkous waves may occur, ''rarefaction shocks'', split waves, and composites. The anomalous waves may appear in numerical calculations as waves smeared out by either too much artificial viscosity or insufficient resolution. In addition, the equation of state may lead to instabilities of fluid flow. Since these anomalous effects due to the equation of state occur for the continuum equations, they can be expected to occur for all computational algorithms. The equation of state may be characterized by three dimensionless variables: the adiabatic exponent γ, the Grueneisen coefficient Γ, and the fundamental derivative G. The fluid flow anomalies occur when inequalities relating these variables are violated. 18 refs
Deduction of Einstein equation from homogeneity of Riemann spacetime
Ni, Jun
The symmetry of spacetime translation leads to the energy-momentum conservation. However, the Lagrange depends on spacetime coordinates, which makes the symmetry of spacetime translation different with other symmetry invariant explicitly under symmetry transformation. We need an equation to guarantee the symmetry of spacetime translation. In this talk, I will show that the Einstein equation can be deduced purely from the general covariant principle and the homogeneity of spacetime in the frame of quantum field theory. The Einstein equation is shown to be the equation to guarantee the symmetry of spacetime translation. Gravity is an apparent force due to the curvature of spacetime resulted from the conservation of energy-momentum. In the action of quantum field, only electroweak-strong interactions appear with curved spacetime metric determined by the Einstein equation.. The general covariant principle and the homogeneity of spacetime are merged into one basic principle: Any Riemann spacetime metric guaranteeing the energy-momentum conservation are equivalent, which can be called as the conserved general covariant principle. [4pt] [1] Jun Ni, Chin. Phys. Lett. 28, 110401 (2011).
Riemann-Christoffel Tensor in Differential Geometry of Fractional Order Application to Fractal Space-Time
Jumarie, Guy
By using fractional differences, one recently proposed an alternative to the formulation of fractional differential calculus, of which the main characteristics is a new fractional Taylor series and its companion Rolle's formula which apply to non-differentiable functions. The key is that now we have at hand a differential increment of fractional order which can be manipulated exactly like in the standard Leibniz differential calculus. Briefly the fractional derivative is the quotient of fractional increments. It has been proposed that this calculus can be used to construct a differential geometry on manifold of fractional order. The present paper, on the one hand, refines the framework, and on the other hand, contributes some new results related to arc length of fractional curves, area on fractional differentiable manifold, covariant fractal derivative, Riemann-Christoffel tensor of fractional order, fractional differential equations of fractional geodesic, strip modeling of fractal space time and its relation with Lorentz transformation. The relation with Nottale's fractal space-time theory then appears in quite a natural way.
On membrane interactions and a three-dimensional analog of Riemann surfaces
Kovacs, Stefano [Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4 (Ireland); ICTP South American Institute for Fundamental Research, IFT-UNESP,São Paulo, SP 01440-070 (Brazil); Sato, Yuki [National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwartersrand,Wits 2050 (South Africa); Shimada, Hidehiko [Okayama Institute for Quantum Physics,Okayama (Japan)
Membranes in M-theory are expected to interact via splitting and joining processes. We study these effects in the pp-wave matrix model, in which they are associated with transitions between states in sectors built on vacua with different numbers of membranes. Transition amplitudes between such states receive contributions from BPS instanton configurations interpolating between the different vacua. Various properties of the moduli space of BPS instantons are known, but there are very few known examples of explicit solutions. We present a new approach to the construction of instanton solutions interpolating between states containing arbitrary numbers of membranes, based on a continuum approximation valid for matrices of large size. The proposed scheme uses functions on a two-dimensional space to approximate matrices and it relies on the same ideas behind the matrix regularisation of membrane degrees of freedom in M-theory. We show that the BPS instanton equations have a continuum counterpart which can be mapped to the three-dimensional Laplace equation through a sequence of changes of variables. A description of configurations corresponding to membrane splitting/joining processes can be given in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann surface, consisting of multiple copies of �{sup 3} connected via a generalisation of branch cuts. We discuss various general features of our proposal and we also present explicit analytic solutions.
Kovacs, Stefano; Sato, Yuki; Shimada, Hidehiko
Membranes in M-theory are expected to interact via splitting and joining processes. We study these effects in the pp-wave matrix model, in which they are associated with transitions between states in sectors built on vacua with different numbers of membranes. Transition amplitudes between such states receive contributions from BPS instanton configurations interpolating between the different vacua. Various properties of the moduli space of BPS instantons are known, but there are very few known examples of explicit solutions. We present a new approach to the construction of instanton solutions interpolating between states containing arbitrary numbers of membranes, based on a continuum approximation valid for matrices of large size. The proposed scheme uses functions on a two-dimensional space to approximate matrices and it relies on the same ideas behind the matrix regularisation of membrane degrees of freedom in M-theory. We show that the BPS instanton equations have a continuum counterpart which can be mapped to the three-dimensional Laplace equation through a sequence of changes of variables. A description of configurations corresponding to membrane splitting/joining processes can be given in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann surface, consisting of multiple copies of �"3 connected via a generalisation of branch cuts. We discuss various general features of our proposal and we also present explicit analytic solutions.
The Equation Δ u + ∇φ· ∇u = 8πc(1-heu) on a Riemann Surface
Wang Meng
Let M be a compact Riemann surface, h(x) a positive smooth function on M, and φ(x) a smooth function on M which satisfies that ∫ M e φ dV g = 1. In this paper, we consider the functional J(u) = 2 1 ∫ M |∇u| 2 e φ dV g +8πc ∫ M ue φ dV g -8πclog ∫ M he u+φ dV g . We give a sufficient condition under which J achieves its minimum for c ≤ inf xelement ofM Φ(x). (author)
The continuous determination of spacetime geometry by the Riemann curvature tensor
Rendall, A.D.
It is shown that generically the Riemann tensor of a Lorentz metric on an n-dimensional manifold (n ≥ 4) determines the metric up to a constant factor and hence determines the associated torsion-free connection uniquely. The resulting map from Riemann tensors to connections is continuous in the Whitney Csup(∞) topology but, at least for some manifolds, constant factors cannot be chosen so as to make the map from Riemann tensors to metrics continuous in that topology. The latter map is, however, continuous in the compact open Csup(∞) topology so that estimates of the metric and its derivatives on a compact set can be obtained from similar estimates on the curvature and its derivatives. (author)
Non-uniqueness of admissible weak solutions to the Riemann problem for isentropic Euler equations
Chiodaroli, Elisabetta; Kreml, Ondřej
We study the Riemann problem for multidimensional compressible isentropic Euler equations. Using the framework developed in Chiodaroli et al (2015 Commun. Pure Appl. Math. 68 1157–90), and based on the techniques of De Lellis and Székelyhidi (2010 Arch. Ration. Mech. Anal. 195 225–60), we extend the results of Chiodaroli and Kreml (2014 Arch. Ration. Mech. Anal. 214 1019–49) and prove that it is possible to characterize a set of Riemann data, giving rise to a self-similar solution consisting of one admissible shock and one rarefaction wave, for which the problem also admits infinitely many admissible weak solutions.
Modular transformations of conformal blocks in WZW models on Riemann surfaces of higher genus
Miao Li; Ming Yu.
We derive the modular transformations for conformal blocks in Wess-Zumino-Witten models on Riemann surfaces of higher genus. The basic ingredient consists of using the Chern-Simons theory developed by Witten. We find that the modular transformations generated by Dehn twists are linear combinations of Wilson line operators, which can be expressed in terms of braiding matrices. It can also be shown that modular transformation matrices for g > 0 Riemann surfaces depend only on those for g ≤ 3. (author). 13 refs, 15 figs
A fast Cauchy-Riemann solver. [differential equation solution for boundary conditions by finite difference approximation
Ghil, M.; Balgovind, R.
The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.
Exact Riemann solutions of the Ripa model for flat and non-flat bottom topographies
Rehman, Asad; Ali, Ishtiaq; Qamar, Shamsul
This article is concerned with the derivation of exact Riemann solutions for Ripa model considering flat and non-flat bottom topographies. The Ripa model is a system of shallow water equations accounting for horizontal temperature gradients. In the case of non-flat bottom topography, the mass, momentum and energy conservation principles are utilized to relate the left and right states across the step-type bottom topography. The resulting system of algebraic equations is solved iteratively. Different numerical case studies of physical interest are considered. The solutions obtained from developed exact Riemann solvers are compared with the approximate solutions of central upwind scheme.
Riemann problems and their application to ultra-relativistic heavy ion collisions
Plohr, B.J.; Sharp, D.H.
Heavy ion collisions at sufficiently high energies to form quark-gluon plasma are considered. The phase transformation from a quark-gluon phase to hadrons as the nuclear matter cools is modeled as a hydrodynamical flow. Nonlinear waves are the predominant feature of this type of flow and the Riemann problem of a relativistic gas undergoing a phase transformation is explored as a method to numerically model this phase transition process in nuclear matter. The solution of the Riemann problem is outlined and results of preliminary numerical computations of the flow are presented. 10 refs., 2 figs
Reduction of 4-dim self dual super Yang-Mills onto super Riemann surfaces
Mendoza, A.; Restuccia, A.; Martin, I.
Recently self dual super Yang-Mills over a super Riemann surface was obtained as the zero set of a moment map on the space of superconnections to the dual of the super Lie algebra of gauge transformations. We present a new formulation of 4-dim Euclidean self dual super Yang-Mills in terms of constraints on the supercurvature. By dimensional reduction we obtain the same set of superconformal field equations which define self dual connections on a super Riemann surface. (author). 10 refs
Super-quasi-conformal transformation and Schiffer variation on super-Riemann surface
Takahasi, Wataru
A set of equations which characterizes the super-Teichmueller deformations is proposed. It is a supersymmetric extension of the Beltrami equation. Relations between the set of equations and the Schiffer variations with the KN bases are discussed. This application of the KN bases shows the powerfulness of the KN theory in the study of super-Riemann surfaces. (author)
The Great Gorilla Jump: An Introduction to Riemann Sums and Definite Integrals
Sealey, Vicki; Engelke, Nicole
The great gorilla jump is an activity designed to allow calculus students to construct an understanding of the structure of the Riemann sum and definite integral. The activity uses the ideas of position, velocity, and time to allow students to explore familiar ideas in a new way. Our research has shown that introducing the definite integral as…
Riemann type algebraic structures and their differential-algebraic integrability analysis
Prykarpatsky A.K.
Full Text Available The differential-algebraic approach to studying the Lax type integrability of generalized Riemann type equations is devised. The differentiations and the associated invariant differential ideals are analyzed in detail. The approach is also applied to studying the Lax type integrability of the well known Korteweg-de Vries dynamical system.
Infinite conformal symmetries and Riemann-Hilbert transformation in super principal chiral model
Hao Sanru; Li Wei
This paper shows a new symmetric transformation - C transformation in super principal chiral model and discover an infinite dimensional Lie algebra related to the Virasoro algebra without central extension. By using the Riemann-Hilbert transformation, the physical origination of C transformation is discussed
Superconformal algebra and central extension of meromorphic vector fields with multipoles on super-Riemann sphere
The superconformal algebras of meromorphic vector fields with multipoles, the central extension and the relevant abelian differential of the third kind on super Riemann sphere were constructed. The background of our theory is concerned with the interaction of closed superstrings. (author). 9 refs
Seeley-De Witt coefficients in a Riemann-Cartan manifold
Cognola, G.; Zerbini, S.; Istituto Nazionale di Fisica Nucleare, Povo
A new derivation of the first two coefficients of the heat kernel expansion for a second-order elliptic differential operator on a Riemann-Cartan manifold with arbitrary torsion is given. The expressions are presented in a very compact and tractable form useful for physical applications. Comparisons with other similar results that appeared in the literature are briefly discussed. (orig.)
Regular Riemann-Hilbert transforms, Baecklund transformations and hidden symmetry algebra for some linearization systems
Chau Ling-Lie; Ge Mo-Lin; Teh, Rosy.
The Baecklund Transformations and the hidden symmetry algebra for Self-Dual Yang-Mills Equations, Landau-Lifshitz equations and the Extended Super Yang-Mills fields (N>2) are discussed on the base of the Regular Riemann-Hilbert Transform and the linearization equations. (author)
Representation theory of current algebra and conformal field theory on Riemann surfaces
Yamada, Yasuhiko
We study conformal field theories with current algebra (WZW-model) on general Riemann surfaces based on the integrable representation theory of current algebra. The space of chiral conformal blocks defined as solutions of current and conformal Ward identities is shown to be finite dimensional and satisfies the factorization properties. (author)
Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces
Varaksin, O.L.; Firstov, V.V.; Shapovalov, A.V.; Shirokov, I.V.
The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented
Classical and quantum Liouville theory on the Riemann sphere with n>3 punctures (III)
Shen Jianmin; Sheng Zhengmao; Wang Zhonghua
We study the Classical and Quantum Liouville theory on the Riemann sphere with n>3 punctures. We get the quantum exchange algebra relations between the chiral components in the Liouville theory from our assumption on the principle of quantization. (author). 5 refs
Fourier-Laplace transform of irreducible regular differential systems on the Riemann sphere
Sabbah, C
It is shown that the Fourier-Laplace transform of an irreducible regular differential system on the Riemann sphere underlies a polarizable regular twistor D-module if one considers only the part at finite distance. The associated holomorphic bundle defined away from the origin of the complex plane is therefore equipped with a natural harmonic metric having a tame behaviour near the origin
Modification of the Riemann problem and the application for the boundary conditions in computational fluid dynamics
Kyncl Martin
Full Text Available We work with the system of partial differential equations describing the non-stationary compressible turbulent fluid flow. It is a characteristic feature of the hyperbolic equations, that there is a possible raise of discontinuities in solutions, even in the case when the initial conditions are smooth. The fundamental problem in this area is the solution of the so-called Riemann problem for the split Euler equations. It is the elementary problem of the one-dimensional conservation laws with the given initial conditions (LIC - left-hand side, and RIC - right-hand side. The solution of this problem is required in many numerical methods dealing with the 2D/3D fluid flow. The exact (entropy weak solution of this hyperbolical problem cannot be expressed in a closed form, and has to be computed by an iterative process (to given accuracy, therefore various approximations of this solution are being used. The complicated Riemann problem has to be further modified at the close vicinity of boundary, where the LIC is given, while the RIC is not known. Usually, this boundary problem is being linearized, or roughly approximated. The inaccuracies implied by these simplifications may be small, but these have a huge impact on the solution in the whole studied area, especially for the non-stationary flow. Using the thorough analysis of the Riemann problem we show, that the RIC for the local problem can be partially replaced by the suitable complementary conditions. We suggest such complementary conditions accordingly to the desired preference. This way it is possible to construct the boundary conditions by the preference of total values, by preference of pressure, velocity, mass flow, temperature. Further, using the suitable complementary conditions, it is possible to simulate the flow in the vicinity of the diffusible barrier. On the contrary to the initial-value Riemann problem, the solution of such modified problems can be written in the closed form for some
Selberg trace formula for bordered Riemann surfaces: Hyperbolic, elliptic and parabolic conjugacy classes, and determinants of Maass-Laplacians
Bolte, J.
The Selberg trace formula for automorphic forms of weight m ε- Z, on bordered Riemann surfaces is developed. The trace formula is formulated for arbitrary Fuchsian groups of the first kind which include hyperbolic, elliptic and parabolic conjugacy classes. In the case of compact bordered Riemann surfaces we can explicitly evaluate determinants of Maass-Laplacians for both Dirichlet and Neumann boundary-conditions, respectively. Some implications for the open bosonic string theory are mentioned. (orig.)
Generalization of the fejer-hadamard type inequalities for p-convex functions via k-fractional integrals
Ghulam Farid
Full Text Available The aim of this paper is to obtain some more general fractional integral inequalities of Fejer Hadamard type for p-convex functions via Riemann-Liouville k-fractional integrals. Also in particular fractional inequalities for p-convex functions via Riemann-Liouville fractional integrals have been deduced.
A variational principle giving gravitational 'superpotentials', the affine connection, Riemann tensor, and Einstein field equations
A first-order Lagrangian is given, from which follow the definitions of the fully covariant form of the Riemann tensor Rsub(μνkappalambda) in terms of the affine connection and metric; the definition of the affine connection in terms of the metric; the Einstein field equations; and the definition of a set of gravitational 'superpotentials' closely connected with the Komar conservation laws (Phys. Rev.; 113:934 (1959)). Substitution of the definition of the affine connection into this Lagrangian results in a second-order Lagrangian, from which follow the definition of the fully covariant Riemann tensor in terms of the metric, the Einstein equations, and the definition of the gravitational 'superpotentials'. (author)
An Exact, Compressible One-Dimensional Riemann Solver for General, Convex Equations of State
Kamm, James Russell [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
This note describes an algorithm with which to compute numerical solutions to the one- dimensional, Cartesian Riemann problem for compressible flow with general, convex equations of state. While high-level descriptions of this approach are to be found in the literature, this note contains most of the necessary details required to write software for this problem. This explanation corresponds to the approach used in the source code that evaluates solutions for the 1D, Cartesian Riemann problem with a JWL equation of state in the ExactPack package [16, 29]. Numerical examples are given with the proposed computational approach for a polytropic equation of state and for the JWL equation of state.
Ship-induced solitary Riemann waves of depression in Venice Lagoon
Parnell, Kevin E.; Soomere, Tarmo; Zaggia, Luca; Rodin, Artem; Lorenzetti, Giuliano; Rapaglia, John; Scarpa, Gian Marco
We demonstrate that ships of moderate size, sailing at low depth Froude numbers (0.37–0.5) in a navigation channel surrounded by shallow banks, produce depressions with depths up to 2.5 m. These depressions (Bernoulli wakes) propagate as long-living strongly nonlinear solitary Riemann waves of depression substantial distances into Venice Lagoon. They gradually become strongly asymmetric with the rear of the depression becoming extremely steep, similar to a bore. As they are dynamically similar, air pressure fluctuations moving over variable-depth coastal areas could generate meteorological tsunamis with a leading depression wave followed by a devastating bore-like feature. - Highlights: • Unprecedently deep long-living ship-induced waves of depression detected. • Such waves are generated in channels with side banks under low Froude numbers. • The propagation of these waves is replicated using Riemann waves. • Long-living waves of depression form bore-like features at rear slope
Instanton calculus without equations of motion: semiclassics from monodromies of a Riemann surface
Gulden, Tobias; Janas, Michael; Kamenev, Alex
Instanton calculations in semiclassical quantum mechanics rely on integration along trajectories which solve classical equations of motion. However in systems with higher dimensionality or complexified phase space these are rarely attainable. A prime example are spin-coherent states which are used e.g. to describe single molecule magnets (SMM). We use this example to develop instanton calculus which does not rely on explicit solutions of the classical equations of motion. Energy conservation restricts the complex phase space to a Riemann surface of complex dimension one, allowing to deform integration paths according to Cauchy's integral theorem. As a result, the semiclassical actions can be evaluated without knowing actual classical paths. Furthermore we show that in many cases such actions may be solely derived from monodromy properties of the corresponding Riemann surface and residue values at its singular points. As an example, we consider quenching of tunneling processes in SMM by an applied magnetic field.
Parnell, Kevin E. [College of Marine and Environmental Sciences and Centre for Tropical Environmental and Sustainability Sciences, James Cook University, Queensland 4811 (Australia); Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn (Estonia); Soomere, Tarmo, E-mail: [email protected] [Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn (Estonia); Estonian Academy of Sciences, Kohtu 6, 10130 Tallinn (Estonia); Zaggia, Luca [Institute of Marine Sciences, National Research Council, Castello 2737/F, 30122 Venice (Italy); Rodin, Artem [Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn (Estonia); Lorenzetti, Giuliano [Institute of Marine Sciences, National Research Council, Castello 2737/F, 30122 Venice (Italy); Rapaglia, John [Sacred Heart University Department of Biology, 5151 Park Avenue, Fairfield, CT 06825 (United States); Scarpa, Gian Marco [Università Ca' Foscari, Dorsoduro 3246, 30123 Venice (Italy)
We demonstrate that ships of moderate size, sailing at low depth Froude numbers (0.37–0.5) in a navigation channel surrounded by shallow banks, produce depressions with depths up to 2.5 m. These depressions (Bernoulli wakes) propagate as long-living strongly nonlinear solitary Riemann waves of depression substantial distances into Venice Lagoon. They gradually become strongly asymmetric with the rear of the depression becoming extremely steep, similar to a bore. As they are dynamically similar, air pressure fluctuations moving over variable-depth coastal areas could generate meteorological tsunamis with a leading depression wave followed by a devastating bore-like feature. - Highlights: • Unprecedently deep long-living ship-induced waves of depression detected. • Such waves are generated in channels with side banks under low Froude numbers. • The propagation of these waves is replicated using Riemann waves. • Long-living waves of depression form bore-like features at rear slope.
Flux quantization and quantum mechanics on Riemann surfaces in an external magnetic field
Bolte, J.; Steiner, F.
We investigate the possibility to apply an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces we find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume we obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial. (orig.)
Integral relations for invariants constructed from three Riemann tensors and their applications in quantum gravity
van Nieuwenhuizen, P.; Wu, C.C.
The lowest order quantum corrections to pure gravitation are finite because there exists an integral relation between products of two Riemann tensors (the Gauss--Bonnet theorem). In this article several algebraic and integral relations are determined between products of three Riemann tensors in four- and six-dimensional spacetime. In both cases, one is left with only one invariant when R/sub μ//sub ν/=0, viz., ∫ (-g) 1 / 2 (R/sub b//sub β//sub μ//sub ν/R/sup μ//sup ν//sup rho//sup sigma/R/sub rho//sub sigma/ /sup α//sup β/).It is explicitly shown that this invariant does not vanish, even when R/sub μ//sub ν/=0. Consequently, the two-loop quantum corrections to pure gravitation will only be finite if, due to miraculous cancellation, the coefficient of this invariant vanishes
Multi-loop string amplitudes and Riemann surfaces
Taylor, J.G.
The paper was presented at the workshop on 'Supersymmetry and its applications', Cambridge, United Kingdom, 1985. Super-string theory is discussed under the following topic headings: the functional approach to the string amplitude, Rieman surfaces, the determinants Δsub(epsilon)(1) and Δsub(epsilon)(2), Green's functions, total amplitude, and divergence analysis. (U.K.)
Quantum Riemann surfaces. Pt. 1. The unit disc
Klimek, S.; Lesniewski, A. (Harvard Univ., Cambridge, MA (United States))
We construct a non-commutative C{sup *}-algebra C{sub {mu}}(anti U) which is a quantum deformation of the algebra of continuous functions on the closed unit disc anti U.C{sub {mu}}(anti U) is generated by the Toeplitz operators on a suitable Hilbert space of holomorphic functions on U. (orig.).
Jet Riemann-Lagrange Geometry Applied to Evolution DEs Systems from Economy
Neagu, Mircea
The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet Yang-Mills energies, starting from some given non-linear evolution DEs systems modelling economic phenomena, like the Kaldor model of the bussines cycle or the Tobin-Benhabib-Miyao model regarding the role of money on economic growth.
Temperature duality on Riemann surface and cosmological solutions for genus g = 1 and 2
Yan Jun; Wang Shunjin
A bosonic string model at finite temperature on the gravitation g μν and the dilaton φ background field is examined. Moreover, the duality relation of energy momentum tensor on high genus Riemann surface is derived. At the same time, the temperature duality invariance for the action of string gas matter is proved in 4-D Robertson-Walker metric, the string cosmological solutions and temperature duality of the equations of motion for genus g = 1 and 2 are also investigated
Stability of the isentropic Riemann solutions of the full multidimensional Euler system
Feireisl, Eduard; Kreml, Ondřej; Vasseur, A.
Ro�. 47, �. 3 (2015), s. 2416-2425 ISSN 0036-1410 R&D Projects: GA ČR GA13-00522S EU Projects: European Commission(XE) 320078 - MATHEF Institutional support: RVO:67985840 Keywords : Euler system * isentropic solutions * Riemann problem * rarefaction wave Subject RIV: BA - General Mathematics Impact factor: 1.486, year: 2015 http://epubs.siam.org/doi/abs/10.1137/140999827
Codomains for the Cauchy-Riemann and Laplace operators in �2
Lloyd Edgar S. Moyo
Full Text Available A codomain for a nonzero constant-coefficient linear partial differential operator P(∂ with fundamental solution E is a space of distributions T for which it is possible to define the convolution E*T and thus solving the equation P(∂S=T. We identify codomains for the Cauchy-Riemann operator in �2 and Laplace operator in �2 . The convolution is understood in the sense of the S′-convolution.
Representation of symmetric metric connection via Riemann-Christoffel curvature tensor
Selikhov, A.V.
Bivector σ-bar μ ν ' which is the Jacoby matrix of the transformation to the Riemanian coordinates is considered in the paper. Basing on the dual nature of σ-bar μ ν ' the representation of metric connection (Christoffel symbols) have been obtained at the Riemanian coordinates via Riemann-Christoffel curvature tensor; the covariant conserved four-momentum in the general theory of relativity have been constructed. 11 refs
The mass-damped Riemann problem and the aerodynamic surface force calculation for an accelerating body
Tan, Zhiqiang; Wilson, D.; Varghese, P.L.
We consider an extension of the ordinary Riemann problem and present an efficient approximate solution that can be used to improve the calculations of aerodynamic forces on an accelerating body. The method is demonstrated with one-dimensional examples where the Euler equations and the body motion are solved in the non-inertial co-ordinate frame fixed to the accelerating body. 8 refs., 6 figs
A novel supersymmetry in 2-dimensional Yang-Mills theory on Riemann surfaces
Soda, Jiro
We find a novel supersymmetry in 2-dimensional Maxwell and Yang-Mills theories. Using this supersymmetry, it is shown that the 2-dimensional Euclidean pure gauge theory on a closed Riemann surface Σ can be reduced to a topological field theory which is the 3-dimensional Chern-Simons gauge theory in the special space-time topology Σ x R. Related problems are also discussed. (author)
Cálculo de áreas mediante la suma de Riemann con la TI-83
Lupiáñez, José Luis
En este artículo presentamos una actividad para introducir el cálculo del área que encierra una curva, basada en la Suma de Riemann, y que puede realizarse con la calculadora TI-83. El planteamiento de la actividad permite estudiar varias funciones sin perder tiempo en tediosos cálculos, con idea de observar lo acertado de este método de aproximación.
Vertex operators, non-abelian orbifolds and the Riemann-Hilbert problem
Gato, B.; Massachusetts Inst. of Tech., Cambridge
We show how to construct the oscillator part of vertex operators for the bosonic string moving on non-abelian orbifolds, using the conserved charges method. When the three-string vertices are twisted by non-commuting group elements, the construction of the conserved charges becomes the Riemann-Hilbert problem with monodromy matrices given by the twists. This is solvable for any given configuration and any non-abelian orbifold. (orig.)
Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations
Chiodaroli, E.; Kreml, Ondřej
Ro�. 31, �. 4 (2018), s. 1441-1460 ISSN 0951-7715 R&D Projects: GA ČR(CZ) GJ17-01694Y Institutional support: RVO:67985840 Keywords : Riemann problem * non-uniqueness * weak solutions Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.767, year: 2016 http://iopscience.iop.org/ article /10.1088/1361-6544/aaa10d/meta
Ro�. 31, �. 4 (2018), s. 1441-1460 ISSN 0951-7715 R&D Projects: GA ČR(CZ) GJ17-01694Y Institutional support: RVO:67985840 Keywords : Riemann problem * non-uniqueness * weak solutions Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.767, year: 2016 http://iopscience.iop.org/article/10.1088/1361-6544/aaa10d/meta
Integrable equations, addition theorems, and the Riemann-Schottky problem
Buchstaber, Viktor M; Krichever, I M
The classical Weierstrass theorem claims that, among the analytic functions, the only functions admitting an algebraic addition theorem are the elliptic functions and their degenerations. This survey is devoted to far-reaching generalizations of this result that are motivated by the theory of integrable systems. The authors discovered a strong form of the addition theorem for theta functions of Jacobian varieties, and this form led to new approaches to known problems in the geometry of Abelian varieties. It is shown that strong forms of addition theorems arise naturally in the theory of the so-called trilinear functional equations. Diverse aspects of the approaches suggested here are discussed, and some important open problems are formulated.
Modelling the transition between fixed and mobile bed conditions in two-phase free-surface flows: The Composite Riemann Problem and its numerical solution
Rosatti, Giorgio; Zugliani, Daniel
In a two-phase free-surface flow, the transition from a mobile-bed condition to a fixed-bed one (and vice versa) occurs at a sharp interface across which the relevant system of partial differential equations changes abruptly. This leads to the possibility of conceiving a new type of Riemann Problem (RP), which we have called Composite Riemann Problem (CRP), where not only the initial constant values of the variables but also the system of equations change from left to right of a discontinuity. In this paper, we present a strategy for solving a CRP by reducing it to a standard RP of a single, composite system of equations. This can be obtained by combining the two original systems by means of a suitable weighting function, namely the erodibility variable, and the introduction of an appropriate differential equation for this quantity. In this way, the CRP problem can be analyzed theoretically with standard methods, and the features of the solutions can be clearly identified. In particular, a stationary contact wave is able to correctly describe the sharp transition between mobile- and fixed-bed conditions. A finite volume scheme based on the Multiple Averages Generalized Roe approach (Rosatti and Begnudelli (2013) [22]) was used to numerically solve the fixed-mobile CRP. Several test cases demonstrate the effectiveness, exact well balanceness and high accuracy of the scheme when applied to problems that fall within the physical range of applicability of the relevant mathematical model.
Multidimensional Riemann problem with self-similar internal structure. Part II - Application to hyperbolic conservation laws on unstructured meshes
Balsara, Dinshaw S.; Dumbser, Michael
Multidimensional Riemann solvers that have internal sub-structure in the strongly-interacting state have been formulated recently (D.S. Balsara (2012, 2014) [5,16]). Any multidimensional Riemann solver operates at the grid vertices and takes as its input all the states from its surrounding elements. It yields as its output an approximation of the strongly interacting state, as well as the numerical fluxes. The multidimensional Riemann problem produces a self-similar strongly-interacting state which is the result of several one-dimensional Riemann problems interacting with each other. To compute this strongly interacting state and its higher order moments we propose the use of a Galerkin-type formulation to compute the strongly interacting state and its higher order moments in terms of similarity variables. The use of substructure in the Riemann problem reduces numerical dissipation and, therefore, allows a better preservation of flow structures, like contact and shear waves. In this second part of a series of papers we describe how this technique is extended to unstructured triangular meshes. All necessary details for a practical computer code implementation are discussed. In particular, we explicitly present all the issues related to computational geometry. Because these Riemann solvers are Multidimensional and have Self-similar strongly-Interacting states that are obtained by Consistency with the conservation law, we call them MuSIC Riemann solvers. (A video introduction to multidimensional Riemann solvers is available on http://www.elsevier.com/xml/linking-roles/text/html". The MuSIC framework is sufficiently general to handle general nonlinear systems of hyperbolic conservation laws in multiple space dimensions. It can also accommodate all self-similar one-dimensional Riemann solvers and subsequently produces a multidimensional version of the same. In this paper we focus on unstructured triangular meshes. As examples of different systems of conservation laws we
Study of the Riemann problem and construction of multidimensional Godunov-type schemes for two-phase flow models
Toumi, I.
This thesis is devoted to the study of the Riemann problem and the construction of Godunov type numerical schemes for one or two dimensional two-phase flow models. In the first part, we study the Riemann problem for the well-known Drift-Flux, model which has been widely used for the analysis of thermal hydraulics transients. Then we use this study to construct approximate Riemann solvers and we describe the corresponding Godunov type schemes for simplified equation of state. For computation of complex two-phase flows, a weak formulation of Roe's approximate Riemann solver, which gives a method to construct a Roe-averaged jacobian matrix with a general equation of state, is proposed. For two-dimensional flows, the developed methods are based upon an approximate solver for a two-dimensional Riemann problem, according to Harten-Lax-Van Leer principles. The numerical results for standard test problems show the good behaviour of these numerical schemes for a wide range of flow conditions [fr
Riemann solvers for multi-component gas mixtures with temperature dependent heat capacities
Beccantini, A.
This thesis represents a contribution to the development of upwind splitting schemes for the Euler equations for ideal gaseous mixtures and their investigation in computing multidimensional flows in irregular geometries. In the preliminary part we develop and investigate the parameterization of the shock and rarefaction curves in the phase space. Then, we apply them to perform some field-by-field decompositions of the Riemann problem: the entropy-respecting one, the one which supposes that genuinely-non-linear (GNL) waves are both shocks (shock-shock one) and the one which supposes that GNL waves are both rarefactions (rarefaction-rarefaction one). We emphasize that their analysis is fundamental in Riemann solvers developing: the simpler the field-by-field decomposition, the simpler the Riemann solver based on it. As the specific heat capacities of the gases depend on the temperature, the shock-shock field-by-field decomposition is the easiest to perform. Then, in the second part of the thesis, we develop an upwind splitting scheme based on such decomposition. Afterwards, we investigate its robustness, precision and CPU-time consumption, with respect to some of the most popular upwind splitting schemes for polytropic/non-polytropic ideal gases. 1-D test-cases show that this scheme is both precise (exact capturing of stationary shock and stationary contact) and robust in dealing with strong shock and rarefaction waves. Multidimensional test-cases show that it suffers from some of the typical deficiencies which affect the upwind splitting schemes capable of exact capturing stationary contact discontinuities i.e the developing of non-physical instabilities in computing strong shock waves. In the final part, we use the high-order multidimensional solver here developed to compute fully-developed detonation flows. (author)
A family of high-order gas-kinetic schemes and its comparison with Riemann solver based high-order methods
Ji, Xing; Zhao, Fengxiang; Shyy, Wei; Xu, Kun
Most high order computational fluid dynamics (CFD) methods for compressible flows are based on Riemann solver for the flux evaluation and Runge-Kutta (RK) time stepping technique for temporal accuracy. The advantage of this kind of space-time separation approach is the easy implementation and stability enhancement by introducing more middle stages. However, the nth-order time accuracy needs no less than n stages for the RK method, which can be very time and memory consuming due to the reconstruction at each stage for a high order method. On the other hand, the multi-stage multi-derivative (MSMD) method can be used to achieve the same order of time accuracy using less middle stages with the use of the time derivatives of the flux function. For traditional Riemann solver based CFD methods, the lack of time derivatives in the flux function prevents its direct implementation of the MSMD method. However, the gas kinetic scheme (GKS) provides such a time accurate evolution model. By combining the second-order or third-order GKS flux functions with the MSMD technique, a family of high order gas kinetic methods can be constructed. As an extension of the previous 2-stage 4th-order GKS, the 5th-order schemes with 2 and 3 stages will be developed in this paper. Based on the same 5th-order WENO reconstruction, the performance of gas kinetic schemes from the 2nd- to the 5th-order time accurate methods will be evaluated. The results show that the 5th-order scheme can achieve the theoretical order of accuracy for the Euler equations, and present accurate Navier-Stokes solutions as well due to the coupling of inviscid and viscous terms in the GKS formulation. In comparison with Riemann solver based 5th-order RK method, the high order GKS has advantages in terms of efficiency, accuracy, and robustness, for all test cases. The 4th- and 5th-order GKS have the same robustness as the 2nd-order scheme for the capturing of discontinuous solutions. The current high order MSMD GKS is a
Loss of hyperbolicity changes the number of wave groups in Riemann problems
Vítor Matos; Julio D. Silva; Dan Marchesin
Themain goal of ourwork is to showthat there exists a class of 2×2 Riemann problems for which the solution comprises a singlewave group for an open set of initial conditions. This wave group comprises a 1-rarefaction joined to a 2-rarefaction, not by an intermediate state, but by a doubly characteristic shock, 1-left and 2-right characteristic. In order to ensure that perturbations of initial conditions do not destroy the adjacency of the waves, local transversality between a composite curve ...
The motion of a classical spinning point particle in a Riemann-Cartan space-time
Amorim, R.
A consistent set of equations of motion for classical charged point particles with spin and magnetic dipole moment in a Riemann-Cartan space-time is generated from a generalized Lagrangean formalism. The equations avoid the spurius free helicoidal solutions and at the same time conserve the canonical condition of normalization of the 4-velocity. The 4-velocity and the mechanical moment are paralell in this theory, where the condition of orthogonality between the spin and the 4-velocity is treated as a non-holonomic one. (Author) [pt
A Riemann-Hilbert approach to the inverse problem for the Stark operator on the line
Its, A.; Sukhanov, V.
The paper is concerned with the inverse scattering problem for the Stark operator on the line with a potential from the Schwartz class. In our study of the inverse problem, we use the Riemann-Hilbert formalism. This allows us to overcome the principal technical difficulties which arise in the more traditional approaches based on the Gel'fand-Levitan-Marchenko equations, and indeed solve the problem. We also produce a complete description of the relevant scattering data (which have not been obtained in the previous works on the Stark operator) and establish the bijection between the Schwartz class potentials and the scattering data.
The transition from regular to irregular motions, explained as travel on Riemann surfaces
We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a three-body problem in the (complex) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology-illustrating the onset in a deterministic context of irregular motions-is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere
Arithmetical Fourier and Limit values of elliptic modular functions
In order to remove singularities, Riemann used a well-known device of taking the odd part (3.2) or an alternate sum (3.3) to be stated in §3. In §2 of this note we shall reveal that the limit values of elliptic modular functions in Riemann's fragment II evaluated by the differences of polyloga- rithm function l1(x) of order 1 (cf.
Do extended objects move along the geodesics in the Riemann space-time
Denisov, V.I.; Logunov, A.A.; Mestvirishvili, M.A.
Movement of an extended self-gravitating body in the gravitational field of another distant body is studied in the postnewtonian approximation of arbitrary metrical gravitational theory. Comparison of the mass center acceleration of the extended body with the acceleration of a point body moving in the Riemann space-time, the metrics of which is formally equivalent to the metrics of two moving extended bodies, shows that in any metrical gravitation theory with conservation laws of energy and momentum of the matter and gravitational field taken together, the mass center of the extended body does not, in general case, move along the geodesics of the Riemann space-time. Application of the general formulas obtained to the system Sun-Earth combined with the experimental data of the lunar laser ranging, shows that the Earth in its orbital motion is oscillating with respect to reference geodesics, with the period about one hour and the amplitude not less than 10 -2 cm. This amplitude is of the postnewtonian magnitude and as a consequence, the deviation of the Earth movement from the geodesical movement can be observed in the experiment possessing the postnewtonian accuracy. The difference between the acceleration of the Earth mass center and that of a test body in the postnewtonian approximation is equal to 10 -7 part of the Earth acceleration. The ratio of the passive gravitational mass of the Earth (defined according to Will) and its inert mass differs from 1 by 10 -8 approximately [ru
Riemann Integration
and that this should be true, no matter how the in- terval [a, b] is subdivided. ..... Moreover, J: 1 is the unique number with this property. We do not know which ..... as some of our previous demonstrations illustrate, the details of the argument ...
Approximate Riemann solvers and flux vector splitting schemes for two-phase flow; Solveurs de Riemann approches et schemas de decentrement de flux pour les ecoulements diphasiques
Toumi, I.; Kumbaro, A.; Paillere, H
These course notes, presented at the 30. Von Karman Institute Lecture Series in Computational Fluid Dynamics, give a detailed and through review of upwind differencing methods for two-phase flow models. After recalling some fundamental aspects of two-phase flow modelling, from mixture model to two-fluid models, the mathematical properties of the general 6-equation model are analysed by examining the Eigen-structure of the system, and deriving conditions under which the model can be made hyperbolic. The following chapters are devoted to extensions of state-of-the-art upwind differencing schemes such as Roe's Approximate Riemann Solver or the Characteristic Flux Splitting method to two-phase flow. Non-trivial steps in the construction of such solvers include the linearization, the treatment of non-conservative terms and the construction of a Roe-type matrix on which the numerical dissipation of the schemes is based. Extension of the 1-D models to multi-dimensions in an unstructured finite volume formulation is also described; Finally, numerical results for a variety of test-cases are shown to illustrate the accuracy and robustness of the methods. (authors)
Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg-de Vries hydrodynamical equations
Prykarpatsky, Anatoliy K [Department of Mining Geodesy, AGH University of Science and Technology, Cracow 30059 (Poland); Artemovych, Orest D [Department of Algebra and Topology, Faculty of Mathematics and Informatics of the Vasyl Stefanyk Pre-Carpathian National University, Ivano-Frankivsk (Ukraine); Popowicz, Ziemowit [Institute of Theoretical Physics, University of Wroclaw (Poland); Pavlov, Maxim V, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Department of Mathematical Physics, P.N. Lebedev Physical Institute, 53 Leninskij Prospekt, Moscow 119991 (Russian Federation)
A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax-type integrability of the well-known Korteweg-de Vries dynamical system.
Prykarpatsky, Anatoliy K; Artemovych, Orest D; Popowicz, Ziemowit; Pavlov, Maxim V
Existence and Solution-representation of IVP for LFDE with Generalized Riemann-Liouville fractional derivatives and $n$ terms
Kim, Myong-Ha; Ri, Guk-Chol; O, Hyong-Chol
This paper provides the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski's type. We prove that the initial value problem has the solution of if and only if some initial values should be zero.
Differential Galois theory through Riemann-Hilbert correspondence an elementary introduction
Sauloy, Jacques
Differential Galois theory is an important, fast developing area which appears more and more in graduate courses since it mixes fundamental objects from many different areas of mathematics in a stimulating context. For a long time, the dominant approach, usually called Picard-Vessiot Theory, was purely algebraic. This approach has been extensively developed and is well covered in the literature. An alternative approach consists in tagging algebraic objects with transcendental information which enriches the understanding and brings not only new points of view but also new solutions. It is very powerful and can be applied in situations where the Picard-Vessiot approach is not easily extended. This book offers a hands-on transcendental approach to differential Galois theory, based on the Riemann-Hilbert correspondence. Along the way, it provides a smooth, down-to-earth introduction to algebraic geometry, category theory and tannakian duality. Since the book studies only complex analytic linear differential equat...
A boundary-fitted staggered difference method for incompressible flow using Riemann geometry
Koshizuka, Seiichi; Kondo, Shunsuke; Oka, Yoshiaki.
A boundary-fitted staggered difference method (BFSDM) is investigated for incompressible flow in nuclear plants. BFSDM employs control cells for scalars, staggered location of velocity components, and integrated formulation of div=0. Governing equations are written as coordinate-free forms using Riemann geometry. Flow velocity is represented with contravariant physical components in the present method. Connection terms emerge as source terms in the coordinate-free governing equations. These terms are studied from the viewpoints of physical meaning, numerical stability, and conservative property. Some flows on a round or slant boundary are solved using boundary-fitted curvilinear (BFC) grids and rectangular grids to compare the present method and the rectangular-type (R-type) staggered difference method (SDM). Supercomputing of the present method, including vector processing, is also discussed compared with the R-type method. (author)
Riemann-Liouville integrals of fractional order and extended KP hierarchy
Kamata, Masaru; Nakamula, Atsushi
An attempt to formulate the extensions of the KP hierarchy by introducing fractional-order pseudo-differential operators is given. In the case of the extension with the half-order pseudo-differential operators, a system analogous to the supersymmetric extensions of the KP hierarchy is obtained. Unlike the supersymmetric extensions, no Grassmannian variable appears in the hierarchy considered here. More general hierarchies constructed by the 1/Nth-order pseudo-differential operators, their integrability and the reduction procedure are also investigated. In addition to finding the new extensions of the KP hierarchy, a brief introduction to the Riemann-Liouville integral is provided to yield a candidate for the fractional-order pseudo-differential operators
Scattering analysis of asymmetric metamaterial resonators by the Riemann-Hilbert approach
Kaminski, Piotr Marek; Ziolkowski, Richard W.; Arslanagic, Samel
This work presents an analytical treatment of an asymmetric metamaterial-based resonator excited by an electric line source, and explores its beam shaping capabilities. The resonator consists of two concentric cylindrical material layers covered with an infinitely thin conducting shell with an ap......This work presents an analytical treatment of an asymmetric metamaterial-based resonator excited by an electric line source, and explores its beam shaping capabilities. The resonator consists of two concentric cylindrical material layers covered with an infinitely thin conducting shell...... with an aperture. Exact analytical solution of the problem is derived; it is based on the n-series approach which is casted into the equivalent Riemann-Hilbert problem. The examined configuration leads to large enhancements of the radiated field and to steerable Huygens-like directivity patterns. Particularly...
Eigenfunctions and Eigenvalues for a Scalar Riemann-Hilbert Problem Associated to Inverse Scattering
Pelinovsky, Dmitry E.; Sulem, Catherine
A complete set of eigenfunctions is introduced within the Riemann-Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schrödinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation.
Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations
Li Jiequan; Li Qibing; Xu Kun
The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an
On the fractional calculus of Besicovitch function
Liang Yongshun
Relationship between fractional calculus and fractal functions has been explored. Based on prior investigations dealing with certain fractal functions, fractal dimensions including Hausdorff dimension, Box dimension, K-dimension and Packing dimension is shown to be a linear function of order of fractional calculus. Both Riemann-Liouville fractional calculus and Weyl-Marchaud fractional derivative of Besicovitch function have been discussed.
Implicit approximate Riemann solver for two fluid two phase flow models
Raymond, P.; Toumi, I.; Kumbaro, A.
This paper is devoted to the description of new numerical methods developed for the numerical treatment of two phase flow models with two velocity fields which are now widely used in nuclear engineering for design or safety calculations. These methods are finite volumes numerical methods and are based on the use of Approximate Riemann Solver's concepts in order to define convective flux versus mean cell quantities. The first part of the communication will describe the numerical method for a three dimensional drift flux model and the extensions which were performed to make the numerical scheme implicit and to have fast running calculations of steady states. Such a scheme is now implemented in the FLICA-4 computer code devoted to 3-D steady state and transient core computations. We will present results obtained for a steady state flow with rod bow effect evaluation and for a Steam Line Break calculation were the 3-D core thermal computation was coupled with a 3-D kinetic calculation and a thermal-hydraulic transient calculation for the four loops of a Pressurized Water Reactor. The second part of the paper will detail the development of an equivalent numerical method based on an approximate Riemann Solver for a two fluid model with two momentum balance equations for the liquid and the gas phases. The main difficulty for these models is due to the existence of differential modelling terms such as added mass effects or interfacial pressure terms which make hyperbolic the model. These terms does not permit to write the balance equations system in a conservative form, and the classical theory for discontinuity propagation for non-linear systems cannot be applied. Meanwhile, the use of non-conservative products theory allows the study of discontinuity propagation for a non conservative model and this will permit the construction of a numerical scheme for two fluid two phase flow model. These different points will be detailed in that section which will be illustrated by
Formulation of dynamical theory of X-ray diffraction for perfect crystals in the Laue case using the Riemann surface.
Saka, Takashi
The dynamical theory for perfect crystals in the Laue case was reformulated using the Riemann surface, as used in complex analysis. In the two-beam approximation, each branch of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg condition and the reflection strength. By representing these parameters on complex planes, these characteristics can be graphically depicted on the Riemann surface. In the conventional case, the absorption is small and the real part of the reflection strength is large, so the formulation is the same as the traditional analysis. However, when the real part of the reflection strength is small or zero, the two branches of the dispersion surface cross, and the dispersion relationship becomes similar to that of the Bragg case. This is because the geometrical relationships among the parameters are similar in both cases. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters. Furthermore, the present method analytically revealed many characteristic features of the dispersion surface and will be quite instructive for further numerical calculations of rocking curves.
Cylinder renormalization for Siegel disks and a constructive Measurable Riemann Mapping Theorem
Gaydashev, D G
The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not been proved yet. The present work considers a cylinder renormalization - a novel type of renormalization for holomorphic maps with a Siegel disk which is better suited for a hyperbolicity proof. A key element of a cylinder renormalization of a holomorphic map is a conformal isomorphism of a dynamical quotient of a subset of $\\field{C}$ to a bi-infinite cylinder $\\field{C} / \\field{Z}$. A construction of this conformal isomorphism is an implicit procedure which can be performed using the Measurable Riemann Mapping Theorem. We present a constructive proof of the Mea...
Approximate Riemann solvers and flux vector splitting schemes for two-phase flow
Toumi, I.; Kumbaro, A.; Paillere, H.
A geometric construction of the Riemann scalar curvature in Regge calculus
McDonald, Jonathan R.; Miller, Warner A.
The Riemann scalar curvature plays a central role in Einstein's geometric theory of gravity. We describe a new geometric construction of this scalar curvature invariant at an event (vertex) in a discrete spacetime geometry. This allows one to constructively measure the scalar curvature using only clocks and photons. Given recent interest in discrete pre-geometric models of quantum gravity, we believe is it ever so important to reconstruct the curvature scalar with respect to a finite number of communicating observers. This derivation makes use of a new fundamental lattice cell built from elements inherited from both the original simplicial (Delaunay) spacetime and its circumcentric dual (Voronoi) lattice. The orthogonality properties between these two lattices yield an expression for the vertex-based scalar curvature which is strikingly similar to the corresponding hinge-based expression in Regge calculus (deficit angle per unit Voronoi dual area). In particular, we show that the scalar curvature is simply a vertex-based weighted average of deficits per weighted average of dual areas.
3D staggered Lagrangian hydrodynamics scheme with cell-centered Riemann solver-based artificial viscosity
Loubere, Raphael; Maire, Pierre-Henri; Vachal, Pavel
The aim of the present work is the 3D extension of a general formalism to derive a staggered discretization for Lagrangian hydrodynamics on unstructured grids. The classical compatible discretization is used; namely, momentum equation is discretized using the fundamental concept of subcell forces. Specific internal energy equation is obtained using total energy conservation. The subcell force is derived by invoking the Galilean invariance and thermodynamic consistency. A general form of the subcell force is provided so that a cell entropy inequality is satisfied. The subcell force consists of a classical pressure term plus a tensorial viscous contribution proportional to the difference between the node velocity and the cell-centered velocity. This cell-centered velocity is an extra degree of freedom solved with a cell-centered approximate Riemann solver. The second law of thermodynamics is satisfied by construction of the local positive definite subcell tensor involved in the viscous term. A particular expression of this tensor is proposed. A more accurate extension of this discretization both in time and space is also provided using a piecewise linear reconstruction of the velocity field and a predictor-corrector time discretization. Numerical tests are presented in order to assess the efficiency of this approach in 3D. Sanity checks show that the 3D extension of the 2D approach reproduces 1D and 2D results. Finally, 3D problems such as Sedov, Noh, and Saltzman are simulated. (authors)
McDonald, Jonathan R; Miller, Warner A
The Riemann scalar curvature plays a central role in Einstein's geometric theory of gravity. We describe a new geometric construction of this scalar curvature invariant at an event (vertex) in a discrete spacetime geometry. This allows one to constructively measure the scalar curvature using only clocks and photons. Given recent interest in discrete pre-geometric models of quantum gravity, we believe is it ever so important to reconstruct the curvature scalar with respect to a finite number of communicating observers. This derivation makes use of a new fundamental lattice cell built from elements inherited from both the original simplicial (Delaunay) spacetime and its circumcentric dual (Voronoi) lattice. The orthogonality properties between these two lattices yield an expression for the vertex-based scalar curvature which is strikingly similar to the corresponding hinge-based expression in Regge calculus (deficit angle per unit Voronoi dual area). In particular, we show that the scalar curvature is simply a vertex-based weighted average of deficits per weighted average of dual areas
On the properties of torsions in Riemann-Cartan space-times
Baker, W.M.; Atkins, W.K.; Davis, W.R.
This paper is the first paper in a series of three papers dealing with the physical properties of torsions in Riemann-Cartan space-times (U 4 ). Paper one deals with the particular types of torsion that can be associated with the U 4 reinterpretation of a special class of null electromagnetic solutions of the standard form of Einstein's equations. In particular, for plane null electromagnetic solutions, three types of torsion solutions are associated with this type of reinterpretation. Two of these solutions, the trivector and semi-symmetric torsions, although rather special, serve as examples of what could be done to find the associated torsions in terms of simple requirements on identities in U 4 . The third class is obtained by relating the contorsion to the Lanczos ''spin'' tensor. Paper two, dealing with gravitational radiation, provides the proper background relating to the physical significance of the Lanczos tensor. This series of papers is primarily concerned with the question of the possible physical role of all types of torsion, compatible with an extension or an U 4 reinterpretation of Einstein's theory, consistent with the broadest possible interpretation of the present form of the Einstein-Cartan-Sciama-Kibble theory. However, in paper three some consideration will be given on theories with simpler metrical generalizations of U 4 and the related types of torsion. We emphasize that the content of paper one and two should be viewed mainly as special formal results that introduce the more general considerations of paper three
Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
Duca, Vittorio Del [Institute for Theoretical Physics, ETH Zürich,Hönggerberg, 8093 Zürich (Switzerland); Druc, Stefan; Drummond, James [School of Physics & Astronomy, University of Southampton,Highfield, Southampton, SO17 1BJ (United Kingdom); Duhr, Claude [Theoretical Physics Department, CERN,Route de Meyrin, CH-1211 Geneva 23 (Switzerland); Center for Cosmology, Particle Physics and Phenomenology (CP3),Université catholique de Louvain,Chemin du Cyclotron 2, 1348 Louvain-La-Neuve (Belgium); Dulat, Falko [SLAC National Accelerator Laboratory, Stanford University,Stanford, CA 94309 (United States); Marzucca, Robin [Center for Cosmology, Particle Physics and Phenomenology (CP3),Université catholique de Louvain,Chemin du Cyclotron 2, 1348 Louvain-La-Neuve (Belgium); Papathanasiou, Georgios [SLAC National Accelerator Laboratory, Stanford University,Stanford, CA 94309 (United States); Verbeek, Bram [Center for Cosmology, Particle Physics and Phenomenology (CP3),Université catholique de Louvain,Chemin du Cyclotron 2, 1348 Louvain-La-Neuve (Belgium)
We show that scattering amplitudes in planar N=4 Super Yang-Mills in multi-Regge kinematics can naturally be expressed in terms of single-valued iterated integrals on the moduli space of Riemann spheres with marked points. As a consequence, scattering amplitudes in this limit can be expressed as convolutions that can easily be computed using Stokes' theorem. We apply this framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove that at L loops all MHV amplitudes are determined by amplitudes with up to L+4 external legs. We also investigate non-MHV amplitudes, and we show that they can be obtained by convoluting the MHV results with a certain helicity flip kernel. We classify all leading singularities that appear at LLA in the Regge limit for arbitrary helicity configurations and any number of external legs. Finally, we use our new framework to obtain explicit analytic results at LLA for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to eight external legs and four loops.
AdS5 solutions from M5-branes on Riemann surface and D6-branes sources
Bah, Ibrahima [Department of Physics and Astronomy, University of Southern California,Los Angeles, CA 90089 (United States); Institut de Physique Théorique, CEA/Saclay,91191 Gif-sur-Yvette (France)
We describe the gravity duals of four-dimensional N=1 superconformal field theories obtained by wrapping M5-branes on a punctured Riemann surface. The internal geometry, normal to the AdS{sub 5} factor, generically preserves two U(1)s, with generators (J{sup +},J{sup −}), that are fibered over the Riemann surface. The metric is governed by a single potential that satisfies a version of the Monge-Ampère equation. The spectrum of N=1 punctures is given by the set of supersymmetric sources of the potential that are localized on the Riemann surface and lead to regular metrics near a puncture. We use this system to study a class of punctures where the geometry near the sources corresponds to M-theory description of D6-branes. These carry a natural (p,q) label associated to the circle dual to the killing vector pJ{sup +}+qJ{sup −} which shrinks near the source. In the generic case the world volume of the D6-branes is AdS{sub 5}×S{sup 2} and they locally preserve N=2 supersymmetry. When p=−q, the shrinking circle is dual to a flavor U(1). The metric in this case is non-degenerate only when there are co-dimension one sources obtained by smearing M5-branes that wrap the AdS{sub 5} factor and the circle dual the superconformal R-symmetry. The D6-branes are extended along the AdS{sub 5} and on cups that end on the co-dimension one branes. In the special case when the shrinking circle is dual to the R-symmetry, the D6-branes are extended along the AdS{sub 5} and wrap an auxiliary Riemann surface with an arbitrary genus. When the Riemann surface is compact with constant curvature, the system is governed by a Monge-Ampère equation.
Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme
Doliwa, A.; Grinevich, P.; Nieszporski, M.; Santini, P. M.
We present the sublattice approach, a procedure to generate, from a given integrable lattice, a sublattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sublattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable discrete geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). Therefore an integrable, at one energy, discretization of elliptic two-dimensional operators is considered. We use the sublattice point of view to derive, from the Darboux transformations and superposition formulas of the discrete Moutard equation, the Darboux transformations and superposition formulas of the self-adjoint 5-point scheme. We also construct, from algebro-geometric solutions of the discrete Moutard equation, algebro-geometric solutions of the self-adjoint 5-point scheme. In particular, we show that the corresponding restrictions on the finite-gap data are of the same type as those for the fixed energy problem for the two-dimensional Schroedinger operator. We finally use these solutions to construct explicit examples of discrete holomorphic and harmonic functions, as well as examples of quadrilateral surfaces in R 3
Note on asymptotic series expansions for the derivative of the Hurwitz zeta function and related functions
Rudaz, S.
Asymptotic series for the Hurwitz zeta function, its derivative, and related functions (including the Riemann zeta function of odd integer argument) are derived as an illustration of a simple, direct method of broad applicability, inspired by the calculus of finite differences
Algebraic functions
Bliss, Gilbert Ames
This book, immediately striking for its conciseness, is one of the most remarkable works ever produced on the subject of algebraic functions and their integrals. The distinguishing feature of the book is its third chapter, on rational functions, which gives an extremely brief and clear account of the theory of divisors.... A very readable account is given of the topology of Riemann surfaces and of the general properties of abelian integrals. Abel's theorem is presented, with some simple applications. The inversion problem is studied for the cases of genus zero and genus unity. The chapter on t
Adaptive spacetime method using Riemann jump conditions for coupled atomistic-continuum dynamics
Kraczek, B.; Miller, S.T.; Haber, R.B.; Johnson, D.D.
We combine the Spacetime Discontinuous Galerkin (SDG) method for elastodynamics with the mathematically consistent Atomistic Discontinuous Galerkin (ADG) method in a new scheme that concurrently couples continuum and atomistic models of dynamic response in solids. The formulation couples non-overlapping continuum and atomistic models across sharp interfaces by weakly enforcing jump conditions, for both momentum balance and kinematic compatibility, using Riemann values to preserve the characteristic structure of the underlying hyperbolic system. Momentum balances to within machine-precision accuracy over every element, on each atom, and over the coupled system, with small, controllable energy dissipation in the continuum region that ensures numerical stability. When implemented on suitable unstructured spacetime grids, the continuum SDG model offers linear computational complexity in the number of elements and powerful adaptive analysis capabilities that readily bridge between atomic and continuum scales in both space and time. A special trace operator for the atomic velocities and an associated atomistic traction field enter the jump conditions at the coupling interface. The trace operator depends on parameters that specify, at the scale of the atomic spacing, the position of the coupling interface relative to the atoms. In a key finding, we demonstrate that optimizing these parameters suppresses spurious reflections at the coupling interface without the use of non-physical damping or special boundary conditions. We formulate the implicit SDG-ADG coupling scheme in up to three spatial dimensions, and describe an efficient iterative solution scheme that outperforms common explicit schemes, such as the Velocity Verlet integrator. Numerical examples, in 1dxtime and employing both linear and nonlinear potentials, demonstrate the performance of the SDG-ADG method and show how adaptive spacetime meshing reconciles disparate time steps and resolves atomic-scale signals in
Riemann solvers for multi-component gas mixtures with temperature dependent heat capacities; Solveurs de riemann pour des melanges de gaz parfaits avec capacites calorifiques dependant de la temperature
Beccantini, A
Rational Functions with a General Distribution of Poles on the Real Line Orthogonal with Respect to Varying Exponential Weights: I
McLaughlin, K. T.-R.; Vartanian, A. H.; Zhou, X.
Orthogonal rational functions are characterized in terms of a family of matrix Riemann-Hilbert problems on R, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures which solve the energy minimisation problems are established. These measures are used to derive a family of 'model' matrix Riemann-Hilbert problems which are amenable to asymptotic analysis via the Deift-Zhou non-linear steepest-descent method
On a Nonlocal Ostrovsky-Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability
Jolanta Golenia
Full Text Available Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The well defined regularization of the model is constructed and its Lax type integrability is discussed. A generalized hydrodynamical Riemann type system is considered, infinite hierarchies of conservation laws, related compatible Poisson structures and a Lax type representation for the special case N=3 are constructed.
Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo
Bui-Thanh, T.; Girolami, M.
We consider the Riemann manifold Hamiltonian Monte Carlo (RMHMC) method for solving statistical inverse problems governed by partial differential equations (PDEs). The Bayesian framework is employed to cast the inverse problem into the task of statistical inference whose solution is the posterior distribution in infinite dimensional parameter space conditional upon observation data and Gaussian prior measure. We discretize both the likelihood and the prior using the H1-conforming finite element method together with a matrix transfer technique. The power of the RMHMC method is that it exploits the geometric structure induced by the PDE constraints of the underlying inverse problem. Consequently, each RMHMC posterior sample is almost uncorrelated/independent from the others providing statistically efficient Markov chain simulation. However this statistical efficiency comes at a computational cost. This motivates us to consider computationally more efficient strategies for RMHMC. At the heart of our construction is the fact that for Gaussian error structures the Fisher information matrix coincides with the Gauss-Newton Hessian. We exploit this fact in considering a computationally simplified RMHMC method combining state-of-the-art adjoint techniques and the superiority of the RMHMC method. Specifically, we first form the Gauss-Newton Hessian at the maximum a posteriori point and then use it as a fixed constant metric tensor throughout RMHMC simulation. This eliminates the need for the computationally costly differential geometric Christoffel symbols, which in turn greatly reduces computational effort at a corresponding loss of sampling efficiency. We further reduce the cost of forming the Fisher information matrix by using a low rank approximation via a randomized singular value decomposition technique. This is efficient since a small number of Hessian-vector products are required. The Hessian-vector product in turn requires only two extra PDE solves using the adjoint
(Anti-) selfdual Riemann curvature tensor in four spacelike compactified dimensions, O5 isometry group and chiral fermion zero modes
Minkowski, P.
The metric and contorsion tensors are constructed which yield a combined Riemann curvature tensor of the form Rsup(+-)sub(μνsigmatau)=(1/2a 2 )(gsub(μsigma)gsub(νtau) - gsub(μtau)gsub(νsigma)+-√g epsilonsub(μνsigmatau)). The metric with euclidean signature (++++) describes a sphere S 4 with radius a, i.e. admits the isometry group O5. For selfdual (antiselfdual) curvature tensor the contorsion tensor is given by the antiselfdual (selfdual) instanton configuration with respect to the spin gauge group SU2sub(R) (SU2sub(L)). The selfdual (antiselfdual) Riemann tensor admits two covariantly constant right-handed (left-handed) spin 1/2 fermion zero modes, one J=1/2 and one J=3/2 right-handed (left-handed) multiplet corresponding to L=1, transforming as a pseudoreal representation of O4 (SU2sub(R(L))). The hermitean Dirac equation retains only the two constant chiral modes. (orig.)
A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces
Pioline, Boris
The Kawazumi-Zhang invariant $\\varphi$ for compact genus-two Riemann surfaces was recently shown to be a eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior of $\\varphi$ in the non-separating degeneration limit, it is shown that $\\varphi$ is equal to the Theta lift of the unique (up to normalization) weak Jacobi form of weight $-2$. This identification provides the complete Fourier-Jacobi expansion of $\\varphi$ near the non-separating node, gives full control on the asymptotics of $\\varphi$ in the various degeneration limits, and provides a efficient numerical procedure to evaluate $\\varphi$ to arbitrary accuracy. It also reveals a mock-type holomorphic Siegel modular form of weight $-2$ underlying $\\varphi$. From the general relation between the Faltings invariant, the Kawazumi-Zhang invariant and the discriminant for hyperelliptic Riemann surfaces, a Theta lift representation for the Faltings invariant in genus two ...
La notion husserlienne de multiplicité : au-delà de Cantor et Riemann
Carlo Ierna
Full Text Available En raison du rôle changeant qu'il joue dans les différents ouvrages de Husserl, le concept de Mannigfaltigkeit afait l'objet de nombreuses interprétations. La présence de ce terme a notamment induit en erreur plusieurs commentateurs, qui ont cru en déterminer l'origine dans les années de Halle, à l'époque où Husserl, ami et collègue de Cantor, rédigeait la Philosophie de l'arithmétique. Mais force est de constater qu'à cette époque Husserl s'était déjà ouvertement éloigné de la définition cantorienne de Mannigfaltigkeit en s'approchant plutôt de Riemann, comme le montrent les nombreuses études et leçons qui lui sont consacrées. La Mannigfaltigkeitslehre de Husserl semble donc plus proche de la topologie que de la théorie des ensembles de Cantor. Ainsi, dans les Prolégomènes, Husserl introduit l'idée d'une Mannigfaltigkeitslehre pure en tant qu'entreprise méta-théorique dont le but est d'étudier les relations entre théories, à savoir la manière par laquelle une théorie est dérivée ou fondée à partir d'une autre. Dès lors, lorsque Husserl affirme que le meilleur exemple d'une telle théorie pure des multiplicités se trouve dans les mathématiques, cela risque donc de prêter à confusion. En effet, la théorie pure des théories ne saurait être simplement identifiée aux mathématiques qui relèvent de la topologie, mais considérée en tant que mathesis universalis. Bien qu'une telle position ne fût sans doute pas entièrement claire en 1900-01, Husserl ne tardera pas à relier explicitement théorie des multiplicités et mathesis universalis.En ce sens, la mathesis universalis, théorie des théories en général, est une discipline formelle, apriori et analytique qui a pour but l'analyse des catégories sémantiques suprêmes et des catégories d'objets qui leur sont corrélées. Dans cet article j'essayerai de comprendre le développement de la notion de
Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers
Several advances have been reported in the recent literature on divergence-free finite volume schemes for Magnetohydrodynamics (MHD). Almost all of these advances are restricted to structured meshes. To retain full geometric versatility, however, it is also very important to make analogous advances in divergence-free schemes for MHD on unstructured meshes. Such schemes utilize a staggered Yee-type mesh, where all hydrodynamic quantities (mass, momentum and energy density) are cell-centered, while the magnetic fields are face-centered and the electric fields, which are so useful for the time update of the magnetic field, are centered at the edges. Three important advances are brought together in this paper in order to make it possible to have high order accurate finite volume schemes for the MHD equations on unstructured meshes. First, it is shown that a divergence-free WENO reconstruction of the magnetic field can be developed for unstructured meshes in two and three space dimensions using a classical cell-centered WENO algorithm, without the need to do a WENO reconstruction for the magnetic field on the faces. This is achieved via a novel constrained L2-projection operator that is used in each time step as a postprocessor of the cell-centered WENO reconstruction so that the magnetic field becomes locally and globally divergence free. Second, it is shown that recently-developed genuinely multidimensional Riemann solvers (called MuSIC Riemann solvers) can be used on unstructured meshes to obtain a multidimensionally upwinded representation of the electric field at each edge. Third, the above two innovations work well together with a high order accurate one-step ADER time stepping strategy, which requires the divergence-free nonlinear WENO reconstruction procedure to be carried out only once per time step. The resulting divergence-free ADER-WENO schemes with MuSIC Riemann solvers give us an efficient and easily-implemented strategy for divergence-free MHD on
On calculation of zeta function of integral matrix
Janá�ek, Jiří
Ro�. 134, �. 1 (2009), s. 49-58 ISSN 0862-7959 R&D Projects: GA AV ČR(CZ) IAA100110502 Institutional research plan: CEZ:AV0Z50110509 Keywords : Epstein zeta function * integral lattice * Riemann theta function Subject RIV: BA - General Mathematics
Effective action and β-functions for the heterotic string
Foakes, A.P.; Mohammedi, N.; Ross, D.A.
The results of the calculation of the metric β-function for the heterotic string sigma model up to three loops are presented and it is shown that although this β-function is non vanishing it is compatible with an O((α') 2 ) effective action in which there are no terms cubic in the Riemann tensor or gauge field strength. (orig.)
Extension problem for generalized multi-monogenic functions in Clifford analysis
Tran Quyet Thang.
The main purpose of this paper is to extend some properties of multi-monogenic functions, which is a generalization of monogenic functions in higher dimensions, for a class of functions satisfying Vekua-type generalized Cauchy-Riemann equations in Clifford Analysis. It is proved that the Hartogs theorem is valid for these functions. (author). 7 refs
Efficient analytical implementation of the DOT Riemann solver for the de Saint Venant-Exner morphodynamic model
Carraro, F.; Valiani, A.; Caleffi, V.
Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant-Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.
Fractional derivative of the Hurwitz ζ-function and chaotic decay to zero
C. Cattani
Full Text Available In this paper the fractional order derivative of a Dirichlet series, Hurwitz zeta function and Riemann zeta function is explicitly computed using the Caputo fractional derivative in the Ortigueira sense. It is observed that the obtained results are a natural generalization of the integer order derivative. Some interesting properties of the fractional derivative of the Riemann zeta function are also investigated to show that there is a chaotic decay to zero (in the Gaussian plane and a promising expression as a complex power series.
On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking
Manakov, S V; Santini, P M
We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev-Petviashvili (dKP) equation. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann-Hilbert problems on the real axis. In this paper, we make use of the nonlinear Riemann-Hilbert problem of dKP (i) to construct the longtime behaviour of the solutions of its Cauchy problem; (ii) to characterize a class of implicit solutions; (iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking
Manakov, S V [Landau Institute for Theoretical Physics, Moscow (Russian Federation); Santini, P M [Dipartimento di Fisica, Universita di Roma ' La Sapienza' , and Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, Piazz.le Aldo Moro 2, I-00185 Rome (Italy)
We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev-Petviashvili (dKP) equation. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann-Hilbert problems on the real axis. In this paper, we make use of the nonlinear Riemann-Hilbert problem of dKP (i) to construct the longtime behaviour of the solutions of its Cauchy problem; (ii) to characterize a class of implicit solutions; (iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.
Extending the Riemann-Solver-Free High-Order Space-Time Discontinuous Galerkin Cell Vertex Scheme (DG-CVS) to Solve Compressible Magnetohydrodynamics Equations
Ideal Magnetohydrodynamics,� J. Com- put. Phys., Vol. 153, No. 2, 1999, pp. 334–352. [14] Tang, H.-Z. and Xu, K., "A high-order gas -kinetic method for...notwithstanding any other provision of law , no person shall be subject to any penalty for failing to comply with a collection of information if it does...Riemann-solver-free spacetime discontinuous Galerkin method for general conservation laws to solve compressible magnetohydrodynamics (MHD) equations. The
Functional geometric method for solving free boundary problems for harmonic functions
Demidov, Aleksander S [M. V. Lomonosov Moscow State University, Moscow (Russian Federation)
A survey is given of results and approaches for a broad spectrum of free boundary problems for harmonic functions of two variables. The main results are obtained by the functional geometric method. The core of these methods is an interrelated analysis of the functional and geometric characteristics of the problems under consideration and of the corresponding non-linear Riemann-Hilbert problems. An extensive list of open questions is presented. Bibliography: 124 titles.
Complex function theory
Sarason, Donald
Complex Function Theory is a concise and rigorous introduction to the theory of functions of a complex variable. Written in a classical style, it is in the spirit of the books by Ahlfors and by Saks and Zygmund. Being designed for a one-semester course, it is much shorter than many of the standard texts. Sarason covers the basic material through Cauchy's theorem and applications, plus the Riemann mapping theorem. It is suitable for either an introductory graduate course or an undergraduate course for students with adequate preparation. The first edition was published with the title Notes on Co
Changing gauge of second type for the election of Landau and tangential Cauchy-Riemann equations
Laville, Guy
For any type choice of gauge it is possible to get an equality between the Hamiltonian of a charge particle and an operator of second order which is associated with the boundary values of holomorphic functions of two complex variables [fr
Evaluation of integrals with hypergeometric and logarithmic functions
Sofo Anthony
Full Text Available We provide an explicit analytical representation for a number of logarithmic integrals in terms of the Lerch transcendent function and other special functions. The integrals in question will be associated with both alternating harmonic numbers and harmonic numbers with positive terms. A few examples of integrals will be given an identity in terms of some special functions including the Riemann zeta function. In general none of these integrals can be solved by any currently available mathematical package.
The Riemann Surface of Static Limit Dispersion Relation and Projective Spaces
Majewski, M; Meshcheryakov, D V; Tran Quang Tuyet
The rigorous Bogoliubov's prove of the dispersion relations (DR) for pion-nucleon scattering is a good foundation for the static models. DR contain the small parameter (ratio of the pion-nucleon masses). The static models arise when this parameter goes to zero. The S-matrix in the static models has a block structure. Each block of the S-matrix has a finite order N\\times N and is a matrix of meromorphic functions of the light particle energy \\omega in the complex plane with cuts (-\\inf,-1], [+1, +\\inf). In the elastic case, it reduces to N functions S_{i}(\\omega) connected by N\\times N the crossing-symmetry matrix A. The unitarity and the crossing symmetry are the base for the system of nonlinear boundary value problems. It defines the analytical continuation of S_{i}(\\omega) from the physical sheet to the unphysical ones and can be treated as a system of nonlinear difference equations. The problem is solvable for any 2\\times 2 crossing-symmetry matrix A that permits one to calculate the Regge trajectories for...
Majewski, M.; Meshcheryakov, V.A.; Meshcheryakov, D.V.; Tran Quang Tuyet
The rigorous Bogolyubov's proof of the dispersion relations (DR) for pion-nucleon scattering is a good foundation for the static models. DR contain a small parameter (ratio of the pion-nucleon masses). The static models arise when this parameter goes to zero. The S-matrix in the static models has a block structure. Each block of the S-matrix has a finite order NxN and is a matrix of meromorphic functions of the light particle energy ω in the complex plane with cuts (-∞, -1], [+1,+∞). In the elastic case, it reduces to N functions S i (ω) connected by the NxN crossing-symmetry matrix A. The unitarity and the crossing symmetry are the base for the system of nonlinear boundary value problems. It defines the analytical continuation of S i (ω) from the physical sheet to the unphysical ones and can be treated as a system of nonlinear difference equations. The problem is solvable for any 2x2 crossing-symmetry matrix A that permits one to calculate the Regge trajectories for the SU(2) static model. It is shown that global analyses of this system can be carried out effectively in projective spaces P N-1 and P N . The connection between the spaces P N-1 and P N is discussed. Some particular solutions of the system are found
Controlling lightwave in Riemann space by merging geometrical optics with transformation optics.
Liu, Yichao; Sun, Fei; He, Sailing
In geometrical optical design, we only need to choose a suitable combination of lenses, prims, and mirrors to design an optical path. It is a simple and classic method for engineers. However, people cannot design fantastical optical devices such as invisibility cloaks, optical wormholes, etc. by geometrical optics. Transformation optics has paved the way for these complicated designs. However, controlling the propagation of light by transformation optics is not a direct design process like geometrical optics. In this study, a novel mixed method for optical design is proposed which has both the simplicity of classic geometrical optics and the flexibility of transformation optics. This mixed method overcomes the limitations of classic optical design; at the same time, it gives intuitive guidance for optical design by transformation optics. Three novel optical devices with fantastic functions have been designed using this mixed method, including asymmetrical transmissions, bidirectional focusing, and bidirectional cloaking. These optical devices cannot be implemented by classic optics alone and are also too complicated to be designed by pure transformation optics. Numerical simulations based on both the ray tracing method and full-wave simulation method are carried out to verify the performance of these three optical devices.
Two-loop superstring partition function
Morozov, A.Y.
Is it possible to choose the odd moduli on super-Riemann surfaces of genus p≥2 in such a way that the corresponding contributions to the superstring partition function vanish before the integration over the space of the moduli? It is shown that, at least for p = 2, the answer to this question is affirmative, and in this case the odd moduli should be localized at branch points
Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator
Owolabi, Kolade M.
In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain (-∞, ∞), which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.
Isomonodromic tau-functions from Liouville conformal blocks
Iorgov, N.; Lisovyy, O.
The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with SL(2,C)-valued monodromy on Riemann surfaces of genus zero with n punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at c=1. This implies a similar representation for the isomonodromic tau-function. In the case n=4 we thereby get a proof of the relation between tau-functions and conformal blocks discovered in O. Gamayun, N. Iorgov, and O. Lisovyy (2012). We briefly discuss a possible application of our results to the study of relations between certain N=2 supersymmetric gauge theories and conformal field theory.
Bernoulli numbers and zeta functions
Arakawa, Tsuneo; Kaneko, Masanobu
Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of ...
Riemann-Hypothesis Millennium-Problem(MP) Physics Proof via CATEGORY-SEMANTICS(C-S)/F =C Aristotle SQUARE-of-OPPOSITION(SoO) DEduction-LOGIC DichotomY
Baez, Joao-Joan; Lapidaryus, Michelle; Siegel, Edward Carl-Ludwig
Riemann-hypothesis physics-proof combines: Siegel-Antono®-Smith[AMS Joint Mtg.(2002)- Abs.973-03-126] digits on-average statistics HIll[Am. J. Math 123, 3, 887(1996)] logarithm-function's (1,0)- xed-point base =units =scale-invariance proven Newcomb [Am. J. Math. 4, 39(1881)]-Weyl[Goett. Nachr.(1914); Math. Ann.7, 313(1916)]-Benford[Proc. Am. Phil. Soc. 78, 4, 51(1938)]-law [Kac,Math. of Stat.-Reasoning(1955); Raimi, Sci. Am. 221, 109(1969)] algebraic-inversion to ONLY Bose-Einstein quantum-statistics(BEQS) with digit d = 0 gapFUL Bose-Einstein Condensation(BEC) insight that digits are quanta are bosons because bosons are and always were quanta are and always were digits, via Siegel-Baez category-semantics tabular list-format matrix truth-table analytics in Plato-Aristotle classic ''square-of-opposition'' : FUZZYICS =CATEGORYICS/Category-Semantics, with Goodkind Bose-Einstein Condensation (BEC) ABOVE ground-state with/and Rayleigh(cut-limit of ''short-cut method''1870)-Polya(1922)-''Anderson''(1958) localization [Doyle and Snell,Random-Walks and Electrical-Networks, MAA(1981)-p.99-100!!!] in Brillouin[Wave-Propagation in Periodic-Structures(1946) Dover(1922)]-Hubbard-Beeby[J.Phys.C(1967)] Siegel[J.Nonxline-Sol.40,453(1980)] generalized-disorder collective-boson negative-dispersion mode-softening universality-principle(G...P) first use of the ``square-of-opposition'' in physics since Plato and Aristote!!!
Robust fractional order differentiators using generalized modulating functions method
Liu, Dayan; Laleg-Kirati, Taous-Meriem
This paper aims at designing a fractional order differentiator for a class of signals satisfying a linear differential equation with unknown parameters. A generalized modulating functions method is proposed first to estimate the unknown parameters, then to derive accurate integral formulae for the left-sided Riemann-Liouville fractional derivatives of the studied signal. Unlike the improper integral in the definition of the left-sided Riemann-Liouville fractional derivative, the integrals in the proposed formulae can be proper and be considered as a low-pass filter by choosing appropriate modulating functions. Hence, digital fractional order differentiators applicable for on-line applications are deduced using a numerical integration method in discrete noisy case. Moreover, some error analysis are given for noise error contributions due to a class of stochastic processes. Finally, numerical examples are given to show the accuracy and robustness of the proposed fractional order differentiators.
Integral approximants for functions of higher monodromic dimension
Baker, G.A. Jr.
In addition to the description of multiform, locally analytic functions as covering a many sheeted version of the complex plane, Riemann also introduced the notion of considering them as describing a space whose ''monodromic'' dimension is the number of linearly independent coverings by the monogenic analytic function at each point of the complex plane. I suggest that this latter concept is natural for integral approximants (sub-class of Hermite-Pade approximants) and discuss results for both ''horizontal'' and ''diagonal'' sequences of approximants. Some theorems are now available in both cases and make clear the natural domain of convergence of the horizontal sequences is a disk centered on the origin and that of the diagonal sequences is a suitably cut complex-plane together with its identically cut pendant Riemann sheets. Some numerical examples have also been computed.
Liu, Dayan
On Montgomery's pair correlation conjecture to the zeros of Riedmann zeta function
Li, Pei
In this thesis, we are interested in Montgomery's pair correlation conjecture which is about the distribution of.the spacings between consecutive zeros of the Riemann Zeta function. Our goal is to explain and study Montgomery's pair correlation conjecture and discuss its connection with the random matrix theory. In Chapter One, we will explain how to define the Ftiemann Zeta function by using the analytic continuation. After this, several classical properties of the Ftiemann Zeta function wil...
Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions
Zamolodchikov, A.B.
A multipoint conformal block of Ramond states of the two-dimensional free scalar field is calculated. This function is related to the free energy of the scalar field on the hyperelliptic Riemann surface under a particular choice of boundary conditions. Being compactified on the circle this field leads to the crossing symmetric correlation functions with a discrete spectrum of scale dimensions. These functions are supposed to describe multipoint spin correlations of the critical Ashkin-Teller model. (orig.)
Exp-function method for solving fractional partial differential equations.
Zheng, Bin
We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.
1-loop partition function in AdS{sub 3}/CFT{sub 2}
Chen, Bin [Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University,5 Yiheyuan Rd, Beijing 100871 (China); Collaborative Innovation Center of Quantum Matter,5 Yiheyuan Rd, Beijing 100871 (China); Center for High Energy Physics, Peking University,5 Yiheyuan Rd, Beijing 100871 (China); Wu, Jie-qiang [Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University,5 Yiheyuan Rd, Beijing 100871 (China)
The 1-loop partition function of the handlebody solutions in the AdS{sub 3} gravity have been derived some years ago using the heat kernel techniques and the method of images. In the semiclassical limit, such partition function should correspond to the order O(c{sup 0}) part in the partition function of dual conformal field theory(CFT) on the boundary Riemann surface. The higher genus partition function could be computed by the multi-point functions in the Riemann sphere via sewing prescription. In the large central charge limit, the CFT is effectively free in the sense that to the leading order of c the multi-point function is further simplified to be a summation over the products of two-point functions of single-particle states. Correspondingly in the bulk, the graviton is freely propagating without interaction. Furthermore the product of the two-point functions may define the links, each of which is in one-to-one correspondence with the conjugacy class of the Schottky group of the Riemann surface. Moreover, the value of a link is determined by the multiplier of the element in the conjugacy class. This allows us to reproduce exactly the gravitational 1-loop partition function. The proof can be generalized to the higher spin gravity and its dual CFT.
Mayer Transfer Operator Approach to Selberg Zeta Function
Momeni, Arash; Venkov, Alexei
. In a special situation the dynamical zeta function is defined for a geodesic flow on a hyperbolic plane quotient by an arithmetic cofinite discrete group. More precisely, the flow is defined for the corresponding unit tangent bundle. It turns out that the Selberg zeta function for this group can be expressed...... in terms of a Fredholm determinant of a classical transfer operator of the flow. The transfer operator is defined in a certain space of holomorphic functions and its matrix representation in a natural basis is given in terms of the Riemann zeta function and the Euler gamma function....
Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations
Ahmet Bekir
Full Text Available The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.
Scalar field Green functions on causal sets
Nomaan Ahmed, S; Surya, Sumati; Dowker, Fay
We examine the validity and scope of Johnston's models for scalar field retarded Green functions on causal sets in 2 and 4 dimensions. As in the continuum, the massive Green function can be obtained from the massless one, and hence the key task in causal set theory is to first identify the massless Green function. We propose that the 2d model provides a Green function for the massive scalar field on causal sets approximated by any topologically trivial 2-dimensional spacetime. We explicitly demonstrate that this is indeed the case in a Riemann normal neighbourhood. In 4d the model can again be used to provide a Green function for the massive scalar field in a Riemann normal neighbourhood which we compare to Bunch and Parker's continuum Green function. We find that the same prescription can also be used for de Sitter spacetime and the conformally flat patch of anti-de Sitter spacetime. Our analysis then allows us to suggest a generalisation of Johnston's model for the Green function for a causal set approximated by 3-dimensional flat spacetime. (paper)
Two-point correlation function for Dirichlet L-functions
Bogomolny, E.; Keating, J. P.
The two-point correlation function for the zeros of Dirichlet L-functions at a height E on the critical line is calculated heuristically using a generalization of the Hardy-Littlewood conjecture for pairs of primes in arithmetic progression. The result matches the conjectured random-matrix form in the limit as E → ∞ and, importantly, includes finite-E corrections. These finite-E corrections differ from those in the case of the Riemann zeta-function, obtained in Bogomolny and Keating (1996 Phys. Rev. Lett. 77 1472), by certain finite products of primes which divide the modulus of the primitive character used to construct the L-function in question.
Bogomolny, E; Keating, J P
The two-point correlation function for the zeros of Dirichlet L-functions at a height E on the critical line is calculated heuristically using a generalization of the Hardy–Littlewood conjecture for pairs of primes in arithmetic progression. The result matches the conjectured random-matrix form in the limit as E → ∞ and, importantly, includes finite-E corrections. These finite-E corrections differ from those in the case of the Riemann zeta-function, obtained in Bogomolny and Keating (1996 Phys. Rev. Lett. 77 1472), by certain finite products of primes which divide the modulus of the primitive character used to construct the L-function in question. (paper)
Renormalization in quantum field theory and the Riemann-Hilbert problem. I. Hopf algebra structure of graphs and the main theorem
Connes, A.; Kreimer, D.
This paper gives a complete selfcontained proof of our result (1999) showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which is commutative asan algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra G whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of H. We show then that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop γ(z) element of G, z element of C, where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ + of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. (orig.)
Integration a functional approach
Bichteler, Klaus
This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory. From this point of view Lebesgue's integral can be had as a rather straightforward, even simplistic, extension of Riemann's integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood's observations rather than being conveyed authoritatively through definitions of (sigma)-algebras and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It might...
Orthogonal Polynomials and Special Functions
Assche, Walter
The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. The volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring only a basic knowledge of analysis and algebra, and each includes many exercises.
The method of images and Green's function for spherical domains
Gutkin, Eugene; Newton, Paul K
Motivated by problems in electrostatics and vortex dynamics, we develop two general methods for constructing Green's function for simply connected domains on the surface of the unit sphere. We prove a Riemann mapping theorem showing that such domains can be conformally mapped to the upper hemisphere. We then categorize all domains on the sphere for which Green's function can be constructed by an extension of the classical method of images. We illustrate our methods by several examples, such as the upper hemisphere, geodesic triangles, and latitudinal rectangles. We describe the point vortex motion in these domains, which is governed by a Hamiltonian determined by the Dirichlet Green's function
A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism
Balsara, Dinshaw S.; Amano, Takanobu; Garain, Sudip; Kim, Jinho
In various astrophysics settings it is common to have a two-fluid relativistic plasma that interacts with the electromagnetic field. While it is common to ignore the displacement current in the ideal, classical magnetohydrodynamic limit, when the flows become relativistic this approximation is less than absolutely well-justified. In such a situation, it is more natural to consider a positively charged fluid made up of positrons or protons interacting with a negatively charged fluid made up of electrons. The two fluids interact collectively with the full set of Maxwell's equations. As a result, a solution strategy for that coupled system of equations is sought and found here. Our strategy extends to higher orders, providing increasing accuracy. The primary variables in the Maxwell solver are taken to be the facially-collocated components of the electric and magnetic fields. Consistent with such a collocation, three important innovations are reported here. The first two pertain to the Maxwell solver. In our first innovation, the magnetic field within each zone is reconstructed in a divergence-free fashion while the electric field within each zone is reconstructed in a form that is consistent with Gauss' law. In our second innovation, a multidimensionally upwinded strategy is presented which ensures that the magnetic field can be updated via a discrete interpretation of Faraday's law and the electric field can be updated via a discrete interpretation of the generalized Ampere's law. This multidimensional upwinding is achieved via a multidimensional Riemann solver. The multidimensional Riemann solver automatically provides edge-centered electric field components for the Stokes law-based update of the magnetic field. It also provides edge-centered magnetic field components for the Stokes law-based update of the electric field. The update strategy ensures that the electric field is always consistent with Gauss' law and the magnetic field is
Balsara, Dinshaw S., E-mail: [email protected] [Physics Department, University of Notre Dame (United States); Amano, Takanobu, E-mail: [email protected] [Department of Earth and Planetary Science, University of Tokyo, Tokyo 113-0033 (Japan); Garain, Sudip, E-mail: [email protected] [Physics Department, University of Notre Dame (United States); Kim, Jinho, E-mail: [email protected] [Physics Department, University of Notre Dame (United States)
A Walsh Function Module Users' Manual
Gnoffo, Peter A.
The solution of partial differential equations (PDEs) with Walsh functions offers new opportunities to simulate many challenging problems in mathematical physics. The approach was developed to better simulate hypersonic flows with shocks on unstructured grids. It is unique in that integrals and derivatives are computed using simple matrix multiplication of series representations of functions without the need for divided differences. The product of any two Walsh functions is another Walsh function - a feature that radically changes an algorithm for solving PDEs. A FORTRAN module for supporting Walsh function simulations is documented. A FORTRAN code is also documented with options for solving time-dependent problems: an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the usage of the Walsh function module including such features as operator overloading, Fast Walsh Transforms in multi-dimensions, and a Fast Walsh reciprocal.
Particle in the magnetic field: 2D Riemann spherical space and complex analogue of the Poincare half-plane
Red'kov, V.M.; Ovsiyuk, E.M.; Ishkhanyan, A.M.
The Schrodinger particle on the background of the 2D space of the constant positive curvature S 2 , a sphere in the 3D Euclidean space, is considered in the external magnetic field. By analogy with the case of the hyperbolic Lobachevsky plane H 2 , where quasi-Cartesian coordinates exist with the realization of H 2 as the Poincare half-plane, a specific system of quasi-Cartesian coordinates (x, y) in S 2 is introduced. It turns out that it is possible only if the two coordinates are complex and obey an additional restriction in order to present a real 2D space. The Schrodinger equation is solved using the method of separation of the variables in the both coordinate systems, cylindrical and quasi-Cartesian, the energy spectrum is the same. For parametrization of the space S 2 , one can use the coordinates (x, x*) or (y, y*), however, in this case the separability of the variables in the wave functions is lost. Constructed solutions may be of interest for describing charged particles in magnetic fields in the context of cosmological models, and for simulating the behavior of the particles in a specific field-configuration in the nano-physics. (authors)
String operator formalism and functional intergal in the holomorphic representation
Losev, A.S.; Morozov, A.Yu.; Rislyj, A.A.; Shatashvili, S.L.
Connection between the continual integral over open Riemann surfaces and the operator formalism on closed Riemann surfaces is discussed. States of the operator formalism are the holomorphic representation of the continual integral
Geometric function theory: a modern view of a classical subject
Crowdy, Darren
Geometric function theory is a classical subject. Yet it continues to find new applications in an ever-growing variety of areas such as modern mathematical physics, more traditional fields of physics such as fluid dynamics, nonlinear integrable systems theory and the theory of partial differential equations. This paper surveys, with a view to modern applications, open problems and challenges in this subject. Here we advocate an approach based on the use of the Schottky–Klein prime function within a Schottky model of compact Riemann surfaces. (open problem)
Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method
Rahmatullah
Full Text Available We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses. Keywords: Exp-function method, New exact traveling wave solutions, Modified Riemann-Liouville derivative, Fractional complex transformation, Fractional order Boussinesq-like equations, Symbolic computation
Conditions for bound states in a periodic linear chain, and the spectra of a class of Toeplitz operators in terms of polylogarithm functions
Prunele, E de
Conditions for bound states for a periodic linear chain are given within the framework of an exactly solvable non-relativistic quantum-mechanical model in three-dimensional space. These conditions express the strength parameter in terms of the distance between two consecutive centres of the chain, and of the range interaction parameter. This expression can be formulated in terms of polylogarithm functions, and, in some particular cases, in terms of the Riemann zeta function. An interesting mathematical result is that these expressions also correspond to the spectra of Toeplitz complex symmetric operators. The non-trivial zeros of the Riemann zeta function are interpreted as multiple points, at the origin, of the spectra of these Toeplitz operators
Certain fractional integral formulas involving the product of generalized Bessel functions.
Baleanu, D; Agarwal, P; Purohit, S D
We apply generalized operators of fractional integration involving Appell's function F 3(·) due to Marichev-Saigo-Maeda, to the product of the generalized Bessel function of the first kind due to Baricz. The results are expressed in terms of the multivariable generalized Lauricella functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.
Certain Fractional Integral Formulas Involving the Product of Generalized Bessel Functions
Baleanu, D.; Agarwal, P.; Purohit, S. D.
We apply generalized operators of fractional integration involving Appell's function F 3(·) due to Marichev-Saigo-Maeda, to the product of the generalized Bessel function of the first kind due to Baricz. The results are expressed in terms of the multivariable generalized Lauricella functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions. PMID:24379745
Dynamics on the Riemann Sphere
Petersen, Carsten Lunde
Collection of research papers on holomorphic dynamical systems with an introduction to Bodil Branners works on the field. Contributions: John Milnor: On Lattès Maps. Carsten Lunde Petersen and Tan Lei: Branner-Hubbard motions and attracting dynamics. Artur Avila and Mikhail Lyubich: Examples...
Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution - Part II, higher order FVTD schemes
Balsara, Dinshaw S.; Garain, Sudip; Taflove, Allen; Montecinos, Gino
The Finite Difference Time Domain (FDTD) scheme has served the computational electrodynamics community very well and part of its success stems from its ability to satisfy the constraints in Maxwell's equations. Even so, in the previous paper of this series we were able to present a second order accurate Godunov scheme for computational electrodynamics (CED) which satisfied all the same constraints and simultaneously retained all the traditional advantages of Godunov schemes. In this paper we extend the Finite Volume Time Domain (FVTD) schemes for CED in material media to better than second order of accuracy. From the FDTD method, we retain a somewhat modified staggering strategy of primal variables which enables a very beneficial constraint-preservation for the electric displacement and magnetic induction vector fields. This is accomplished with constraint-preserving reconstruction methods which are extended in this paper to third and fourth orders of accuracy. The idea of one-dimensional upwinding from Godunov schemes has to be significantly modified to use the multidimensionally upwinded Riemann solvers developed by the first author. In this paper, we show how they can be used within the context of a higher order scheme for CED. We also report on advances in timestepping. We show how Runge-Kutta IMEX schemes can be adapted to CED even in the presence of stiff source terms brought on by large conductivities as well as strong spatial variations in permittivity and permeability. We also formulate very efficient ADER timestepping strategies to endow our method with sub-cell resolving capabilities. As a result, our method can be stiffly-stable and resolve significant sub-cell variation in the material properties within a zone. Moreover, we present ADER schemes that are applicable to all hyperbolic PDEs with stiff source terms and at all orders of accuracy. Our new ADER formulation offers a treatment of stiff source terms that is much more efficient than previous ADER
Hexagonalization of correlation functions
Fleury, Thiago [Instituto de Física Teórica, UNESP - University Estadual Paulista,ICTP South American Institute for Fundamental Research,Rua Dr. Bento Teobaldo Ferraz 271, 01140-070, São Paulo, SP (Brazil); Komatsu, Shota [Perimeter Institute for Theoretical Physics,31 Caroline St N Waterloo, Ontario N2L 2Y5 (Canada)
We propose a nonperturbative framework to study general correlation functions of single-trace operators in N=4 supersymmetric Yang-Mills theory at large N. The basic strategy is to decompose them into fundamental building blocks called the hexagon form factors, which were introduced earlier to study structure constants using integrability. The decomposition is akin to a triangulation of a Riemann surface, and we thus call it hexagonalization. We propose a set of rules to glue the hexagons together based on symmetry, which naturally incorporate the dependence on the conformal and the R-symmetry cross ratios. Our method is conceptually different from the conventional operator product expansion and automatically takes into account multi-trace operators exchanged in OPE channels. To illustrate the idea in simple set-ups, we compute four-point functions of BPS operators of arbitrary lengths and correlation functions of one Konishi operator and three short BPS operators, all at one loop. In all cases, the results are in perfect agreement with the perturbative data. We also suggest that our method can be a useful tool to study conformal integrals, and show it explicitly for the case of ladder integrals.
A pedagogical introduction to theta functions
Koh, I.G.; Shin, H.J.
This paper reports on revolutions in physics that have been frequently accompanied by new developments in mathematics. In seventeenth century, Newton has initiated a program of describing celestial motion by classical mechanics. Integral and differential calculus was essential tool. Orbits of the moon and the earth are given by solving the differential equation of Newton's equation. Imagine a situation where one tries to solve such orbits without integral and differential calculus. Similar revolutions in understanding quantum gravity and in making deep connections between statistical and string physics are under progresses. One of indispensible tools are the theory of theta functions on Riemann surfaces. Since the literature of theta functions is mainly written by professional mathematician, physicists feel somewhat uneasy to begin to read a long chapters of lemmas and theorems, but it is now generally accepted that theta function is essential in understanding two-dimensional conformal field theory as the integral and differential calculus was indispensible in Newtonian mechanics
Signed zeros of Gaussian vector fields - density, correlation functions and curvature
Foltin, G
We calculate correlation functions of the (signed) density of zeros of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann Cartan or Riemannian manifold. As an application, we discuss one- and two-point functions. The zeros of a two-dimensional Gaussian vector field model the distribution of topological defects in the high-temperature phase of two-dimensional systems with orientational degrees of freedom, such as superfluid films, thin superconductors and liquid crystals.
Generalized fractional integration of the \\overline{H}-function
Praveen Agarwal
Full Text Available A significantly large number of earlier works on the subject of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, et cetera. In the present paper, we study and develop the generalized fractional integral operators given by Saigo. First, we establish two Theorems that give the images of the product of H-function and a general class of polynomials inSaigo operators. On account of the general nature of the Saigo operators, H-function and a general class of polynomials a large number of new and known Images involving Riemann-Liouville and Erdélyi-Kober fractional integral operators and several special functions notably generalized Wright hypergeometric function, generalized Wright-Bessel function, the polylogarithm and Mittag-Leffler functions follow as special cases of our main findings.
Analytic structure and power series expansion of the Jost function for the two-dimensional problem
Rakityansky, S A; Elander, N
For a two-dimensional quantum-mechanical problem, we obtain a generalized power series expansion of the S-matrix that can be done near an arbitrary point on the Riemann surface of the energy, similar to the standard effective-range expansion. In order to do this, we consider the Jost function and analytically factorize its momentum dependence that causes the Jost function to be a multi-valued function. The remaining single-valued function of the energy is then expanded in the power series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain a semi-analytic expression for the Jost function (and therefore for the S-matrix) near an arbitrary point on the Riemann surface and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles. The method is applied to a model similar to those used in the theory of quantum dots. (paper)
Justification of the zeta-function renormalization in rigid string model
Nesterenko, V.V.; Pirozhenko, I.G.
A consistent procedure for regularization of divergences and for the subsequent renormalization of the string tension is proposed in the framework of the one-loop calculation of the interquark potential generated by the Polyakov-Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein-Hurwitz zeta-functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy
Singular integral equations boundary problems of function theory and their application to mathematical physics
Muskhelishvili, N I
Singular integral equations play important roles in physics and theoretical mechanics, particularly in the areas of elasticity, aerodynamics, and unsteady aerofoil theory. They are highly effective in solving boundary problems occurring in the theory of functions of a complex variable, potential theory, the theory of elasticity, and the theory of fluid mechanics.This high-level treatment by a noted mathematician considers one-dimensional singular integral equations involving Cauchy principal values. Its coverage includes such topics as the Hölder condition, Hilbert and Riemann-Hilbert problem
Genus two partition functions of extremal conformal field theories
Gaiotto, Davide; Yin Xi
Recently Witten conjectured the existence of a family of 'extremal' conformal field theories (ECFTs) of central charge c = 24k, which are supposed to be dual to three-dimensional pure quantum gravity in AdS 3 . Assuming their existence, we determine explicitly the genus two partition functions of k = 2 and k = 3 ECFTs, using modular invariance and the behavior of the partition function in degenerating limits of the Riemann surface. The result passes highly nontrivial tests and in particular provides a piece of evidence for the existence of the k = 3 ECFT. We also argue that the genus two partition function of ECFTs with k ≤ 10 are uniquely fixed (if they exist)
Hexagon functions and the three-loop remainder function
Dixon, Lance J.; Drummond, James M.; von Hippel, Matt; Pennington, Jeffrey
We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar NN = 4 super-Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to -7.
Lecture notes: string theory and zeta-function
Toppan, Francesco [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil). E-mail: [email protected]
These lecture notes are based on a revised and LaTexed version of the Master thesis defended at ISAS. The research part being omitted, they included a review of the bosonic closed string a la Polyakov and of the one-loop background field method of quantisation defined through the zeta-function. In an appendix some basic features of the Riemann zeta-function are also reviewed. The pedagogical aspects of the material here presented are particularly emphasized. These notes are used, together with the Scherk's article in Rev. Mod. Phys. and the first volume of the Polchinski book, for the mini-course on String Theory (16-hours of lectures) held at CBPF. In this course the Green-Schwarz-Witten two-volumes book is also used for consultative purposes. (author)
Fractionalization of the complex-valued Brownian motion of order n using Riemann-Liouville derivative. Applications to mathematical finance and stochastic mechanics
The (complex-valued) Brownian motion of order n is defined as the limit of a random walk on the complex roots of the unity. Real-valued fractional noises are obtained as fractional derivatives of the Gaussian white noise (or order two). Here one combines these two approaches and one considers the new class of fractional noises obtained as fractional derivative of the complex-valued Brownian motion of order n. The key of the approach is the relation between differential and fractional differential provided by the fractional Taylor's series of analytic function f(z+h)=E α (h α D z α ).f(z), where E α is the Mittag-Leffler function on the one hand, and the generalized Maruyama's notation, on the other hand. Some questions are revisited such as the definition of fractional Brownian motion as integral w.r.t. (dt) α , and the exponential growth equation driven by fractional Brownian motion, to which a new solution is proposed. As a first illustrative example of application, in mathematical finance, one proposes a new approach to the optimal management of a stochastic portfolio of fractional order via the Lagrange variational technique applied to the state moment dynamical equations. In the second example, one deals with non-random Lagrangian mechanics of fractional order. The last example proposes a new approach to fractional stochastic mechanics, and the solution so obtained gives rise to the question as to whether physical systems would not have their own internal random times
Zeta functions for the spectrum of the non-commutative harmonic oscillators
Ichinose, T
This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in \\cite{PW1, 2}. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at $s=1$, and further that it has a zero at all non-positive even integers, i.e. at $s=0$ and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.
Bosonization on higher genus Riemann surfaces
Alvarez-Gaume, L.; Moore, G.; Nelson, P.; Vafa, C.
We prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions. The theories have fields of arbitrary spin on compact surfaces with any number of handles. Global considerations required that we add new topological terms to the bosonic action. The proof that our prescritpion is correct relies on methods of complex algebraic geometry. (orig.)
The Einstein tensor characterizing some Riemann spaces
Rahman, M.S.
A formal definition of the Einstein tensor is given. Mention is made of how this tensor plays a role of expressing certain conditions in a precise form. The cases of reducing the Einstein tensor to a zero tensor are studied on its merit. A lucid account of results, formulated as theorems, on Einstein symmetric and Einstein recurrent spaces is then presented. (author). 5 refs
Factorizations of rational matrix functions with application to discrete isomonodromic transformations and difference Painleve equations
Dzhamay, Anton
We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painleve equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D Arinkin and A Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.
The Positive Properties of Green's Function for Fractional Differential Equations and Its Applications
Fuquan Jiang
Full Text Available We consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem: D0+αu(t+f(t,u(t+e(t=0,0<1,u(0=u'(0=⋯=u(n-2(0=0,u(1=βu(η, where n-1Riemann-Liouville derivative. Here our nonlinearity f may be singular at u=0. As applications of Green's function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.
An introduction to bicomplex variables and functions; Introduzione ai sistemi ipercomplessi commutativi a quattro unita
Catoni, Francesco; Cannata, Roberto [ENEA, Centro Ricerche Casaccia, Rome (Italy). Servizio Centralizzato Informatica e Reti; Catoni, Vincenzo; Zampetti, Paolo [ENEA, Centro Ricerche Casaccia, Rome (Italy). Unita tecnico scientifica Fonti Rinnovabili e Cicli Energetici Innovativi
The commutative quaternions introduced by C. Segre are similar to the Hamilton quaternions but, thanks to their commutativity, allow to introduce the functions. This property opens new ways far applications. [Italian] E noto da un teorema di Scheffers che per i sistemi ipercomplessi com-mutativi esiste il calcolo integrodifferenziale che permette di definire le loro funzioni in modo perfettamente analogo alle funzioni di variabili complesse. Questa proprieta rende questi sistemi, potenzialmente utilizzabili in nuovi campi, rispetto ai quaternioni noncommutativi di Hamilton. In questo lavoro introduciamo due di questi sistemi con le loro proprieta algebriche, le condizioni differenziali (condizioni di Cauchy-Riemann generalizzate) a cui devono soddisfare le loro funzioni e, infine, sono date le espressioni delle funzioni elementari.
Identification of fractional order systems using modulating functions method
The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville fractional derivatives. First, a new fractional integration by parts formula involving the fractional derivative of a modulating function is given. Then, we apply this formula to a fractional order system, for which the fractional derivatives of the input and the output can be transferred into the ones of the modulating functions. By choosing a set of modulating functions, a linear system of algebraic equations is obtained. Hence, the unknown parameters of a fractional order system can be estimated by solving a linear system. Using this method, we do not need any initial values which are usually unknown and not equal to zero. Also we do not need to estimate the fractional derivatives of noisy output. Moreover, it is shown that the proposed estimators are robust against high frequency sinusoidal noises and the ones due to a class of stochastic processes. Finally, the efficiency and the stability of the proposed method is confirmed by some numerical simulations.
Some Families of the Incomplete H-Functions and the Incomplete \\overline H -Functions and Associated Integral Transforms and Operators of Fractional Calculus with Applications
Srivastava, H. M.; Saxena, R. K.; Parmar, R. K.
Our present investigation is inspired by the recent interesting extensions (by Srivastava et al. [35]) of a pair of the Mellin-Barnes type contour integral representations of their incomplete generalized hypergeometric functions p γ q and p Γ q by means of the incomplete gamma functions γ( s, x) and Γ( s, x). Here, in this sequel, we introduce a family of the relatively more general incomplete H-functions γ p,q m,n ( z) and Γ p,q m,n ( z) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions p Ψ q (γ) [ z] and p Ψ q (Γ) [ z]. The main object of this paper is to study and investigate several interesting properties of these incomplete H-functions, including (for example) decomposition and reduction formulas, derivative formulas, various integral transforms, computational representations, and so on. We apply some substantially general Riemann-Liouville and Weyl type fractional integral operators to each of these incomplete H-functions. We indicate the easilyderivable extensions of the results presented here that hold for the corresponding incomplete \\overline H -functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.
Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions
A probability distribution of fractional (or fractal) order is defined by the measure μ{dx} = p(x)(dx) α , 0 α (D x α h α )f(x) provided by the modified Riemann Liouville definition, one can expand a probability calculus parallel to the standard one. A Fourier's transform of fractional order using the Mittag-Leffler function is introduced, together with its inversion formula; and it provides a suitable generalization of the characteristic function of fractal random variables. It appears that the state moments of fractional order are more especially relevant. The main properties of this fractional probability calculus are outlined, it is shown that it provides a sound approach to Fokker-Planck equation which are fractional in both space and time, and it provides new results in the information theory of non-random functions.
Rahmatullah; Ellahi, Rahmat; Mohyud-Din, Syed Tauseef; Khan, Umar
We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses.
Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations.
Li, Q; He, Y L; Wang, Y; Tao, W Q
A coupled double-distribution-function lattice Boltzmann method is developed for the compressible Navier-Stokes equations. Different from existing thermal lattice Boltzmann methods, this method can recover the compressible Navier-Stokes equations with a flexible specific-heat ratio and Prandtl number. In the method, a density distribution function based on a multispeed lattice is used to recover the compressible continuity and momentum equations, while the compressible energy equation is recovered by an energy distribution function. The energy distribution function is then coupled to the density distribution function via the thermal equation of state. In order to obtain an adjustable specific-heat ratio, a constant related to the specific-heat ratio is introduced into the equilibrium energy distribution function. Two different coupled double-distribution-function lattice Boltzmann models are also proposed in the paper. Numerical simulations are performed for the Riemann problem, the double-Mach-reflection problem, and the Couette flow with a range of specific-heat ratios and Prandtl numbers. The numerical results are found to be in excellent agreement with analytical and/or other solutions.
Supersymmetric partition functions and the three-dimensional A-twist
Closset, Cyril [Theory Department, CERN,CH-1211, Geneva 23 (Switzerland); Kim, Heeyeon [Perimeter Institute for Theoretical Physics,31 Caroline Street North, Waterloo, N2L 2Y5, Ontario (Canada); Willett, Brian [Kavli Institute for Theoretical Physics, University of California,Santa Barbara, CA 93106 (United States)
We study three-dimensional N=2 supersymmetric gauge theories on M{sub g,p}, an oriented circle bundle of degree p over a closed Riemann surface, Σ{sub g}. We compute the M{sub g,p} supersymmetric partition function and correlation functions of supersymmetric loop operators. This uncovers interesting relations between observables on manifolds of different topologies. In particular, the familiar supersymmetric partition function on the round S{sup 3} can be understood as the expectation value of a so-called "fibering operator� on S{sup 2}×S{sup 1} with a topological twist. More generally, we show that the 3d N=2 supersymmetric partition functions (and supersymmetric Wilson loop correlation functions) on M{sub g,p} are fully determined by the two-dimensional A-twisted topological field theory obtained by compactifying the 3d theory on a circle. We give two complementary derivations of the result. We also discuss applications to F-maximization and to three-dimensional supersymmetric dualities.
The bosonic thermal Green function, its dual, and the fermion correlators of the massive Thirring model at finite temperature
Mondaini, Leonardo; Marino, E.C.
Full text: Despite the fact that quantum field theories are usually formulated in coordinate space, calculations, in both T = 0 and T ≠0 cases, are almost always performed in momentum space. However, when we are faced with the exact calculation of correlation functions we are naturally led to the problem of finding closed-form expressions for Green functions in coordinate space. In the present work, we derive an exact closed-form representation for the Euclidian thermal Green function of the two-dimensional (2D) free massless scalar field in coordinate space. This can be interpreted as the real part of a complex analytic function of a variable that conformally maps the infinite strip -∞ < x < ∞ (0 < τ < β of the z = x + iτ (τ: imaginary time) plane into the upper-half-plane. Use of the Cauchy-Riemann conditions, then allows us to identify the dual thermal Green function as the imaginary part of that function. Using both the thermal Green function and its dual, we obtain an explicit series expression for the fermionic correlation functions of the massive Thirring model (MTM) at a finite temperature. (author)
More on zeta-function regularization of high-temperature expansions
Actor, A.
A recent paper using the Riemann ζ-function to regularize the (divergent) coefficients occurring in the high-temperature expansions of one-loop thermodynamic potentials is extended. This method proves to be a powerful tool for converting Dirichlet-type series Σ m a m (x i )/m s into power series in the dimensionless parameters x i . The coefficients occurring in the power series are (proportional to) ζ-functions evaluated away from their poles - this is where the regularization occurs. High-temperature expansions are just one example of this highly-nontrivial rearrangement of Dirichlet series into power series form. We discuss in considerable detail series in which a m (x i ) is a product of trigonometric, algebraic and Bessel function factors. The ζ-function method is carefully explained, and a large number of new formulae are provided. The means to generalize these formulae are also provided. Previous results on thermodynamic potentials are generalized to include a nonzero constant term in the gauge potential (time component) which can be used to probe the electric sector of temperature gauge theories. (author)
Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series
Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.
Asymptotic expansion of a partition function related to the sinh-model
Borot, Gaëtan; Kozlowski, Karol K
This book elaborates on the asymptotic behaviour, when N is large, of certain N-dimensional integrals which typically occur in random matrices, or in 1+1 dimensional quantum integrable models solvable by the quantum separation of variables. The introduction presents the underpinning motivations for this problem, a historical overview, and a summary of the strategy, which is applicable in greater generality. The core aims at proving an expansion up to o(1) for the logarithm of the partition function of the sinh-model. This is achieved by a combination of potential theory and large deviation theory so as to grasp the leading asymptotics described by an equilibrium measure, the Riemann-Hilbert approach to truncated Wiener-Hopf in order to analyse the equilibrium measure, the Schwinger-Dyson equations and the boostrap method to finally obtain an expansion of correlation functions and the one of the partition function. This book is addressed to researchers working in random matrices, statistical physics or integra...
Generalized functions
Gelfand, I M; Graev, M I; Vilenkin, N Y; Pyatetskii-Shapiro, I I
Volume 1 is devoted to basics of the theory of generalized functions. The first chapter contains main definitions and most important properties of generalized functions as functional on the space of smooth functions with compact support. The second chapter talks about the Fourier transform of generalized functions. In Chapter 3, definitions and properties of some important classes of generalized functions are discussed; in particular, generalized functions supported on submanifolds of lower dimension, generalized functions associated with quadratic forms, and homogeneous generalized functions are studied in detail. Many simple basic examples make this book an excellent place for a novice to get acquainted with the theory of generalized functions. A long appendix presents basics of generalized functions of complex variables.
Quantum kinetic field theory in curved spacetime: Covariant Wigner function and Liouville-Vlasov equations
Calzetta, E.; Habib, S.; Hu, B.L.
We consider quantum fields in an external potential and show how, by using the Fourier transform on propagators, one can obtain the mass-shell constraint conditions and the Liouville-Vlasov equation for the Wigner distribution function. We then consider the Hadamard function G 1 (x 1 ,x 2 ) of a real, free, scalar field in curved space. We postulate a form for the Fourier transform F/sup (//sup Q//sup )/(X,k) of the propagator with respect to the difference variable x = x 1 -x 2 on a Riemann normal coordinate centered at Q. We show that F/sup (//sup Q//sup )/ is the result of applying a certain Q-dependent operator on a covariant Wigner function F. We derive from the wave equations for G 1 a covariant equation for the distribution function and show its consistency. We seek solutions to the set of Liouville-Vlasov equations for the vacuum and nonvacuum cases up to the third adiabatic order. Finally we apply this method to calculate the Hadamard function in the Einstein universe. We show that the covariant Wigner function can incorporate certain relevant global properties of the background spacetime. Covariant Wigner functions and Liouville-Vlasov equations are also derived for free fermions in curved spacetime. The method presented here can serve as a basis for constructing quantum kinetic theories in curved spacetime or for near-uniform systems under quasiequilibrium conditions. It can also be useful to the development of a transport theory of quantum fields for the investigation of grand unification and post-Planckian quantum processes in the early Universe
Riemann-Cartan geometry of nonlinear disclination mechanics
Yavari, A.; Goriely, A.
In the continuous theory of defects in nonlinear elastic solids, it is known that a distribution of disclinations leads, in general, to a non-trivial residual stress field. To study this problem, we consider the particular case of determining
Riemann Geometric Color-Weak Compensationfor Individual Observers
Kojima, Takanori; Mochizuki, Rika; Lenz, Reiner; Chao, Jinhui
We extend a method for color weak compensation based on the criterion of preservation of subjective color differences between color normal and color weak observers presented in [2]. We introduce a new algorithm for color weak compensation using local affine maps between color spaces of color normal and color weak observers. We show howto estimate the local affine map and how to determine correspondences between the origins of local coordinates in color spaces of color normal and color weak ob...
An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces
Schlichenmaier, Martin
This book gives an introduction to modern geometry. Starting from an elementary level the author develops deep geometrical concepts, playing an important role nowadays in contemporary theoretical physics. He presents various techniques and viewpoints, thereby showing the relations between the alternative approaches. At the end of each chapter suggestions for further reading are given to allow the reader to study the touched topics in greater detail. This second edition of the book contains two additional more advanced geometric techniques: (1) The modern language and modern view of Algebraic Geometry and (2) Mirror Symmetry. The book grew out of lecture courses. The presentation style is therefore similar to a lecture. Graduate students of theoretical and mathematical physics will appreciate this book as textbook. Students of mathematics who are looking for a short introduction to the various aspects of modern geometry and their interplay will also find it useful. Researchers will esteem the book as reliable ...
Yavari, A.
In the continuous theory of defects in nonlinear elastic solids, it is known that a distribution of disclinations leads, in general, to a non-trivial residual stress field. To study this problem, we consider the particular case of determining the residual stress field of a cylindrically symmetric distribution of parallel wedge disclinations. We first use the tools of differential geometry to construct a Riemannian material manifold in which the body is stress-free. This manifold is metric compatible, has zero torsion, but has non-vanishing curvature. The problem then reduces to embedding this manifold in Euclidean 3-space following the procedure of a classical nonlinear elastic problem. We show that this embedding can be elegantly accomplished by using Cartan\\'s method of moving frames and compute explicitly the residual stress field for various distributions in the case of a neo-Hookean material. © 2012 The Author(s).
A Polyakov action on Riemann surfaces. Pt. 2
The model independent study of the Polyakov action is continued on an arbitrary compact surface without boundary of genus larger than 2 as the general solution of the relevant conformal Ward identity. A general formula for the Polyakov action and an explicit calculation of the energy-momentum tensor density is provided. The general geometric setting of the construction is described in detail. It is shown that the Polyakov action defines a distribution of finite dimensional directions in the holomorphic tangent bundle of the manifold of Beltrami differentials. It is further argued that motions parallel to such distribution correspond to Polyakov's SL(2,C) symmetry transformations. Owing to the existence of renormalization ambiguities on a topologically non-trivial surface, the energy-momentum tensor needs not be invariant under the full SL(2,C) symmetry. The residual SL(2,C) symmetry is characterized geometrically. (author) 31 refs
Beltrami algebra and symmetry of Beltrami equation on Riemann surfaces
Guo Hanying; Xu Kaiwen; Shen Jianmin; Wang Shikun
It is shown that the Beltrami equation has an infinite dimensional symmetry, namely the Beltrami algebra, on its solution spaces. The Beltrami algebra with central extension and its supersymmetric version are explicitly found. (author). 12 refs
Functional Boxplots
Sun, Ying
This article proposes an informative exploratory tool, the functional boxplot, for visualizing functional data, as well as its generalization, the enhanced functional boxplot. Based on the center outward ordering induced by band depth for functional data, the descriptive statistics of a functional boxplot are: the envelope of the 50% central region, the median curve, and the maximum non-outlying envelope. In addition, outliers can be detected in a functional boxplot by the 1.5 times the 50% central region empirical rule, analogous to the rule for classical boxplots. The construction of a functional boxplot is illustrated on a series of sea surface temperatures related to the El Niño phenomenon and its outlier detection performance is explored by simulations. As applications, the functional boxplot and enhanced functional boxplot are demonstrated on children growth data and spatio-temporal U.S. precipitation data for nine climatic regions, respectively. This article has supplementary material online. © 2011 American Statistical Association.
Staircase functions, spectral regidity and a rule for quantizing chaos
Aurich, R.; Steiner, F.
Considering the Selberg trace formula as an exact version of Gutzwiller's semiclassical periodic-orbit theory in the case of the free motion on compact Riemann surfaces with constant negative curvature (Hadamard-Gutzwiller model), we study two complementary basic problems in quantum chaology: the computation of the calssical staircase N(l), the number of periodic orbits with length shorter than l, in terms of the quantal energy spectrum {E n }, the computation of the spectral staircase N (E), the number of quantal energies below the energy E, in terms of the length spectrum {l n } of the classical periodic orbits. A formulation of the periodic-orbit theory is presented which is intrinsically unsmoothed, but for which an effective smoothing arises from the limited 'input data', i.e. from the limited knowledge of the periodic orbits in the case of N(E) and the limited knowledge of quantal energies in the case of N(l). Based on the periodic-orbit formula for N(E), we propose a new rule for quantizing chaos, which simply states that the quantal energies are determined by the zeros of the function ξ 1 (E) = cos (πN(E)). The formulas for N(l) and N(E) as well as the new quantization condition are tested numerically. Furthermore, it is shown that the staircase N(E) computed from the length spectrum yields (up to a constant) a good description of the spectral rigidity Δ 3 (L), being the first numerical attempt to compute a statistical property of the quantal energy spectrum of a chaotic system from classical periodic orbits. (orig.)
Orbit Functions
Anatoliy Klimyk
Full Text Available In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space E_n are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be described. An orbit function is the contribution to an irreducible character of a compact semisimple Lie group G of rank n from one of its Weyl group orbits. It is shown that values of orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group in the entire Euclidean space E_n. Orbit functions are solutions of the corresponding Laplace equation in E_n, satisfying the Neumann condition on the boundary of F. Orbit functions determine a symmetrized Fourier transform and a transform on a finite set of points.
Conference on Commutative rings, integer-valued polynomials and polynomial functions
Frisch, Sophie; Glaz, Sarah; Commutative Algebra : Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions
This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non- Noetherian ring theory as well as integer-valued polynomials and functions. Specific topics include: · Homological dimensions of Prüfer-like rings · Quasi complete rings · Total graphs of rings · Properties of prime ideals over various rings · Bases for integer-valued polynomials · Boolean subrings · The portable property of domains · Probabilistic topics in Intn(D) · Closure operations in Zariski-Riemann spaces of valuation domains · Stability of do...
Functional displays
Angelis De, F.; Haentjens, J.
The Functional Displays are directly derived from the Man-Machine Design key document: Function-Based Task Analysis. The presentation defines and describes the goals-means structure of the plant function along with applicable control volumes and parameters of interest. The purpose of the subject is to show, through an example of a preliminary design, what the main parts of a function are. (3 figs.)
Chitil, Olaf
Functional programming is a programming paradigm like object-oriented programming and logic programming. Functional programming comprises both a specific programming style and a class of programming languages that encourage and support this programming style. Functional programming enables the programmer to describe an algorithm on a high-level, in terms of the problem domain, without having to deal with machine-related details. A program is constructed from functions that only map inputs to ...
Functionalized amphipols
Della Pia, Eduardo Antonio; Hansen, Randi Westh; Zoonens, Manuela
Amphipols are amphipathic polymers that stabilize membrane proteins isolated from their native membrane. They have been functionalized with various chemical groups in the past years for protein labeling and protein immobilization. This large toolbox of functionalized amphipols combined with their...... surfaces for various applications in synthetic biology. This review summarizes the properties of functionalized amphipols suitable for synthetic biology approaches....
Lightness functions
Campi, Stefano; Gardner, Richard; Gronchi, Paolo
Variants of the brightness function of a convex body K in n-dimensional Euclidean are investigated. The Lambertian lightness function L(K; v , w ) gives the total reflected light resulting from illumination by a light source at infinity in the direction w that is visible when looking...... in the direction v . The partial brightness function R( K ; v , w ) gives the area of the projection orthogonal to v of the portion of the surface of K that is both illuminated by a light source from the direction w and visible when looking in the direction v . A class of functions called lightness functions...... is introduced that includes L(K;.) and R(K;.) as special cases. Much of the theory of the brightness function like uniqueness, stability, and the existence and properties of convex bodies of maximal and minimal volume with finitely many function values equal to those of a given convex body, is extended...
Kantorovich, L V
Functional Analysis examines trends in functional analysis as a mathematical discipline and the ever-increasing role played by its techniques in applications. The theory of topological vector spaces is emphasized, along with the applications of functional analysis to applied analysis. Some topics of functional analysis connected with applications to mathematical economics and control theory are also discussed. Comprised of 18 chapters, this book begins with an introduction to the elements of the theory of topological spaces, the theory of metric spaces, and the theory of abstract measure space
Sun, Ying; Genton, Marc G.
data, the descriptive statistics of a functional boxplot are: the envelope of the 50% central region, the median curve, and the maximum non-outlying envelope. In addition, outliers can be detected in a functional boxplot by the 1.5 times the 50% central
Functional coma.
Ludwig, L; McWhirter, L; Williams, S; Derry, C; Stone, J
Functional coma - here defined as a prolonged motionless dissociative attack with absent or reduced response to external stimuli - is a relatively rare presentation. In this chapter we examine a wide range of terms used to describe states of unresponsiveness in which psychologic factors are relevant to etiology, such as depressive stupor, catatonia, nonepileptic "pseudostatus," and factitious disorders, and discuss the place of functional or psychogenic coma among these. Historically, diagnosis of functional coma has sometimes been reached after prolonged investigation and exclusion of other diagnoses. However, as is the case with other functional disorders, diagnosis should preferably be made on the basis of positive findings that provide evidence of inconsistency between an apparent comatose state and normal waking nervous system functioning. In our review of physical signs, we find some evidence for the presence of firm resistance to eye opening as reasonably sensitive and specific for functional coma, as well as the eye gaze sign, in which patients tend to look to the ground when turned on to one side. Noxious stimuli such as Harvey's sign (application of high-frequency vibrating tuning fork to the nasal mucosa) can also be helpful, although patients with this disorder are often remarkably unresponsive to usually painful stimuli, particularly as more commonly applied using sternal or nail bed pressure. The use of repeated painful stimuli is therefore not recommended. We also discuss the role of general anesthesia and other physiologic triggers to functional coma. © 2016 Elsevier B.V. All rights reserved.
Rhinoplasty (Functional)
... RESOURCES Medical Societies Patient Education About this Website Font Size + - Home > TREATMENTS > Rhinoplasty (Functional) Nasal/Sinus Irrigation ... performed to restore breathing, it typically necessitates some type of change to the appearance of the nose. ...
Functional tremor.
Schwingenschuh, P; Deuschl, G
Functional tremor is the commonest reported functional movement disorder. A confident clinical diagnosis of functional tremor is often possible based on the following "positive" criteria: a sudden tremor onset, unusual disease course, often with fluctuations or remissions, distractibility of the tremor if attention is removed from the affected body part, tremor entrainment, tremor variability, and a coactivation sign. Many patients show excessive exhaustion during examination. Other somatizations may be revealed in the medical history and patients may show additional functional neurologic symptoms and signs. In cases where the clinical diagnosis remains challenging, providing a "laboratory-supported" level of certainty aids an early positive diagnosis. In rare cases, in which the distinction from Parkinson's disease is difficult, dopamine transporter single-photon emission computed tomography (DAT-SPECT) can be indicated. © 2016 Elsevier B.V. All rights reserved.
Because chemicals can adversely affect cognitive function in humans, considerable effort has been made to characterize their effects using animal models. Information from such models will be necessary to: evaluate whether chemicals identified as potentially neurotoxic by screenin...
Functional unparsing
Danvy, Olivier
A string-formatting function such as printf in C seemingly requires dependent types, because its control string determines the rest of its arguments. Examples: formula here We show how changing the representation of the control string makes it possible to program printf in ML (which does not allow...... dependent types). The result is well typed and perceptibly more efficient than the corresponding library functions in Standard ML of New Jersey and in Caml....
A string-formatting function such as printf in C seemingly requires dependent types, because its control string determines the rest of its arguments. We show how changing the representation of the control string makes it possible to program printf in ML (which does not allow dependent types......). The result is well typed and perceptibly more efficient than the corresponding library functions in Standard ML of New Jersey and in Caml....
Overlap functions
Bustince, H.; Fernández, J.; Mesiar, Radko; Montero, J.; Orduna, R.
Ro�. 72, 3-4 (2010), s. 1488-1499 ISSN 0362-546X R&D Projects: GA ČR GA402/08/0618 Institutional research plan: CEZ:AV0Z10750506 Keywords : t-norm * Migrative property * Homogeneity property * Overlap function Subject RIV: BA - General Mathematics Impact factor: 1.279, year: 2010 http://library.utia.cas.cz/separaty/2009/E/mesiar-overlap functions.pdf
Bessel functions
Nambudiripad, K B M
After presenting the theory in engineers' language without the unfriendly abstraction of pure mathematics, several illustrative examples are discussed in great detail to see how the various functions of the Bessel family enter into the solution of technically important problems. Axisymmetric vibrations of a circular membrane, oscillations of a uniform chain, heat transfer in circular fins, buckling of columns of varying cross-section, vibrations of a circular plate and current density in a conductor of circular cross-section are considered. The problems are formulated purely from physical considerations (using, for example, Newton's law of motion, Fourier's law of heat conduction electromagnetic field equations, etc.) Infinite series expansions, recurrence relations, manipulation of expressions involving Bessel functions, orthogonality and expansion in Fourier-Bessel series are also covered in some detail. Some important topics such as asymptotic expansions, generating function and Sturm-Lioville theory are r...
Functional inequalities
Ghoussoub, Nassif
The book describes how functional inequalities are often manifestations of natural mathematical structures and physical phenomena, and how a few general principles validate large classes of analytic/geometric inequalities, old and new. This point of view leads to "systematic" approaches for proving the most basic inequalities, but also for improving them, and for devising new ones--sometimes at will and often on demand. These general principles also offer novel ways for estimating best constants and for deciding whether these are attained in appropriate function spaces. As such, improvements of Hardy and Hardy-Rellich type inequalities involving radially symmetric weights are variational manifestations of Sturm's theory on the oscillatory behavior of certain ordinary differential equations. On the other hand, most geometric inequalities, including those of Sobolev and Log-Sobolev type, are simply expressions of the convexity of certain free energy functionals along the geodesics on the Wasserstein manifold of...
Kleibeuker, JH; Thijs, JC
Purpose of review Functional dyspepsia is a common disorder, most of the time of unknown etiology and with variable pathophysiology. Therapy has been and still is largely empirical. Data from recent studies provide new clues for targeted therapy based on knowledge of etiology and pathophysiologic
Fani Nolimal
Full Text Available The author first defines literacy as the ability of co-operation in all fields of life and points at the features of illiterate or semi-literate individuals. The main stress is laid upon the assessment of literacy and illiteracy. In her opinion the main weak ness of this kind of evaluation are its vague psycho-metric characteristics, which leads to results valid in a single geographical or cultural environment only. She also determines the factors causing illiteracy, and she states that the level of functional literacy is more and more becoming a national indicator of successfulness.
Lung function
Sorichter, S.
The term lung function is often restricted to the assessment of volume time curves measured at the mouth. Spirometry includes the assessment of lung volumes which can be mobilised with the corresponding flow-volume curves. In addition, lung volumes that can not be mobilised, such as the residual volume, or only partially as FRC and TLC can be measured by body plethysmography combined with the determination of the airway resistance. Body plethysmography allows the correct positioning of forced breathing manoeuvres on the volume-axis, e.g. before and after pharmacotherapy. Adding the CO single breath transfer factor (T LCO ), which includes the measurement of the ventilated lung volume using He, enables a clear diagnosis of different obstructive, restrictive or mixed ventilatory defects with and without trapped air. Tests of reversibility and provocation, as well as the assessment of inspiratory mouth pressures (PI max , P 0.1 ) help to classify the underlying disorder and to clarify treatment strategies. For further information and to complete the diagnostic of disturbances of the ventilation, diffusion and/or perfusion (capillar-)arterial bloodgases at rest and under physical strain sometimes amended by ergospirometry are recommended. Ideally, lung function measurements are amended by radiological and nuclear medicine techniques. (orig.) [de
Functional phlebology
Weber, J.; May, R.; Biland, L.; Endert, G.; Gottlob, R.; Justich, E.; Luebcke, P.; Mignon, G.; Moltz, L.; Partsch, H.; Petter, A.; Ritter, H.; Soerensen, R.; Widmer, L.K.; Widmer, M.T.; Zemp, E.
The book presents a complete survey of the problems occurring in the venous system of the legs, pelvis, and abdomen. The material is arranged in the following main chapters: (1) Introduction to the phlebology of the low-pressure system in the lower part of the body; (2) Phlebographic methods; (3) Instrumented function studies and methods; (4) Pathologic findings; (5) Diagnostic methods and vein therapy; (6) Interventional radiology; (7) Expert opinions on venous lesions including insurance aspects. The first chapter encompasses a section briefly discussing the available instrumented diagnostic imaging methods. In view of the novel imaging methods, namely digital subtraction phlebology, sonography, CT and MRI, the classical phlebography remains the gold standard, so to speak: all currently available phlebographic methods for imaging the venes in the legs, pelvis and abdomen are explained and comparatively evaluated. Instrumented function tests such as Doppler effect ultrasound testing, plethysmography, peripheral and central phlebodynamometry (venous pressure measurement) are analysed for their diagnostic value and as alternative or supplementing techniques in comparison to phlebology. (orig./MG) With 843 figs., 101 tabs [de
Functional Credentials
Deuber Dominic
Full Text Available A functional credential allows a user to anonymously prove possession of a set of attributes that fulfills a certain policy. The policies are arbitrary polynomially computable predicates that are evaluated over arbitrary attributes. The key feature of this primitive is the delegation of verification to third parties, called designated verifiers. The delegation protects the privacy of the policy: A designated verifier can verify that a user satisfies a certain policy without learning anything about the policy itself. We illustrate the usefulness of this property in different applications, including outsourced databases with access control. We present a new framework to construct functional credentials that does not require (non-interactive zero-knowledge proofs. This is important in settings where the statements are complex and thus the resulting zero-knowledge proofs are not efficient. Our construction is based on any predicate encryption scheme and the security relies on standard assumptions. A complexity analysis and an experimental evaluation confirm the practicality of our approach.
Functional Angioplasty
Rohit Tewari
Full Text Available Coronary angiography underestimates or overestimates lesion severity, but still remains the cornerstone in the decision making for revascularization for an overwhelming majority of interventional cardiologists. Guidelines recommend and endorse non invasive functional evaluation ought to precede revascularization. In real world practice, this is adopted in less than 50% of patients who go on to have some form of revascularization. Fractional flow reserve (FFR is the ratio of maximal blood flow in a stenotic coronary relative to maximal flow in the same vessel, were it normal. Being independent of changes in heart rate, BP or prior infarction; and take into account the contribution of collateral blood flow. It is a majorly specific index with a reasonably high sensitivity (88%, specificity (100%, positive predictive value (100%, and overall accuracy (93%. Whilst FFR provides objective determination of ischemia and helps select appropriate candidates for revascularization (for both CABG and PCI in to cath lab itself before intervention, whereas intravascular ultrasound/optical coherence tomography guidance in PCI can secure the procedure by optimizing stent expansion. Functional angioplasty simply is incorporating both intravascular ultrasound and FFR into our daily Intervention practices.
Cavalieri Integration | Ackermann | Quaestiones Mathematicae
We also present two methods of evaluating a Cavalieri integral by first transforming it to either an equivalent Riemann or Riemann-Stieltjes integral by using special transformation functions h(x) and its inverse g(x), respectively. Interestingly enough it is often very difficult to find the transformation function h(x), whereas it is ...
Thyroid Function Tests
... Home » Thyroid Function Tests Leer en Español Thyroid Function Tests FUNCTION HOW DOES THE THYROID GLAND FUNCTION? ... Cancer Thyroid Nodules in Children and Adolescents Thyroid Function Tests Resources Thyroid Function Tests Brochure PDF En ...
Park, J. Y.; Hong, G. W.; Lee, H. J.
Development of fabrication process of functional ceramic materials, evaluation of characteristics and experiments for understanding of irradiation behavior of ceramics were carried out for application of ceramics to the nuclear industry. The developed processes were the SiC surface coating technology with large area for improvement of wear resistance and corrosion resistance, the fabrication technology of SiC composites for excellent irradiation resistance, performance improvement technology of SiC fiber and nano-sized powder processing by combustion ignition and spray. Typical results were CVD SiC coating with diameter of 25cm and thickness of 100μm, highly dense SiC composite by F-CVI, heat-treating technology of SiC fiber using B4C power, and nano-sized powders of ODS-Cu, Li-based breeding materials, Ni-based metal powders with primary particle diameter of 20∼50nm. Furthermore, test equipment, data productions and damage evaluations were performed to understand corrosion resistance and wear resistance of alumina, silicon carbide and silicon nitride under PWR or PHWR operation conditions. Experimental procedures and basic technologies for evaluation of irradiation behavior were also established. Additionally, highly reactive precursor powders were developed by various technologies and the powders were applied to the fabrication of 100 m long Ag/Bi-2223 multi-filamentary wires. High Tc magnets and fly wheel for energy storage were developed, as well
Special functions & their applications
Lebedev, N N
Famous Russian work discusses the application of cylinder functions and spherical harmonics; gamma function; probability integral and related functions; Airy functions; hyper-geometric functions; more. Translated by Richard Silverman.
Functional Programming in R
Mailund, Thomas
Master functions and discover how to write functional programs in R. In this book, you'll make your functions pure by avoiding side-effects; you'll write functions that manipulate other functions, and you'll construct complex functions using simpler functions as building blocks. In Functional...... Programming in R, you'll see how we can replace loops, which can have side-effects, with recursive functions that can more easily avoid them. In addition, the book covers why you shouldn't use recursion when loops are more efficient and how you can get the best of both worlds. Functional programming...... functions by combining simpler functions. You will: Write functions in R including infix operators and replacement functions Create higher order functions Pass functions to other functions and start using functions as data you can manipulate Use Filer, Map and Reduce functions to express the intent behind...
Resummed coefficient function for the shape function
Aglietti, U.
We present a leading evaluation of the resummed coefficient function for the shape function. It is also shown that the coefficient function is short-distance-dominated. Our results allow relating the shape function computed on the lattice to the physical QCD distributions.
Time functions function best as functions of multiple times
Desain, P.; Honing, H.
This article presents an elegant way of representing control functions at an abstractlevel. It introduces time functions that have multiple times as arguments. In this waythe generalized concept of a time function can support absolute and relative kinds of time behavior. Furthermore the
Wave-function functionals for the density
Slamet, Marlina; Pan Xiaoyin; Sahni, Viraht
We extend the idea of the constrained-search variational method for the construction of wave-function functionals ψ[χ] of functions χ. The search is constrained to those functions χ such that ψ[χ] reproduces the density �(r) while simultaneously leading to an upper bound to the energy. The functionals are thereby normalized and automatically satisfy the electron-nucleus coalescence condition. The functionals ψ[χ] are also constructed to satisfy the electron-electron coalescence condition. The method is applied to the ground state of the helium atom to construct functionals ψ[χ] that reproduce the density as given by the Kinoshita correlated wave function. The expectation of single-particle operators W=Σ i r i n , n=-2,-1,1,2, W=Σ i δ(r i ) are exact, as must be the case. The expectations of the kinetic energy operator W=-(1/2)Σ i ∇ i 2 , the two-particle operators W=Σ n u n , n=-2,-1,1,2, where u=|r i -r j |, and the energy are accurate. We note that the construction of such functionals ψ[χ] is an application of the Levy-Lieb constrained-search definition of density functional theory. It is thereby possible to rigorously determine which functional ψ[χ] is closer to the true wave function.
Nonlocal kinetic energy functionals by functional integration
Mi, Wenhui; Genova, Alessandro; Pavanello, Michele
Since the seminal studies of Thomas and Fermi, researchers in the Density-Functional Theory (DFT) community are searching for accurate electron density functionals. Arguably, the toughest functional to approximate is the noninteracting kinetic energy, Ts[�], the subject of this work. The typical paradigm is to first approximate the energy functional and then take its functional derivative, δ/Ts[� ] δ � (r ) , yielding a potential that can be used in orbital-free DFT or subsystem DFT simulations. Here, this paradigm is challenged by constructing the potential from the second-functional derivative via functional integration. A new nonlocal functional for Ts[�] is prescribed [which we dub Mi-Genova-Pavanello (MGP)] having a density independent kernel. MGP is constructed to satisfy three exact conditions: (1) a nonzero "Kinetic electron" arising from a nonzero exchange hole; (2) the second functional derivative must reduce to the inverse Lindhard function in the limit of homogenous densities; (3) the potential is derived from functional integration of the second functional derivative. Pilot calculations show that MGP is capable of reproducing accurate equilibrium volumes, bulk moduli, total energy, and electron densities for metallic (body-centered cubic, face-centered cubic) and semiconducting (crystal diamond) phases of silicon as well as of III-V semiconductors. The MGP functional is found to be numerically stable typically reaching self-consistency within 12 iterations of a truncated Newton minimization algorithm. MGP's computational cost and memory requirements are low and comparable to the Wang-Teter nonlocal functional or any generalized gradient approximation functional.
Generalized Probability Functions
Alexandre Souto Martinez
Full Text Available From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs. A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.
Functionality and homogeneity.
Functionality and homogeneity are two of the five Sustainable Safety principles. The functionality principle aims for roads to have but one exclusive function and distinguishes between traffic function (flow) and access function (residence). The homogeneity principle aims at differences in mass,
Extraocular muscle function testing
... medlineplus.gov/ency/article/003397.htm Extraocular muscle function testing To use the sharing features on this page, please enable JavaScript. Extraocular muscle function testing examines the function of the eye muscles. ...
Congenital platelet function defects
... pool disorder; Glanzmann's thrombasthenia; Bernard-Soulier syndrome; Platelet function defects - congenital ... Congenital platelet function defects are bleeding disorders that cause reduced platelet function. Most of the time, people with these disorders have ...
Hepatic (Liver) Function Panel
... Educators Search English Español Blood Test: Hepatic (Liver) Function Panel KidsHealth / For Parents / Blood Test: Hepatic (Liver) ... kidneys ) is working. What Is a Hepatic (Liver) Function Panel? A liver function panel is a blood ...
Platelet Function Tests
... Patient Resources For Health Professionals Subscribe Search Platelet Function Tests Send Us Your Feedback Choose Topic At ... Also Known As Platelet Aggregation Studies PFT Platelet Function Assay PFA Formal Name Platelet Function Tests This ...
On Poisson functions
Terashima, Yuji
In this paper, defining Poisson functions on super manifolds, we show that the graphs of Poisson functions are Dirac structures, and find Poisson functions which include as special cases both quasi-Poisson structures and twisted Poisson structures.
Investigating body function
Monks, R.; Riley, A.L.M.
This invention relates to the investigation of body function, especially small bowel function but also liver function, using bile acids and bile salts or their metabolic precursors labelled with radio isotopes and selenium or tellurium. (author)
Functional bowel disorders and functional abdominal pain
Thompson, W; Longstreth, G; Drossman, D; Heaton, K; Irvine, E; Muller-Lissner, S
The Rome diagnostic criteria for the functional bowel disorders and functional abdominal pain are used widely in research and practice. A committee consensus approach, including criticism from multinational expert reviewers, was used to revise the diagnostic criteria and update diagnosis and treatment recommendations, based on research results. The terminology was clarified and the diagnostic criteria and management recommendations were revised. A functional bowel disorder (FBD) is diagnosed ...
Functional microorganisms for functional food quality.
Gobbetti, M; Cagno, R Di; De Angelis, M
Functional microorganisms and health benefits represent a binomial with great potential for fermented functional foods. The health benefits of fermented functional foods are expressed either directly through the interactions of ingested live microorganisms with the host (probiotic effect) or indirectly as the result of the ingestion of microbial metabolites synthesized during fermentation (biogenic effect). Since the importance of high viability for probiotic effect, two major options are currently pursued for improving it--to enhance bacterial stress response and to use alternative products for incorporating probiotics (e.g., ice cream, cheeses, cereals, fruit juices, vegetables, and soy beans). Further, it seems that quorum sensing signal molecules released by probiotics may interact with human epithelial cells from intestine thus modulating several physiological functions. Under optimal processing conditions, functional microorganisms contribute to food functionality through their enzyme portfolio and the release of metabolites. Overproduction of free amino acids and vitamins are two classical examples. Besides, bioactive compounds (e.g., peptides, γ-amino butyric acid, and conjugated linoleic acid) may be released during food processing above the physiological threshold and they may exert various in vivo health benefits. Functional microorganisms are even more used in novel strategies for decreasing phenomenon of food intolerance (e.g., gluten intolerance) and allergy. By a critical approach, this review will aim at showing the potential of functional microorganisms for the quality of functional foods.
The Interpretive Function
Agerbo, Heidi
Approximately a decade ago, it was suggested that a new function should be added to the lexicographical function theory: the interpretive function(1). However, hardly any research has been conducted into this function, and though it was only suggested that this new function was relevant...... to incorporate into lexicographical theory, some scholars have since then assumed that this function exists(2), including the author of this contribution. In Agerbo (2016), I present arguments supporting the incorporation of the interpretive function into the function theory and suggest how non-linguistic signs...... can be treated in specific dictionary articles. However, in the current article, due to the results of recent research, I argue that the interpretive function should not be considered an individual main function. The interpretive function, contrary to some of its definitions, is not connected...
Every storage function is a state function
Trentelman, H.L.; Willems, J.C.
It is shown that for linear dynamical systems with quadratic supply rates, a storage function can always be written as a quadratic function of the state of an associated linear dynamical system. This dynamical system is obtained by combining the dynamics of the original system with the dynamics of
Persistent Functional Languages: Toward Functional Relational Databases
Wevers, L.
Functional languages provide new approaches to concurrency control, based on techniques such as lazy evaluation and memoization. We have designed and implemented a persistent functional language based on these ideas, which we plan to use for the implementation of a relational database system. With
Museums and Their Functions.
Osborne, Harold
Historical background concerning the nature and function of museums is provided, and the aesthetic functions of museums are discussed. The first major aesthetic function of museums is to preserve the artistic heritage of mankind and to make it widely available. The second major function is patronage. (RM)
Hierarchical wave functions revisited
Li Dingping.
We study the hierarchical wave functions on a sphere and on a torus. We simplify some wave functions on a sphere or a torus using the analytic properties of wave functions. The open question, the construction of the wave function for quasi electron excitation on a torus, is also solved in this paper. (author)
Master functions and discover how to write functional programs in R. In this book, you'll make your functions pure by avoiding side-effects; you'll write functions that manipulate other functions, and you'll construct complex functions using simpler functions as building blocks. In Functional...... Programming in R, you'll see how we can replace loops, which can have side-effects, with recursive functions that can more easily avoid them. In addition, the book covers why you shouldn't use recursion when loops are more efficient and how you can get the best of both worlds. Functional programming...... is a style of programming, like object-oriented programming, but one that focuses on data transformations and calculations rather than objects and state. Where in object-oriented programming you model your programs by describing which states an object can be in and how methods will reveal or modify...
Teager Correlation Function
Bysted, Tommy Kristensen; Hamila, R.; Gabbouj, M.
A new correlation function called the Teager correlation function is introduced in this paper. The connection between this function, the Teager energy operator and the conventional correlation function is established. Two applications are presented. The first is the minimization of the Teager error...... norm and the second one is the use of the instantaneous Teager correlation function for simultaneous estimation of TDOA and FDOA (Time and Frequency Difference of Arrivals)....
Properties of Ambiguity Functions
Mulcahy-Stanislawczyk, John
The use of ambiguity functions in radar signal design and analysis is very common. Understanding the various properties and meanings of ambiguity functions allow a signal designer to understand the time delay and doppler shift properties of a given signal. Through the years, several different versions of the ambiguity function have been used. Each of these functions essentially have the same physical meaning; however, the use of different functions makes it difficult to be sure that certai...
Ergotic / epistemic / semiotic functions
Luciani , Annie
International audience; Claude Cadoz has introduced a typology of human-environment relation, identifying three functions. This typology allows characterizing univocally, i.e. in a non-redundant manner, the computer devices and interfaces that allow human to interact with environment through and by computers. These three functions are: the epistemic function, the semiotic function, the ergotic function. Conversely to the terms epistemic and semiotic that are usual, the term ergotic has been s...
Variational functionals which admit discontinuous trial functions
Nelson, P. Jr.
It is argued that variational synthesis with discontinuous trial functions requires variational principles applicable to equations involving operators acting between distinct Hilbert spaces. A description is given of a Roussopoulos-type variational principle generalized to cover this situation. This principle is suggested as the basis for a unified approach to the derivation of variational functionals. In addition to esthetics, this approach has the advantage that the mathematical details increase the understanding of the derived functional, particularly the sense in which a synthesized solution should be regarded as an approximation to the true solution. By way of illustration, the generalized Roussopoulos principle is applied to derive a class of first-order diffusion functionals which admit trial functions containing approximations at an interface. These ''asymptotic'' interface quantities are independent of the limiting approximations from either side and permit use of different trial spectra at and on either side of an interface. The class of functionals derived contains as special cases both the Lagrange multiplier method of Buslik and two functionals of Lambropoulos and Luco. Some numerical results for a simple two-group model confirm that the ''multipliers'' can closely approximate the appropriate quantity in the region near an interface. (U.S.)
A Blue Lagoon Function
Markvorsen, Steen
We consider a specific function of two variables whose graph surface resembles a blue lagoon. The function has a saddle point $p$, but when the function is restricted to any given straight line through $p$ it has a {\\em{strict local minimum}} along that line at $p$.......We consider a specific function of two variables whose graph surface resembles a blue lagoon. The function has a saddle point $p$, but when the function is restricted to any given straight line through $p$ it has a {\\em{strict local minimum}} along that line at $p$....
Quality function deployment
This book indicates quality function deployment with quality and deployment of quality function, process and prospect of quality function deployment and development, product process and conception of quality table, deployment of quality demand, design of quality table and application of concurrent multi design, progress design and quality development, main safe part and management of important function part, quality development and deployment of method of construction, quality deployment and economics, total system of quality function deployment and task of quality function deployment in the present and future.
Functional Object Analysis
Raket, Lars Lau
We propose a direction it the field of statistics which we will call functional object analysis. This subfields considers the analysis of functional objects defined on continuous domains. In this setting we will focus on model-based statistics, with a particularly emphasis on mixed......-effect formulations, where the observed functional signal is assumed to consist of both fixed and random functional effects. This thesis takes the initial steps toward the development of likelihood-based methodology for functional objects. We first consider analysis of functional data defined on high...
High spin structure functions
Khan, H.
This thesis explores deep inelastic scattering of a lepton beam from a polarized nuclear target with spin J=1. After reviewing the formation for spin-1/2, the structure functions for a spin-1 target are defined in terms of the helicity amplitudes for forward compton scattering. A version of the convolution model, which incorporates relativistic and binding energy corrections is used to calculate the structure functions of a neutron target. A simple parameterization of these structure functions is given in terms of a few neutron wave function parameters and the free nucleon structure functions. This allows for an easy comparison of structure functions calculated using different neutron models. (author)
From functional architecture to functional connectomics.
Reid, R Clay
"Receptive Fields, Binocular Interaction and Functional Architecture in the Cat's Visual Cortex" by Hubel and Wiesel (1962) reported several important discoveries: orientation columns, the distinct structures of simple and complex receptive fields, and binocular integration. But perhaps the paper's greatest influence came from the concept of functional architecture (the complex relationship between in vivo physiology and the spatial arrangement of neurons) and several models of functionally specific connectivity. They thus identified two distinct concepts, topographic specificity and functional specificity, which together with cell-type specificity constitute the major determinants of nonrandom cortical connectivity. Orientation columns are iconic examples of topographic specificity, whereby axons within a column connect with cells of a single orientation preference. Hubel and Wiesel also saw the need for functional specificity at a finer scale in their model of thalamic inputs to simple cells, verified in the 1990s. The difficult but potentially more important question of functional specificity between cortical neurons is only now becoming tractable with new experimental techniques. Copyright © 2012 Elsevier Inc. All rights reserved.
Functional Median Polish
This article proposes functional median polish, an extension of univariate median polish, for one-way and two-way functional analysis of variance (ANOVA). The functional median polish estimates the functional grand effect and functional main factor effects based on functional medians in an additive functional ANOVA model assuming no interaction among factors. A functional rank test is used to assess whether the functional main factor effects are significant. The robustness of the functional median polish is demonstrated by comparing its performance with the traditional functional ANOVA fitted by means under different outlier models in simulation studies. The functional median polish is illustrated on various applications in climate science, including one-way and two-way ANOVA when functional data are either curves or images. Specifically, Canadian temperature data, U. S. precipitation observations and outputs of global and regional climate models are considered, which can facilitate the research on the close link between local climate and the occurrence or severity of some diseases and other threats to human health. © 2012 International Biometric Society.
Hearing the music of the primes: auditory complementarity and the siren song of zeta
Berry, M V
A counting function for the primes can be rendered as a sound signal whose harmonies, spanning the gamut of musical notes, are the Riemann zeros. But the individual primes cannot be discriminated as singularities in this 'music', because the intervals between them are too short. Conversely, if the prime singularities are detected as a series of clicks, the Riemann zeros correspond to frequencies too low to be heard. The sound generated by the Riemann zeta function itself is very different: a rising siren howl, which can be understood in detail from the Riemann–Siegel formula. (fast track communication)
Operations Between Functions
Gardner, Richard J.; Kiderlen, Markus
A structural theory of operations between real-valued (or extended-real-valued) functions on a nonempty subset A of Rn is initiated. It is shown, for example, that any operation ∗ on a cone of functions containing the constant functions, which is pointwise, positively homogeneous, monotonic......, and associative, must be one of 40 explicitly given types. In particular, this is the case for operations between pairs of arbitrary, or continuous, or differentiable functions. The term pointwise means that (f ∗g)(x) = F(f(x), g(x)), for all x ∈ A and some function F of two variables. Several results in the same...... spirit are obtained for operations between convex functions or between support functions. For example, it is shown that ordinary addition is the unique pointwise operation between convex functions satisfying the identity property, i.e., f ∗ 0 = 0 ∗ f = f, for all convex f, while other results classify Lp...
Kidney function tests
Kidney function tests are common lab tests used to evaluate how well the kidneys are working. Such tests include: ... Oh MS, Briefel G. Evaluation of renal function, water, electrolytes ... and Management by Laboratory Methods . 23rd ed. Philadelphia, ...
... digest food, store energy, and remove poisons. Liver function tests are blood tests that check to see ... as hepatitis and cirrhosis. You may have liver function tests as part of a regular checkup. Or ...
Functionalized diamond nanoparticles
Beaujuge, Pierre M.; El Tall, Omar; Raja, Inam U.
A diamond nanoparticle can be functionalized with a substituted dienophile under ambient conditions, and in the absence of catalysts or additional reagents. The functionalization is thought to proceed through an addition reaction.
Smart hydrogel functional materials
Chu, Liang-Yin; Ju, Xiao-Jie
This book systematically introduces smart hydrogel functional materials with the configurations ranging from hydrogels to microgels. It serves as an excellent reference for designing and fabricating artificial smart hydrogel functional materials.
Integrals of Bessel functions
Babusci, D.; Dattoli, G.; Germano, B.; Martinelli, M. R.; Ricci, P. E.
We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique we propose is based on the formal reduction of these family of functions to Gaussians.
Functional Python programming
Lott, Steven
This book is for developers who want to use Python to write programs that lean heavily on functional programming design patterns. You should be comfortable with Python programming, but no knowledge of functional programming paradigms is needed.
Beaujuge, Pierre M.
Ecological Functions of Landscapes
Kiryushin, V. I.
Ecological functions of landscapes are considered a system of processes ensuring the development, preservation, and evolution of ecosystems and the biosphere as a whole. The concept of biogeocenosis can be considered a model that integrates biotic and environmental functions. The most general biogeocenotic functions specify the biodiversity, biotic links, self-organization, and evolution of ecosystems. Close interaction between biocenosis and the biotope (ecotope) is ensured by the continuous exchange of matter, energy, and information. Ecotope determines the biocenosis. The group of ecotopic functions includes atmospheric (gas exchange, heat exchange, hydroatmospheric, climate-forming), lithospheric (geodynamic, geophysical, and geochemical), hydrologic and hydrogeologic functions of landscape and ecotopic functions of soils. Bioecological functions emerge as a result of the biotope and ecotope interaction; these are the bioproductive, destructive, organoaccumulative, biochemical (gas, concentration, redox, biochemical, biopedological), pedogenetic, and energy functions
polish is demonstrated by comparing its performance with the traditional functional ANOVA fitted by means under different outlier models in simulation studies. The functional median polish is illustrated on various applications in climate science
Hybrid functional pseudopotentials
Yang, Jing; Tan, Liang Z.; Rappe, Andrew M.
The consistency between the exchange-correlation functional used in pseudopotential construction and in the actual density functional theory calculation is essential for the accurate prediction of fundamental properties of materials. However, routine hybrid density functional calculations at present still rely on generalized gradient approximation pseudopotentials due to the lack of hybrid functional pseudopotentials. Here, we present a scheme for generating hybrid functional pseudopotentials, and we analyze the importance of pseudopotential density functional consistency for hybrid functionals. For the PBE0 hybrid functional, we benchmark our pseudopotentials for structural parameters and fundamental electronic gaps of the Gaussian-2 (G2) molecular dataset and some simple solids. Our results show that using our PBE0 pseudopotentials in PBE0 calculations improves agreement with respect to all-electron calculations.
Photon structure function - theory
Bardeen, W.A.
The theoretical status of the photon structure function is reviewed. Particular attention is paid to the hadronic mixing problem and the ability of perturbative QCD to make definitive predictions for the photon structure function. 11 references
Monotone Boolean functions
Korshunov, A D
Monotone Boolean functions are an important object in discrete mathematics and mathematical cybernetics. Topics related to these functions have been actively studied for several decades. Many results have been obtained, and many papers published. However, until now there has been no sufficiently complete monograph or survey of results of investigations concerning monotone Boolean functions. The object of this survey is to present the main results on monotone Boolean functions obtained during the last 50 years
and chebyshev functions
Mohsen Razzaghi
Full Text Available A direct method for finding the solution of variational problems using a hybrid function is discussed. The hybrid functions which consist of block-pulse functions plus Chebyshev polynomials are introduced. An operational matrix of integration and the integration of the cross product of two hybrid function vectors are presented and are utilized to reduce a variational problem to the solution of an algebraic equation. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Functional Task Test (FTT)
Bloomberg, Jacob J.; Mulavara, Ajitkumar; Peters, Brian T.; Rescheke, Millard F.; Wood, Scott; Lawrence, Emily; Koffman, Igor; Ploutz-Snyder, Lori; Spiering, Barry A.; Feeback, Daniel L.;
This slide presentation reviews the Functional Task Test (FTT), an interdisciplinary testing regimen that has been developed to evaluate astronaut postflight functional performance and related physiological changes. The objectives of the project are: (1) to develop a set of functional tasks that represent critical mission tasks for the Constellation Program, (2) determine the ability to perform these tasks after space flight, (3) Identify the key physiological factors that contribute to functional decrements and (4) Use this information to develop targeted countermeasures.
Pseudolinear functions and optimization
Mishra, Shashi Kant
Pseudolinear Functions and Optimization is the first book to focus exclusively on pseudolinear functions, a class of generalized convex functions. It discusses the properties, characterizations, and applications of pseudolinear functions in nonlinear optimization problems.The book describes the characterizations of solution sets of various optimization problems. It examines multiobjective pseudolinear, multiobjective fractional pseudolinear, static minmax pseudolinear, and static minmax fractional pseudolinear optimization problems and their results. The authors extend these results to locally
Photon wave function
Bialynicki-Birula, Iwo
Photon wave function is a controversial concept. Controversies stem from the fact that photon wave functions can not have all the properties of the Schroedinger wave functions of nonrelativistic wave mechanics. Insistence on those properties that, owing to peculiarities of photon dynamics, cannot be rendered, led some physicists to the extreme opinion that the photon wave function does not exist. I reject such a fundamentalist point of view in favor of a more pragmatic approach. In my view, t...
On Functional Calculus Estimates
Schwenninger, F.L.
This thesis presents various results within the field of operator theory that are formulated in estimates for functional calculi. Functional calculus is the general concept of defining operators of the form $f(A)$, where f is a function and $A$ is an operator, typically on a Banach space. Norm
Phylogenetic molecular function annotation
Engelhardt, Barbara E; Jordan, Michael I; Repo, Susanna T; Brenner, Steven E
It is now easier to discover thousands of protein sequences in a new microbial genome than it is to biochemically characterize the specific activity of a single protein of unknown function. The molecular functions of protein sequences have typically been predicted using homology-based computational methods, which rely on the principle that homologous proteins share a similar function. However, some protein families include groups of proteins with different molecular functions. A phylogenetic approach for predicting molecular function (sometimes called 'phylogenomics') is an effective means to predict protein molecular function. These methods incorporate functional evidence from all members of a family that have functional characterizations using the evolutionary history of the protein family to make robust predictions for the uncharacterized proteins. However, they are often difficult to apply on a genome-wide scale because of the time-consuming step of reconstructing the phylogenies of each protein to be annotated. Our automated approach for function annotation using phylogeny, the SIFTER (Statistical Inference of Function Through Evolutionary Relationships) methodology, uses a statistical graphical model to compute the probabilities of molecular functions for unannotated proteins. Our benchmark tests showed that SIFTER provides accurate functional predictions on various protein families, outperforming other available methods.
New Similarity Functions
Yazdani, Hossein; Ortiz-Arroyo, Daniel; Kwasnicka, Halina
spaces, in addition to their similarity in the vector space. Prioritized Weighted Feature Distance (PWFD) works similarly as WFD, but provides the ability to give priorities to desirable features. The accuracy of the proposed functions are compared with other similarity functions on several data sets....... Our results show that the proposed functions work better than other methods proposed in the literature....
Two Functions of Language
Feldman, Carol Fleisher
Author advocates the view that meaning is necessarily dependent upon the communicative function of language and examines the objections, particularly those of Noam Chomsky, to this view. Argues that while Chomsky disagrees with the idea that communication is the essential function of language, he implicitly agrees that it has a function.…
Automatic differentiation of functions
Douglas, S.R.
Automatic differentiation is a method of computing derivatives of functions to any order in any number of variables. The functions must be expressible as combinations of elementary functions. When evaluated at specific numerical points, the derivatives have no truncation error and are automatically found. The method is illustrated by simple examples. Source code in FORTRAN is provided
Nonparametric Transfer Function Models
Liu, Jun M.; Chen, Rong; Yao, Qiwei
In this paper a class of nonparametric transfer function models is proposed to model nonlinear relationships between 'input' and 'output' time series. The transfer function is smooth with unknown functional forms, and the noise is assumed to be a stationary autoregressive-moving average (ARMA) process. The nonparametric transfer function is estimated jointly with the ARMA parameters. By modeling the correlation in the noise, the transfer function can be estimated more efficiently. The parsimonious ARMA structure improves the estimation efficiency in finite samples. The asymptotic properties of the estimators are investigated. The finite-sample properties are illustrated through simulations and one empirical example. PMID:20628584
Weakly clopen functions
Son, Mi Jung; Park, Jin Han; Lim, Ki Moon
We introduce a new class of functions called weakly clopen function which includes the class of almost clopen functions due to Ekici [Ekici E. Generalization of perfectly continuous, regular set-connected and clopen functions. Acta Math Hungar 2005;107:193-206] and is included in the class of weakly continuous functions due to Levine [Levine N. A decomposition of continuity in topological spaces. Am Math Mon 1961;68:44-6]. Some characterizations and several properties concerning weakly clopenness are obtained. Furthermore, relationships among weak clopenness, almost clopenness, clopenness and weak continuity are investigated
Implementing function spreadsheets
Sestoft, Peter
: that of turning an expression into a named function. Hence they proposed a way to define a function in terms of a worksheet with designated input and output cells; we shall call it a function sheet. The goal of our work is to develop implementations of function sheets and study their application to realistic...... examples. Therefore, we are also developing a simple yet comprehensive spreadsheet core implementation for experimentation with this technology. Here we report briefly on our experiments with function sheets as well as other uses of our spreadsheet core implementation....
Transfer function combinations
Zhou, Liang; Schott, Mathias; Hansen, Charles
Direct volume rendering has been an active area of research for over two decades. Transfer function design remains a difficult task since current methods, such as traditional 1D and 2D transfer functions, are not always effective for all data sets. Various 1D or 2D transfer function spaces have been proposed to improve classification exploiting different aspects, such as using the gradient magnitude for boundary location and statistical, occlusion, or size metrics. In this paper, we present a novel transfer function method which can provide more specificity for data classification by combining different transfer function spaces. In this work, a 2D transfer function can be combined with 1D transfer functions which improve the classification. Specifically, we use the traditional 2D scalar/gradient magnitude, 2D statistical, and 2D occlusion spectrum transfer functions and combine these with occlusion and/or size-based transfer functions to provide better specificity. We demonstrate the usefulness of the new method by comparing to the following previous techniques: 2D gradient magnitude, 2D occlusion spectrum, 2D statistical transfer functions and 2D size based transfer functions. © 2012 Elsevier Ltd.
Zhou, Liang
Functional Maximum Autocorrelation Factors
Larsen, Rasmus; Nielsen, Allan Aasbjerg
MAF outperforms the functional PCA in concentrating the interesting' spectra/shape variation in one end of the eigenvalue spectrum and allows for easier interpretation of effects. Conclusions. Functional MAF analysis is a useful methods for extracting low dimensional models of temporally or spatially......Purpose. We aim at data where samples of an underlying function are observed in a spatial or temporal layout. Examples of underlying functions are reflectance spectra and biological shapes. We apply functional models based on smoothing splines and generalize the functional PCA in......\\verb+~+\\$\\backslash\\$cite{ramsay97} to functional maximum autocorrelation factors (MAF)\\verb+~+\\$\\backslash\\$cite{switzer85,larsen2001d}. We apply the method to biological shapes as well as reflectance spectra. {\\$\\backslash\\$bf Methods}. MAF seeks linear combination of the original variables that maximize autocorrelation between...
The Riemann Solution for the Injection of Steam and Nitrogen in a Porous Medium
Lambert, W.; Marchesin, D.; Bruining, J.
We solve the model for the flow of nitrogen, vapor, and water in a porous medium, neglecting compressibility, heat conductivity, and capillary effects. Our choice of injection conditions is determined by the application to clean up polluted sites. We study all mathematical structures, such as
Study of the Issues of Computational Aerothermodynamics Using a Riemann Solver
should satisfy positiv - ity and not violate the Second Law of Thermodynamics (commonly called the entropy condition). Positivity preserving schemes...can be seen that N2 starts to dissociate around 6000 K - 12,000 K. 5.3 Conclusion This work has produced quantitative dividing lines between a regime...This figure provides a great deal of quantitative information and should be very useful to the high-speed flow CFD community. This work has also
Quantum Yang-Mills theory of Riemann surfaces and conformal field theory
Killingback, T.P.
It is shown that Yang-Mills theory on a smooth surface, when suitably quantized, is a topological quantum field theory. This topological gauge theory is intimately related to two-dimensional conformal field theory. It is conjectured that all conformal field theories may be obtained from Yang-Mills theory on smooth surfaces. (orig.)
Differential geometry for physicists and mathematicians moving frames and differential forms : from Euclid past Riemann
Vargas, José G
This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative - almost like a story being told - that does not impede sophistication and deep results. It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas
Geometry of the fundamental interactions on Riemann's legacy to high energy physics and cosmology
Maia, M D
The Yang-Mills theory of gauge interactions is a prime example of interdisciplinary mathematics and advanced physics. Its historical development is a fascinating window into the ongoing struggle of mankind to understand nature. The discovery of gauge fields and their properties is the most formidable landmark of modern physics. The expression of the gauge field strength as the curvature associated to a given connection, places quantum field theory in the same geometrical footing as the gravitational field of general relativity which is naturally written in geometrical terms. The understanding of such geometrical property may help one day to write a unified field theory starting from symmetry principles. Of course, there are remarkable differences between the standard gauge fields and the gravitational field, which must be understood by mathematicians and physicists before attempting such unification. In particular, it is important to understand why gravitation is not a standard gauge field. This book presents...
The holographic dual of a Riemann problem in a large number of dimensions
Herzog, Christopher P.; Spillane, Michael [C.N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy,Stony Brook University, Stony Brook, NY 11794 (United States); Yarom, Amos [Department of Physics, Technion,Haifa 32000 (Israel)
We study properties of a non equilibrium steady state generated when two heat baths are initially in contact with one another. The dynamics of the system we study are governed by holographic duality in a large number of dimensions. We discuss the "phase diagram� associated with the steady state, the dual, dynamical, black hole description of this problem, and its relation to the fluid/gravity correspondence.
Asymptotic behavior of monodromy singularly perturbed differential equations on a Riemann surface
Simpson, Carlos
This book concerns the question of how the solution of a system of ODE's varies when the differential equation varies. The goal is to give nonzero asymptotic expansions for the solution in terms of a parameter expressing how some coefficients go to infinity. A particular classof families of equations is considered, where the answer exhibits a new kind of behavior not seen in most work known until now. The techniques include Laplace transform and the method of stationary phase, and a combinatorial technique for estimating the contributions of terms in an infinite series expansion for the solution. Addressed primarily to researchers inalgebraic geometry, ordinary differential equations and complex analysis, the book will also be of interest to applied mathematicians working on asymptotics of singular perturbations and numerical solution of ODE's.
Riemann solvers and numerical methods for fluid dynamics a practical introduction
Toro, Eleuterio F
High resolution upwind and centred methods are a mature generation of computational techniques applicable to a range of disciplines, Computational Fluid Dynamics being the most prominent. This book gives a practical presentation of this class of techniques.
E-Orbit Functions
Jiri Patera
Full Text Available We review and further develop the theory of $E$-orbit functions. They are functions on the Euclidean space $E_n$ obtained from the multivariate exponential function by symmetrization by means of an even part $W_{e}$ of a Weyl group $W$, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group $W$. The $E$-orbit functions, determined by integral parameters, are invariant withrespect to even part $W^{aff}_{e}$ of the affine Weyl group corresponding to $W$. The $E$-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain $F^{e}$ of the group $W^{aff}_{e}$ (the discrete $E$-orbit function transform.
Antisymmetric Orbit Functions
Full Text Available In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform. Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.
Mathematical modelling and numerical simulation of casting processes
Hattel, Jesper Henri
The control volume method applied to numerical modelling of castning. Analytical solutions based on the error function.Riemann-temperature. Modelling of release of latent heat with the enthalpy method....
B Plant function analysis report
Lund, D.P.
The document contains the functions, function definitions, function interfaces, function interface definitions, Input Computer Automated Manufacturing Definition (IDEFO) diagrams, and a function hierarchy chart that describe what needs to be performed to deactivate B Plant
Rδ-Supercontinuous Functions
Kohli J. K.
Full Text Available A new class of functions called 'Rδ-supercontinuous functions' is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity which already exist in the literature is elaborated. The class of Rδ-supercontinuous functions (Math. Bohem., to appear properly contains the class of Rz-supercontinuous functions which in its turn properly contains the class of Rcl- supercontinuous functions (Demonstratio Math. 46(1 (2013, 229-244 and so includes all Rcl-supercontinuous (≡clopen continuous functions (Applied Gen. Topol. 8(2 (2007, 293-300; Indian J. Pure Appl. Math. 14(6 (1983, 767-772 and is properly contained in the class of R-supercontinuous functions (Demonstratio Math. 43(3 (2010, 703-723.
Intrinsic-density functionals
Engel, J.
The Hohenberg-Kohn theorem and Kohn-Sham procedure are extended to functionals of the localized intrinsic density of a self-bound system such as a nucleus. After defining the intrinsic-density functional, we modify the usual Kohn-Sham procedure slightly to evaluate the mean-field approximation to the functional, and carefully describe the construction of the leading corrections for a system of fermions in one dimension with a spin-degeneracy equal to the number of particles N. Despite the fact that the corrections are complicated and nonlocal, we are able to construct a local Skyrme-like intrinsic-density functional that, while different from the exact functional, shares with it a minimum value equal to the exact ground-state energy at the exact ground-state intrinsic density, to next-to-leading order in 1/N. We briefly discuss implications for real Skyrme functionals
Functional analysis and applications
Siddiqi, Abul Hasan
This self-contained textbook discusses all major topics in functional analysis. Combining classical materials with new methods, it supplies numerous relevant solved examples and problems and discusses the applications of functional analysis in diverse fields. The book is unique in its scope, and a variety of applications of functional analysis and operator-theoretic methods are devoted to each area of application. Each chapter includes a set of problems, some of which are routine and elementary, and some of which are more advanced. The book is primarily intended as a textbook for graduate and advanced undergraduate students in applied mathematics and engineering. It offers several attractive features making it ideally suited for courses on functional analysis intended to provide a basic introduction to the subject and the impact of functional analysis on applied and computational mathematics, nonlinear functional analysis and optimization. It introduces emerging topics like wavelets, Gabor system, inverse pro...
Counting with symmetric functions
Mendes, Anthony
This monograph provides a self-contained introduction to symmetric functions and their use in enumerative combinatorics. It is the first book to explore many of the methods and results that the authors present. Numerous exercises are included throughout, along with full solutions, to illustrate concepts and also highlight many interesting mathematical ideas. The text begins by introducing fundamental combinatorial objects such as permutations and integer partitions, as well as generating functions. Symmetric functions are considered in the next chapter, with a unique emphasis on the combinatorics of the transition matrices between bases of symmetric functions. Chapter 3 uses this introductory material to describe how to find an assortment of generating functions for permutation statistics, and then these techniques are extended to find generating functions for a variety of objects in Chapter 4. The next two chapters present the Robinson-Schensted-Knuth algorithm and a method for proving Pólya's enu...
Relativistic plasma dispersion functions
Robinson, P.A.
The known properties of plasma dispersion functions (PDF's) for waves in weakly relativistic, magnetized, thermal plasmas are reviewed and a large number of new results are presented. The PDF's required for the description of waves with small wave number perpendicular to the magnetic field (Dnestrovskii and Shkarofsky functions) are considered in detail; these functions also arise in certain quantum electrodynamical calculations involving strongly magnetized plasmas. Series, asymptotic series, recursion relations, integral forms, derivatives, differential equations, and approximations for these functions are discussed as are their analytic properties and connections with standard transcendental functions. In addition a more general class of PDF's relevant to waves of arbitrary perpendicular wave number is introduced and a range of properties of these functions are derived
dftools: Distribution function fitting
Obreschkow, Danail
dftools, written in R, finds the most likely P parameters of a D-dimensional distribution function (DF) generating N objects, where each object is specified by D observables with measurement uncertainties. For instance, if the objects are galaxies, it can fit a mass function (D=1), a mass-size distribution (D=2) or the mass-spin-morphology distribution (D=3). Unlike most common fitting approaches, this method accurately accounts for measurement in uncertainties and complex selection functions.
Entropy and wigner functions
Manfredi; Feix
The properties of an alternative definition of quantum entropy, based on Wigner functions, are discussed. Such a definition emerges naturally from the Wigner representation of quantum mechanics, and can easily quantify the amount of entanglement of a quantum state. It is shown that smoothing of the Wigner function induces an increase in entropy. This fact is used to derive some simple rules to construct positive-definite probability distributions which are also admissible Wigner functions.
Manfredi, G.; Feix, M. R.
The properties of an alternative definition of quantum entropy, based on Wigner functions, are discussed. Such definition emerges naturally from the Wigner representation of quantum mechanics, and can easily quantify the amount of entanglement of a quantum state. It is shown that smoothing of the Wigner function induces an increase in entropy. This fact is used to derive some simple rules to construct positive definite probability distributions which are also admissible Wigner functions
Time functions revisited
Fathi, Albert
In this paper we revisit our joint work with Antonio Siconolfi on time functions. We will give a brief introduction to the subject. We will then show how to construct a Lipschitz time function in a simplified setting. We will end with a new result showing that the Aubry set is not an artifact of our proof of existence of time functions for stably causal manifolds.
SPLINE, Spline Interpolation Function
Allouard, Y.
1 - Nature of physical problem solved: The problem is to obtain an interpolated function, as smooth as possible, that passes through given points. The derivatives of these functions are continuous up to the (2Q-1) order. The program consists of the following two subprograms: ASPLERQ. Transport of relations method for the spline functions of interpolation. SPLQ. Spline interpolation. 2 - Method of solution: The methods are described in the reference under item 10
Hadron structure functions
Martin, F.
The x dependence of hadron structure functions is investigated. If quarks can exist in very low mass states (10 MeV for d and u quarks) the pion structure function is predicted to behave like (1-x) and not (1-x) 2 in a x-region around 1. Relativistic and non-relativistic quark bound state pictures of hadrons are considered together with their relation with the Q 2 evolution of structure functions. Good agreement with data is in general obtained
Calculus of bivariant function
PT�ČN�K, Jan
This thesis deals with the introduction of function of two variables and differential calculus of this function. This work should serve as a textbook for students of elementary school's teacher. Each chapter contains a summary of basic concepts and explanations of relationships, then solved model exercises of the topic and finally the exercises, which should solve the student himself. Thesis have transmit to students basic knowledges of differential calculus of functions of two variables, inc...
Functional esophageal disorders
Clouse, R; Richter, J; Heading, R; Janssens, J; Wilson, J
The functional esophageal disorders include globus, rumination syndrome, and symptoms that typify esophageal diseases (chest pain, heartburn, and dysphagia). Factors responsible for symptom production are poorly understood. The criteria for diagnosis rest not only on compatible symptoms but also on exclusion of structural and metabolic disorders that might mimic the functional disorders. Additionally, a functional diagnosis is precluded by the presence of a pathology-based motor disorder or p...
Functional Programming With Relations
While programming in a relational framework has much to offer over the functional style in terms of expressiveness, computing with relations is less efficient, and more semantically troublesome. In this paper we propose a novel blend of the functional and relational styles. We identify a class of "causal relations", which inherit some of the bi-directionality properties of relations, but retain the efficiency and semantic foundations of the functional style.
Photon structure function
Theoretical understanding of the photon structure function is reviewed. As an illustration of the pointlike component, the parton model is briefly discussed. However, the systematic study of the photon structure function is presented through the framework of the operator product expansion. Perturbative QCD is used as the theoretical basis for the calculation of leading contributions to the operator product expansion. The influence of higher order QCD effects on these results is discussed. Recent results for the polarized structure functions are discussed
Nonrespiratory lung function
Isawa, Toyoharu
The function of the lungs is primarily the function as a gas exchanger: the venous blood returning to the lungs is arterialized with oxygen in the lungs and the arterialized blood is sent back again to the peripheral tissues of the whole body to be utilized for metabolic oxygenation. Besides the gas exchanging function which we call ''respiratory lung function'' the lungs have functions that have little to do with gas exchange itself. We categorically call the latter function of the lungs as ''nonrespiratory lung function''. The lungs consist of the conductive airways, the gas exchanging units like the alveoli, and the interstitial space that surrounds the former two compartments. The interstitial space contains the blood and lymphatic capillaries, collagen and elastic fibers and cement substances. The conductive airways and the gas exchanging units are directly exposed to the atmosphere that contains various toxic and nontoxic gases, fume and biological or nonbiological particles. Because the conductive airways are equipped with defense mechanisms like mucociliary clearance or coughs to get rid of these toxic gases, particles or locally produced biological debris, we are usually free from being succumbed to ill effects of inhaled materials. By use of nuclear medicine techniques, we can now evaluate mucociliary clearance function, and other nonrespiratory lung functions as well in vivo
Subordination by convex functions
Rosihan M. Ali
Full Text Available For a fixed analytic function g(z=z+∑n=2∞gnzn defined on the open unit disk and γ<1, let Tg(γ denote the class of all analytic functions f(z=z+∑n=2∞anzn satisfying ∑n=2∞|angn|≤1−γ. For functions in Tg(γ, a subordination result is derived involving the convolution with a normalized convex function. Our result includes as special cases several earlier works.
Renormalization Group Functional Equations
Curtright, Thomas L
Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories. With minimal assumptions, the methods produce continuous flows from step-scaling {\\sigma} functions, and lead to exact functional relations for the local flow {\\beta} functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of {\\sigma} are sometimes not true fixed points under continuous changes in scale, and zeroes of {\\beta} do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.
Perceptual Audio Hashing Functions
Emin Anarım
Full Text Available Perceptual hash functions provide a tool for fast and reliable identification of content. We present new audio hash functions based on summarization of the time-frequency spectral characteristics of an audio document. The proposed hash functions are based on the periodicity series of the fundamental frequency and on singular-value description of the cepstral frequencies. They are found, on one hand, to perform very satisfactorily in identification and verification tests, and on the other hand, to be very resilient to a large variety of attacks. Moreover, we address the issue of security of hashes and propose a keying technique, and thereby a key-dependent hash function.
Isawa, Toyoharu [Tohoku University Research Institute for Chest Disease and Cancer, Sendai (Japan)
The function of the lungs is primarily the function as a gas exchanger: the venous blood returning to the lungs is arterialized with oxygen in the lungs and the arterialized blood is sent back again to the peripheral tissues of the whole body to be utilized for metabolic oxygenation. Besides the gas exchanging function which we call ''respiratory lung function'' the lungs have functions that have little to do with gas exchange itself. We categorically call the latter function of the lungs as ''nonrespiratory lung function''. The lungs consist of the conductive airways, the gas exchanging units like the alveoli, and the interstitial space that surrounds the former two compartments. The interstitial space contains the blood and lymphatic capillaries, collagen and elastic fibers and cement substances. The conductive airways and the gas exchanging units are directly exposed to the atmosphere that contains various toxic and nontoxic gases, fume and biological or nonbiological particles. Because the conductive airways are equipped with defense mechanisms like mucociliary clearance or coughs to get rid of these toxic gases, particles or locally produced biological debris, we are usually free from being succumbed to ill effects of inhaled materials. By use of nuclear medicine techniques, we can now evaluate mucociliary clearance function, and other nonrespiratory lung functions as well in vivo.
Control functions in MFM
Lind, Morten
Multilevel Flow Modeling (MFM) has been proposed as a tool for representing goals and functions of complex industrial plants and suggested as a basis for reasoning about control situations. Lind presents an introduction to MFM but do not describe how control functions are used in the modeling....... The purpose of the present paper is to serve as a companion paper to this introduction by explaining the basic principles used in MFM for representation of control functions. A theoretical foundation for modeling control functions is presented and modeling examples are given for illustration....
Regulated functions and integrability
Ján Gun�aga
Full Text Available Properties of functions defined on a bounded closed interval, weaker than continuity, have been considered by many mathematicians. Functions having both sides limits at each point are called regulated and were considered by J. Dieudonné [2], D. Fraňková [3] and others (see for example S. Banach [1], S. Saks [8]. The main class of functions we deal with consists of piece-wise constant ones. These functions play a fundamental role in the integration theory which had been developed by Igor Kluvanek (see Š. Tkacik [9]. We present an outline of this theory.
Introduction to functional methods
Faddeev, L.D.
The functional integral is considered in relation to Feynman diagrams and phase space. The holomorphic form of the functional integral is then discussed. The main problem of the lectures, viz. the construction of the S-matrix by means of the functional integral, is considered. The functional methods described explicitly take into account the Bose statistics of the fields involved. The different procedure used to treat fermions is discussed. An introduction to the problem of quantization of gauge fields is given. (B.R.H.)
The gamma function
Artin, Emil
This brief monograph on the gamma function was designed by the author to fill what he perceived as a gap in the literature of mathematics, which often treated the gamma function in a manner he described as both sketchy and overly complicated. Author Emil Artin, one of the twentieth century's leading mathematicians, wrote in his Preface to this book, ""I feel that this monograph will help to show that the gamma function can be thought of as one of the elementary functions, and that all of its basic properties can be established using elementary methods of the calculus."" Generations of teachers
Normal Functions As A New Way Of Defining Computable Functions
Leszek Dubiel
Full Text Available Report sets new method of defining computable functions. This is formalization of traditional function descriptions, so it allows to define functions in very intuitive way. Discovery of Ackermann function proved that not all functions that can be easily computed can be so easily described with Hilbert's system of recursive functions. Normal functions lack this disadvantage.
Full Text Available Report sets new method of defining computable functions. This is formalization of traditional function descriptions, so it allows to define functions in very intuitive way. Discovery of Ackermann function proved that not all functions that can be easily computed can be so easily described with Hilbert's system of recursive functions. Normal functions lack this disadvantage.
Function spaces, 1
Pick, Luboš; John, Oldrich; Fucík, Svatopluk
This is the first part of the second revised and extended edition of a well established monograph. It is an introduction to function spaces defined in terms of differentiability and integrability classes. It provides a catalogue of various spaces and benefits as a handbook for those who use function spaces to study other topics such as partial differential equations. Volum
F-supercontinuous functions
J.K. Kohli
Full Text Available A strong variant of continuity called 'F-supercontinuity' is introduced. The class of F-supercontinuous functions strictly contains the class of z-supercontinuous functions (Indian J. Pure Appl. Math. 33 (7 (2002, 1097–1108 which in turn properly contains the class of cl-supercontinuous functions ( clopen maps (Appl. Gen. Topology 8 (2 (2007, 293–300; Indian J. Pure Appl. Math. 14 (6 (1983, 762–772. Further, the class of F-supercontinuous functions is properly contained in the class of R-supercontinuous functions which in turn is strictly contained in the class of continuous functions. Basic properties of F-supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity, which already exist in the mathematical literature, is elaborated. If either domain or range is a functionally regular space (Indagationes Math. 15 (1951, 359–368; 38 (1976, 281–288, then the notions of continuity, F-supercontinuity and R-supercontinuity coincide.
Photon strength functions
Bergqvist, I.
Methods for extracting photon strength functions are briefly discussed. We follow the Brink-Axel approach to relate the strength functions to the giant resonances observed in photonuclear work and summarize the available data on the E1, E2 and M1 resonances. Some experimental and theoretical problems are outlined. (author)
A Functional HAZOP Methodology
Liin, Netta; Lind, Morten; Jensen, Niels
A HAZOP methodology is presented where a functional plant model assists in a goal oriented decomposition of the plant purpose into the means of achieving the purpose. This approach leads to nodes with simple functions from which the selection of process and deviation variables follow directly...
Voos, Avery; Pelphrey, Kevin
Functional magnetic resonance imaging (fMRI), with its excellent spatial resolution and ability to visualize networks of neuroanatomical structures involved in complex information processing, has become the dominant technique for the study of brain function and its development. The accessibility of in-vivo pediatric brain-imaging techniques…
Cryptographic Hash Functions
Gauravaram, Praveen; Knudsen, Lars Ramkilde
functions, also called message authentication codes (MACs) serve data integrity and data origin authentication in the secret key setting. The building blocks of hash functions can be designed using block ciphers, modular arithmetic or from scratch. The design principles of the popular Merkle...
Functional Assessment Inventory Manual.
Crewe, Nancy M.; Athelstan, Gary T.
This manual, which provides extensive new instructions for administering the Functional Assessment Inventory (FAI), is intended to enable counselors to begin using the inventory without undergoing any special training. The first two sections deal with the need for functional assessment and issues in the development and use of the inventory. The…
Functional and cognitive grammars
Institute of Scientific and Technical Information of China (English)
Anna Siewierska
This paper presents a comprehensive review of the functional approach and cognitive approach to the nature of language and its relation to other aspects of human cognition. The paper starts with a brief discussion of the origins and the core tenets of the two approaches in Section 1. Section 2 discusses the similarities and differences between the three full-fledged structural functional grammars subsumed in the functional approach: Halliday's Systemic Functional Grammar (SFG), Dik's Functional Grammar (FG), and Van Valin's Role and Reference Grammar (RRG). Section 3 deals with the major features of the three cognitive frameworks: Langacker's Cognitive Grammar (CG), Goldberg's Cognitive Construction Grammar (CCG), and Croft's Radical Construction Grammar (RCG). Section 4 compares the two approaches and attempts to provide a unified functional-cognitive grammar. In the last section, the author concludes the paper with remarks on the unidirectional shift from functional grammar to cognitive grammar that may indicate a reinterpretation of the traditional relationship between functional and cognitive models of grammar.
Heterogeneity in kinesin function
Reddy, Babu J N; Tripathy, Suvranta; Vershinin, Michael; Tanenbaum, Marvin E; Xu, Jing; Mattson-Hoss, Michelle; Arabi, Karim; Chapman, Dail; Doolin, Tory; Hyeon, Changbong; Gross, Steven P
The kinesin family proteins are often studied as prototypical molecular motors; a deeper understanding of them can illuminate regulation of intracellular transport. It is typically assumed that they function identically. Here we find that this assumption of homogeneous function appears incorrect:
Thermal dielectric function
Moneta, M.
Thermal dielectric functions ε(k,ω) for homogeneous electron gas were determined and discussed. The ground state of the gas is described by the Fermi-Dirac momentum distribution. The low and high temperature limits of ε(k,ω) were related to the Lindhard dielectric function and to ε(k, omega) derived for Boltzmann and for classical momentum distributions, respectively. (author)
Monadic Functional Reactive Programming
A.J. van der Ploeg (Atze); C Shan
htmlabstractFunctional Reactive Programming (FRP) is a way to program reactive systems in functional style, eliminating many of the problems that arise from imperative techniques. In this paper, we present an alternative FRP formulation that is based on the notion of a reactive computation: a
The Grindahl Hash Functions
Knudsen, Lars Ramkilde; Rechberger, Christian; Thomsen, Søren Steffen
to the state. We propose two concrete hash functions, Grindahl-256 and Grindahl-512 with claimed security levels with respect to collision, preimage and second preimage attacks of 2^128 and 2^256, respectively. Both proposals have lower memory requirements than other hash functions at comparable speeds...
Neurophysiology of functional imaging
van Eijsden, Pieter; Hyder, Fahmeed; Rothman, Douglas L.; Shulman, Robert G.
The successes of PET and fMRI in non-invasively localizing sensory functions had encouraged efforts to transform the subjective concepts of cognitive psychology into objective physical measures. The assumption was that mental functions could be decomposed into non-overlapping, context-independent
properties and luminosity functions
Hektor Monteiro
Full Text Available In this article, we present an investigation of a sample of 1072 stars extracted from the Villanova Catalog of Spectroscopically Identified White Dwarfs (2005 on-line version, studying their distribution in the Galaxy, their physical properties and their luminosity functions. The distances and physical properties of the white dwarfs are determined through interpolation of their (B-V or (b-y colors in model grids. The solar position relative to the Galactic plane, luminosity function, as well as separate functions for each white dwarf spectral type are derived and discussed. We show that the binary fraction does not vary significantly as a function of distance from the Galactic disk out to 100 pc. We propose that the formation rates of DA and non-DAs have changed over time and/or that DAs evolve into non-DA types. The luminosity functions for DAs and DBs have peaks possibly related to a star burst event.
The triad value function
Vedel, Mette
the triad value function. Next, the applicability and validity of the concept is examined in a case study of four closed vertical supply chain triads. Findings - The case study demonstrates that the triad value function facilitates the analysis and understanding of an apparent paradox; that distributors...... are not dis-intermediated in spite of their limited contribution to activities in the triads. The results indicate practical adequacy of the triad value function. Research limitations/implications - The triad value function is difficult to apply in the study of expanded networks as the number of connections...... expands exponentially with the number of ties in the network. Moreover, it must be applied in the study of service triads and open vertical supply chain triads to further verify the practical adequacy of the concept. Practical implications - The triad value function cannot be used normatively...
Pair Correlation Function Integrals
Wedberg, Nils Hejle Rasmus Ingemar; O'Connell, John P.; Peters, Günther H.J.
We describe a method for extending radial distribution functions obtained from molecular simulations of pure and mixed molecular fluids to arbitrary distances. The method allows total correlation function integrals to be reliably calculated from simulations of relatively small systems. The long......-distance behavior of radial distribution functions is determined by requiring that the corresponding direct correlation functions follow certain approximations at long distances. We have briefly described the method and tested its performance in previous communications [R. Wedberg, J. P. O'Connell, G. H. Peters......, and J. Abildskov, Mol. Simul. 36, 1243 (2010); Fluid Phase Equilib. 302, 32 (2011)], but describe here its theoretical basis more thoroughly and derive long-distance approximations for the direct correlation functions. We describe the numerical implementation of the method in detail, and report...
Managing Functional Power
Rosenstand, Claus Andreas Foss; Laursen, Per Kyed
How does one manage functional power relations between leading functions in vision driven digital media creation, and this from idea to master during the creation cycle? Functional power is informal, and it is understood as roles, e.g. project manager, that provide opportunities to contribute...... to the product quality. The area of interest is the vision driven digital media industry in general; however, the point of departure is the game industry due to its aesthetic complexity. The article's contribution to the area is a power graph, which shows the functional power of the leading functions according...... to a general digital media creation cycle. This is used to point out potential power conflicts and their consequences. It is concluded that there is normally more conflict potential in vision driven digital media creation than in digital media creation in general or in software development. The method...
Functional data analysis
Ramsay, J O
Scientists today collect samples of curves and other functional observations. This monograph presents many ideas and techniques for such data. Included are expressions in the functional domain of such classics as linear regression, principal components analysis, linear modelling, and canonical correlation analysis, as well as specifically functional techniques such as curve registration and principal differential analysis. Data arising in real applications are used throughout for both motivation and illustration, showing how functional approaches allow us to see new things, especially by exploiting the smoothness of the processes generating the data. The data sets exemplify the wide scope of functional data analysis; they are drwan from growth analysis, meterology, biomechanics, equine science, economics, and medicine. The book presents novel statistical technology while keeping the mathematical level widely accessible. It is designed to appeal to students, to applied data analysts, and to experienced researc...
The function of introns
Liran eCarmel
Full Text Available The intron-exon architecture of many eukaryotic genes raises the intriguing question of whether this unique organization serves any function, or is it simply a result of the spread of functionless introns in eukaryotic genomes. In this review, we show that introns in contemporary species fulfill a broad spectrum of functions, and are involved in virtually every step of mRNA processing. We propose that this great diversity of intronic functions supports the notion that introns were indeed selfish elements in early eukaryotes, but then independently gained numerous functions in different eukaryotic lineages. We suggest a novel criterion of evolutionary conservation, dubbed intron positional conservation, which can identify functional introns.
A phased translation function
Read, R.J.; Schierbeek, A.J.
A phased translation function, which takes advantage of prior phase information to determine the position of an oriented mulecular replacement model, is examined. The function is the coefficient of correlation between the electron density computed with the prior phases and the electron density of the translated model, evaluated in reciprocal space as a Fourier transform. The correlation coefficient used in this work is closely related to an overlap function devised by Colman, Fehlhammer and Bartels. Tests with two protein structures, one of which was solved with the help of the phased translation function, show that little phase information is required to resolve the translation problem, and that the function is relatively insensitive to misorientation of the model. (orig.)
Submodular functions and optimization
Fujishige, Satoru
It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Since the publication of the 1st edition of this book fifteen years ago, submodular functions have been showing further increasing importance in optimization, combinatorics, discrete mathematics, algorithmic computer science, and algorithmic economics, and there have been made remarkable developments of theory and algorithms in submodular functions. The 2nd edition of the book supplements the 1st edition with a lot of remarks and with new two chapters: "Submodular Function Minimization" and "Discrete Convex Analysis." The present 2nd edition is still a unique book on submodular functions, which is essential to students and researchers interested in combinatorial optimization, discrete mathematics, and discrete algorithms in the fields of mathematics, operations research, computer science, and economics. Key features: - Self-contained exposition of the theory of submodular ...
Time Functions as Utilities
Minguzzi, E.
Every time function on spacetime gives a (continuous) total preordering of the spacetime events which respects the notion of causal precedence. The problem of the existence of a (semi-)time function on spacetime and the problem of recovering the causal structure starting from the set of time functions are studied. It is pointed out that these problems have an analog in the field of microeconomics known as utility theory. In a chronological spacetime the semi-time functions correspond to the utilities for the chronological relation, while in a K-causal (stably causal) spacetime the time functions correspond to the utilities for the K + relation (Seifert's relation). By exploiting this analogy, we are able to import some mathematical results, most notably Peleg's and Levin's theorems, to the spacetime framework. As a consequence, we prove that a K-causal (i.e. stably causal) spacetime admits a time function and that the time or temporal functions can be used to recover the K + (or Seifert) relation which indeed turns out to be the intersection of the time or temporal orderings. This result tells us in which circumstances it is possible to recover the chronological or causal relation starting from the set of time or temporal functions allowed by the spacetime. Moreover, it is proved that a chronological spacetime in which the closure of the causal relation is transitive (for instance a reflective spacetime) admits a semi-time function. Along the way a new proof avoiding smoothing techniques is given that the existence of a time function implies stable causality, and a new short proof of the equivalence between K-causality and stable causality is given which takes advantage of Levin's theorem and smoothing techniques.
Multidisciplinary team functioning.
Kovitz, K E; Dougan, P; Riese, R; Brummitt, J R
This paper advocates the need to move beyond interdisciplinary team composition as a minimum criterion for multidisciplinary functioning in child abuse treatment. Recent developments within the field reflect the practice of shared professional responsibility for detection, case management and treatment. Adherence to this particular model for intervention requires cooperative service planning and implementation as task related functions. Implicitly, this model also carries the potential to incorporate the supportive functioning essential to effective group process. However, explicit attention to the dynamics and process of small groups has been neglected in prescriptive accounts of multidisciplinary child abuse team organization. The present paper therefore focuses upon the maintenance and enhancement aspects of multidisciplinary group functioning. First, the development and philosophy of service for the Alberta Children's Hospital Child Abuse Program are reviewed. Second, composition of the team, it's mandate for service, and the population it serves are briefly described. Third, the conceptual framework within which the program functions is outlined. Strategies for effective group functioning are presented and the difficulties encountered with this model are highlighted. Finally, recommendations are offered for planning and implementing a multidisciplinary child abuse team and for maintaining its effective group functioning.
The Enzyme Function Initiativeâ€
Gerlt, John A.; Allen, Karen N.; Almo, Steven C.; Armstrong, Richard N.; Babbitt, Patricia C.; Cronan, John E.; Dunaway-Mariano, Debra; Imker, Heidi J.; Jacobson, Matthew P.; Minor, Wladek; Poulter, C. Dale; Raushel, Frank M.; Sali, Andrej; Shoichet, Brian K.; Sweedler, Jonathan V.
The Enzyme Function Initiative (EFI) was recently established to address the challenge of assigning reliable functions to enzymes discovered in bacterial genome projects; in this Current Topic we review the structure and operations of the EFI. The EFI includes the Superfamily/Genome, Protein, Structure, Computation, and Data/Dissemination Cores that provide the infrastructure for reliably predicting the in vitro functions of unknown enzymes. The initial targets for functional assignment are selected from five functionally diverse superfamilies (amidohydrolase, enolase, glutathione transferase, haloalkanoic acid dehalogenase, and isoprenoid synthase), with five superfamily-specific Bridging Projects experimentally testing the predicted in vitro enzymatic activities. The EFI also includes the Microbiology Core that evaluates the in vivo context of in vitro enzymatic functions and confirms the functional predictions of the EFI. The deliverables of the EFI to the scientific community include: 1) development of a large-scale, multidisciplinary sequence/structure-based strategy for functional assignment of unknown enzymes discovered in genome projects (target selection, protein production, structure determination, computation, experimental enzymology, microbiology, and structure-based annotation); 2) dissemination of the strategy to the community via publications, collaborations, workshops, and symposia; 3) computational and bioinformatic tools for using the strategy; 4) provision of experimental protocols and/or reagents for enzyme production and characterization; and 5) dissemination of data via the EFI's website, enzymefunction.org. The realization of multidisciplinary strategies for functional assignment will begin to define the full metabolic diversity that exists in nature and will impact basic biochemical and evolutionary understanding, as well as a wide range of applications of central importance to industrial, medicinal and pharmaceutical efforts. PMID
The Enzyme Function Initiative.
Gerlt, John A; Allen, Karen N; Almo, Steven C; Armstrong, Richard N; Babbitt, Patricia C; Cronan, John E; Dunaway-Mariano, Debra; Imker, Heidi J; Jacobson, Matthew P; Minor, Wladek; Poulter, C Dale; Raushel, Frank M; Sali, Andrej; Shoichet, Brian K; Sweedler, Jonathan V
The Enzyme Function Initiative (EFI) was recently established to address the challenge of assigning reliable functions to enzymes discovered in bacterial genome projects; in this Current Topic, we review the structure and operations of the EFI. The EFI includes the Superfamily/Genome, Protein, Structure, Computation, and Data/Dissemination Cores that provide the infrastructure for reliably predicting the in vitro functions of unknown enzymes. The initial targets for functional assignment are selected from five functionally diverse superfamilies (amidohydrolase, enolase, glutathione transferase, haloalkanoic acid dehalogenase, and isoprenoid synthase), with five superfamily specific Bridging Projects experimentally testing the predicted in vitro enzymatic activities. The EFI also includes the Microbiology Core that evaluates the in vivo context of in vitro enzymatic functions and confirms the functional predictions of the EFI. The deliverables of the EFI to the scientific community include (1) development of a large-scale, multidisciplinary sequence/structure-based strategy for functional assignment of unknown enzymes discovered in genome projects (target selection, protein production, structure determination, computation, experimental enzymology, microbiology, and structure-based annotation), (2) dissemination of the strategy to the community via publications, collaborations, workshops, and symposia, (3) computational and bioinformatic tools for using the strategy, (4) provision of experimental protocols and/or reagents for enzyme production and characterization, and (5) dissemination of data via the EFI's Website, http://enzymefunction.org. The realization of multidisciplinary strategies for functional assignment will begin to define the full metabolic diversity that exists in nature and will impact basic biochemical and evolutionary understanding, as well as a wide range of applications of central importance to industrial, medicinal, and pharmaceutical efforts. Â
Polysheroidal periodic functions
Truskova, N.F.
Separation of variables in the Helmholtz N-dimensional (N≥4) equation in polyspheroidal coordinate systems leads to the necessity of solving equations going over into equations for polyspheroidal periodic functions used for solving the two-centre problem in quantum mechanics, the three-body problem with Coulomb interaction, etc. For these functions the expansions are derived in terms of the Jacobi polynomials and Bessel functions. Their basic properties, asymptotics are considered. The algorithm of their computer calculations is developed. The results of numerical calculations are given
cl-Supercontinuous Functions
D. Singh
Full Text Available Basic properties of cl-supercontinuity, a strong variant of continuity, due to Reilly and Vamanamurthy [Indian J. Pure Appl. Math., 14 (1983, 767–772], who call such maps clopen continuous, are studied. Sufficient conditions on domain or range for a continuous function to be cl-supercontinuous are observed. Direct and inverse transfer of certain topological properties under cl-supercontinuous functions are studied and existence or nonexistence of certain cl-supercontinuous function with specified domain or range is outlined.
Mean-periodic functions
Carlos A. Berenstein
Full Text Available We show that any mean-periodic function f can be represented in terms of exponential-polynomial solutions of the same convolution equation f satisfies, i.e., u∗f=0(μ∈E′(�n. This extends to n-variables the work of L. Schwartz on mean-periodicity and also extends L. Ehrenpreis' work on partial differential equations with constant coefficients to arbitrary convolutors. We also answer a number of open questions about mean-periodic functions of one variable. The basic ingredient is our work on interpolation by entire functions in one and several complex variables.
Functional Amyloids in Reproduction.
Hewetson, Aveline; Do, Hoa Quynh; Myers, Caitlyn; Muthusubramanian, Archana; Sutton, Roger Bryan; Wylie, Benjamin J; Cornwall, Gail A
Amyloids are traditionally considered pathological protein aggregates that play causative roles in neurodegenerative disease, diabetes and prionopathies. However, increasing evidence indicates that in many biological systems nonpathological amyloids are formed for functional purposes. In this review, we will specifically describe amyloids that carry out biological roles in sexual reproduction including the processes of gametogenesis, germline specification, sperm maturation and fertilization. Several of these functional amyloids are evolutionarily conserved across several taxa, including human, emphasizing the critical role amyloids perform in reproduction. Evidence will also be presented suggesting that, if altered, some functional amyloids may become pathological.
THE PSEUDO-SMARANDACHE FUNCTION
David Gorski
The Pseudo-Smarandache Function is part of number theory. The function comes from the Smarandache Function. The Pseudo-Smarandache Function is represented by Z(n) where n represents any natural number.
Coded Network Function Virtualization
Al-Shuwaili, A.; Simone, O.; Kliewer, J.
Network function virtualization (NFV) prescribes the instantiation of network functions on general-purpose network devices, such as servers and switches. While yielding a more flexible and cost-effective network architecture, NFV is potentially limited by the fact that commercial off......-the-shelf hardware is less reliable than the dedicated network elements used in conventional cellular deployments. The typical solution for this problem is to duplicate network functions across geographically distributed hardware in order to ensure diversity. In contrast, this letter proposes to leverage channel...... coding in order to enhance the robustness on NFV to hardware failure. The proposed approach targets the network function of uplink channel decoding, and builds on the algebraic structure of the encoded data frames in order to perform in-network coding on the signals to be processed at different servers...
Functional Use Database (FUse)
U.S. Environmental Protection Agency — There are five different files for this dataset: 1. A dataset listing the reported functional uses of chemicals (FUse) 2. All 729 ToxPrint descriptors obtained from...
Contributing to Functionality
Törpel, Bettina
The objective of this paper is the design of computer supported joint action spaces. It is argued against a view of functionality as residing in computer applications. In such a view the creation of functionality is equivalent to the creation of computer applications. Functionality, in the view...... advocated in this paper, emerges in the specific dynamic interplay of actors, objectives, structures, practices and means. In this view, functionality is the result of creating, harnessing and inhabiting computer supported joint action spaces. The successful creation and further development of a computer...... supported joint action space comprises a whole range of appropriate design contributions. The approach is illustrated by the example of the creation of the computer supported joint action space "exchange network of voluntary union educators". As part of the effort a group of participants created...
... Liver Function Tests Clinical Trials Liver Transplant FAQs Medical Terminology Diseases of the Liver Alagille Syndrome Alcohol-Related ... the Liver The Progression of Liver Disease FAQs Medical Terminology HOW YOU CAN HELP Sponsorship Ways to Give ...
Introduction to structure functions
Kwiecinski, J.
The theory of deep inelastic scattering structure functions is reviewed with an emphasis put on the QCD expectations of their behaviour in the region of small values of Bjorken parameter x. (author). 56 refs
Bioprinting: Functional droplet networks
Durmus, Naside Gozde; Tasoglu, Savas; Demirci, Utkan
Tissue-mimicking printed networks of droplets separated by lipid bilayers that can be functionalized with membrane proteins are able to spontaneously fold and transmit electrical currents along predefined paths.
Center for Functional Nanomaterials
Federal Laboratory Consortium — The Center for Functional Nanomaterials (CFN) explores the unique properties of materials and processes at the nanoscale. The CFN is a user-oriented research center...
density functional theory approach
YOGESH ERANDE
Jul 27, 2017 ... a key role in all optical switching devices, since their optical properties can be .... optimized in the gas phase using Density Functional Theory. (DFT).39 The ...... The Mediation of Electrostatic Effects by Sol- vents J. Am. Chem.
Reasoning about Function Objects
Nordio, Martin; Calcagno, Cristiano; Meyer, Bertrand; Müller, Peter; Tschannen, Julian
Modern object-oriented languages support higher-order implementations through function objects such as delegates in C#, agents in Eiffel, or closures in Scala. Function objects bring a new level of abstraction to the object-oriented programming model, and require a comparable extension to specification and verification techniques. We introduce a verification methodology that extends function objects with auxiliary side-effect free (pure) methods to model logical artifacts: preconditions, postconditions and modifies clauses. These pure methods can be used to specify client code abstractly, that is, independently from specific instantiations of the function objects. To demonstrate the feasibility of our approach, we have implemented an automatic prover, which verifies several non-trivial examples.
Normal Functioning Family
... Spread the Word Shop AAP Find a Pediatrician Family Life Medical Home Family Dynamics Adoption & Foster Care ... Español Text Size Email Print Share Normal Functioning Family Page Content Article Body Is there any way ...
Fundamentals of functional analysis
Farenick, Douglas
This book provides a unique path for graduate or advanced undergraduate students to begin studying the rich subject of functional analysis with fewer prerequisites than is normally required. The text begins with a self-contained and highly efficient introduction to topology and measure theory, which focuses on the essential notions required for the study of functional analysis, and which are often buried within full-length overviews of the subjects. This is particularly useful for those in applied mathematics, engineering, or physics who need to have a firm grasp of functional analysis, but not necessarily some of the more abstruse aspects of topology and measure theory normally encountered. The reader is assumed to only have knowledge of basic real analysis, complex analysis, and algebra. The latter part of the text provides an outstanding treatment of Banach space theory and operator theory, covering topics not usually found together in other books on functional analysis. Written in a clear, concise manner,...
Smooth functions statistics
Arnold, V.I.
To describe the topological structure of a real smooth function one associates to it the graph, formed by the topological variety, whose points are the connected components of the level hypersurface of the function. For a Morse function, such a graph is a tree. Generically, it has T triple vertices, T + 2 endpoints, 2T + 2 vertices and 2T + 1 arrows. The main goal of the present paper is to study the statistics of the graphs, corresponding to T triple points: what is the growth rate of the number φ(T) of different graphs? Which part of these graphs is representable by the polynomial functions of corresponding degree? A generic polynomial of degree n has at most (n - 1) 2 critical points on R 2 , corresponding to 2T + 2 = (n - 1) 2 + 1, that is to T = 2k(k - 1) saddle-points for degree n = 2k
Structure function monitor
McGraw, John T [Placitas, NM; Zimmer, Peter C [Albuquerque, NM; Ackermann, Mark R [Albuquerque, NM
Methods and apparatus for a structure function monitor provide for generation of parameters characterizing a refractive medium. In an embodiment, a structure function monitor acquires images of a pupil plane and an image plane and, from these images, retrieves the phase over an aperture, unwraps the retrieved phase, and analyzes the unwrapped retrieved phase. In an embodiment, analysis yields atmospheric parameters measured at spatial scales from zero to the diameter of a telescope used to collect light from a source.
Sexual Function Across Aging.
Clayton, Anita H; Harsh, Veronica
Women experience multiple changes in social and reproductive statuses across the life span which can affect sexual functioning. Various phases of the sexual response cycle may be impacted and can lead to sexual dysfunction. Screening for sexual problems and consideration of contributing factors such as neurobiology, reproductive life events, medical problems, medication use, and depression can help guide appropriate treatment and thereby improve the sexual functioning and quality of life of affected women. Treatment options include psychotropic medications, hormone therapy, and psychotherapy.
Inequalities for Humbert functions
Ayman Shehata
Full Text Available This paper is motivated by an open problem of Luke's theorem. We consider the problem of developing a unified point of view on the theory of inequalities of Humbert functions and of their general ratios are obtained. Some particular cases and refinements are given. Finally, we obtain some important results involving inequalities of Bessel and Whittaker's functions as applications.
Functionally graded materials
Mahamood, Rasheedat Modupe
This book presents the concept of functionally graded materials as well as their use and different fabrication processes. The authors describe the use of additive manufacturing technology for the production of very complex parts directly from the three dimension computer aided design of the part by adding material layer after layer. A case study is also presented in the book on the experimental analysis of functionally graded material using laser metal deposition process.
[Functional (psychogenic) vertigo].
Diukova, G M; Zamergrad, M V; Golubev, V L; Adilova, S M; Makarov, S A
Psychogenic (functional) vertigo is in second place by frequency after benign positional paroxysmal vertigo. It is often difficult to make the diagnosis, diagnostic program is expensive and traditional treatment often is not effective. This literature review covers current concepts on the terminology, clinical signs, pathogenesis and treatment approaches with regard to functional vertigo. Special attention is given to cerebral mechanisms of the pathogenesis including cognitive aspects.
NEUROFEEDBACK USING FUNCTIONAL SPECTROSCOPY
Hinds, Oliver; Wighton, Paul; Tisdall, M. Dylan; Hess, Aaron; Breiter, Hans; van der Kouwe, André
Neurofeedback based on real-time measurement of the blood oxygenation level-dependent (BOLD) signal has potential for treatment of neurological disorders and behavioral enhancement. Commonly employed methods are based on functional magnetic resonance imaging (fMRI) sequences that sacrifice speed and accuracy for whole-brain coverage, which is unnecessary in most applications. We present multi-voxel functional spectroscopy (MVFS): a system for computing the BOLD signal from multiple volumes of...
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
Website Policies/Important Links | CommonCrawl |
Shilov boundary
In functional analysis, a branch of mathematics, the Shilov boundary is the smallest closed subset of the structure space of a commutative Banach algebra where an analog of the maximum modulus principle holds. It is named after its discoverer, Georgii Evgen'evich Shilov.
Precise definition and existence
Let ${\mathcal {A}}$ be a commutative Banach algebra and let $\Delta {\mathcal {A}}$ be its structure space equipped with the relative weak*-topology of the dual ${\mathcal {A}}^{*}$. A closed (in this topology) subset $F$ of $\Delta {\mathcal {A}}$ is called a boundary of ${\mathcal {A}}$ if $ \max _{f\in \Delta {\mathcal {A}}}|f(x)|=\max _{f\in F}|f(x)|$ for all $x\in {\mathcal {A}}$. The set $ S=\bigcap \{F:F{\text{ is a boundary of }}{\mathcal {A}}\}$ is called the Shilov boundary. It has been proved by Shilov[1] that $S$ is a boundary of ${\mathcal {A}}$.
Thus one may also say that Shilov boundary is the unique set $S\subset \Delta {\mathcal {A}}$ which satisfies
1. $S$ is a boundary of ${\mathcal {A}}$, and
2. whenever $F$ is a boundary of ${\mathcal {A}}$, then $S\subset F$.
Examples
Let $\mathbb {D} =\{z\in \mathbb {C} :|z|<1\}$ :|z|<1\}} be the open unit disc in the complex plane and let ${\mathcal {A}}=H^{\infty }(\mathbb {D} )\cap {\mathcal {C}}({\bar {\mathbb {D} }})$ be the disc algebra, i.e. the functions holomorphic in $\mathbb {D} $ and continuous in the closure of $\mathbb {D} $ with supremum norm and usual algebraic operations. Then $\Delta {\mathcal {A}}={\bar {\mathbb {D} }}$ and $S=\{|z|=1\}$.
References
• "Bergman-Shilov boundary", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Notes
1. Theorem 4.15.4 in Einar Hille, Ralph S. Phillips: Functional analysis and semigroups. -- AMS, Providence 1957.
See also
• James boundary
• Furstenberg boundary
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
Spectral theory and *-algebras
Basic concepts
• Involution/*-algebra
• Banach algebra
• B*-algebra
• C*-algebra
• Noncommutative topology
• Projection-valued measure
• Spectrum
• Spectrum of a C*-algebra
• Spectral radius
• Operator space
Main results
• Gelfand–Mazur theorem
• Gelfand–Naimark theorem
• Gelfand representation
• Polar decomposition
• Singular value decomposition
• Spectral theorem
• Spectral theory of normal C*-algebras
Special Elements/Operators
• Isospectral
• Normal operator
• Hermitian/Self-adjoint operator
• Unitary operator
• Unit
Spectrum
• Krein–Rutman theorem
• Normal eigenvalue
• Spectrum of a C*-algebra
• Spectral radius
• Spectral asymmetry
• Spectral gap
Decomposition
• Decomposition of a spectrum
• Continuous
• Point
• Residual
• Approximate point
• Compression
• Direct integral
• Discrete
• Spectral abscissa
Spectral Theorem
• Borel functional calculus
• Min-max theorem
• Positive operator-valued measure
• Projection-valued measure
• Riesz projector
• Rigged Hilbert space
• Spectral theorem
• Spectral theory of compact operators
• Spectral theory of normal C*-algebras
Special algebras
• Amenable Banach algebra
• With an Approximate identity
• Banach function algebra
• Disk algebra
• Nuclear C*-algebra
• Uniform algebra
• Von Neumann algebra
• Tomita–Takesaki theory
Finite-Dimensional
• Alon–Boppana bound
• Bauer–Fike theorem
• Numerical range
• Schur–Horn theorem
Generalizations
• Dirac spectrum
• Essential spectrum
• Pseudospectrum
• Structure space (Shilov boundary)
Miscellaneous
• Abstract index group
• Banach algebra cohomology
• Cohen–Hewitt factorization theorem
• Extensions of symmetric operators
• Fredholm theory
• Limiting absorption principle
• Schröder–Bernstein theorems for operator algebras
• Sherman–Takeda theorem
• Unbounded operator
Examples
• Wiener algebra
Applications
• Almost Mathieu operator
• Corona theorem
• Hearing the shape of a drum (Dirichlet eigenvalue)
• Heat kernel
• Kuznetsov trace formula
• Lax pair
• Proto-value function
• Ramanujan graph
• Rayleigh–Faber–Krahn inequality
• Spectral geometry
• Spectral method
• Spectral theory of ordinary differential equations
• Sturm–Liouville theory
• Superstrong approximation
• Transfer operator
• Transform theory
• Weyl law
• Wiener–Khinchin theorem
| Wikipedia |
\begin{definition}[Definition:Square Number/Definition 4]
Square numbers are defined as the sequence:
:$\forall n \in \N: S_n = \map P {4, n} = \begin{cases}
0 & : n = 0 \\
\map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0
\end{cases}$
where $\map P {k, n}$ denotes the $k$-gonal numbers.
\end{definition} | ProofWiki |
\begin{document}
\title{Introducing the Qplex: A Novel Arena for Quantum Theory}
\author{Marcus Appleby} \affiliation{Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, Australia} \author{Christopher A.\ Fuchs} \affiliation{Physics Department, University of Massachusetts Boston, Boston, MA 02125, USA} \affiliation{Max Planck Institute for Quantum Optics, 85748 Garching, Germany} \author{Blake C.\ Stacey} \affiliation{Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA}
\author{Huangjun Zhu}
\affiliation{Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany}
\date{\today}
\begin{abstract} We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, ``Unperformed experiments have no results.'' The tools of quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres's dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, hypothetical and mutually exclusive experiments ought to mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. Along the way, we derive a condition for the existence of a $d$-dimensional SIC. \end{abstract}
\maketitle
\setcounter{assump}{-1}
\section{Introduction}
The arena for standard probability theory is the probability simplex---that is, for a trial of $n$ possible outcomes, the continuous set $\Delta_n$ of all $n$-vectors ${p}\,$ with nonnegative entries $p(i)$ satisfying $\sum_i p(i)=1$. But what is the arena for quantum theory? The answer to this question depends upon how one views quantum theory. If, for instance, one views it as a noncommutative {\it generalization\/} of probability theory, then the arena could be the convex sets of density operators and positive-operator-valued measures over a complex Hilbert space. In contrast, Refs.\ \cite{fuchs2013,fuchs2009,Voldemort} have argued that quantum theory is not so much a generalization of probability theory as an {\it addition\/} to it. This means that standard probability theory is never invalidated, but that further rules must be added to it when the subject matter concerns measurements on quantum systems. One implication of this is that behind every application of quantum theory is a more basic simplex, which through a not-yet-completely-understood consistency requirement, gets trimmed or cropped to a convex subset isomorphic to the usual space of quantum states~\cite{fuchs2002}.\footnote{See also \cite[p.\ 487]{fuchs2010} for the historical roots of this idea.} In the specific context formalized below, we call an arena of this sort---a suitably cropped simplex as the starting point for a full-fledged derivation of quantum theory---a {\it qplex}. In a slogan: If the simplex is the starting point for probability theory, the qplex is the starting point for the quantum.
The introduction of a more basic simplex surrounding the qplex, however, should not be construed as a capitulation to the idea of a hidden-variable theory. Rather it is an attempt to bring to the front of the formalism a foundational idea nicely captured by Asher Peres's famous quip ``unperformed experiments have no outcomes''~\cite{Peres78}. Here the simplex stands for the outcomes of an experiment that will never be done, but could have been done. How is probability theory all by itself to connect the one experiment to the other? It has no tools for it. But quantum theory does, through the Born rule, when suitably rewritten in the language of the qplex. From this point of view, the meaning of the Born rule for probabilities in any actual experiment is that ``behind'' the experiment is a different, hypothetical experiment whose probabilities {\it must be taken into account\/} in the calculation.
To be concrete, let us rewrite quantum theory in a language that would make this apparent {\it were the right mathematical tool available}. Consider the setting of a finite $d$-level quantum system, and suppose that one of the elusive symmetric informationally complete quantum measurements~\cite{zauner1999,renes2004} exists for it. We shall call such an object a ``SIC'' for short. A SIC is a set of $d^2$ rank-one projection operators $\Pi_i=|\psi_i\rangle\langle\psi_i|$ such that \begin{equation} \Tr (\Pi_k \Pi_l) = \frac{d\delta_{kl} + 1}{d + 1}\;. \label{eq:SICOverlaps} \end{equation} For such a set of operators, one can prove that if they exist at all, they must be linearly independent, and rescaling each to $\frac{1}{d}\Pi_i$, they collectively give an informationally complete positive-operator-valued measure (POVM), i.e., $\sum_i\frac{1}{d}\Pi_i=I$. Thus, for any quantum state $\rho$, a SIC can be used to specify a measurement for which the probabilities of outcomes $p(i)$ specify $\rho$ itself. That is, if \begin{equation} p(i) = \frac{1}{d} \Tr(\rho \Pi_i)\;, \label{eq:SicProbabilities} \end{equation} then \begin{eqnarray} \rho
&=& \sum_{i=1}^{d^2} \left[(d+1) p(i) - \frac{1}{d}\right]\! \Pi_i \label{eq:rhoTermsProbs}\\
&=& (d + 1) \sum_{i=1}^{d^2} p(i) \Pi_i - I. \end{eqnarray}
Is it always possible to write a quantum state like this?\footnote{To
our knowledge the first person to write down this expression was the
Cornell University undergraduate Gabriel G.\ Plunk in an attachment
to a 18 June 2002 email to one of us (CAF), though it went
undiscovered for many years. See
Ref.~\cite[pp.\ 472--474]{Fuchs2014}.} Unfortunately, to date, analytic proofs of SIC existence have only been found in dimensions 2--21, 24, 28, 30, 31, 35, 37, 39, 43 and 48~\cite{appleby2016,
applebyprivate}. However, very high-precision numerical approximations (many to 8,000 and 16,000 digits) have been discovered for all dimensions 2 to 147 without exception, plus some dimensions sporadically beyond that---168, 172, 195, 199, 228, 259, 323, at last count~\cite{scott2010, scottprivate, hoangprivate}. In general, the mood of the community is that a SIC should exist in every finite dimension $d$, but we call the SICs ``elusive'' because in more than 18 years of effort no one has ever proven it. See Ref.~\cite{Galois} for an extensive bibliography on the subject. For the purpose of the present discussion, let us suppose that at least one SIC can be found in any finite dimension $d$.
One can now see how to express quantum-state space as a proper subset $Q$ of a probability simplex $\Delta_{d^2}$ over $d^2$ outcomes. That it cannot be the full simplex comes about from the following consideration: For any ${p}\in \Delta_{d^2}$, Eq.~(\ref{eq:rhoTermsProbs}) gives a Hermitian operator $\rho$ with trace~1, but the operator may not be positive-semidefinite as is required of a density operator. Instead, the density operators correspond to a convex subset specified by its extreme points, the pure states $\rho^2 = \rho$. Thanks to an observation by Jones, Flammia and Linden~\cite{flammia2004,jones2005}, we can also characterize pure states as those Hermitian matrices satisfying \begin{equation} \Tr\rho^2 = \Tr\rho^3 = 1. \end{equation} This expression of purity yields two conditions on the probability distributions ${p}$ \cite{fuchs2009,appleby2011,fuchs2013}. First, \begin{equation} \sum_{i=1}^{d^2} p(i)^2 = \frac{2}{d(d+1)}, \label{eq:purity1} \end{equation} and second, \begin{equation} \sum_{ijk} c_{ijk}\, p(i) p(j) p(k) = \frac{d+7}{(d+1)^3}, \label{eq:purity2} \end{equation} where we have defined the real-valued, completely symmetric three-index tensor \begin{equation} c_{ijk} = \hbox{Re}\, \Tr (\Pi_i \Pi_j \Pi_k). \label{eq:c-tensor} \end{equation} The full state space $Q$ is the convex hull of probability distributions satisfying Eqs.~(\ref{eq:purity1}) and (\ref{eq:purity2}).
So the claim can be made true, but what a strange-looking set the quantum states become when written in these terms! What could account for it except already knowing the full-blown quantum theory as usually formulated?
Nevertheless, every familiar operation in the textbook quantum formalism has its translation into the language of this underlying probability simplex, properly restricted to the subset $Q$. For example, given a quantum state $\rho$, one uses the Born rule to calculate the probabilities an experiment will yield its various outcomes with. Using the SIC representation, the description of the measuring apparatus becomes an ordinary set of conditional probabilities, $r(j|i)$. For instance, for a POVM defined by the set of effects \begin{equation} \{E_1,\ldots,E_n\},\ \sum_j E_j = I, \end{equation} the Born rule tells us the probabilities $q(j)$ for its outcomes are \begin{equation} q(j) = \Tr (\rho E_j), \end{equation} but this can be reexpressed as \begin{equation}
q(j) = \sum_i \left[(d+1) p(i) - \frac{1}{d}\right] r(j|i), \label{eq:SICMeasProbs} \end{equation} where \begin{equation}
r(j|i) = \Tr (E_j\Pi_i) \end{equation} meets the criteria for a conditional probability distribution.
In Ref.\ \cite{fuchs2013}, the simple form in Eq.~(\ref{eq:SICMeasProbs}) was considered so evocative of the usual law of total probability from standard probability theory, and seemingly so basic to Peres's ``unperformed experiments have no outcomes'' considerations, that it was dubbed the \emph{urgleichung}---or German for ``primal equation.''
Similarly, if we have a quantum state $\rho$ encoding our expectations for the SIC measurement on some system at time $t = 0$, we can evolve that state forward to deduce what we should expect at a later time, $t = \tau$. In textbook language, we relate these two quantum states by a quantum channel---in the simplest case, by a unitary operation: \begin{equation} \rho' = U \rho U^\dag. \end{equation} Let the SIC representation of $\rho$ be $p(i)$, and let the SIC representation of $\rho'$ be $p'(j)$. We translate the unitary $U$ into SIC language by calculating \begin{equation}
u(j|i) = \frac{1}{d}\Tr\left(U\Pi_i U^\dag \Pi_j\right). \end{equation} The object $u$ is a $d^2 \times d^2$ doubly stochastic matrix~\cite{DStoch}. But now, something fascinating happens. The two quantum states $p(i)$ and $p'(j)$ are related according to \begin{equation}
p'(j) = \sum_i \left[(d+1) p(i) - \frac{1}{d}\right] u(j|i), \label{eq:SICTimeEvol} \end{equation} an expression identical in form to Eq.\ (\ref{eq:SICMeasProbs}).
Formulas (\ref{eq:SICMeasProbs}) and (\ref{eq:SICTimeEvol}) may be compared with {\it what would have been given\/} by the standard law of total probability \begin{equation}
q(j) = \sum_i p(i) r(j|i), \label{eq:ClassMeasProbs} \end{equation} and the standard rule for stochastic evolution, \begin{equation}
p'(j) = \sum_i p(i) u(j|i), \label{eq:ClassTimeEvol} \end{equation} were they applicable. This emphasizes again that the quantum laws are different but, in the setting of a SIC-induced simplex, intriguingly similar to their classical counterparts.
This leads one to wonder whether, or to what extent, these very special forms Eqs.\ (\ref{eq:SICMeasProbs}) and (\ref{eq:SICTimeEvol}) might imply the very arena $Q$ in which they are valid. This is the program laid out in Refs.\ \cite{fuchs2013,fuchs2009,Voldemort} and a key motivation for the geometric studies of Refs.\ \cite{appleby2011,appleby2011b,GroupAlg}. Here we will carry the program much further than previously.
Another familiar operation in the standard language of quantum theory is the Hilbert--Schmidt inner product between two quantum states, $\Tr(\rho\sigma)$. Using the SIC representations of $\rho$ and $\sigma$ as probability vectors ${p}$ and ${s}$, it is straightforward to show that \begin{equation} \Tr(\rho\sigma) = d(d+1) \inprod{p}{s} - 1. \end{equation} Because the inner product of any two quantum states $\rho$ and $\sigma$ is bounded between 0 and 1, we know that \begin{equation} \frac{1}{d(d+1)} \leq \inprod{p}{s}
\leq \frac{2}{d(d+1)}. \label{eq:p-dot-s-bounds} \end{equation} We designate these the \emph{fundamental inequalities.} The upper bound is simply the quadratic constraint we saw already in Eq.~(\ref{eq:purity1}), but the lower bound imposes new and surprisingly intricate conditions on the vectors that can be admissible states.
We will say that two vectors ${p}$ and ${s}$ in the probability simplex $\Delta_{d^2}$ are \emph{consistent} if their inner product obeys both inequalities in Eq.~(\ref{eq:p-dot-s-bounds}). If we have a subset of the probability simplex in which every pair of vectors obeys those bounds, we call it a \emph{germ}: It is an entity from which a larger structure can grow. If including one additional vector in a germ could make that set inconsistent, then that germ is said to be a \emph{maximal.} We will see that a maximal germ is one way to define a \emph{qplex.}
Any quantum state space in SIC representation is a qplex. However, the converse is not true: There exist qplexes that are not equivalent to quantum state space. That said, any qplex is already a mathematically rich structure. A primary goal of this paper is to use that richness and identify an extra condition which can be imposed upon a qplex, such that satisfying that constraint will make the qplex into a quantum state space.
In Section~\ref{sec:basic} we see how quantum physics furnishes a new way that probability assignments can mesh together, a way not foreseen in classical thinking. This will lead us from very general considerations to the specific definition of a qplex. In Section~\ref{sec:polarity} we apply a tool from the theory of polytopes~\cite{grun,zieg} to derive a number of basic results about the geometry of an arbitrary qplex. Among other applications, we find a simple, intuitively appealing proof that a polytope embedded in quantum state space cannot contain the in-sphere of quantum state space.
Sections~\ref{sec:intg} and~\ref{sec:qgroups} are the core of the paper. In almost every geometrical problem, a study of the symmetries of the object or objects of interest plays an essential role. However, it turns out that qplexes\ have the unusual property that the symmetry group, instead of having to be imposed from the outside, is contained internally to the structure. In this they might be compared with elliptic curves~\cite{elliptic3}. In spite of the extreme simplicity of the defining equation \begin{equation} y^2 = x^3 + a x + b, \end{equation} elliptic curves have managed to remain at the cutting edge of mathematics for two millennia, from the work of Diophantus down to the present day. They play an important role in, for example, the recent proof of Fermat's last theorem~\cite{wiles}. One of the reasons for their high degree of mathematical importance is the fact that they carry within themselves a concealed group. Qplexes\ have a similar property. In Sections~\ref{sec:intg} and~\ref{sec:qgroups} we describe this property, and examine its implications.
In Section~\ref{sec:intg} we present our main application. We apply the results established in the previous section to the SIC existence problem and show that SIC existence in dimension $d$ is equivalent to the existence of a certain kind of subgroup of the real orthogonal group in dimension $d^2-1$. We presented this result in a previous publication~\cite{GroupAlg}, where we derived it by more conventional means. In this paper, we describe the way we originally proved it, using the qplex\ formulation. This is because we believe the method of proof is at least as interesting as the result itself.
In Section~\ref{sec:character} we turn to the problem of identifying the ``missing assumption'' which will serve to pick out quantum state space uniquely from the set of all qplexes. Of course, as is usual in such cases, there is more than one possibility. We identify one such assumption: the requirement that the symmetry group contain a subgroup isomorphic to the projective unitary group. This is a useful result because it means that we have a complete characterization of quantum state space in probabilistic terms. It also has an important corollary: That SIC existence in dimension $d$ is equivalent to the existence of a certain kind of subgroup of the real orthogonal group in dimension $d^2-1$.
Finally, we wrap up in Section~\ref{sec:future} with list of several possible directions for future investigations. If this research program is on the right track, it is imperative that a more basic path from qplex to quantum state space be found. There is plenty of work to do here.
\section{The Basic Scheme} \label{sec:basic} The urgleichung (\ref{eq:SICMeasProbs}) and the inequalities (\ref{eq:p-dot-s-bounds}) are not independent. In this section, we will start with a generalized form of the urgleichung and, making a few additional assumptions, derive the fundamental inequalities. This is, strictly speaking, not necessary for the mathematical developments in the later sections of the paper. One can assume the fundamental inequalities as a starting point and then proceed from that premise. In fact, we will later see that using that approach, one can derive as consequences the assumptions we will invoke here. Speaking in general terms, we can think of this section as proving the ``if'' direction, and the following section as proving ``only if.'' One benefit of deriving the fundamental inequalities in this manner is to help compare and contrast our reconstruction of quantum theory with other approaches~\cite{transcript, coecke2016, HoehnWever, Masanes, Hardy01,
Schack03, Barnum}. These other reconstructions are \emph{operational} in character: They take, as fundamental conceptual ingredients, laboratory procedures like ``preparations'' and ``tests.'' Our language in this section will have a similar tone. However, we will keep Peres' dictum that ``unperformed experiments have no results'' at the forefront of our considerations.
Our first step is to understand how the urgleichung is an example of this principle. To do so, we consider the following scenario~\cite{Voldemort, transcript}.
\begin{figure}
\caption{ Analysing one scenario in terms
of another: An agent Alice intends to perform an experiment on the
ground, whose outcomes she labels with the index $j$. The other
index, $i$, labels the outcomes of a ``Bureau of Standards''
measurement which Alice \emph{could} carry out, but which remains
unperformed. Classical physics and quantum physics both allow for
Bureau of Standards measurements, experiments that are
\emph{informationally complete} in the following sense. If Alice
has a set of probabilities $p(i)$ for the Bureau of Standards
measurement outcomes, she can calculate the proper set of
probabilities $q(j)$ for the outcomes of the ground measurement,
using the conditional probabilities $r(j|i)$.}
\label{fig:ground-versus-sky}
\end{figure}
Fix a dimension $d \geq 2$, and consider a system to which we will ascribe a quantum state in $d$-dimensional Hilbert space $\mathcal{H}_d$. We will investigate this system by means of two measuring devices, which we model in the standard way by POVMs. One measuring device is a SIC measurement, defined by a set of $d^2$ rank-1 projection operators $\{\Pi_i\}$. The effects which comprise this POVM are the operators rescaled by the dimension: \begin{equation} E_i = \frac{1}{d} \Pi_i. \end{equation} We will refer to this as the ``Bureau of Standards'' measurement. It is helpful to imagine this measuring device as being located in some comparatively inaccessible place: perhaps inside a vault, or secured in an airship floating through the sky. An agent \emph{can} take her system of interest to the Bureau of Standards device, but she has good reason to want to bypass that step. The other measurement is an arbitrary POVM, whose effects we denote by~$F_j$.
As illustrated in Figure~\ref{fig:ground-versus-sky}, we will consider two experimental scenarios, which we will call the ``ground path'' and the ``sky path.'' If we follow the ground path, we take our system of interest directly to the $\{F_j\}$ measuring device, which we will call the measurement on the ground. If we instead follow the sky path, we will take our system to the Bureau of Standards measurement, physically obtain a result by performing that measurement, and then come back down for the second stage, where we conduct the measurement on the ground.
Suppose that Alice follows the sky path in Figure~\ref{fig:ground-versus-sky}. That is, she physically takes her system of interest and performs the Bureau of Standards measurement upon it. Then, she returns the system to the ground and conducts the measurement $\{F_j\}$ there. Before carrying out the Bureau of Standards measurement, she has some expectations for what might happen, which she encodes as a probability distribution $p(i)$. Obtaining an outcome $i$, she updates her state assignment for the system to the operator $\Pi_i$. Her expectations for the outcome of the ground measurement will then be the conditional probabilities
$r(j|i)$. Prior to performing the Bureau of Standards measurement, Alice assigns the probability \begin{equation}
\hbox{Prob}(j) = \sum_i p(i) r(j|i) \label{eq:naive-LoTP} \end{equation} to the event of obtaining outcome $j$ when she brings the system back down to the ground and performs the second measurement in the sequence.
Classical intuition suggests that Alice should use the same expression for computing the probability of outcome $j$ on the ground even if she goes directly to the ground experiment and does not perform the measurement in the sky. If $p(i)$ is the probability that she
\emph{would} obtain outcome $i$ \emph{were she to perform} the sky measurement, and $r(j|i)$ is the conditional probability for outcome $j$ \emph{if} the event $i$ \emph{were to occur} in the sky, then it is almost instinctive to calculate the probability of~$j$ by summing
$p(i) r(j|i)$. Mathematically, this is \emph{not necessarily
correct,} because the ground path and the sky path are two different physical scenarios. If $C_1$ and $C_2$ are two background conditions, then nothing in probability theory forces $\hbox{Prob}(j|C_1) =
\hbox{Prob}(j|C_2)$. Writing $q(j)$ for the probability of obtaining $j$ by following the ground path, we have that \begin{equation} q(j) \hbox{ is not necessarily equal to }
\sum_i p(i) r(j|i). \end{equation} It is merely the assumption that an informationally complete measurement must be measuring some pre-existing physical property of the system that leads Alice to use Eq.~(\ref{eq:naive-LoTP}) even when she does not physically obtain an outcome in the sky. In other words, using Eq.~(\ref{eq:naive-LoTP}) to calculate $q(j)$ amounts to assuming that the measurement outcome $i$ is \emph{as good as
existing,} even when it remains completely counterfactual.
Probability theory itself does not tell us how to find $q(j)$ in terms of~$p(i)$ and $r(j|i)$. Classical intuition suggests one way of augmenting the abstract formalism of probability theory: using Eq.~(\ref{eq:naive-LoTP}). The crucial point is that \emph{quantum
theory gives us an alternative.} It is simply to use the Born rule, in the form of the urgleichung.
The Born-rule probability for obtaining the outcome with index $j$ is \begin{equation} q(j) = \Tr(\rho F_j) = \sum_i \left[(d+1) p(i) - \frac{1}{d}\right]
r(j|i), \end{equation} where \begin{equation}
r(j|i) = \Tr (\Pi_i F_j). \end{equation}
Note that $r(j|i)$ is also the probability that the Born rule would tell us to assign to the outcome $j$ if our quantum state assignment for the system were $\Pi_i$.
Probability theory is a way to augment our raw experiences of life: It provides a means to manage our expectations carefully. In turn, quantum theory augments the mathematics of probability, furnishing links between quantities that, considering only the formalism of probability theory, would be unrelated. These new relationships are quantitatively precise, but at variance with classical intuition, reflecting the principle that unperformed experiments have no outcomes.
We now explore the consequences of relating mutually exclusive hypothetical scenarios by the urgleichung. Using seven assumptions, of which the urgleichung is the most radical, we will arrive at the fundamental inequalities (\ref{eq:p-dot-s-bounds}). Because the constants $d^2$ and $d+1$ and $1/d$ look rather arbitrary at first glance, we will begin with a more general expression.
\begin{assump} \label{assump:urgleichung}
The Generalized Urgleichung. Given a Bureau of Standards probability distribution $\{p(i): i = 1,\ldots,N\}$, and a matrix of conditional probabilities $r(j|i)$, we compute the probabilities for an experiment on the ground by means of \begin{equation}
q(j) = \sum_{i=1}^N \left[\alpha p(i) - \beta\right] r(j|i). \label{eq:gen-urgleichung} \end{equation} \end{assump} In what follows, this will be our primary means of relating one probability distribution to another. The basic normalization requirements are \begin{equation}
\sum_i p(i) = 1,\ \sum_j r(j|i) = 1,\ \sum_j q(j) = 1. \end{equation} Normalization relates the constants $\alpha$, $\beta$ and $N$: \begin{equation} \alpha = N\beta + 1. \end{equation}
We denote the set of valid states ${p}$ by~$\mathcal{P}$, and the set of valid measurements by~$\mathcal{R}$. For any ${p} \in \mathcal{P}$ and any
$r(j|i) \in \mathcal{R}$, the vector ${q}$ calculated using the urgleichung is a proper probability distribution.
If we take any $r \in \mathcal{R}$ and sum over both indices, we find that \begin{equation}
\sum_{i,j} r(j|i) = \sum_i 1 = N. \end{equation}
\begin{assump} \label{assump:max} Maximality. The set of all states $\mathcal{P}$ and the set of all measurements $\mathcal{R}$ together have the property that no element can be added to either without introducing an inconsistency, i.e., a pair $(p \in \mathcal{P}, r \in \mathcal{R})$ for which the urgleichung yields an invalid probability. \end{assump}
It is sometimes helpful to write the urgleichung in vector notation: \begin{equation} {q} = rM{p}. \end{equation}
Here, $r$ is a matrix whose $(j,i)$ entry is given by $r(j|i)$, and $M$ is a linear combination of the identity matrix $I$ and the matrix whose elements all equal 1, the so-called Hadamard identity $J$: \begin{equation} M = \alpha I - \beta J. \end{equation}
Assumptions \ref{assump:urgleichung} and \ref{assump:max} imply a fair bit about the structure of~$\mathcal{P}$ and $\mathcal{R}$. \begin{lemma} \label{lm:convex} The set $\mathcal{P}$ of all states and the set $\mathcal{R}$ of all measurements are both convex and closed. \end{lemma} \begin{proof} Let $p_1,p_2 \in \mathcal{P}$, and for any $r \in \mathcal{R}$, define \ea{ q_1 &= rMp_1, & q_2 &= rMp_2. } By assumption, both $q_1$ and $q_2$ are valid probability vectors (i.e., they are normalized, and all their entries are nonnegative). Define \begin{equation} p_\lambda = \lambda p_1 + (1-\lambda)p_2. \end{equation} Then \begin{equation} q_\lambda = rMp_\lambda = \lambda q_1 + (1-\lambda)q_2. \end{equation} This is a convex combination of points in the probability simplex, and as such it also belongs to the probability simplex. By assumption, this holds true for every $r \in \mathcal{R}$, and so by maximality, $p_\lambda \in \mathcal{P}$. The proofs of the convexity of~$\mathcal{R}$ and of closure work analogously. \end{proof}
Consider the case where the ground and sky measurements are the same. In that scenario, we have ${q} = {p}$, and so the measurement matrix must be the inverse of~$M$: \begin{equation} r_F = M^{-1} = \frac{1}{\alpha}I + \frac{\beta}{\alpha}J. \label{eq:r_F} \end{equation} Note that we have to include $r_F$ within $\mathcal{R}$ by the maximality assumption.
The urgleichung is one way that quantum theory builds upon the mathematics of probability, interconnecting our previsions for different experiments, previsions that basic probability theory alone would leave separate. Quantum theory augments the probability formalism in another fashion as well, and it is to that which we now turn.
Our next assumption will establish that the set of measurements $\mathcal{R}$ can be constructed from the set of states $\mathcal{P}$. On a purely mathematical level, we could justify this by saying that we wish to build the most parsimonious theory possible upon the urgleichung, and so we simplify matters by having one fundamental set instead of two. As far as constructing a mathematical theory goes, this is certainly a legitimate way to begin. We can, however, provide a more physical motivation than that.
Probability theory, intrinsically, assumes very little about the structure of event spaces. With it, we can for example discuss rolling a die and recording the side that lands facing up; we say that the realm of possible outcomes for this experiment is the set $\{1,2,3,4,5,6\}$. In this experiment, the outcome ``1'' is no more \emph{like} the outcome ``2'' than it is \emph{like} the outcome ``6''. We can ascribe probabilities to these six potential events without imposing a similarity metric upon the realm of outcomes. We use integers as labels, but we care hardly at all about the number-theoretic properties of those integers. When we roll the die, we are indifferent to the fact that 5 is prime and 6 is perfect. Nor is the event of observing a particular integer in this experiment related, necessarily, to the event of observing that same integer in a \emph{different} experiment.
When Alice first learns probability theory, she picks up this habit of tagging events with integers. If Alice considers a long catalogue of experiments that she could perform, she might label the possible outcomes of the first experiment by the integers from 1 to~$N_1$, the outcomes of the second experiment by the integers $\{1,\ldots,N_2\}$ and so on. But, in general, Alice has the freedom to permute these labels as she pleases. She does not have to regard the experience of obtaining $j = 17$ in one experiment as similar to the experience of obtaining $j = 17$ in any other.
But what if Alice wants more structure than this? When Alice contemplates an experiment that she might carry out, she considers a set of possible outcomes for it, \emph{i.e.,} a realm of potential experiences which that action might elicit. She can assign each of those potential experiences a label drawn from whatever mathematical population she desires. Her \emph{index set} for a given experiment can be a subset of whatever population she finds convenient. When Alice adopts the urgleichung as an empirically-motivated addition to the bare fundamentals of probability theory, does she, by that act, also gain a natural collection of mathematical entities from which to build index sets?
In fact, she has just such a collection at hand: She can use the set of valid states, $\mathcal{P}$!
To consider the matter more deeply, we ask the following question: Under what conditions would Alice consider two outcomes of two different experiments to be equivalent? For example, Alice contemplates two experiments she might feasibly perform, which she describes by two matrices $r$ and $r'$. When would Alice treat an outcome $j$ of experiment $r$ to be equivalent to an outcome $j'$ of~$r'$~\cite{fuchs2002}? Generally, the tools she has on hand to make such a judgment are her probability ascriptions for those outcomes. If her overall mesh of beliefs is that her probability of experiencing $j$ upon enacting $r$ is the same as her probability for finding $j'$ when enacting $r'$, no matter what her state assignment ${p}$, then she has good grounds to call $j$ and $j'$ equivalent. In order to satisfy $q(j) = q'(j')$ for all $p \in \mathcal{P}$, the measurement matrices $r$ and $r'$ must obey \begin{equation}
r(j|i) = r'(j'|i),\ \forall i. \end{equation}
The simplest way to ensure that this is possible is to build all elements $r$ of the set $\mathcal{R}$ from a common vocabulary. When we construct an element $r \in \mathcal{R}$, we draw each row from a shared pool of ingredients. The natural, parsimonious choice we have on hand for this purpose is the set $\mathcal{P}$. This means that, up to scaling, measurement outcomes are actually identified with points in the probability simplex.
Let $r \in \mathcal{R}$ be a valid measurement. If each row of the matrix
$\{r(j|i)\}$ can also naturally be identified with a vector ${s} \in \mathcal{P}$, then we are led to consider the vector ${s}$ sitting inside $r$ in some fashion. The simplest reasonable relation between ${s}$, which is a vector with $N$ elements, and the measurement matrix $r$, whose rows have length $N$, is to have a row of~$r$ be linearly proportional to~${s}$.
\begin{assump} \label{assump:R-from-P}
Measurement Matrices are Constructed from States. Given any $r \in \mathcal{R}$, we can write a row $\{r(j|i) : i = 1,\ldots,N\}$ as a vector ${s}_j \in \mathcal{P}$, up to a normalization factor: \begin{equation}
r(j|i) = N \gamma_j s_j(i). \label{eq:R-from-P} \end{equation} Furthermore, any state in $\mathcal{P}$ can be used in this manner. \end{assump} For brevity, we will refer to the $s_j$ as ``measurement vectors.'' We will shortly identify the meaning of the constants $\{\gamma_j\}$, which we have written with the prefactor $N$ for later convenience.
\begin{assump} \label{assump:ignorance} Possibility of Maximal Ignorance. The state $c$, defined by \begin{equation} c(i) = \frac{1}{N}\ \forall\ i, \end{equation} belongs to $\mathcal{P}$. \end{assump} This can be deduced from other postulates, but the state $c$ is a useful tool, and it is helpful to point its existence out explicitly. For example, substituting the state of complete ignorance ${c}$ into the urgleichung, we obtain \begin{equation}
q(j) = \frac{1}{N} \sum_i r(j|i). \end{equation}
What is the meaning of the factors $\{\gamma_j\}$? To find out, we apply a measurement $r \in \mathcal{R}$ to the state ${c}$: \begin{equation}
q(j) = \frac{1}{N} \sum_i r(j|i)
= \gamma_j \sum_i s_j(i)
= \gamma_j. \end{equation} The factors $\{\gamma_j\}$ indicate the probability of obtaining the $j^{\rm th}$ outcome on the ground when the agent is completely indifferent to the potential outcomes of the sky experiment.
If the effect of some $r \in \mathcal{R}$, when applied via the urgleichung, is to send ${c}$ to itself, then we have that \begin{equation}
c(j) = \frac{1}{N} = \frac{1}{N} \sum_i r(j|i)
\Rightarrow \sum_i r(j|i) = 1. \end{equation} Combined with the basic normalization requirement for conditional probabilities, this states that a measurement that preserves ${c}$ is represented by a \emph{doubly stochastic} matrix.
\begin{lemma} \label{lm:doubly-stochastic} Measurements that send the state $c$ to itself are represented by doubly stochastic matrices. \end{lemma}
When we postulated the urgleichung, we added structure to the bare essentials of probability theory, and the structure we added related one experiment to another in a way above and beyond basic coherence. With Assumption~\ref{assump:R-from-P}, we are also interrelating different experiments. We can appreciate this in another way by considering what it means for a physical system to be usable as a scientific instrument.
What conditions must an object meet in order to qualify as a piece of laboratory apparatus? Classically, a bare minimum requirement is that the object has a set of distinguishable configurations in which it can exist. These might be positions of a pointer needle, heights of a mercury column, patterns of glowing lights and so forth. The essential point is that the system can be in different configurations at different times: A thermometer that always reports the same temperature is useless. We can label these distinguishable configurations by an index $j$. The \emph{calibration} process for a laboratory instrument is a procedure by which a scientist assigns conditional probabilities $r(j|i)$ to the instrument, relating the readout states $j$ to the inputs $i$. In order to make progress, we habitually assume that nature is not so perverse that the results of the calibration phase become completely irrelevant when we proceed to the next step and apply the instrument to new systems of unknown character.
But what if nature \emph{is} perverse? Not enough so to forbid the possibility of science, but enough to make life interesting. Quantitatively speaking, what if we must modify the everyday assumption that one can carry the results of a calibration process unchanged from one experimental context to another?
\emph{The urgleichung is just such a modification.} The $\{r(j|i)\}$ do not become irrelevant when we move from the sky context to the ground, but we do have to use them in a different way.
In quantum physics, we no longer treat ``measurement'' as a passive reading-off of a specified, pre-existing physical quantity. However, we do still have a counterpart for our classical notion of a system that can qualify as a laboratory apparatus. Instead of asking whether the system can exist in one of multiple possible classical states, we ask whether our overall mesh of beliefs allows us to consistently assign any one of multiple possible catalogues of expectations. That is, if an agent Alice wishes to use a system as a laboratory apparatus, she must be able to say now that she can conceive of ascribing any one of several states to it at a later time. We define a \emph{discrete apparatus} as a physical system with an associated set of states, \begin{equation} \{s_1,\ldots,s_{m}\} \subset \mathcal{P}. \end{equation} The analogue of classical uncertainty about where a pointer might be pointing is the convex combination of the states $\{s_j\}$. Therefore, our basic mental model of a laboratory apparatus is a polytope in~$\mathcal{P}$, with the $\{s_j\}$ as its vertices. Assumption \ref{assump:R-from-P} says that \emph{Alice can pick up any such
apparatus and use it as a ``prosthetic hand'' to enrich her
experience of asking questions of nature.}
We can think of Assumption~\ref{assump:R-from-P} in another way, if we rewrite Eq.~(\ref{eq:R-from-P}) in the following manner: \begin{equation}
s_j(i) = \frac{\left(\frac{1}{N}\right) r(j|i)}{\gamma_j}. \end{equation} Earlier, we noted that $\gamma_j$ is the probability of obtaining the $j^{\rm th}$ outcome on the ground, given complete ignorance about the potential outcomes of the sky experiment. In addition, $1/N$ is the probability assigned to each outcome of the sky experiment by the state of complete ignorance. So, \begin{equation}
s_j(i) = \frac{\hbox{PrCI}(i)\, r(j|i)}{\hbox{PrCI}(j)}, \end{equation}
where the notation ``PrCI'' here indicates a probability assignment given that the state for the sky experiment is $c$. Note that $\hbox{PrCI}(j|i) = r(j|i)$. But this means that the expression on the right-hand side above is just the ordinary Bayes formula for inverting conditional probabilities: \begin{equation}
\hbox{PrCI}(i|j) = \frac{\hbox{PrCI}(i)\, \hbox{PrCI}(j|i)}{\hbox{PrCI}(j)}. \end{equation} Therefore, we can interpret the mathematical relation established in Assumption~\ref{assump:R-from-P} as saying that ``posteriors from maximal ignorance are priors''~\cite{fuchs2013}. For the remainder of this paper, we will not be considering in detail the rules for changing one's probabilities upon new experiences---a rather intricate subject, all things told~\cite{QBist-decoherence, stacey-thesis}. So, we will not stress the ideas of ``priors'' and ``posteriors,'' but it is good to know that this reading of Assumption~\ref{assump:R-from-P} exists.
Writing the urgleichung in terms of the vector ${s}_j$, \begin{align} q(j) &= \sum_i \left[\alpha p(i) - \beta\right] N \gamma_j
s_j(i) \\
&= N\alpha \gamma_j \inprod{p}{s_j} - N\beta\gamma_j. \end{align} The fact that $q(j)$ must be nonnegative for all $j$ implies a lower bound on the scalar product $\inprod{p}{s_j}$: \begin{equation} \inprod{p}{s_j} \geq \frac{\beta}{\alpha}. \label{eq:first-lower-bound} \end{equation}
The measurement described by the matrix $r_F$ in Eq.~(\ref{eq:r_F}) yields, by construction, equal probabilities for all outcomes given the input state ${c}$. That is, it is an experiment with $N$ outcomes, and $\gamma_j = 1/N$ for all of them. Therefore, we can take the rows of~$r_F$ as specifying $N$ special vectors within~$\mathcal{P}$. We have that \begin{equation}
r_F(j|i) = e_j(i), \end{equation} where the vector ${e}_j$ is flat across all but one entries: \begin{equation} e_j(i) = \frac{1}{\alpha}(\delta_{ji} + \beta). \end{equation} We will refer to the vectors $\{{e}_k\}$ as the \emph{basis
distributions.}
What happens if we take a measurement $r \in \mathcal{R}$, and act with it via the urgleichung upon a basis distribution ${e}_k$? The result is straightforwardly computed to be \begin{align} q(j) &= \sum_i \left[\alpha\left(\frac{\beta}{\alpha}
+ \frac{1}{\alpha} \delta_{ik}
\right)
- \beta
\right] r(j|i) \\
&= \beta \sum_i r(j|i) + \sum_i \delta_{ik} r(j|i)
- \beta \sum_i r(j|i) \\
&= r(j|k). \label{eq:r-upon-basis} \end{align} This will be useful later.
Note that the basis distributions all have magnitude equal to \begin{equation} \inprod{e_k}{e_k} = \frac{1 + 2\beta + N\beta^2}{\alpha^2}. \label{eq:basis-purity} \end{equation} This result singles out a \emph{distinguished length scale} in probability space, namely, the radius of the sphere on which all the basis distributions live.
The lower bound (\ref{eq:first-lower-bound}) suggests the following construction. Let $H$ be the hyperplane of vectors in~$\mathbb{R}^N$ that sum to unity: \begin{equation} H =\left\{v \in \fd{R}^{N} \colon \inprod{v}{c} = \frac{1}{N}\right\}. \end{equation} This hyperplane includes the probability simplex. For any set $A$ of probability distributions, consider the set \begin{equation} \dl{A} = \left\{ u \in H \colon \inprod{u}{v} \ge \frac{\beta}{\alpha} \ \forall v \in A\right\}. \end{equation} This set includes all the probability distributions that are consistent with each point in~$A$, with respect to the lower bound we derived from the urgleichung. We will designate the set $\dl{A}$ the \emph{polar} of~$A$, following the terminology for a related concept in geometry~\cite{grun, zieg}. Let $\mathcal{P}$ be the set of all valid states. The set of all measurement vectors that are consistent with these states, with respect to the lower bound, is that portion of the polar of~$\mathcal{P}$ that lies within the probability simplex: \begin{equation} \dl{\mathcal{P}} \cap \Delta = \left\{ s : \inprod{s}{p} \geq \frac{\beta}{\alpha} \forall p \in \mathcal{P} \right\} \cap \Delta. \end{equation} If some $s$ in this set is not in the set $\mathcal{P}$, then some measurement vector does not correspond to a state. Likewise, if some $p \in \mathcal{P}$ is not in this set, then that state cannot correspond to a measurement vector. Both of these cases violate the mapping we have advocated on general conceptual grounds. Therefore, our first three assumptions imply that we consider sets $\mathcal{P}$ for which \begin{equation} \mathcal{P} = \dl{\mathcal{P}} \cap \Delta. \end{equation} We will see momentarily how to simplify this condition, establishing the condition that a state space $\mathcal{P}$ must be self-polar: \begin{equation} \mathcal{P} = \dl{\mathcal{P}}. \end{equation}
In order to prove this proposition, we need to know more about the operation of taking the polar. We can derive the relations we require by adapting some results from the higher-dimensional geometry literature. Gr\"unbaum~\cite{grun} defines the polar of $A\subseteq \fd{R}^{d^2}$ to be the set \begin{equation} A^{\circ} = \{ u \in \fd{R}^{d^2} \colon \inprod{u}{v} \le 1 \ \forall v \in A\}. \end{equation} Our definition of the polar $\dl{A}$ is close enough to this definition of~$A^{\circ}$ that many results about the latter can be carried over with little effort. The properties of the polar $\dl{A}$ are summarized in the following theorem. \begin{theorem} \label{tm:polarity} For all $A\subseteq H$, the polar $\dl{A}$ is a closed, convex set containing $c$. Since we will frequently be invoking the concept of convex hulls, we introduce the notation $\cc(A)$ for the closed, convex hull of the set $A$. We have \ea{ \dl{A} & = \dl{\bigl(\cc(A\cup \{c\})\bigr)}, \\ \ddl{A} &= \cc(A\cup \{c\}), } for all $A\subseteq H$. In particular, $A$ is equal to its double polar $\ddl{A}$ if and only if it is closed, convex and contains $c$.
For all $A$, $B\subseteq H$ \begin{equation} A\subseteq B \implies \dl{B} \subseteq \dl{A}. \end{equation} If $\mathcal{A}$ is an arbitrary family of subsets of $H$ then \ea{ \dl{\left(\bigcup_{A\in \mathcal{A}} A \right)} &= \bigcap_{A\in \mathcal{A}} \dl{A}. \label{eq:dualUnionB} \\ \intertext{If, in addition, $\ddl{A}=A$ for all $A\in\mathcal{A}$ then} \dl{\left(\bigcap_{A\in \mathcal{A}} A \right)} &=\cc\left( \bigcup_{A\in \mathcal{A}} \dl{A}\right). \label{eq:dualIntersectionB} } \end{theorem} \begin{proof} All these properties follow by relating Gr\"unbaum's definition of the polar with ours. Let $f\colon \fd{R}^{N} \to \fd{R}^{N}$ be the affine map defined by \begin{equation} f(u) = N\alpha (u-c), \end{equation} and let $H_0$ be the subspace \begin{equation} H_0 = \{ u\in \fd{R}^{N} \colon \langle u, c \rangle = 0\}. \end{equation} One then has \begin{equation} \dl{A} = f^{-1} \Bigl( \bigl(-f(A)\bigr)^{\circ}\cap H_0\Bigr) \end{equation} for all $A\subseteq H$. With this in hand the theorem becomes a straightforward consequence of textbook results. \end{proof}
Now, consider the relation $\mathcal{P} = \dl{\mathcal{P}} \cap \Delta$, and take the polar of both sides: \begin{equation} \dl{\mathcal{P}} = \dl{\left(\dl{\mathcal{P}} \cap \Delta\right)} = \cc \left(\ddl{\mathcal{P}} \cup \dl{\Delta_N}\right). \end{equation} We know that $\mathcal{P}$ is closed and convex, and that it contains the center point $c$. Therefore, \begin{equation} \ddl{\mathcal{P}} = \mathcal{P}. \end{equation} What is the polar of the probability simplex $\Delta$? In fact, it is the basis simplex $\Delta_{\rm{e}}$. \begin{lemma} \label{lm:polar-basis} The probability simplex and the basis simplex are mutually polar: \ea{ \dl{\Delta} &= \Delta_{\rm{e}}, & \dl{\Delta_{\rm{e}}} &= \Delta. } \end{lemma} \begin{proof} The probability simplex contains normalized vectors, so it lies in the hyperplane $H$, and all of its vectors have wholly nonnegative entries. Let $v_i$ be the $i^{\rm{th}}$ vertex of $\Delta$ (so $v_i(j) = \delta_{ij}$). Then the probability simplex is \ea{ \Delta = \{ u \in H \colon \langle u, v_i\rangle \ge 0 \ \forall i\}. } Let $f\colon H \to H$ be the affine map defined by \begin{equation} f(u) = \frac{1}{\alpha} u + \frac{\beta}{\alpha}. \end{equation} Then $\Delta_{\rm{e}} = f(\Delta)$. It follows that \ea{ \Delta_{\rm{e}} = \left\{u \in H \colon \langle u, v_i \rangle \ge \frac{\beta}{\alpha}
\ \forall i \right\}. } Taking account of Theorem~\ref{tm:polarity} we deduce \ea{ \Delta_{\rm{e}} & = \dl{\{v_i \colon i = 1, \dots, N\}} = \dl{\Delta}. } The fact that $\dl{\Delta_{\rm{e}}} = \Delta$ is an immediate consequence of this and the fact that the double polar of a closed convex set is itself (see Theorem~\ref{tm:polarity}). \end{proof}
\begin{theorem} A state space $\mathcal{P}$ satisfying Assumptions \ref{assump:urgleichung}, \ref{assump:max}, \ref{assump:R-from-P} and \ref{assump:ignorance} is self-polar: \begin{equation} \mathcal{P} = \dl{\mathcal{P}}. \end{equation} \end{theorem} \begin{proof} We already know that \begin{equation} \dl{\mathcal{P}} = \cc\left(\ddl{\mathcal{P}} \cup \dl{\Delta}\right), \end{equation} and now we can say that \begin{equation} \dl{\mathcal{P}} = \cc(\mathcal{P} \cup \Delta_{\rm{e}}). \end{equation} But we established already that $\mathcal{P}$ always contains the basis distributions, and that $\mathcal{P}$ is closed and convex. Therefore, $\mathcal{P}$ is self-polar. \end{proof}
The fact that a state space is self-polar implies the existence of two more distinguished length scales. To see why, it is helpful to work in barycentric coordinates, shifting all our vectors so that the origin lies at the barycenter point of the simplex, the point $c$: \begin{equation} p \to p' = p - c. \end{equation} In these coordinates, our lower bound (\ref{eq:first-lower-bound}) becomes \begin{equation} \inprod{p'}{s'} \geq - \frac{1}{N\alpha}. \end{equation} Any basis distribution $e_j$ satisfies \begin{equation} \inprod{e_j'}{e_j'} = \frac{N-1}{N\alpha^2}. \end{equation} We define the \emph{out-sphere} $S_{\rm{o}}$ to be the sphere centered on the barycenter with radius \begin{equation} r_{\rm{o}}^2 = \frac{N-1}{N\alpha^2}. \end{equation} The ball bounded by $S_{\rm{o}}$ is the \emph{out-ball} $B_{\rm{o}}$. We will see shortly that the polar of the out-ball is a ball centered at the barycenter and having radius \begin{equation} r_{\rm{i}}^2 = \frac{1}{N(N-1)}. \end{equation} We designate this ball the \emph{in-ball} $B_{\rm{i}}$, and its surface is the \emph{in-sphere} $S_{\rm{i}}$. Finally, note that if we take \begin{equation} r_{\rm m}^2 = \frac{1}{N\alpha}, \end{equation} any two points both lying within $r_{\rm m}$ of the barycenter will be consistent with respect to the bound (\ref{eq:first-lower-bound}). This defines the \emph{mid-ball} $B_{\rm m}$ and its surface, the \emph{mid-sphere} $S_{\rm m}$. It follows that \begin{equation} r_{\rm{i}} r_{\rm{o}} = r_{\rm m}^2. \end{equation}
We now prove the fact we stated a moment ago. \begin{lemma} \label{lm:simpballpolars} The out- and in-balls are mutually polar: \ea{ \dl{B_{\rm{o}}} &= B_{\rm{i}}, & \dl{B_{\rm{i}}} &= B_{\rm{o}}. } \end{lemma} \begin{proof} Let $f\colon H\to H$ be the affine map defined by \begin{equation} f(u) = c + \frac{r_{\rm{o}}}{r_{\rm{i}}} (u-c). \end{equation} Then $f(B_{\rm{i}}) = B_{\rm{o}}$. Consequently, given arbitrary $u\in H$, \ea{ & &u &\in \dl{B_{\rm{o}}} && \\ &\iff & \langle u, f(v) \rangle &\ge \frac{\beta}{\alpha} & \forall v& \in B_{\rm{i}} \\ &\iff & \langle u-c, f(v)-c\rangle &\ge -r_{\rm{o}} r_{\rm{i}} &\forall v& \in B_{\rm{i}} \\ &\iff & \langle u-c , v-c\rangle & \ge -r_{\rm{i}}^2 &\forall v&\in B_{\rm{i}} \\ &\iff & u & \in B_{\rm{i}} && } So $\dl{B_{\rm{o}}}=B_{\rm{i}}$. The fact that $\dl{B_{\rm{i}}} = B_{\rm{o}}$ is an immediate consequence of this and the fact that the double polar of a closed convex set is itself. \end{proof}
These distinguished length scales suggest another assumption we ought to make about our state space. Earlier, we stated that the barycenter $c$ must belong to our set of admissible probability distributions. It is natural to ask how far away from complete ignorance we can go before we encounter complications. Can our state space $\mathcal{P}$ contain all the points in a little ball around~$c$? Intuitively, it is hard to see why not. How big can we make that ball around the center point $c$ before we run into trouble? The simplest assumption, in this context, is to postulate that the first complication we encounter is the edge of the probability simplex itself. Where does a sphere centered at~$c$ touch the faces of the simplex? The center of a face of the probability simplex is found by taking the average of $N-1$ of its vertices: \begin{equation} \bar{v}_k(i) = \frac{1}{N-1}(1 - \delta_{ik}). \end{equation} The sphere centered on $c$ that just touches these points has a radius given by \begin{equation} (\bar{v}_k - c)^2 = \frac{1}{N(N-1)}. \end{equation} The in-sphere $S_{\rm{i}}$ is just the \emph{inscribed} sphere of the probability simplex.
\begin{assump} \label{assump:upper-bound} Every state space $\mathcal{P}$ contains the in-ball. \end{assump}
Because the polar of the in-ball is the out-ball, and polarity reverses inclusion, it follows that every self-consistent state space is bounded by the out-sphere. This result has the form of an ``uncertainty principle'': It means that our probability distributions can never become too narrowly focused. For any two points $p$ and $s$ within our state space $\mathcal{P}$, we have \begin{equation} L \leq \inprod{p}{s} \leq U, \end{equation} where the lower and upper bounds are given by \begin{align} L &= -\frac{1}{N\alpha} + \frac{1}{N}, \\ U &= \frac{N-1}{N\alpha^2} + \frac{1}{N}. \end{align}
Recall from Lemma~\ref{lm:polar-basis} that the polar of the probability simplex is the simplex defined by the basis distributions $e_k$, which in barycentric coordinates is seen to be the probability simplex rescaled: \begin{equation} e_k'(i) = e_k(i) - c(i) = \frac{1}{\alpha}\left(\delta_{ik} - c(i)\right). \end{equation}
Call two extremal states $p$ and $s$ in a state space \emph{maximally distant} if they saturate the lower bound: \begin{equation} \inprod{p'}{s'} = -\frac{1}{N\alpha}. \end{equation} Let \begin{equation} \{p'_k : k = 1,\ldots,m \} \end{equation} be a set of Mutually Maximally Distant (MMD) states. That is, for all $k$, \begin{equation} \inprod{p_k'}{p_k'} = r_{\rm{o}}^2, \end{equation} and for $k \neq l$, \begin{equation} \inprod{p_k'}{p_l'} = -r_{\rm m}^2. \end{equation} Construct the vector quantity \begin{equation} V = \sum_k p_k'. \end{equation} From the fact that the magnitude $\inprod{V}{V} \geq 0$, it follows that \begin{equation} m \leq 1 + \frac{r_{\rm{o}}^2}{r_{\rm m}^2}. \end{equation} Substituting in the definitions of the radii, we arrive at the relation \begin{equation} m \leq 1 + \frac{N-1}{\alpha}. \end{equation} Let us now make an assumption: We want this bound to be attainable.
\begin{assump} \label{assump:m-max} A state space $\mathcal{P}$ contains an MMD set of size \begin{equation} m_{\rm max} = 1 + \frac{N-1}{\alpha}. \label{eq:m-max} \end{equation} \end{assump} Note that both $N$ and $m_{\rm max}$ are positive integers by assumption. This means that $\alpha$ must divide $N-1$ neatly.
To set the context for our next assumption, switch back to the original frame. Recall that any two points $p$ and $s$ within our state space $\mathcal{P}$ satisfy \begin{equation} L \leq \inprod{p}{s} \leq U, \end{equation} where the lower and upper bounds are given by \begin{align} L &= -\frac{1}{N\alpha} + \frac{1}{N}, \\ U &= \frac{N-1}{N\alpha^2} + \frac{1}{N}. \end{align} Comparing these two quantities, and using Eq.~(\ref{eq:m-max}) to simplify, we obtain \begin{equation} \frac{U}{L} = 1 + \frac{m_{\rm max}}{\alpha - 1}, \end{equation} where $m_{\rm max}$ is a positive integer. This expression makes it inviting to set the ratio on the right-hand side to unity by fixing \begin{equation} m_{\rm max} = \alpha - 1, \end{equation} and thus $U/L = 2$ is, in a sense, the natural first option to explore.
\begin{assump} \label{assump:upper-and-lower} The upper and lower bounds in the fundamental inequalities are related by \begin{equation} U = 2L. \end{equation} \end{assump}
This lets us solve for $N$ in terms of~$\alpha$: \begin{equation} N = (\alpha - 1)^2. \end{equation} Thanks to our two latest assumptions, we can fix all three parameters in the generalized urgleichung (\ref{eq:gen-urgleichung}) in terms of the maximal size of an MMD set: \begin{equation} N = m_{\rm max}^2,\ \alpha = m_{\rm max} + 1,\ \beta = \frac{1}{m_{\rm max}}. \end{equation} Relabeling $m_{\rm max}$ by $d$ for brevity, we recover the formulas familiar from the SIC representation of quantum state space. Here, the generalized urgleichung takes the specific form \begin{equation} q(j) = \sum_{i}\left[(d+1)p(i)
- \frac{1}{d}\right] r(j|i), \end{equation} and we arrive at the following pair of inequalities: \begin{equation} \frac{1}{d(d+1)} \leq \inprod{p}{s}
\leq \frac{2}{d(d+1)}. \label{eq:germ-defining} \end{equation}
Consequently, the polar of a set $A$ is \begin{equation} \dl{A} = \left\{ u \in H \colon \inprod{u}{v} \ge \frac{1}{d(d+1)} \ \forall v \in A\right\}. \label{eq:our-def-polar} \end{equation}
We now arrive at the definition upon which the rest of our theory will stand.
\begin{definition} A \emph{qplex} is a self-polar subset of the out-ball in the probability simplex $\Delta_{d^2}$, with the parameters in the generalized urgleichung set to $\alpha = (d+1)$ and $\beta = 1/d$. \end{definition}
\section{Fundamental Geometry of Qplexes} \label{sec:polarity} In the previous section, we began with the urgleichung and, making a few assumptions of an operational character, arrived at the double inequality \begin{equation} \frac{1}{d(d+1)} \leq \inprod{p}{s}
\leq \frac{2}{d(d+1)}. \label{eq:double-inequality-repeat} \end{equation} Here, we will take this as established, and we will demonstrate several important geometrical properties of the sets that maximally satisfy it---the qplexes.
A qplex\ is a subset of $\Delta$, the probability simplex in $\fd{R}^{d^2}$ (i.e. the space of probability distributions with $d^2$ outcomes). $\Delta$ is, in turn, a subset of the hyperplane \begin{equation} H =\left\{u \in \fd{R}^{d^2} \colon \inprod{u}{c} = \frac{1}{d^2}\right\}, \end{equation} where $\inprod{\cdot}{\cdot}$ denotes the usual scalar product on $\fd{R}^{d^2}$ and \begin{equation} c = \begin{pmatrix} \frac{1}{d^2} & \dots & \frac{1}{d^2}\end{pmatrix}^{\rm{T}} \end{equation} is the barycenter of $\Delta$.
It is important to appreciate the geometrical relationships between the four sets $\Delta, \Delta_{\rm{e}}, B_{\rm{o}}, B_{\rm{i}}$. Specializing our results from the previous section, we have \ea{ e_i - c &= \frac{1}{d+1}(v_i - c), \\ r_{\rm{i}} & = \frac{1}{d-1} r_{\rm{o}} . } So the basis simplex is obtained from the probability simplex by scaling by a factor $1/(d+1)$, while the in-ball is obtained from the out-ball by scaling by a factor $1/(d-1)$. In particular $B_{\rm{i}}=B_{\rm{o}}$ when $d=2$, but is otherwise strictly smaller. We have \begin{equation}
\inprod{e_j}{e_k} = \frac{d\delta_{jk} + d + 2}{d(d+1)^2}. \label{eq:basis-purity-special} \end{equation}
If $d=2$ then \begin{equation} \Delta_{\rm{e}} \subseteq B_{\rm{i}}=B_{\rm{o}} \subseteq \Delta. \end{equation} If $d>2$ then one still has \begin{equation} \Delta_{\rm{e}} \cup B_{\rm{i}} \subseteq \Delta \cap B_{\rm{o}} \end{equation} but \ea{ \Delta_{\rm{e}} &\nsubseteq B_{\rm{i}}, & B_{\rm{o}} \nsubseteq \Delta. } The first of these statements is an immediate consequence of the foregoing. To prove the second observe that $e_i\in \Delta_{\rm{e}}$ but $\notin B_{\rm{i}}$, while $c + (r_{\rm{o}}/r_{\rm{i}})(\bar{e}_i-c) \in B_{\rm{o}}$ but $\notin \Delta$.
These facts are perhaps most easily appreciated by examining the diagram in Fig.~\ref{figSimpsAndBalls}. Observe, however, that the metric relations are impossible to reproduce in a 2-dimensional diagram. So, although Fig.~\ref{figSimpsAndBalls} reproduces the inclusion relations, and points of contact, it badly misrepresents the sizes of the sets $\Delta_{\rm{e}},B_{\rm{i}}$ in comparison to the sets $\Delta,B_{\rm{o}}$. \begin{figure}\label{figSimpsAndBalls}
\end{figure}
General properties of qplexes include the following: \begin{itemize} \item Any qplex is convex and closed, and is thus the convex hull of
its extremal points.
\item Because a qplex is self-polar, it can be thought of as the
intersection of half-spaces. Each half-space is defined, per
Eq.~(\ref{eq:our-def-polar}), by a hyperplane that is composed of
points all maximally distant from an extreme point of the qplex.
\item For every extreme point of a qplex, there exists at least one
point that is maximally distant to it, in the sense of saturating
the lower bound in Eq.~(\ref{eq:double-inequality-repeat}).
\item Call a vector ${p} \in Q$ a \emph{pure} vector if
$\inprod{p}{p} = 2/(d(d+1))$. Any set of pure vectors that pairwise
saturate the lower bound of the consistency condition
(\ref{eq:double-inequality-repeat}) contains no more than $d$
elements.
\item Suppose we have a qplex $Q$ that is a polytope, \emph{i.e.,} the
convex hull of a finite set of vertices. Because all qplexes
contain the basis distributions, this polytope must have at least
$d^2$ vertices. The polar of each extreme point is a half-space
bounded by a hyperplane, all of the points on which are maximally
distant from that extreme point. The intersection of the
half-spaces defined by all these hyperplanes forms a polytope. By
self-polarity, this polytope is identical to~$Q$. It follows that
each extreme point of~$Q$ must lie on at least $d^2 - 1$ such
hyperplanes. Therefore, each vertex of~$Q$ is maximally distant
from at least $d^2 - 1$ other extreme points.
\item It follows from the above that a qplex cannot be a simplex.
Consequently, any point in the interior of a qplex can be written in
more than one way as a convex combination of points on the boundary.
This is a generalization of the result that any mixed quantum state
has multiple convex decompositions into different sets of pure
states, a theorem that has historically been of some significance in
interpreting the quantum formalism~\cite{Jaynes1957, Ochs1981,
CFS2001, stacey-VN}. Also, a result of Pl\'avala implies that any
qplex admits incompatible measurements~\cite{Plavala2016}.
\item If $Q$ is a qplex, then no vector $p \in Q$ can have an element
whose value exceeds $1/d$.
\item The total number of zero-valued entries in any vector belonging
to a qplex is bounded above by $d(d-1)/2$.
\end{itemize}
A SIC representation of a quantum state space is a qplex with a continuous set of pure points. All qplexes with this property enjoy an interesting geometrical relation with the polytopes that can be inscribed within them.
\begin{theorem} If $Q$ is a qplex\ that contains an infinite number
of pure points, then any polytope inscribed in~$Q$ cannot contain
the in-sphere $S_{\rm{i}}$. \end{theorem} \begin{proof} Suppose that $P$ is a polytope inscribed in $Q$ that contains the in-sphere $S_{\rm{i}}$. Recall that the polarity operation reverses inclusion (Theorem~\ref{tm:polarity}), so the polar polytope $\dl{P}$ of~$P$ must contain the polar $\dl{Q}$ of~$Q$. But all qplexes are self-polar, so $Q \subset \dl{P}$. Likewise, because the polar of the in-ball $B_{\rm{i}}$ is the out-ball $B_{\rm{o}}$, it follows that $\dl{P}$ is contained within the out-sphere $S_{\rm{o}}$. Consequently, $Q$ can have only a finite number of pure points. \end{proof}
Let us consider the two-outcome measurement $r_{{s}}$ defined by rescaling a state ${s} \in Q$: \begin{equation}
r_{{s}}(0|i) = d^2 \gamma_0 s(i),\ \gamma_0 = \frac{1}{d}. \end{equation}
We fix the other row of the matrix $r_{{s}}(j|i)$ by normalization: \begin{equation}
r(0|i) + r(1|i) = 1. \end{equation} Does this actually define a legitimate measurement? Because $\inprod{p}{s}$ is always bounded above and below for any vector ${p} \in \mathcal{P}$, then applying $r_{{s}}$ to any ${p} \in \mathcal{P}$ via the urgleichung will yield a valid probability vector ${q}$. Therefore, $r_{{s}}$ defined in this way is indeed a member of~$\mathcal{R}$.
What's more, if we apply $r_{{s}}$ to the state ${s}$ itself, then we can be \emph{certain} about the outcome, if ${s}$ lies on the same sphere as the basis distributions. In such a case, we have $q(0) = 1$. If Alice ascribes a state having this magnitude to a system, she is asserting her confidence that performing a particular experiment will have a specific result. But certainty about \emph{one} experiment does not, and indeed cannot, imply certainty about \emph{all.} Even when Alice is certain about what would happen should she perform the experiment $r_{{s}}$, she is necessarily uncertain about what would happen if she brought the Bureau of Standards measurement down to the ground and applied it.
Note that when we apply $r_{{s}}$ to a state ${p}$, we compute \begin{equation} q(0) = d(d+1)\inprod{p}{s} - 1. \label{eq:p-s-dual} \end{equation} The bound established by Assumption~\ref{assump:upper-bound} implies that we can associate the factor $d$ just as well with ${s}$ or with ${p}$. That is, both $r_{{s}}$ and $r_{{p}}$ are valid measurements within $\mathcal{R}$, and we obtain the same probability $q(0)$ when we apply $r_{{s}}$ to~${p}$ as we would if we applied $r_{{p}}$ to the state ${s}$.
This is a point worth considering in depth. With Assumption~\ref{assump:R-from-P}, we introduced a relation between the set of all states and the set of all measurements. Now, thanks to the additional assumptions we have invoked since then, we have a more specific correspondence between the two sets: For every pure state, there is a binary measurement for which that state, and no other state, implies certainty. This result depends upon our assumption that departures from complete ignorance are minimally constrained, or equivalently, that the basis distributions are extremal. As a consequence, we know that we can take any valid state $s$ and scale by a factor $d$ to create a row in a measurement matrix. In the language of Asher Peres, the fact that we can interpret Eq.~(\ref{eq:p-s-dual}) as $r_s$ applied to~$p$ or as $r_p$ applied to~$s$, for any states $p$ and $s$, is the reciprocity of ``preparations'' and ``tests''~\cite{PeresBook}.
This reciprocity is an important concept for many mathematical treatments of quantum physics. For example, it is one of the primary axioms in Haag's formulation~\cite{Haag, Araki}. To those who apply category theory to quantum mechanics, it is the reason why they construct ``dagger-categories,'' and how the basic idea of an inner product is introduced into their diagrammatic language~\cite{coecke2016}.
Next, we consider sets which are related to qplexes.
\begin{definition} A subset $A$ of the probability simplex $\Delta$ is a \emph{germ} if it satisfies the fundamental inequalities (\ref{eq:germ-defining}) for all $p$, $s\in A$. \end{definition}
\begin{definition} A germ is \emph{maximal} if no point can be added to it without violating the fundamental inequalities (\ref{eq:germ-defining}). \end{definition}
We start by proving two results about germs that follow from the Cauchy--Schwarz inequality. Originally, these theorems were proved for qplexes~\cite{fuchs2009, appleby2011, fuchs2013}, but they apply more broadly.
\begin{theorem} If $G$ is a germ, then no vector $p \in G$ can have an element
whose value exceeds $1/d$. \end{theorem} \begin{proof} Let $p \in G$ be a point on the out-sphere. Assume without loss of generality that $p(0) \geq p(i)$. Then \begin{equation} \frac{2}{d(d+1)} = p(0)^2 + \sum_{i=1}^{d^2-1} p(i)^2, \end{equation} and using the Cauchy--Schwarz inequality, \begin{equation} \frac{2}{d(d+1)} \geq p(0)^2 + \frac{1}{d^2-1} \left(\sum_{i=1}^{d^2-1} p(i)\right)^2. \end{equation} By normalization, we can simplify the sum in the last term, yielding \begin{equation} \frac{2}{d(d+1)} \geq p(0)^2 + \frac{1}{d^2-1} \left(1 - p(0)\right)^2. \end{equation} Thus, \begin{equation} p(0) \leq \frac{1}{d}, \end{equation} with equality if and only if all the other $p(i)$ are equal, in which case, normalization forces them to take the value $1/(d(d+1))$. \end{proof} \begin{remark} If the germ $G$ contains the basis distributions, this result also follows from \begin{equation} \inprod{p}{e_k} = \frac{1}{d(d+1)} + \frac{p_k}{d+1} \leq \frac{2}{d(d+1)}. \end{equation} \end{remark}
\begin{theorem} The total number of zero-valued entries in any vector belonging
to a germ is bounded above by $d(d-1)/2$. \end{theorem} \begin{proof} Let $G$ be a germ and choose $p \in G$. Square the basic normalization condition to find \begin{equation} \left( \sum_i p(i) \right)^2 = 1. \end{equation} Apply the Cauchy--Schwarz inequality to show, writing $n_0$ for the number of zero-valued elements in $p$, \begin{equation} (d^2 - n_0) \sum_{\{i:p(i) >0\}} p(i)^2 \geq
\left( \sum_{\{i:p(i) >0\}} p(i) \right)^2 = 1. \end{equation} Consequently, \begin{equation} n_0 \leq d^2 - \frac{d(d+1)}{2} = \frac{d(d-1)}{2}. \label{eq:weak-zeros-bound} \end{equation} \end{proof}
It follows from Zorn's lemma~\cite{Zorn} that every germ is contained in at least one maximal germ. In other words, we can extend any germ in at least one way to form a set that is also a germ, but which admits no further consistent extension. Adding any new point to a maximal germ implies that some pair of points will violate the inequalities (\ref{eq:germ-defining}). Every qplex is a germ, but the converse is not true. Using the theory of polarity, we will show that any maximal germ is a self-polar subset of the out-ball. That is, a maximal germ is a qplex, and in fact, any qplex is also a maximal germ.
It is an immediate consequence of the definition that if $G$ is an arbitrary germ then \begin{equation} G \subseteq \Delta \cap B_{\rm{o}}, \label{eq:gmInclude} \end{equation} where $B_{\rm{o}}$ is the out-ball: \ea{ B_{\rm{o}} &= \left\{u \in H \colon \inprod{u}{u} \le \frac{2}{d(d+1)} \right\} \\
& = \left\{ u\in H \colon \| u- c\| \le r_{\rm{o}} \right\}. }
Taking polars on both sides of Eq.~(\ref{eq:gmInclude}) and taking account of what polarity does to inclusion and intersection (Theorem~\ref{tm:polarity}), we find \begin{equation} \cc (\dl{\Delta} \cup \dl{B_{\rm{o}}}) \subseteq \dl{G} \label{eq:gmIncludeStar} \end{equation} for every germ $G$. Recall from Lemma~\ref{lm:polar-basis} that the polar of~$\Delta$ is the basis simplex $\Delta_{\rm{e}}$, and by Lemma~\ref{lm:simpballpolars} we know that the polar of the out-ball $B_{\rm{o}}$ is the in-ball $B_{\rm{i}}$. Therefore, \begin{equation} \cc (\Delta_{\rm{e}} \cup B_{\rm{i}}) \subseteq \dl{G}. \end{equation}
We are now able to prove \begin{theorem} \label{tm:qplexPolarity} Let $A$ be a subset of $\Delta\cap B_{\rm{o}}$. Then \begin{enumerate} \item $A$ is a germ if and only if $A \subseteq \dl{A}$. \item $A$ is a maximal germ if and only if $A = \dl{A}$. \end{enumerate} Therefore, the terms ``maximal germ'' and ``qplex'' are equivalent. \end{theorem} \begin{proof} The first statement is an immediate consequence of the definition. To prove the second statement we need to do a little work. This is because it is not immediately apparent that if $A$ is a maximal germ then $\dl{A}\subseteq B_{\rm{o}}$.
Suppose that $A$ is a maximal germ. We know from the first part of the theorem that $A \subseteq \dl{A}$. To prove the reverse inclusion let $u\in \dl{A}$ be arbitrary. In order to show that $u\in A$ first consider the vector \begin{equation}
\tilde{u} = c - \frac{r_{\rm{i}}}{\|u-c\|} (u-c). \end{equation} We have, for all $v\in A$, \ea{ -r_{\rm{i}} r_{\rm{o}} \le \inprod{\tilde{u}-c}{v-c} \le \rir_{\rm{o}}, \\ \intertext{implying} \frac{1}{d(d+1)} \le \inprod{\tilde{u}}{v} \le \frac{2}{d(d+1)}. } Also \ea{ \frac{1}{d(d+1)} \le \inprod{\tilde{u}}{\tilde{u}} = \frac{1}{d^2-1} \le \frac{2}{d(d+1)}. } So $\tilde{u} \in A$. We now use this to show that $u \in A$. In fact \ea{
-r_{\rm{i}} \|u-c\| &= \inprod{u-c}{\tilde{u}-c} \nonumber \\ &= \inprod{u}{\tilde{u}} - \frac{1}{d^2} \nonumber \\ & \ge -r_{\rm{i}} r_{\rm{o}} } implying $u \in B_{\rm{o}}$. Consequently \ea{ \langle u,v \rangle &= \inprod{u-c}{v-c} + \frac{1}{d^2} \nonumber \\ &\le r_{\rm{o}}^2 + \frac{1}{d^2} \nonumber \\ & = \frac{2}{d(d+1)} } for all $v\in A$. The fact that $u\in \dl{A}$ means \begin{equation} \inprod{u}{v} \ge \frac{1}{d(d+1)} \end{equation} for all $v \in A$. Finally \begin{equation}
\frac{1}{d(d+1)} \le \|u-c\|^2 +\frac{1}{d^2} = \inprod{u}{u} \le \frac{2}{d(d+1)}. \end{equation} So $u\in A$. This completes the proof that if $A$ is a maximal germ then $A=\dl{A}$. The converse statement, that if $A=\dl{A}$ then $A$ is a maximal germ, is an immediate consequence of the definition. \end{proof}
Let us note that this theorem means, in particular, that every maximal germ contains the basis simplex. If we start with the fundamental inequalities and assume that the set of points satisfying them is maximal, then that set turns out to be self-polar. Because the state space is contained within the probability simplex, the state space must \emph{contain} the \emph{polar of} the probability simplex, which by Lemma~\ref{lm:polar-basis} is the basis simplex. In earlier papers on germs~\cite{fuchs2009, appleby2011, fuchs2013}, the existence of the basis distributions was an extra assumption in addition to maximality; here, using the concept of polarity, we have been able to derive it.
Let us also note, as another consequence of this theorem, that if $Q$ is a maximal germ, and if $q$ is any element of $Q$, then there exists a measurement $r$ and index $a$ such that $q=s_a$, where $s_a$ is the distribution \begin{equation}
s_a(j) = \frac{r(a|j)}{\sum_k r(a|k)}. \end{equation} This too was something that was assumed in older work~\cite{fuchs2009, appleby2011, fuchs2013}, but which we are now in a position to derive. To see that it is true observe that the statement is trivial if $q=c$ (simply take $r$ to be the one-outcome measurement). If, on the other hand, $q\neq c$ we can define
\begin{equation} q' = c-\frac{r_{\rm{i}}}{\|q-c\|}(q-c). \end{equation} By construction $q' \in S_{\rm{i}}$. So it follows from the theorem that $q'\in Q$. Consequently, if we define \ea{
r(a|i) = \begin{cases}
\frac{d^2r_{\rm{i}}}{\|q-c\|+r_{\rm{i}}} q(i) \qquad & a = 1 \\
\frac{d^2\|q-c\|}{\|q-c\|+r_{\rm{i}}} q'(i) \qquad & a=2 \end{cases} } then $r$ describes a two-outcome measurement such that \ea{
\frac{r(1|i)}{\sum_k r(1|k)} &= q(i), & \frac{r(2|i)}{\sum_k r(2|k)} &= q'(i). }
At this stage, we turn to the question of what germs can have in common, and how they can differ. In order to develop this topic, we introduce some more definitions. Given an arbitrary germ $G$, let $\mathcal{Q}_G$ denote the set of all qplexes\ containing $G$ (necessarily nonempty, as we noted above).
\begin{definition} The \emph{stem} of a germ $G$ is the set \begin{equation} \mathscr{S}(G) = \bigcap_{Q \in \mathcal{Q}_G} Q, \end{equation} and the \emph{envelope} of $G$ is the set \begin{equation} \mathscr{E}(G) = \bigcup_{Q \in \mathcal{Q}_G} Q. \end{equation} \end{definition} When $G$ is the empty set $\mathcal{Q}_{\emptyset}$ is the set of all qplexes, without restriction. In that case we omit the subscript and simply denote it $\mathcal{Q}$. Similarly we write $\mathscr{S}(\emptyset) = \mathscr{S}$ and $\mathscr{E}(\emptyset) = \mathscr{E}$. We will refer to $\mathscr{S}$ and $\mathscr{E}$ as the principal stem\ and envelope.
\begin{theorem} Let $G$ be a germ. Then \ea{ \mathscr{S}(G) &= \cc(\Delta_{\rm{e}} \cup B_{\rm{i}} \cup G), \label{eq:trnExpn} \\ \mathscr{E}(G) &= \Delta \cap B_{\rm{o}} \cap \dl{G}. \label{eq:envExpn} } In particular, $\mathscr{S}(G)$ and $\mathscr{E}(G)$ are mutually polar. \end{theorem} \begin{proof} If $Q$ is a qplex containing $G$ we must have $Q =\dl{Q} \subseteq \Delta \cap B_{\rm{o}} \cap \dl{G}$. So \begin{equation} \mathscr{E}(G) \subseteq \Delta\cap B_{\rm{o}} \cap \dl{G}. \end{equation} On the other hand if $p$ is any point in $\Delta \cap B_{\rm{o}} \cap \dl{G}$ then $G \cup \{p\}$ is a germ, and so must be contained in some $Q\in \mathcal{Q}_G$. The second statement now follows.
To prove the first statement we take duals on both sides of \begin{equation} \bigcup_{Q \in \mathcal{Q}_G} Q = \Delta \cap B_{\rm{o}} \cap \dl{G}. \end{equation} We find \ea{ \mathscr{S} (G)& = \cc\Bigl(\Delta_{\rm{e}} \cup B_{\rm{i}} \cup \cc\bigl(G\cup \{c\}\bigr)\Bigr) \nonumber \\ &= \cc(\Delta_{\rm{e}} \cup B_{\rm{i}} \cup G). } \end{proof} \begin{corollary} \label{cor:mainTrEnv} The principal stem\ and envelope\ are given by \ea{ \mathscr{S} &= \cc(\Delta_{\rm{e}} \cup B_{\rm{i}}), \\ \mathscr{E} &= \Delta \cap B_{\rm{o}}. } \end{corollary} \begin{proof} Immediate. \end{proof} This result is illustrated schematically in Figure~\ref{figCoreEnvelope}.
\begin{figure}\label{figCoreEnvelope}
\end{figure}
\begin{corollary} \label{cr:germExtend} Let $G$ be a closed, convex germ containing $\mathscr{S}$. Then \begin{align} \mathscr{S}(G) & = G & \mathscr{E}(G) & = \dl{G} \end{align} Moreover, given arbitrary $p\in \dl{G}$ such that $p \notin G$ there exist qplexes\ $Q_1$, $Q_2$ containing $G$ such that \begin{align} p &\notin Q_1 & p& \in Q_2 \end{align} \end{corollary} \begin{proof} Immediate. \end{proof} Every germ can be extended to a qplex. It is natural to ask how many ways there are of performing the extension. The following theorem provides a partial answer to that question. \begin{theorem} Let $G$ be a closed, convex germ containing $\mathscr{S}$. If $G$ is not already a qplex, then there are uncountably many qplexes\ containing $G$. \end{theorem} \begin{proof} It will be convenient to begin by introducing some notation. Given any two points $p_1$, $p_2\in H$ we define \ea{ [p_1,p_2]&=\{\lambda p_1 + (1-\lambda)p_2 \colon 0 \le \lambda \le 1\} \\ (p_1,p_2]&=\{\lambda p_1 + (1-\lambda)p_2 \colon 0 < \lambda \le 1\} \\ [p_1,p_2)&=\{\lambda p_1 + (1-\lambda)p_2 \colon 0 \le \lambda < 1\} \\ (p_1,p_2)&=\{\lambda p_1 + (1-\lambda)p_2 \colon 0 < \lambda <1\} }
Turning to the proof, suppose that
$G$ is not a qplex. Then we can choose $p \in \dl{G}$ such that $p \notin G$. Let $q$ be the point where $[c,p]$ meets the boundary of $G$. We will show that, for each $s \in (q,p)$ there exists a qplex\ $Q_s$ such that $G \cup [c,s] \subseteq Q_s$ and $(s,p] \cap Q_s = \emptyset$. The result will then follow since $Q_s \neq Q_{s'}$ if $s\neq s'$.
To construct the qplex\ $Q_s$ for given $s\in (q,p)$ observe that it follows from the basic theory of convex sets~\cite{grun} that there exists a hyperplane through $s$ and not intersecting $G$. This means we can choose $u\in H$ such that \ea{ \langle u, v \rangle & < \langle u, s\rangle =1 } for all $v\inG$. Observe that for all $t \in (s,p)$ we have \ea{ t = \lambda s + (1-\lambda) c } for some $\lambda >1$ and, consequently, \ea{ \langle u, t \rangle = \frac{(d^2-1)\lambda + 1}{d^2} > 1. } Let \ea{ u' = \left(1 + \frac{1}{(d+1)(d^2-1)}\right) c - \frac{1}{(d+1)(d^2-1)} u } and let $A=\cc\bigl( G \cup \{s\}\bigr)$. Then it is easily seen that $u'\in \dl{A}$ while \ea{ \langle u' , t \rangle & < \frac{1}{d(d+1)} } for all $t \in (s,p)$. $A$ is a closed, convex germ containing $\mathscr{S}$, so it follows from Corollary~\ref{cr:germExtend} that there exists a qplex\ $Q_s$ containing $A$ and $u'$. By construction $t \notin Q_s$ for all $t\in (s,p)$, so $Q_s$ has the required properties. \end{proof} The result just proved shows that there exist uncountably many qplexes. However, we would like to know a little more: namely, how many qplexes\ there are which are geometrically distinct. We now prove a series of results leading to Theorem~\ref{thm:infNonIsomorphic}, which states that there are uncountably many qplexes\ which are not isomorphic to each other, or to quantum state space.
\begin{definition} Let $s\in H$ be arbitrary. We define the polar point of $s$ to be the point \ea{
\dl{s} &= c - \frac{r_{\rm{o}}r_{\rm{i}}}{\|s-c\|^2} (s-c) \\ \intertext{and the polar hyperplane of $s$ to be the set} H_s &=\left\{u \in H \colon \langle u, s\rangle = \frac{1}{d(d+1)}\right\}. } \end{definition}Observe that $\ddl{s} = s$ for all $s$, and \ea{ \langle \dl{s} , s\rangle = \frac{1}{d(d+1)} } (so $\dl{s} \in H_s$). The relations between the polar $\dl{\{s\}}$, the polar point $\dl{s}$ and the polar hyperplane $H_s$ are depicted in Fig.~\ref{fig:PolarPoint}. \begin{figure}\label{fig:PolarPoint}
\end{figure} It follows from these definitions that if $s$ is any point on $S_{\rm{o}}$ (respectively $S_{\rm{i}}$), then $\dl{s}$ is on $S_{\rm{i}}$ (respectively $S_{\rm{o}}$).
\begin{theorem} \label{thm:dualOfPtOnSo} Let $G$ be a closed germ, and let $s$ be any point on $S_{\rm{o}}$. Then $s\in G$ if and only if $\dl{s}$ is on the boundary of $\dl{G}$. \end{theorem} \begin{remark} Specializing to the case when $G$ is a qplex, the theorem says that the points where the boundary of $G$ touches the out-sphere are antipodal to the points where it touches the in-sphere. This is a subtle property of quantum state space~\cite{Kimura2005, Appleby2007}. \end{remark} \begin{proof} Suppose $s\in G$. Then it follows that $\dl{s} \in S_{\rm{i}}$. The fact that $G$ is a germ means $G \subseteq\mathscr{E}$, implying $\mathscr{S} \subseteq \dl{G}$. So $\dl{s} \in \dl{G}$. Moreover, if we define \ea{ t_n = \frac{n+1}{n} \dl{s} -\frac{1}{n} c, } then \ea{ \langle t_n , s\rangle &= \frac{1}{d(d+1)} - \frac{1}{nd^2(d+1)} < \frac{1}{d(d+1)} } for all $n$. So $t_n$ is a sequence outside $\dl{G}$ converging to $\dl{s}$. We conclude that $\dl{s}$ is on the boundary of $\dl{G}$.
Conversely, suppose $\dl{s}$ is on the boundary of $\dl{G}$. Then we can choose a sequence $t_n \notin \dl{G}$ such that $t_n \to \dl{s}$. For each $n$ there must exist $p_n\in G$ such that \ea{ \langle t_n , p_n \rangle < \frac{1}{d(d+1)}. } Since $G$ is closed and bounded it is compact. A theorem of point set topology has it that in a compact set, every sequence contains a convergent subsequence. Therefore, we can choose a convergent subsequence $p_{n_j} \to p\in G$. Also, the fact that $t_{n_j} \to \dl{s}$ means $\dl{t}_{n_j} \to s$. So \begin{equation}
\| p - s\|^2 = \lim_j \left( \| p_{n_j} - \dl{t}_{n_j}\|^2 \right). \end{equation} We can expand the quantity inside the limit as \begin{equation}
\| p_{n_j} - \dl{t}_{n_j}\|^2
=
\|p_{n_j}-c\|^2 + \|\dl{t}_{n_j}-c\|^2 - 2 \inprod{\dl{t}_{n_j}-c}{p_{n_j}-c}. \end{equation} In turn, we have that \begin{equation}
\lim_j \left( \| p_{n_j} - \dl{t}_{n_j}\|^2 \right)
\leq
\|p-c\|^2 - r_{\rm{o}}^2. \end{equation} Because $p$ is contained in the out-ball, its distance from $c$ has to be less than $r_{\rm{o}}$, meaning that \begin{equation}
\| p - s\|^2 \leq \|p-c\|^2 - r_{\rm{o}}^2 \leq 0. \end{equation} So $p$ coincides with $s$, which consequently belongs to $G$. \end{proof} We next prove two results which show that we can restrict our attention to the out-sphere when trying to establish the existence of non-isomorphic qplexes. The first of these, Lemma~\ref{lm:extendSoconst}, is a technical result which will also be used in Section~\ref{sec:qgroups}. \begin{lemma} \label{lm:extendSoconst} Let $G$ be a closed germ containing $\mathscr{S}$, and let \ea{ C = G \cup (\dl{G}\cap B_{\rm m}) } where $B_{\rm m}$ is the mid-ball. Then $C$ is a closed germ such that $C'\cap S_{\rm{o}} = G \cap S_{\rm{o}}$ for every germ $C'$ containing $C$. \end{lemma} \begin{proof} It follows from Theorem~\ref{tm:polarity} and Lemma~\ref{lm:simpballpolars} that \ea{ \dl{C} &= \dl{G} \cap \cc(G \cup B_{\rm m}), } from which one sees that $C\subseteq \dl{C}$. Moreover, the fact that $\mathscr{S} \subseteq G$ means $\dl{G} \subseteq \Delta \cap B_{\rm{o}}$, implying $C\subseteq \Delta\capB_{\rm{o}}$. So $C$ is a closed germ containing $G$. Let $C'$ be any germ containing $C$. It is immediate that $G\cap S_{\rm{o}} = C \cap S_{\rm{o}} \subseteq C'\capS_{\rm{o}}$. Suppose, on the other hand, that $s$ is a point on $S_{\rm{o}}$ not belonging to $G$. Let $\dl{G}_{\rm{b}}$, $C_{\rm{b}}$ be the boundaries of $\dl{G}$, $C$ respectively. The fact that $C\cap B_{\rm m} = \dl{G}\cap B_{\rm m}$ is easily seen to imply $\dl{G}_{\rm{b}}\cap S_{\rm{i}} = C_{\rm{b}}\capS_{\rm{i}}$. So it follows from Theorem~\ref{thm:dualOfPtOnSo} that $\dl{s}\notin C_{\rm{b}}$. Since $S_{\rm{i}} \subseteq C$ this means $\dl{s}$ must lie in the interior of $C$. So there exists $\lambda > 1$ such that \begin{equation} t = \lambda \dl{s} + (1-\lambda) s \end{equation} is in $C$. Since \ea{ \langle t, s \rangle =\frac{2-\lambda}{d(d+1)} < \frac{1}{d(d+1)}, } it follows that $s \notin \dl{C}$. Consequently $s\notin C'$. \end{proof} \begin{theorem} \label{thm:bdryTheoremA} Let $G$ be a closed germ containing the vertices of the basis simplex. Then there exists a qplex\ $Q$ such that $Q\cap S_{\rm{o}}=G\cap S_{\rm{o}}$. \end{theorem} \begin{proof} Let \ea{ \tilde{G} = \cc(G\cup \Delta_{\rm{e}} \cup B_{\rm{i}}). } Then $\tilde{G}$ is a closed germ containing $\mathscr{S}$. Moreover $\tilde{G}\cap S_{\rm{o}} = G \cap S_{\rm{o}}$. Let \ea{ C &= \tilde{G} \cup \left(\dl{\tilde{G}}\cap B_{\rm m}\right). } Then it follows from Lemma~\ref{lm:extendSoconst} that $C$ is a germ and that $Q\cap S_{\rm{o}} = G\cap S_{\rm{o}}$ for any qplex\ $Q$ containing $C$. \end{proof} Before proving Theorem~\ref{thm:infNonIsomorphic} we need to give a sharp definition of what it means for two qplexes\ to be isomorphic. \begin{definition} \label{def:qplexIsomorphism} We say that two qplexes\ $Q$ and $Q'$ are isomorphic if and only if there exists a linear bijection $f\colon \fd{R}^{d^2} \to \fd{R}^{d^2}$ such that \begin{enumerate} \item $Q'=f(Q)$. \item For all $q_1$, $q_2 \in Q$ \ea{ \langle f(q_1),f(q_2) \rangle = \langle q_1, q_2\rangle. } \end{enumerate} \end{definition}
We are now ready to prove the final result of this section. \begin{theorem} \label{thm:infNonIsomorphic} There exist uncountably many qplexes\ which are not isomorphic to each other. \end{theorem} \begin{proof} We have \ea{ \langle e_i , e_j\rangle = \frac{d\delta_{ij} + d + 2}{d(d+1)^2} } for all $i$, $j$ (c.f. Eq.~(\ref{eq:basis-purity-special})). So if we define \ea{ p_{\theta}& = c + \cos \theta (e_1-c) + \sin \theta (e_2-c) \\ G_{\theta} &= \{p_{\theta} , e_1, \dots, e_{d^2}\} } then, for sufficiently small $\epsilon$, the set $G_{\theta}$ is a germ for all $\theta \in (0,\epsilon)$. It follows from Theorem~\ref{thm:bdryTheoremA} that we can choose qplexes\ $Q_{\theta}$ such that $Q_{\theta} \cap S_{\rm{o}} = G_{\theta} \cap S_{\rm{o}} = G_{\theta}$. By construction, the scalar products $\inprod{p_{\theta}}{e_j}$ are different for different choices of~$\theta$, and so qplexes\ $Q_{\theta}$ corresponding to different values of $\theta$ are non-isomorphic. Moreover, the fact that the intersection with $S_{\rm{o}}$ is finite means that $Q_{\theta}$ is non-isomorphic to quantum state space for all $\theta$. \end{proof} So far we have been focussing on qplexes\ in general. However, it seems to us that the method of analysis employed is a potentially insightful way of thinking about the geometry of quantum state space.
\section{Type-preserving measurements} \label{sec:intg} We now come to the central result of this paper. We will show that the symmetry group of a qplex\ can be identified with a set of measurements, which in turn can be identified with a set of regular simplices within the qplex whose vertices all lie on the out-sphere.
Let $Q$ be a qplex\ and $r$ a measurement with $n$ outcomes. For each $q\in Q$, let $q_r$ be the distribution given by the urgleichung, Eq.~(\ref{eq:SICMeasProbs}). Then the map $q \to q_r$ takes $Q$ to \begin{equation} Q_r = \{q_r \colon q\in Q\} \subseteq \Delta_n \end{equation} (where $\Delta_n$ is the $n-1$ dimensional probability simplex). We refer to $Q_r$ as the measurement set, and the map $q\to q_r$ as the measurement map.
We are interested in measurements having $d^2$ outcomes for which the measurement set is another qplex. We will refer to such measurements as type-preserving. We are particularly interested in the case when the measurement set is $Q$ itself, in which case we will say that the measurement is $Q$-preserving.
Let $r$ be an arbitrary measurement. Then it is easily seen that the urgleichung can be written in the alternative form \begin{equation} q_r(i) = \sum_j R_{ij} q(j), \end{equation} where
\begin{equation} R_{ij} = (d+1) r(i|j) -\frac{1}{d} \sum_k r(i|k). \label{eq:RmtDef} \end{equation} We refer to $R$ as the stretched measurement matrix. Note that Eq.~(\ref{eq:RmtDef}) can be inverted:
\begin{equation} r(i|j) = \frac{1}{d+1} \left( R_{ij} + \frac{1}{d} \sum_k R_{ik}\right). \end{equation} So the stretched measurement matrix uniquely specifies the measurement.
Now specialize to the case of a type-preserving measurement. In that case it turns out that $R$ must be an orthogonal matrix. To see this we begin by observing that, since the basis simplex belongs to both $Q$ and $Q_r$, there must exist $s_i \in Q$, $s'_i\in Q_r$ such that \ea{ Rs_i &= e_i, & Re_i &= s'_i. \label{eq:sspdef} } We then have \begin{lemma} \label{lm:typPMeasDetREq1} Let $R$, $s_i$, $s'_i$ be as above. Then \begin{enumerate} \item $\det R = \pm 1$. \item $s_i$, $s'_i\in S_{\rm{o}}$ for all $i$. \item $\cc(\{s_i\})$ and $\cc(\{s'_i\})$ are regular simplices. \end{enumerate} \end{lemma} \begin{proof} The proof is based on the fact~\cite{VolSimp} that the simplices of maximal volume within a ball are precisely the regular simplices with vertices on the sphere that bounds the ball. The desired result follows from considering the simplex formed by the $s_i$ and the origin (and the corresponding simplex formed by the $s'_i$ and the origin). \end{proof}
To complete the proof that $R$ is an orthogonal matrix, we observe that maps from regular simplices to regular simplices are orthogonal. From this, we can derive the following theorem. \begin{theorem} \label{thm:TPreserveMapTheorem1} Let $Q$ be a qplex, and let $R$ be the stretched measurement matrix of a type-preserving measurement. Then $R$ is an orthogonal matrix such that $Rc=c$. Moreover there exists a regular simplex with vertices $s_i \in Q\capS_{\rm{o}}$ such that \ea{ R_{ij} &= (d+1)s_i(j) -\frac{1}{d}, \label{eq:MeasMatTermsSimp} \\ Rs_i &= e_i, \\ (Re_i)(j) &= s_j(i). } \end{theorem} \begin{remark} We will refer to $\cc(\{s_i\})$ as the measurement simplex. \end{remark}
For a given qplex\ $Q$ define \begin{enumerate} \item $\typ{Q}$ to be the class of type-preserving measurements. \item $\sym{Q}$ to be the class of regular simplices with vertices in $Q\cap S_{\rm{o}}$. \item $\ort{Q}$ to be the class of orthogonal matrices $R$ such that $RQ$ is a qplex. \end{enumerate} The previous theorem states that to each element of $\typ{Q}$ there corresponds an element of $\sym{Q}$ and an element of $\ort{Q}$. The next theorem we prove states that the correspondences are in fact bijective, so that we can identify the three classes $\typ{Q}$, $\sym{Q}$ and $\ort{Q}$. \begin{theorem} \label{thm:RegSimpIsTPreserve} Let $Q$ be a $qplex$ and let $s_i\in Q\cap S_{\rm{o}}$ be the vertices of a regular simplex $\Delta_s$. Then $\Delta_s$ is the measurement simplex of a type-preserving measurement. Likewise, if $R$ is an orthogonal matrix such that $RQ$ is also a qplex, then $R$ is the stretched measurement matrix for a type-preserving measurement. \end{theorem} \begin{proof} Define \ea{
r(i|j) = s_i(j). }
It is immediate that the $r(i|j)$ are the conditional probabilities defining a measurement with stretched measurement matrix \begin{equation} R_{ij} = (d+1)s_i(j) -\frac{1}{d}. \end{equation} We need to show that the measurement is type-preserving. In other words, we need to show that the set $RQ$ is a qplex. For all $q\in Q$ \ea{ (Rq)(i) &= (d+1) \langle s_i , q\rangle - \frac{1}{d}, } from which it follows \ea{ (Rq)(i) &\ge 0 & \sum_i (R q)(i) &= 1. } So $RQ\subseteq \Delta$. Also, it follows from the same considerations that led to Theorem~\ref{thm:TPreserveMapTheorem1} that $R$ is orthogonal. The defining condition of a germ, Eq.~(\ref{eq:germ-defining}), is invariant under orthogonal transformations. Therefore, $RQ$ is a qplex.
We now prove the other direction of the correspondence. Let $R$ be an orthogonal matrix such that $RQ$ is a qplex. We know that the basis distributions $e_i$ must belong to $RQ$. So, there exist $s_i\in Q$ such that \begin{equation} e_i = Rs_i. \label{eq:biRsi} \end{equation} Since $\det R = \pm 1$ we have, by the same argument used to prove Lemma~\ref{lm:typPMeasDetREq1}, that the $s_i\in S_{\rm{o}}$ and are the vertices of a regular simplex. It now follows from the considerations above that $\cc(\{ s_i\})$ is the measurement simplex of a type-preserving measurement, with stretched measurement matrix \begin{equation} R'_{ij} = (d+1) s_i(j) -\frac{1}{d}. \end{equation}
By multiplying both sides of Eq.~(\ref{eq:biRsi}) by $R^{\rm T}$, we find that \ea{ s_i(j) &= \sum_k R^{\rm{T}}_{jk} e_i(k) \nonumber \\ &=\frac{1}{d+1} R_{ij} + \frac{1}{d(d+1)} \sum_{k} R_{kj}. \label{eq:MeasMatTermsSimpCalc} } Summing over $i$ on both sides of this equation we find $ \sum_k R_{kj} = 1 $ for all $j$ and, consequently, $R=R'$. \end{proof}
At this stage, we recall our definition of an isomorphism between qplexes: Two qplexes\ $Q$ and $Q'$ are isomorphic if and only if there exists an inner-product-preserving map $f\colon \fd{R}^{d^2} \to \fd{R}^{d^2}$ that sends $Q$ to $Q'$.
\begin{theorem} Let $Q$, $Q'$ be qplexes. Then $Q$ and $Q'$ are isomorphic if and only if there is a type-preserving measurement on $Q$ such that $Q' = RQ$, where $R$ is the stretched measurement matrix. \end{theorem} \begin{proof} Sufficiency is immediate. To prove necessity suppose that $f\colon Q \to Q'$ is an isomorphism. The fact that $f$ preserves scalar products on a set which spans $\fd{R}^{d^2}$ means that it must be represented by an orthogonal matrix. The claim now follows from Theorem~\ref{thm:RegSimpIsTPreserve}. \end{proof} So far we have been looking at type-preserving measurements in general. Let us now focus on the special case of $Q$-preserving measurements. Suppose that we have two such measurements, with measurement matrices $R$, $R'$. Then $RR'$ is also an orthogonal matrix, with the property that $RR'Q = Q$. So it follows from Theorem~\ref{thm:RegSimpIsTPreserve} that $RR'$ is the stretched measurement matrix for a $Q$-preserving measurement. Similarly with $R^{\rm{T}}$, the inverse. In short, the $Q$-preserving measurement maps form a group. For ease of reference let us give it a name: \begin{definition} Let $Q$ be a qplex. The preservation group of $Q$, denoted $\qgp{Q}$, is the group of type-preserving measurement maps between $Q$ and itself. \end{definition}
The elements of $\qgp{Q}$ are symmetries of $Q$. The question naturally arises, whether they comprise \emph{all} the symmetries. The above considerations are not sufficient to answer that question because they leave open the possibility that $Q$ is invariant under orthogonal transformations which do not fix the origin of $\fd{R}^{d^2}$. The following theorem eliminates that possibility. \begin{theorem} \label{tm:preservation-symmetry} Let $Q$ be a qplex. Then the preservation group is the symmetry group of $Q$. \end{theorem} \begin{proof} The symmetry group of a subset of a normed vector space is defined to be the group of isometries of the set. It has been shown above that every $Q$-preserving measurement map is an isometry of $Q$. We need to show the converse. Let $f$ be an isometry of $Q$. It follows from Theorem~\ref{thm:TPreserveMapTheorem1} that $f(c) = c$.
Now define a map $\tilde{f}\colon Q-c \to Q-c$ by \ea{ \tilde{f}(u) = f(u+c) - c }
One easily sees that $\|\tilde{f}(u)\| = \|u\|$ for all $u \in Q-c$. Consequently \begin{equation} \tilde{f}(u) = T u \end{equation} for some orthogonal transformation $T$ of the subspace $H-c$. We may extend $T$ to an orthogonal transformation $R$ of the whole space $\fd{R}^{d^2}$ by defining $Rc = c$. It is then immediate that $RQ = Q$. The result now follows by Theorem~\ref{thm:RegSimpIsTPreserve}.
\end{proof}
\section{From preservation group to qplex} \label{sec:qgroups} In this section we ask what conditions a subgroup of $\Ot(d^2)$ must satisfy in order to be the preservation group of some qplex. This will lead us to the question of when symmetries are powerful enough to determine a qplex essentially uniquely. Let $Q$ be a qplex and $\mathcal{G}$ be its preservation group. Under what conditions can $Q$ be maximally symmetric, in the sense that $\mathcal{G}$ is not a proper subgroup of the symmetry group of any qplex? The answer will turn out to depend upon how the group $\mathcal{G}$ acts on the basis simplex.
Quantum state space has the property that any pure state can be mapped to any other pure state by some unitary operation, that is, by some symmetry of the state space. Indeed, given any pure state, the set of all pure states is the orbit of the original state under the action of the symmetry group. This leads us to consider the general question of qplexes whose extremal points form a single orbit under the action of the qplex's symmetries. One can prove that if $Q$ is such a qplex, then the symmetry group of~$Q$ is maximal, and furthermore, any other qplex $Q'$ with the same symmetry group is identical to~$Q$.
Given a group $\mathcal{G} \subseteq \Ot(d^2)$, can $\mathcal{G}$ be the preservation group of a qplex? It is easy to find a necessary condition. Following our previous paper \cite{GroupAlg}, we introduce the concept of a stochastic subgroup: \begin{definition} A subgroup $\mathcal{G}\subseteq \Ot(d^2)$ is stochastic if, for all $R\in \mathcal{G}$, \begin{equation} R_{ij} \ge -\frac{1}{d} \quad \forall i,j \quad\quad \hbox{ and } \quad\quad R c = c. \end{equation} \end{definition} Equivalently, we may say that a subgroup $\mathcal{G}\subseteq \Ot(d^2)$ is stochastic if every matrix in $\mathcal{G}$ is of the form \begin{equation} R_{ij} = (d+1) S_{ij} - \frac{1}{d}, \end{equation} where $S_{ij} = s_i(j)$ is a doubly-stochastic matrix (hence the name). It can then be seen from Theorem~\ref{thm:TPreserveMapTheorem1} that every preservation group is a stochastic subgroup of $\Ot(d^2)$.
It is natural to ask whether the condition is sufficient as well as necessary, so that every stochastic subgroup of $\Ot(d^2)$ is the preservation group of some qplex. We have not been able to answer this question in full generality. However, we have obtained some partial results. We can show that any stochastic subgroup $\mathcal{G} \subseteq \Ot(d^2)$ is at least contained in the preservation group of some qplex. To see why, we start with a preliminary result. \begin{lemma} \label{thm:assocgerm} Let $\mathcal{G}$ be a stochastic subgroup of $\Ot(d^2)$. For each $R\in \mathcal{G}$ define the vectors $s^R_i$ by applying $R$ to the basis distributions: \ea{ s^R_i(j) = \frac{1}{d(d+1)}\left( dR_{ij} + 1\right). } Then $s^R_i\in \Delta\cap S_{\rm{o}}$ for all $i$ and $\cc(\{s^R_i\})$ is a regular simplex. Moreover \ea{ G = \{s^R_i \colon R \in \mathcal{G}, \ i = 1, \dots, d^2\} } is a germ. \end{lemma} \begin{proof} Straightforward consequence of the definitions. \end{proof} \begin{definition} Let $\mathcal{G}$ be a stochastic subgroup of $\Ot(d^2)$. The orbital germ is the orbit of the basis distributions under the action of~$\mathcal{G}$, that is, the set $G$ specified in the statement of Lemma~\ref{thm:assocgerm}. \end{definition} \begin{theorem} \label{thm:StochSgpContQgp} Let $\mathcal{G}$ be a stochastic subgroup of $\Ot(d^2)$. Then there exists a qplex\ $Q$ such that $\mathcal{G} \subseteq \qgp{Q}$. \end{theorem} \begin{proof} Let $G$ be the orbital germ of $\mathcal{G}$, and let $\mathcal{A}_G$ be the set of all germs $P$ such that \begin{enumerate} \item $P$ contains $G$. \item $RP=P$ for all $R \in \mathcal{G}$. \end{enumerate} It follows from Zorn's lemma that $\mathcal{A}_G$ contains at least one maximal element. Let $Q$ be such a maximal element. Observe that if $P$ is in $\mathcal{A}_G$ then its convex closure is also in $\mathcal{A}_G$; consequently $Q$ must be convex and closed. Observe, also, that if $R$ is any element of $\mathcal{G}$, then $c$ is in the interior of the simplex $\cc(\{s^R_i\})$; consequently $c$ is in the interior of $Q$.
We claim that $Q$ is in fact a qplex. For suppose it were not. Then we could choose $p \in \Delta\cap B_{\rm{o}} \cap \dl{Q}$ such that $p\notin Q$. For each $\lambda$ in the closed interval $[0,1]$ define $p_{\lambda}=\lambda p + (1-\lambda) c$. The fact that $Q$ is closed, convex together with the fact that $c$ is in the interior of $Q$ means that there exists $\lambda_0 \in (0,1)$ such that $p_{\lambda} \in Q$ if and only if $\lambda \in [0,\lambda_0]$. We have \ea{ \langle p , Rp_{\lambda} \rangle \ge \frac{1}{d(d+1)} } for all $R \in \mathcal{G}$, $\lambda \in [0,\lambda_0]$. Consequently \ea{ \langle p_{\lambda} , R p_{\lambda} \rangle \ge \frac{1}{d(d+1)} + \frac{1-\lambda}{d^2(d+1)} } for all $\lambda \in [0,\lambda_0]$, $R\in \mathcal{G}$. By continuity this inequality must hold for all $\lambda \in [0,\lambda_0]$, $R\in \bar{\mathcal{G}}$, where $\bar{\mathcal{G}}$ is the closure of $\mathcal{G}$ in $\Ot(d^2)$. It follows that there must exist a fixed number $\mu \in (\lambda_0, 1]$ such that \ea{ \langle p_{\mu} , R p_{\mu}\rangle \ge \frac{1}{d(d+1)} } for all $R \in \mathcal{G}$. For suppose that were not the case. Then we could choose a sequence $\nu_n \downarrow \lambda_0$, and a sequence $R_n \in \mathcal{G}$, such that \ea{ \langle p_{\nu_n} , R_n p_{\nu_n}\rangle < \frac{1}{d(d+1)} } for all $n$. The group $\bar{\mathcal{G}}$ is compact (because $\Ot(d^2)$ is compact~\cite{CompactGroup}) as is the closed interval $\left[0,\frac{1}{d(d+1)}\right]$. Consequently we can choose a subsequence $n_j$ such that $R_{n_j} \to \bar{R}\in \bar{\mathcal{G}}$ and \ea{ \langle p_{\nu_{n_j}} , R_{n_j} p_{\nu_{n_j}}\rangle \to a } for some $a \in \left[0,\frac{1}{d(d+1)}\right]$. But this would imply that \ea{ \langle p_{\lambda_0} , \bar{R} p_{\lambda_0} \rangle = a \le \frac{1}{d(d+1)} } ---which is a contradiction.
Now consider the set \ea{ Q' = Q \cup \{R p_{\mu} \colon R \in \mathcal{G}\}. } Observe that \ea{ (Rp_{\mu})(i) &= (d+1)\langle s^R_i , p_{\mu}\rangle - \frac{1}{d} \ge 0 } for all $i$ and all $R\in \mathcal{G}$ (because $p_{\mu} \in \dl{Q}\subseteq \dl{G}$). So $Q' \subseteq \Delta$. It is immediate that $Q'\subseteq B_{\rm{o}}$ and $Q'\subseteq Q^{\prime *}$. So $Q'$ is a germ such that $RQ' = Q'$ for all $R\in \mathcal{G}$, and which is strictly larger than $Q$---which is a contradiction.
It is now immediate that $\mathcal{G}$ is a subgroup of $\qgp{Q}$. \end{proof} We can make stronger statements if we introduce some new concepts. \begin{definition} A stochastic subgroup $\mathcal{G}\subseteq \Ot(d^2)$ is maximal if it is not contained in any larger stochastic subgroup. \end{definition} \begin{definition} A stochastic subgroup $\mathcal{G}\subseteq \Ot(d^2)$ is strongly maximal if it is maximal and if, in addition, the closed convex hull of the orbital germ is a qplex. \end{definition} We then have the following results. \begin{corollary} \label{cor:maxStochQplex} Let $\mathcal{G}$ be a maximal stochastic subgroup of $\Ot(d^2)$. Then there exists a qplex\ $Q$ such that $\mathcal{G}=\qgp{Q}$. \end{corollary} \begin{proof} Immediate consequence of Theorem~\ref{thm:StochSgpContQgp}. \end{proof} \begin{theorem} Let $\mathcal{G}$ be a strongly maximal stochastic subgroup of $\Ot(d^2)$ and let $G$ be the orbital germ. Then $\cc(G)$ is the unique qplex\ $Q$ such that $\mathcal{G}=\qgp{Q}$. \end{theorem} \begin{proof} We know from Corollary~\ref{cor:maxStochQplex} that there exists at least one qplex\ $Q$ such that $\mathcal{G}=\qgp{Q}$. If $Q$, $Q'$ are qplexes\ such that $\mathcal{G}=\qgp{Q} = \qgp{Q'}$ then $Q$, $Q'$ must both contain $\cc(G)$, where $G$ is the orbital germ. Since $\cc(G)$ is a qplex we must have $Q = \cc(G) = Q'$. \end{proof}
This brings us back to the claim we made at the beginning of this section.
\begin{corollary} If $Q$ is a qplex whose extreme points form a single orbit under the action of the preservation group, then the preservation group of~$Q$ is strongly maximal. \end{corollary} \begin{proof} Let $Q$ be a qplex and $\mathcal{G}$ be its preservation group. Assume that the extremal points form a single orbit under the action of~$\mathcal{G}$. The basis distributions are among the extremal points, so all extremal points are on the same orbit as any basis distribution. In other words, the orbital germ is the set of extreme points. Suppose that $Q'$ is a qplex whose preservation group contains $\mathcal{G}$. Then $Q'$ contains all the extremal points of~$Q$, and thus, $Q'$ contains $Q$. But a qplex is a maximal germ, so we must have $Q' = Q$. \end{proof}
\section{Characterizing qplexes\ isomorphic to quantum state space}
\label{sec:character} We are, of course, most interested in qplexes\ corresponding to SIC measurements. In this section, we will define what it means for a qplex\ to be isomorphic to quantum state space. We will prove that if $Q$ is a qplex\ isomorphic to quantum state space, then its preservation group is isomorphic to the projective extended unitary group, essentially the group of all unitaries and anti-unitaries with phase factors quotiented out. Then, we will establish the converse: If the preservation group of a qplex is isomorphic to the projective extended unitary group, then that qplex is isomorphic to quantum state space. This result indicates one way of recovering quantum theory from the urgleichung.
\begin{definition} \label{def:qstpIsomorphism} Let $B_H$ be the space of Hermitian operators on $d$-dimensional Hilbert space and let $S$ be the space of density matrices. We will say that a qplex\ $Q$ is isomorphic to quantum state space if there exists an $\fd{R}$-linear bijection $f\colon B_H \to \fd{R}^{d^2}$ such that \begin{enumerate} \item $Q = f(S)$. \item For all $\rho$, $\rho'\in S$ \ea{ \bigl< f(\rho), f(\rho')\bigr>=\frac{ \Tr(\rho \rho') +1 }{d(d+1)}. \label{eq:qstIsoProp2} } \end{enumerate} A qplex that is isomorphic to quantum state space will be designated a Hilbert qplex. \end{definition} It is straightforward to verify that definitions~\ref{def:qplexIsomorphism} and~\ref{def:qstpIsomorphism} are consistent, in the sense that if $Q$ is a Hilbert qplex, and if $Q'$ is any other qplex, then $Q'$ is a Hilbert qplex if and only if it is isomorphic to $Q$ in the sense of definition~\ref{def:qplexIsomorphism}. \begin{theorem} Let $Q$ be a qplex. Then a map $f\colon S \to Q$ is an isomorphism of quantum state space onto $Q$ if and only if there is a SIC $\Pi_j$ such that \ea{ (f(\rho))(j) = \frac{1}{d} \Tr(\rho \Pi_j) \label{eq:frhojTermsSIC} } for all $j$ and all $\rho\in S$. \end{theorem} \begin{remark} Thus, to each isomorphism of quantum state space onto $Q$, there corresponds a unique SIC. In particular a SIC exists in dimension $d$ if and only if a Hilbert qplex exists in dimension $d$. \end{remark} \begin{proof} Suppose $f\colon S\to Q$ is an isomorphism. Define \begin{equation} \Pi_j = f^{-1}(e_j). \end{equation} Then \ea{ \Tr(\Pi_j\Pi_k) & = d(d+1)\langle e_j, e_k \rangle -1 =\frac{d\delta_{jk} + 1}{d+1}. } So $\Pi_j$ is a SIC. Moreover, for all $\rho\in S$, and all $j$, \ea{ \frac{1}{d} \Tr(\rho \Pi_j) &= (d+1) \langle f(\rho), e_j\rangle -\frac{1}{d} = (f(\rho))(j). } Suppose, on the other hand, $f\colon S\to Q$ is a map for which Eq.~(\ref{eq:frhojTermsSIC}) is satisfied for some SIC $\Pi_j$. Then we can extend $f$ to a linear bijection of $B_H$ onto $\fd{R}^{d^2}$. We know from prior work~\cite{fuchs2009, appleby2011, fuchs2013} that $f(S)$ is a qplex. Since it is contained in $Q$ we must have $f(S) = Q$. Moreover, since \ea{ \rho = \sum_j \left( (d+1) (f(\rho))(j) -\frac{1}{d}\right) \Pi_j, } with a similar expression for $\rho'$, we have \ea{ \Tr(\rho \rho') = d(d+1) \langle f(\rho), f(\rho') \rangle -1 } from which Eq.~(\ref{eq:qstIsoProp2}) follows. \end{proof}
One might wonder if other qplexes, not isomorphic to $Q$ (and we know that these exist, per Theorem~\ref{thm:infNonIsomorphic} and Appendix~\ref{sec:altQplex}), correspond to other informationally complete POVMs. This is not the case. It follows from the foregoing that there is no measurement which will take us from a qplex\ of one kind to a qplex\ of a different, nonisomorphic kind.
Knowing this, let us characterize the preservation group of a Hilbert qplex\ $Q$. We define the extended unitary group, denoted $\EU(d)$, to be the group consisting of all unitary and anti-unitary operators, and the projective extended unitary group, denoted $\PEU(d)$, to be the quotient $\EU(d)/\M(d)$, where $\M(d)$ is the sub-group consisting of all unitaries of the form $e^{i\theta} I$, for some phase $e^{i\theta}$. \begin{theorem} \label{eq:qisoSGpPeU} Let $Q$ be a Hilbert qplex. Then $\qgp{Q}$ is isomorphic to $\PEU(d)$. \end{theorem} \begin{proof} Straightforward consequence of Wigner's theorem~\cite{Wigner}. \end{proof}
We showed in Theorem~\ref{eq:qisoSGpPeU} that if $Q$ is a Hilbert qplex then $\qgp{Q}$ is isomorphic to $\PEU(d)$. Now, we will prove the converse: If $\qgp{Q}$ is isomorphic to $\PEU(d)$, then $Q$ is a Hilbert qplex. It turns out, in fact, that a weaker statement is true: If $\qgp{Q}$ contains a subgroup isomorphic to $\PU(d)$, then $Q$ is a Hilbert qplex.
In the Introduction we remarked on the need for an extra assumption,
additional to the basic definition of a qplex, which will serve to
uniquely pick out those qplexes\ which correspond to quantum state
space. The theorem we will prove momentarily supplies us with one
possible choice for this assumption. As we remarked in the
introduction, there may be others.
As a by-product of this result we obtain a criterion for SIC existence: Namely, a SIC exists in dimension $d$ if and only if $\PU(d)$ is isomorphic to a stochastic subgroup of $\Ot(d^2)$. We proved this result by another method in a previous paper~\cite{GroupAlg}, but this is the route by which we were originally led to it. Indeed, it is hard to see why it should occur to anyone that stochastic subgroups of $\Ot(d^2)$ might be relevant to SIC existence if they were not aware of the role that such subgroups play in the theory of qplexes.
The result depends on the following method for embedding a qplex\ in operator space. The question of whether a SIC exists in every dimension is very hard, and, indeed, is still unsolved. But if one simply asks for a set of operators $\Pi_1, \dots , \Pi_{d^2}$ satisfying the equations \ea{ \Tr(\Pi_j) &= 1, \label{eq:quasiSICDef1} \\ \Tr(\Pi_j \Pi_k) & = \frac{d\delta_{jk} +1}{d+1}, \label{eq:quasiSICDef2} } without imposing any further constraint---in particular, without requiring that the $\Pi_j$ be positive semi-definite---then the problem becomes almost trivial. To see this consider the real Lie algebra $\su(d)$ (i.e.\ the space of trace-zero Hermitian operators). Equipped with the Hilbert--Schmidt inner product \begin{equation} \langle B, B'\rangle = \Tr(B B'), \end{equation} this becomes a $(d^2-1)$-Euclidean space, so the existence of operators $B_1,\dots, B_{d^2}$, each of length 1, and forming the vertices of a regular simplex, is guaranteed. These operators satisfy \begin{equation} \Tr(B_j B_k) = \begin{cases} 1 \qquad & j=k;\\ -\frac{1}{d^2-1} \qquad & j\neq k. \end{cases} \end{equation} If we now define \begin{equation} \Pi_j =\sqrt{\frac{d-1}{d}} B_j + \frac{1}{d} I, \end{equation} then the $\Pi_j$ satisfy Eqs.~(\ref{eq:quasiSICDef1}) and~(\ref{eq:quasiSICDef2}). We will refer to them as a quasi-SIC.
Now let $Q$ be an arbitrary qplex, and for each $q\in Q$ define, by analogy with Eq.~(\ref{eq:rhoTermsProbs}) \ea{ \rho_q = \sum_j \left( (d+1) q(j) -\frac{1}{d}\right) \Pi_j. } If $\Pi_j$ really were a SIC, and if the $q(j)$ really were the outcome probabilities for a measurement with that SIC, then $\rho_q$ would be a density matrix. In general, however, neither of those conditions need hold true. So, $\rho_q$ will typically not be positive semi-definite (though it will be trace-$1$). We will refer to it as a quasi-density matrix. It will also be convenient to define \ea{ S_Q = \{\rho_q\colon q \in Q\}. } We will refer to $S_Q$ as quasi-state space.
It is easily verified that \begin{equation} 0 \le \langle \rho, \rho' \rangle \le 1, \label{eq:UrungleichungForQuasiStSp} \end{equation} for all $\rho, \rho' \in S_Q$, just as is the case for genuine density matrices.
We are now in a position to prove \begin{theorem} \label{thm:QStateCriterion} Let $Q$ be a qplex. Then the following statements are equivalent: \begin{enumerate} \item $\qgp{Q}$ contains a subgroup isomorphic to $\PU(d)$. \item $Q$ is a Hilbert qplex. \end{enumerate} \end{theorem} \begin{proof} The implication $(2)\implies (1)$ is an immediate consequence of Theorem~\ref{eq:qisoSGpPeU}. It remains to prove the implication $(1) \implies (2)$.
Let $\Pi_j$ be a quasi-SIC, and use this quasi-SIC to map the qplex $Q$ into operator space, creating the quasi-state space $S_Q$. The fact that the qplex{} $Q$ contains a subgroup isomorphic to the projective unitary group $\PU(d)$ implies that the quasi-state space $S_Q$ is invariant under unitary transformations. That is, the projective unitary symmetry of one set carries over to the other. This result is fairly natural; for completeness, we provide an explicit proof in Appendix~\ref{sec:UnitaryImplication}.
Suppose $q\in Q\in S_{\rm{o}}$. Then \ea{ \Tr(\rho_q) = \Tr(\rho_q^2) = 1. } Also, it follows from Eq.~(\ref{eq:UrungleichungForQuasiStSp}) and unitary invariance of the quasi-state space that \ea{ 0 \le \Tr(\rho_q U\rho_qU^{\dagger}) \le 1. } for every unitary $U$. By choosing $U$ to give the appropriate permutation of the eigenvalues we deduce that \begin{equation} 0 \le \sum_i \lambda^{\uparrow}_i \lambda^{\downarrow}_i \le 1, \end{equation} where $\lambda^{\uparrow}_i$ (respectively $\lambda^{\downarrow}_i $) are the eigenvalues of $\rho_q$ arranged in increasing (respectively decreasing) order.
We now invoke a lemma proven in~\cite{GroupAlg}. If $\lambda$ is a vector in $\mathbb{R}^d$ such that \begin{equation} \sum_{j=0}^{d-1} \lambda_j = \sum_{j=0}^{d-1} \lambda_j^2 = 1, \end{equation} then \begin{equation} \inprod{\lambda^{\uparrow}}{\lambda^{\downarrow}} \leq 0. \end{equation} The inequality is saturated if and only if $d-1$ entries in $\lambda$ are equal. This can occur when \begin{equation} \lambda^{\downarrow} = (1,0,\ldots,0), \end{equation} or when \begin{equation} \lambda^{\downarrow} = \left(\frac{2}{d},\ldots,\frac{2}{d},
\frac{2}{d} - 1\right). \end{equation}
So we must have \begin{equation} \sum_i \lambda^{\uparrow}_i \lambda^{\downarrow}_i =0. \end{equation} Moreover, the possible solutions for the eigenvalue spectrum $\lambda^{\downarrow}$ imply that either $\rho_q = P$ or $\rho_q = (2/d)I -P$ for some rank-$1$ projector $P$. If $d=2$, then $\rho_q$ is a rank-$1$ projector either way. Otherwise, if $d>2$, suppose $q$, $q'\in Q\in S_{\rm{o}}$ were such that $\rho_q = P$ and $ \rho_{q'} = (2/d)I - P'$ where $P$ and $P'$ are rank-$1$ projectors. In that case there would be a unitary $U$ such that $UP'U^{\dagger} = P$, which would mean, by unitary invariance, that the quasi-state space contained both $P$ and $(2/d)I -P$. But \begin{equation} \Tr\left(P \left( \frac{2}{d}I - P\right) \right) = \frac{2}{d} - 1 < 1, \end{equation} which contradicts Eq.~(\ref{eq:UrungleichungForQuasiStSp}). We conclude that if $d>2$ then, either $\rho_q$ is a rank-$1$ projector for all $q\in Q$, or else $(2/d)I - \rho_q$ is a rank-$1$ projector for all $q\in Q$. In the latter case we may define a new quasi-SIC \begin{equation} \tilde{\Pi}'_j = \frac{2}{d}I - \tilde{\Pi}_j. \end{equation} One easily verifies that the new quasi-state space is also unitarily invariant. Moreover, if we define \ea{ \rho'_q = \sum_j \left( (d+1) q(j) -\frac{1}{d}\right) \tilde{\Pi}'_j, } then \ea{ \rho'_q = \frac{2}{d} I - \rho_q, } implying that $\rho'_q$ is a rank-$1$ projector for all $q\in Q\in S_{\rm{o}}$. There is therefore no loss of generality in assuming that our original quasi-state space is such that $\rho_q$ is a rank-$1$ projector for all $q\in Q\inS_{\rm{o}}$. Since \begin{equation} \rho_{e_i} = \tilde{\Pi}_i, \end{equation} this means in particular that the $\Pi_i$ are rank-$1$ projectors, and therefore constitute a genuine SIC.
Let us note that unitary invariance means that the set $\{\rho_q \colon q \in Q\in S_{\rm{o}}\}$ does not merely consist of rank-$1$ projectors; it actually comprises all the rank-$1$ projectors. It follows, that if $\rho$ is an arbitrary density matrix, and if $q(j) = (1/d) \Tr(\rho \tilde{\Pi}_j)$, then $q$ is a convex combination of points in $Q\in S_{\rm{o}}$, and therefore $q \in Q$. Since the SIC probabilities are a qplex, it follows that $Q$ does not contain any other points than these, and is therefore isomorphic to quantum state space as claimed. \end{proof}
Let us observe that in proving this theorem we have incidentally shown that if there is a qplex\ $Q$ which contains an isomorphic copy of $\PU(d)$, then a SIC exists in dimension $d$. So the theorem has the following corollary: \begin{corollary} The following statements are equivalent: \begin{enumerate} \item $\PU(d)$ is isomorphic to a stochastic subgroup of $\Ot(d^2)$. \item A SIC exists in dimension $d$. \end{enumerate} \end{corollary} \begin{proof} The implication $(2)\implies (1)$ is an immediate consequence of Theorem~\ref{thm:QStateCriterion}. To prove the implication $(1) \implies (2)$, let $\mathcal{G}$ be a stochastic subgroup of $\Ot(d^2)$ which is isomorphic to $\PU(d)$. It follows from Theorem~\ref{thm:StochSgpContQgp} that there exists a qplex\ $Q$ such that $\mathcal{G} \subseteq \qgp{Q}$. In view of Theorem~\ref{thm:QStateCriterion} this implies $Q$ is the set of outcome probabilities for a SIC measurement, which means, in particular, that a SIC must exist in dimension $d$. \end{proof}
\section{Discussion} \label{sec:future} Our investigation of qplexes exists in the context of many years' effort toward the goal of reconstructing quantum theory. Early pioneers of the subject, like Birkhoff and von Neumann, sought a broader mathematical environment in which quantum theory could be seen to dwell. This led to the subjects of quantum logic and Jordan algebras~\cite{Mccrimmon}. However, despite the mathematical developments, the influence on physics---and, indeed, on the philosophy thereof---was rather subdued. The instensely mathematical character of the work may have played a role in this. Moreover, this work predated the invention and integration into physics of information theory, which turned out to be a boon to the reconstruction enterprise. It also predated the theorems of Bell, Kochen and Specker~\cite{mermin1993, mermin1993-erratum}, and thus it could not benefit from their insight into what is robustly strange about quantum physics.
One might say that the ``modern age'' of quantum reconstructions was inaugurated by Rovelli in 1996. He advocated a research program of deriving quantum theory from physical principles, in a manner analogous to the derivation of special relativity's mathematical formalism~\cite{Rovelli1996}. During the same time period, one of the authors (CAF) also began advocating this project~\cite{fuchs2002, fuchs2010, transcript}. An early success was Hardy's ``Quantum theory from five reasonable axioms''~\cite{Hardy01, Schack03}, which pointed out the importance of what we call a Bureau of Standards measurement~\cite[p.\ 368]{Fuchs2014}.
Looking over the papers produced in this ``modern age,'' one technical commonality worth remarking upon is the idea of building up the unitary (or projective unitary) group from a universal gate set~\cite{Masanes, HoehnWever}. This is an idea from the field of quantum computation. For example, it is known that any unitary operator can be broken down into a sequence of two-level unitaries, applied in succession~\cite[p.\ 188]{MikeAndIke}. Also, given a collection of $N$ qubits, all the projective unitaries acting on their joint state space---that is, the group $\PU(2^N)$---can be synthesized using single-qubit unitaries and an entangling gate, like a Controlled \textsc{not} operation, that can be applied to any pair of qubits~\cite{Harrow}. This suggests one way of making progress in the theory of qplexes, by replacing the unitarity assumption.
Recall that in any qplex, a set of mutually maximally distant points can have at most $d$ elements~\cite{appleby2011, transcript}. Thus, although a qplex is originally defined as living within a $d^2$-dimensional space, in a sense it has an ``underlying dimensionality''~\cite{transcript} equal to~$d$. Consider a qplex $Q$, equipped with a set of $d$ mutually maximally distant pure states. What if we require that any $d-1$ of those states defines a structure isomorphic to a smaller qplex? Applying this recursively, we arrive eventually at the condition that any two maximally distant points define a set of probability distributions isomorphic to a qplex with $d = 2$, which is automatically a Hilbert qplex. This is a strong condition, although it makes no direct mention of a particular symmetry group. At the moment, we see no way to satisfy this condition other than having $Q$ be a Hilbert qplex.
Alternatively, one can try to make progress by relaxing the unitarity assumption. For example, instead of imposing a particular symmetry group, what if we seek the qplexes of maximal allowed symmetry? Assuming that a SIC exists in dimension $d$, then a qplex in~$\Delta_{d^2}$ can be at least as symmetric as a Hilbert qplex. We conjecture that no qplex can be more symmetric than a Hilbert qplex, where we quantify the degree of symmetry by, for example, the dimension of the Lie group of qplex-preserving maps. This conjecture leads to another: We suspect that of all the qplexes of a given dimension, the Hilbert qplexes have maximal Euclidean volume.
Another outstanding question is, out of all the conceivable additions one could make to probability theory in order to relate expectations for different hypotheticals, why pick the urgleichung? To our knowledge, no one considered such a relation before quantum mechanics and the SIC representation. And yet, it is a comparatively mild modification of the classical relationship. This is particularly evident when the measurement on the ground is modeled by a set of $d$ orthogonal projectors, \emph{i.e.,} when it is a von Neumann measurement. In that case, \begin{equation}
q(j) = (d+1) \sum_i p(i) r(j|i) - 1. \end{equation} This is just a rescaling and shifting of the classical formula~\cite{transcript}.
In Section~\ref{sec:basic}, we began with a general affine relationship between Bureau of Standards probabilities and the probabilities for other experiments. By invoking a series of assumptions, we narrowed the parameter values in the generalized urgleichung down to those that occur in quantum theory. (Our last assumption, which fixed the upper bound at twice the lower bound, may be related to the choice of complex numbers over real numbers and quaternions for Hilbert-space coordinates~\cite{fuchs2009}. For an unexpected connection between SICs and the normed division algebras, see~\cite{stacey-sporadic, stacey-hoggar}.) This has the appealing feature that a linear stretching is just about the simplest deformation of the classical Law of Total Probability that one can imagine. However, this area is still, to a great extent, unknown territory: Why linearity? Are qualitatively greater departures from classicality mathematically possible?
Many of the quantum reconstruction efforts to date share the feature that they make quantum physics as unremarkable as possible: While the technical steps from axioms to theorems are unassailable, the choice of axioms gives little insight into what is truly strange about quantum phenomena. To borrow a phrase from David Mermin, these re-expressions tend to make quantum theory sound ``benignly humdrum''~\cite{mermin-pillow}.
For example, should one aim to derive quantum theory from the fact that quantum states cannot be cloned? Arguably not: Even classical distributions over phase space are uncloneable~\cite{caves1996}. What about quantum teleportation? At root, teleportation is a protocol for making information about one system instead relevant to another, and it has exact analogues in classical statistical theories~\cite{spekkens2007, bartlett2012, spekkens2014}. In 2003, Clifton, Bub and Halvorson~\cite{clifton2003} proposed a derivation of quantum theory that started with $C^*$ algebras and then added, as postulates, some results of quantum information science, such as the no-broadcasting theorem~\cite{barnum1996}. However, the no-broadcasting theorem---despite its original motivation~\cite[p.\ 2235]{Fuchs2014}---also applies in classical statistical theories~\cite{spekkens2007, bartlett2012, spekkens2014}, and thus seems a poor foundation to build the quantum upon. Overall, it seems that choosing $C^*$ algebras for a starting point implicitly does a great deal of the work already~\cite[p.\ 1125]{Fuchs2014}.
Similarly, a more recent derivation by Chiribella, D'Ariano and Perinotti~\cite{chiribella11} invokes, at a key juncture, the postulate that any mixed state can be treated as a marginal of a pure state ascribed to a larger system. This postulate, the purifiability of mixed states, is an essential ingredient in their recovery of quantum theory. As with the examples above, however, it is also true in classical statistical theories~\cite{spekkens2007, bartlett2012,
spekkens2014, disilvestro2016}. From that perspective, it is consequently a less than fully compelling candidate for the essence of quantumness.
By contrast, we have chosen as our starting point what we consider to be the ``jugular vein'' of quantum strangeness: Theories of intrinsic hidden variables do so remarkably badly at expressing the vitality of quantum physics. The urgleichung is our way of stating this physical characteristic of the natural world in the language of probability. Quantum states, it avers, are catalogues of expectations---but \emph{not} expectations about hidden variables. This view is in line with ``participatory realist'' interpretations of quantum mechanics~\cite{cabello2015, fuchs2016}, like QBism~\cite{fuchs2013,
Voldemort, FMS-AJP} and related approaches~\cite{zeilinger2005, kofler2010,
appleby2013}.
\section{A qplex\ which does not correspond to a SIC measurement} \label{sec:altQplex}
By definition, a qplex is a subset of the probability simplex $\Delta_{d^2}$ such that each pair of points within it satisfy the fundamental inequalities, \begin{equation} \frac{1}{d(d+1)} \leq \sum_i p(i)s(i) \leq \frac{2}{d(d+1)}. \end{equation} We can construct a qplex which is not isomorphic to quantum state space in the following way. Begin with a set $A$ defined by the intersection of the probability simplex with the ball \begin{equation} \sum_i p(i)^2 \leq \frac{2}{d(d+1)}. \end{equation} Our plan is to trim this set down until it becomes a qplex. First, we break $A$ into $d^2!$ regions, which we label $F_k$, for $k = 1,\ldots,d^2!$. We define the region $F_1$ to be all probability vectors in the set $A$ whose entries appear in decreasing magnitude. That is, \begin{equation} F_1 = \left\{ {p} : {p} \in A
\hbox{ and } p(1) \geq p(2) \geq \cdots \geq p(d^2)
\right\}. \end{equation} The region $F_1$ is consistent with the fundamental inequalities, because for every ${p} \in F_1$, \begin{equation} \inprod{p}{p} \geq \inprod{p}{c}
\geq \frac{1}{d^2} > \frac{1}{d(d+1)}. \end{equation} We define the other regions $F_k$ analogously. Because $k$ runs from 1 to $(d^2)!$, it labels the permutations in the symmetric group on $d^2$ elements. Each $F_k$ consists of the vectors obtained by taking the vectors in $F_1$ and permuting the components according to permutation $k$. All of the regions $F_k$ so defined will be internally consistent.
To obtain a qplex $Q$, start with $F_1$ and include all the points from $F_1$ in $Q$. Then, take all the points from $F_2$ that are consistent with all the points in $F_1$, and include them in $Q$. Continue in this manner, adding the points in each $F_k$ that are consistent with every point added to $Q$ so far. The end result will be a qplex that is surely not isomorphic to quantum state space.
\section{Unitary Symmetry of Quasi-State Spaces} \label{sec:UnitaryImplication} Let $\Pi_j$ be a quasi-SIC, as defined in Eqs.~(\ref{eq:quasiSICDef1}) and (\ref{eq:quasiSICDef2}) of the main text. For each $U\in\PU(d)$ we have a matrix $S^U_{jk}$ such that \ea{ U\Pi_jU^{\dagger} = \sum_k S^U_{jk} \Pi_k. } The matrix is given explicitly by \ea{ S^{U}_{jk} &= \frac{d+1}{d} \Tr\bigl( \Pi_k U \Pi_j U^{\dagger}\bigr) - \frac{1}{d}, } from which one sees \ea{ \sum_j S^{U}_{jk} &= 1, & \sum_k S^{U}_{jk} &= 1, } and \ea{ \sum_k S^{U}_{ik}S^{U}_{jk} &= \frac{d+1}{d} \Tr\left(\left( \sum_kS^{U}_{ik}\Pi_k\right) U\Pi_jU^{\dagger}\right)-\frac{1}{d} \nonumber \\ &= \delta_{ij}. } So $S^{U}_{ij}$ is an orthogonal matrix.
We now appeal to the assumption that $\qgp{Q}$ contains a subgroup isomorphic to $\PU(d)$. So for each $U\in \PU(d)$ there exists an orthogonal matrix $R^{U}_{jk} \in \qgp{Q}$. It can be proven that, up to equivalence, the adjoint representation of~$\PU(d)$ is the only nontrivial irreducible representation of~$\PU(d)$ having degree $d^2 - 1$ or smaller, when $d \geq 2$~\cite{GroupAlg}. Thus, the two representations here must be equivalent, so that \ea{ R^U = T S^U T^{-1} \label{eq:equivRepEq} } for all $U$ and some fixed orthogonal matrix $T$. Summing over $k$ on both sides of \ea{ \sum_{j} R^U_{ij} T_{jk} = \sum_{j}T_{ij}S^U_{jk}
} and appealing to the fact that the representations are irreducible on the subspace orthogonal to $c$ we deduce that \ea{ \sum_{j} T_{ij} = t } for some constant $t$, independent of $i$. Similarly \ea{ \sum_{i} T_{ij} = s } for some constant $s$, independent of $i$. Since \ea{ d^2 t = \sum_{ij} T_{ij} = d^2 s, } we must in fact have $s=t$. Multiplying both sides of \ea{ \sum_j T_{ij} = t } by $T_{ik}$ and summing over $i$ we find \begin{equation} 1= t \sum_i T_{ik} = t^2. \end{equation} So $t^2 = \pm 1$. If $t=-1$ we can make the replacement $T\to -T$ without changing Eq.~(\ref{eq:equivRepEq}). We may therefore assume, without loss of generality, \begin{equation} \sum_j T_{ij} =\sum_j T_{ji} = 1 \end{equation} for all $i$. It follows that, if we define \ea{ \tilde{\Pi}_i &= \sum_{j} T_{ij} \Pi_j, } then the $\tilde{\Pi}_i$ are also a quasi-SIC. Moreover \ea{ U\tilde{\Pi}_i U^{\dagger} &=
\sum_{j,k} T\vpu{U}_{ij} S^U_{jk} T^{\rm{T}}_{kl} \tilde{\Pi}_l \nonumber \\ &= \sum_{l} R^U_{il} \tilde{\Pi}_l. } Suppose we now use the $\tilde{\Pi}_i$ to map $Q$ into operator space by defining \begin{equation} \rho_q = \sum_j \left((d+1) q(j) -\frac{1}{d}\right) \tilde{\Pi}_j \end{equation} for all $q\in Q$. It follows from the foregoing that, for all $q\in Q$ \ea{ U\rho_q U^{\dagger} &= \rho_{q'}, } where \ea{ q'(j) = \sum_k R^{U}_{kj}q(k) } is also in $Q$. It follows that the quasi-state space $\{\rho_q \colon q \in Q\}$ is invariant under unitary transformations.
\section{An Alternate Route to the Fundamental Inequalities}
In the main text, we began with the urgleichung and eventually arrived at the fundamental inequalities \begin{equation} \frac{1}{d(d+1)} \leq \inprod{p}{s} \leq \frac{2}{d(d+1)}, \end{equation} proving in Theorem~\ref{tm:qplexPolarity} that a self-polar subset of the out-ball $B_{\rm{o}}$ is a maximal germ. Because Theorem~\ref{tm:qplexPolarity} is an if-and-only-if result, it is natural to wonder if one could argue for the fundamental inequalities from a different premise, in which case self-polarity would be a consequence of assuming maximality.
One counterintuitive feature of quantum theory is that two quantum states can be perfectly distinguishable by a von Neumann measurement, yet less distinguishable by an informationally complete measurement~\cite{CFS2002, stacey-qutrit, stacey-hoggar}. This runs counter to experience with classical probability and stochastic processes, which leads one to think of a non-IC measurement as a coarse-graining (or a convolution by some kernel) of an IC measurement. If hypothesis $A$ is that the system is in region $A$ of phase space, and hypothesis $B$ is that the system is in region $B$, classical intuition says that hypothesis $A$ and $B$ being perfectly distinguishable means that their regions have no overlap. Therefore, if we measure where the system is in phase space---the fundamental classical image of what an IC experiment can be---then some outcomes would be consistent with hypothesis $A$, some with hypothesis $B$, and none with both.
In quantum physics, two pure states being orthogonal means that the overlap of their SIC representations is minimal, but minimal is not zero. If we regard two orthogonal states $\ket{0}$ and $\ket{1}$ as two hypotheses that Alice can entertain about how a system will behave, then there exists some measurement with the property that no outcome is compatible with both hypotheses. Whatever the outcome of that experiment, one hypothesis or the other will be excluded~\cite{CFS2002}. But the two hypotheses $\ket{0}$ and $\ket{1}$ have SIC representations $p_0$ and $p_1$, and $\inprod{p_0}{p_1} = 1/(d(d+1))$. The measurement that defines the SIC representation, although informationally complete, does not itself automatically exclude either hypothesis, because some possible outcomes of it are consistent with both.
With this motivation, we derive quantum state space in the following way. We again postulate a Bureau of Standards measurement, but we assume as little as possible about the meshing of probability distributions. Instead of the urgleichung (\ref{eq:gen-urgleichung}), we merely postulate some functional relation~\cite{qbism-greeks}, \begin{equation}
q(j) = F\left(\{p(i), r(j|i): i = 1,\ldots,N\}\right), \end{equation} with the property that state vectors with nonzero overlap are incompatible hypotheses with respect to some measurement. We assume, then, that the inner product of two state-space vectors is bounded below, and take this as an aspect of quantum strangeness. Then, we assume that certainty is bounded. This is less strange, since even classically, we can imagine a constraint that probability distributions can never get too focused. These two postulates tell us that the inner product of two state vectors lies in the interval $[L, U]$. Note that $U$, being an upper bound on $\inprod{p}{p}$, has an interpretation as an upper bound on an \emph{index of coincidence,} which is inversely related to the \emph{effective population size}~\cite{Leinster12, stacey-thesis, stacey-qutrit}. Imagine an urn filled with marbles in $N$ different colors. We draw a marble at random from the urn, note its color, replace it and draw at random again. If all colors are equally probable, then the probability of obtaining the same color twice in succession is $1/N$. More generally, if the colors are weighted by some probability vector $p$, then the probability of obtaining the same color twice---i.e., a ``coincidence'' of colors---is $\inprod{p}{p}$. So, we can take the reciprocal of this quantity as the effective number of colors present. Regarding the probability vector $p$ as a hypothesis about a system, the effective population size \begin{equation} N_{\rm eff}(p) = \frac{1}{\inprod{p}{p}} \end{equation} is the effective number of experiment outcomes that are compatible with that hypothesis. Given two probability vectors $p$ and $s$, we can take \begin{equation} N_{\rm eff}(p,s) = N_{\rm eff}(p) N_{\rm eff}(s)\,\inprod{p}{s} = \frac{\inprod{p}{s}}{p^2 s^2} \end{equation} as the effective number of outcomes compatible with both hypotheses $p$ and $s$.
By following the logic in Section~\ref{sec:basic}, we can get an upper bound on the size of a Mutually Maximally Distant set. If we postulate that this bound is saturated, we can relate $L$, $U$ and $N$ to the effective dimensionality: \begin{equation} d = 1 + \frac{U - 1/N}{1/N - L}. \end{equation} If we take $L = 0$ in the above expression, which we can heuristically regard as going to the ``classical limit,'' then we end up with $d = NU$. This says that the total number of MMD states is the total size of the sample space ($N$), divided by the area per state, i.e., the effective population size $1/U$.
Instead of taking $L = 0$, if we choose---for whatever reason---that $L = U/2$ and that $N = d^2$, we get the familiar upper and lower bounds that define a germ. Postulating that our state space is maximal then implies that it is self-polar. Because the state space is contained within the probability simplex, it contains the polar of the probability simplex, which is the basis simplex. By Theorem~\ref{tm:preservation-symmetry}, all the isometries of this set are specified by the regular simplices whose vertices are valid states lying on the out-sphere.
Suppose that $p$ and $s$ are two \emph{pure} states. Then \begin{equation} N_{\rm eff}(p) = N_{\rm eff}(s) = \frac{d(d+1)}{2}, \end{equation} and \begin{equation} N_{\rm eff}(p,s) = \frac{d^2(d+1)^2}{4} \inprod{p}{s} \geq \frac{d(d+1)}{4}. \end{equation} Thus, the fundamental inequalities imply that two hypotheses of maximal certainty can only disagree by so much that their overlap is \emph{half} the effective number of outcomes consistent with either hypothesis alone.
We note that Wootters~\cite{Wootters86}, Hardy~\cite{Hardy01} and others~\cite{CFS2001} have used various premises to argue for a relation of the form $N = d^2$. It bears something of the flavor of a classical state space whose points are labeled by discretized position and momentum~\cite{spekkens2014, Weyl27}. (And this resonates sympathetically with the fact that the Weyl--Heisenberg group, which is projectively equivalent to $\mathbb{Z}_d \times \mathbb{Z}_d$, is the canonical way to generate SICs~\cite{Zhu10, appleby2016}.) However, at the moment we find it neither an obvious choice nor a consequence of a uniquely compelling assumption.
\end{document} | arXiv |
Reaction diffusion equation with non-local term arises as a mean field limit of the master equation
Hopf dances near the tips of Busse balloons
February 2012, 5(1): 93-113. doi: 10.3934/dcdss.2012.5.93
Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero
Marie Henry 1, , Danielle Hilhorst 2, and Robert Eymard 3,
CMI, Université de Provence, 39 rue Frédéric Joliot-Curie 13453 Marseille cedex 13
CNRS and Laboratoire de Mathématiques, Université de Paris-Sud 11, F-91405 Orsay Cedex
Université Paris-Est Marne-La-Vallée, 5 bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France
Received June 2009 Revised December 2009 Published February 2011
In this paper we consider a two-phase flow problem in porous media and study its singular limit as the viscosity of the air tends to zero; more precisely, we prove the convergence of subsequences to solutions of a generalized Richards model.
Keywords: existence of weak solutions, Degenerate parabolic-elliptic system, two phase flow in porous media., singular limit.
Mathematics Subject Classification: 35K65, 35D30, 35B2.
Citation: Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93
H. W. Alt and S. Luckhaus, Quasilinear elliptic equations,, Math.-Z., 183 (1983), 311. doi: 10.1007/BF01176474. Google Scholar
H. Brezis, "Analyse Fonctionnelle Théorie et Applications,'', Masson, (1993). Google Scholar
Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution,, J. Differential Equations, 171 (2001), 203. doi: 10.1006/jdeq.2000.3848. Google Scholar
F. Z. Daïm, R. Eymard and D. Hilhorst, Existence of a solution for two phase flow in porous media: the case that the porosity depends on the pressure,, J. Math. Anal. Appl., 326 (2007), 332. doi: 10.1016/j.jmaa.2006.02.082. Google Scholar
C. J. Van Duijn and L. A. Peletier, Nonstationary filtration in partially satured porous media,, Arch. Rational Mech. Anal., 78 (1982), 173. doi: 10.1007/BF00250838. Google Scholar
R. Eymard, M. Gutnic and D. Hilhorst, The finit volume method for an elliptic-parabolic equation,, Acta Mathematica Universitatis Comenianae, 67 (1998), 181. Google Scholar
J. Hulshof and N. Wolanski, Monotone flows in n-dimensional partially saturated porous media: Lipschitz-continuity of the interface,, Arch. Rational Mech. Anal., 102 (1988), 287. doi: 10.1007/BF00251532. Google Scholar
O. A. Ladyhenskaya and N. N. Ural'ceva, "Linear and Quasilinear Elliptic Equations,'', American Mathematical Society, (1964). Google Scholar
O. A. Ladyhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', American Mathematical Society, (1968). Google Scholar
F. Otto, $L^1$-concentration and uniqueness for quasilinear elliptic-parabolic equations,, J. Differential Equations, 131 (1996), 20. doi: 10.1006/jdeq.1996.0155. Google Scholar
I. S. Pop, Error estimates for a time discretization method for the Richard's equation,, Computational Geosciences, 6 (2002), 141. doi: 10.1023/A:1019936917350. Google Scholar
F. A. Radu, I. S. Pop and P. Knabner, Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards' equation,, SIAM Journal on Numerical Analysis, 42 (2004), 1452. doi: 10.1137/S0036142902405229. Google Scholar
Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217
Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007
Yūki Naito, Takasi Senba. Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1861-1880. doi: 10.3934/cpaa.2013.12.1861
Giuseppe Maria Coclite, Helge Holden, Kenneth H. Karlsen. Wellposedness for a parabolic-elliptic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 659-682. doi: 10.3934/dcds.2005.13.659
Yūki Naito, Takasi Senba. Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model. Conference Publications, 2011, 2011 (Special) : 1111-1118. doi: 10.3934/proc.2011.2011.1111
Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011
Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006
Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020
Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81
Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009
Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691
Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013
Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281
Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure & Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014
Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 321-328. doi: 10.3934/dcdss.2020018
Youcef Amirat, Laurent Chupin, Rachid Touzani. Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2445-2464. doi: 10.3934/cpaa.2014.13.2445
Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335
Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23
Bilal Saad, Mazen Saad. Numerical analysis of a non equilibrium two-component two-compressible flow in porous media. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 317-346. doi: 10.3934/dcdss.2014.7.317
Marie Henry Danielle Hilhorst Robert Eymard | CommonCrawl |
\begin{document}
\preprint{APS/123-QED}
\author{S. Flannigan} \affiliation{Department of Physics $\&$ SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom.}
\author{F. Damanet}\affiliation{Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège, B-4000 Liège, Belgium}
\author{A. J. Daley} \affiliation{Department of Physics $\&$ SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom.}
\title{Many-body quantum state diffusion for non-Markovian dynamics in strongly interacting systems}
\date{\today}
\begin{abstract} Capturing non-Markovian dynamics of open quantum systems is generally a challenging problem, especially for strongly-interacting many-body systems. In this work, we combine recently developed non-Markovian quantum state diffusion techniques with tensor network methods to address this challenge. As a first example, we explore a Hubbard-Holstein model with dissipative phonon modes, where this new approach allows us to quantitatively assess how correlations spread in the presence of non-Markovian dissipation in a 1D many-body system. We find regimes where correlation growth can be enhanced by these effects, offering new routes for dissipatively enhancing transport and correlation spreading, relevant for both solid state and cold atom experiments. \end{abstract}
\maketitle
\textit{Introduction.}
In open quantum system dynamics, it is becoming increasingly crucial to consider the effects of non-Markovian dissipation, i.e., dissipation into a spectrally-structured environment which remembers past interactions with the system~\cite{RevModPhys.89.015001}, as demonstrated in many recent quantum devices which are non-Markovian in nature~\cite{PhysRevLett.122.050501,Liu:2011vt,PhysRevX.8.011053}.
While there has been great progress in treating these features computationally~\cite{Ishizaki:2005aa,Kato:2016aa,Strathearn:2018aa,PhysRevResearch.2.013265,PhysRevLett.126.200401,PhysRevLett.113.150403,Hartmann:2017aa}, there has so far been difficulty in generalising these methods for strongly-interacting many-body systems, even in 1D. Here, by hybridizing tensor network and non-Markovian stochastic techniques, we show how to capture the effects of non-Markovian dissipation on the generation of long-range correlations in strongly interacting one-dimensional many-body systems. As an example, we consider a damped form of the Hubbard-Holstein model, which introduces electron-phonon interactions to strongly correlated systems~\cite{Holstein:1959aa,Holstein:2000aa,doi:10.1142/1476}. We find that the growth of pairing correlations can be enhanced by going beyond the Markovian regime and that by controlling the properties of the environment we can tune the correlation spreading in the (electron) system. Our results demonstrate the capabilities of these methods to explore dissipative many-body systems beyond the Born-Markov limit and quantitatively capture their out-of-equilibrium dynamics, as motivated by experimental advances with many-body cavity quantum electrodynamics (QED)~\cite{PhysRevX.8.011002,Kollar:2017vn,PhysRevA.87.043817} and with cold atoms immersed in reservoir gases~\cite{PhysRevLett.94.040404,PhysRevA.84.031602,PhysRevA.95.033610,PhysRevA.98.062106,PhysRevA.101.033612}.
\begin{figure}
\caption{(a) Illustration of the Hubbard model coupled with strength $g$ to independent identical local phonon modes of frequency $\omega$ and damping rate $\kappa$. While the dissipative dynamics of the system made up of the fermions and the phonons (dashed black box) is Markovian, the one of the Hubbard system alone (dashed blue box) is generally non-Markovian. (b) Matrix product state (MPS) representation of the many-body HOPS equations [see Eq.~(\ref{HOPS_MPS})], with local dimension $d$ and hierarchy dimension $k_{\rm max} + 1$ in the usual form but now with an enlarged local dimension $d + k_{\rm max} + 1$. }
\label{NMSystem}
\end{figure}
Large separations of frequency scales in quantum optical systems coupled to their environment have made theoretical tools such as the Gorini, Kossakowski, Sudarshan, Lindblad (GKSL) master equation~\cite{lindblad1976,Gorini:1976aa} invaluable for quantitatively capturing many important experiments. There, the system and environment are weakly coupled and the environment is memory less, satisfying the Born-Markov approximation~\cite{Qnoise,Breur_book}. Reservoir engineering in recent quantum optics experiments, such as using impurities immersed in Bose-Einstein Condensates (BEC) to produce spin-boson models~\cite{PhysRevLett.94.040404,PhysRevA.84.031602,PhysRevA.95.033610,PhysRevA.98.062106,PhysRevA.101.033612} or with multi-mode cavity QED systems~\cite{PhysRevX.8.011002,Kollar:2017vn,PhysRevA.87.043817}, has made it possible to go beyond the Born-Markov regime in systems where microscopic models can still be derived from first principles. This has motivated interest in creating theoretical tools to compute dynamics in these cases. The large size of these systems makes it necessary to trace out the BEC in the former scenario and the cavity modes in the latter, which results in open quantum system descriptions that are generally non-Markovian~\cite{Caldeira:1983wy,RevModPhys.59.1,doi:10.1142/1476,PhysRevA.99.033845,PhysRevResearch.3.L032016}. Simulating these situations is particularly challenging due to the combination of strong interactions generating strongly correlated phases, the many-body system giving rise to an exponentially large Hilbert space and the non-Markovian features requiring the use of an equation of motion that is non-local in time.
Finding the best way to deal with non-Markovian dynamics, the most natural kind of open system dynamics occurring in the solid-state from which our example originates, is an old and difficult problem, and a number of approaches have been developed over the past decades, ranging from non-Markovian master equations ~\cite{Breur_book} to non-Markovian collapse theories~\cite{PhysRevA.80.012116}, collisional models~\cite{Rybar:2012un} and stochastic Schrödinger equations~\cite{DIOSI1997569, Piilo2008PRL} (see~\cite{RevModPhys.89.015001} for a detailed review). More recently, time-evolving matrix product operators (TEMPO)~\cite{Strathearn:2018aa,PhysRevResearch.2.013265,PhysRevLett.126.200401} or hierarchical equations of motion (HEOM)~\cite{Ishizaki:2005aa,Kato:2016aa} have shown remarkable potential for systems with a small Hilbert space, but so far have not been generalised to many-body systems.
To address this challenge here we employ the hierarchy of pure states (HOPS)~\cite{PhysRevLett.113.150403,Hartmann:2017aa}, a non-Markovian quantum state diffusion method, which we have combined with matrix product state (MPS) techniques~\cite{SCHOLLWOCK201196}. We demonstrate applications for this method by exploring dynamics in a modified Hubbard-Holstein model \cite{Holstein:1959aa,Holstein:2000aa}, where we couple strongly interacting fermions to local harmonic oscillator modes that are damped, representative of phonons that have dispersion. We show that non-Markovian dissipation can enhance the short-time dynamical growth of the pairing correlations where we find a qualitative difference compared to the Markovian, but also the phononless cases. This demonstrates that this method allows us to quantitatively simulate the dynamics of strongly correlated one-dimensional open many-body systems well into the non-Markovian and strong coupling regimes.
\textit{The dissipative Hubbard-Holstein model.} We consider the model shown in Fig.~\ref{NMSystem}(a), with fermions in an $M$ site lattice, described by a many-body system Hamiltonian $\hat{H}_s$ where each site is coupled to a local phonon mode similar to the (Hubbard)-Holstein model~\cite{Holstein:1959aa,Holstein:2000aa}. The total Hamiltonian is given by \begin{equation}\label{Open_problem} \hat{H} = \hat{H}_s + \omega \sum_{n=1}^M \hat{a}_n^{\dagger} \hat{a}_n + g\sum_{n=1}^M \Big( \hat{L}_n\hat{a}^{\dagger}_n + \hat{L}_n^{\dagger}\hat{a}_n \Big), \end{equation} where $\hat{a}_n^{\dagger}$ and $\hat{a}_n$ create and destroy a phonon in the $n$th mode and $\hat{L}_n$ are system operators acting on site $n$. We modify the usual Holstein model by going beyond the approximation of dispersionless phonons, taking a next step in better modelling realistic situations with this toy model~\cite{PhysRevLett.120.187003}. We incorporate these effects by modelling each phonon mode as a damped harmonic oscillator, such that we can write the phonon correlation function as, \begin{equation} \label{Ph_Corr}
\alpha_{n}(t-t') = \langle \hat{a}_n(t) \hat{a}_n^{\dagger}(t') \rangle = e^{- \kappa |t-t'| - i \omega (t-t')}, \end{equation} where $\omega$ and $\kappa$ are the phonon frequency and damping rate, respectively.
\textit{Non-Markovian Quantum State Diffusion.} Non-markovian dynamics arise when we trace out part of the system where we do not have a strong separation of frequency scales that satisfy the conditions for the Born-Markov approximation. In principle it is always possible to place the boundary of the system where the dynamics are Markovian. In this case, we could take the fermions and phonon modes as the \textit{system} [dashed black box in Fig.~1(a)], with the phonon damping remaining Markovian~\cite{Daley:2014aa}.
However, in many relevant situations (such as multi-mode cavities described above), it becomes prohibitively expensive computationally to make this choice because of the large local basis. In this particular case, we find it much more convenient to trace out the phonon modes and work with an effective equation of motion for the Hubbard system only [dashed blue box in Fig.~1(a)]. For finite $\kappa$ the resulting correlation function for the phonon modes, Eq.~(\ref{Ph_Corr}), cannot be approximated as a delta function, and so we must use the non-Markovian quantum state diffusion (NMQSD) equation for the dynamics of the reduced system $|\psi(t) \rangle$~\cite{DIOSI1997569, RevModPhys.89.015001},
\begin{equation} \label{NMQSD_eq}
\begin{split}
\partial_t|{\psi}(t) \rangle = & -i \hat{H}_s |\psi(t) \rangle + g\sum_{n=1}^M \hat{L}_n z^*_n(t) |\psi(t) \rangle \\
& - g\sum_{n=1}^M \hat{L}_n^{\dagger} \int_0^t ds \alpha_n^*(t-s) \frac{\delta |\psi(t) \rangle}{\delta z_n^*(s)}, \end{split} \end{equation} where we have introduced a set of stochastic \textit{colored} noise terms $z^*_n(t)$ which upon taking an ensemble average give the correlation function $\mathcal{E}[z_n(t) z^*_{n'}(t')] = \delta_{n,n'} \alpha_n(t-t')$.
\textit{The HOPS + MPS algorithm.} The insight which lead to the HOPS algorithm~\cite{PhysRevLett.113.150403,Hartmann:2017aa} is to introduce a set of auxiliary states which absorb the numerically intractable functional derivatives $\delta/\delta z_n^*(s)$, \begin{equation}
|\psi^{(1,n)}(t)\rangle = D_n(t) |\psi(t) \rangle \equiv \int_0^t ds \alpha_n^*(t-s) \frac{\delta |\psi(t) \rangle}{\delta z_n^*(s)}. \end{equation}
Deriving an equation of motion for this auxiliary state requires the introduction of further auxiliary states defined through $|\psi^{(k,n)}(t)\rangle = [D_n(t)]^k |\psi(t) \rangle$ which give rise to a hierarchical set of equations. In order to write this hierarchy, we find it convenient to include the hierarchy index into the basis states and write a total state for the combined system and auxiliary Hilbert space, \begin{equation}
|{\Psi} (t) \rangle = \sum_{\vec{\mathbf{k}}} C_{\vec{\mathbf{k}}}(t) |\psi^{(\vec{\mathbf{k}})}(t)\rangle \otimes |\vec{\mathbf{k}}\rangle, \end{equation}
where the $C_{\vec{\mathbf{k}}}(t)$ are time-dependent complex numbers and $|\vec{\mathbf{k}}\rangle = | k_1,k_2,\cdots,k_M \rangle = |k_1\rangle \otimes |k_2 \rangle \otimes \cdots \otimes |k_M \rangle$ with each of the $k_n$ running from $0,1,\cdots,\infty$, as we have a hierarchy index for each of the $M$ phonon environment modes. Each hierarchy index is represented as an independent boson mode, see the supplementary material for details. Note that the $|\psi^{(0)}(t)\rangle \otimes |0\rangle = |\psi(t)\rangle$ is the physical system state. This allows us to write the equation of motion for the total state as, \begin{equation}\label{HOPS_MPS} \begin{split}
\partial_t|{\Psi} (t) \rangle = &-i \hat{H}_s|\Psi(t) \rangle + \sum_{n=1}^M \Big( \tilde{z}^*_n(t) g \hat{L}_n - \left( \kappa + i \omega \right) \hat{K}_{n} \\
& + g\hat{L}_n \otimes \hat{K}_{n} \hat{b}_n^{\dagger} - g \left( \hat{L}_n^{\dagger} - \langle \hat{L}_n^{\dagger} \rangle_t \right) \otimes \hat{b}_n \Big) |\Psi(t) \rangle. \end{split} \end{equation}
Note that we time-dependently modify the colored noise according to $\tilde{z}^*_n(t) = z^*_n(t) + g\int_0^t ds \alpha^*_n(t - s) \langle \hat{L}_n^{\dagger} \rangle_s$ with $\langle \hat{L}^{\dagger}_n \rangle_s = \langle {\psi}^{(0)}(s) | \hat{L}^{\dagger}_n | {\psi}^{(0)}(s) \rangle$ thus explicitly taking into account previous states of the system. Note that one has to consider sufficiently small time steps in the numerical resolution of the equation so that the time-dependent terms in Eq.~\ref{HOPS_MPS} can be approximated as constant in time. In this way the non-linear terms $\tilde{z}^*_n(t)$ and $\langle \hat{L}_n^{\dagger} \rangle_t$ are calculated using the state before the time increment.
In the above equation we have introduced the bare operators (ommitting the index $n$) $\hat{b}^{\dagger} |k\rangle = |k+1 \rangle$, $\hat{b} |k \rangle = |k-1 \rangle$ (see Ref.~\cite{arxiv_bare,PhysRevA.60.4083}) and $\hat{K} = \sum_k k |k\rangle \langle k |$. We initialise the hierarchy with $C_\mathbf{0}(0) = 1$ and $C_{|\mathbf{k}|>0}(0) = 0$ and in order to extract observables we use the (normalized) physical system state $O(t) = \langle {\psi}^{(0)}(t) | \hat{O} | {\psi}^{(0)}(t) \rangle$ which we must average over many trajectories with different realisations of the random numbers $z_n^*(t)$, similar to conventional QSD equations~\cite{RevModPhys.89.015001,Qnoise,Daley:2014aa}.
Formally the hierarchy depth is infinite, but the populations of the auxiliary states typically decrease with the hierarchy indices $k_n$, which makes it possible in practice to truncate each hierarchy to some index $k_\mathrm{max}$ (chosen such that the results have converged to a given precision) to render the problem numerically feasible.
In general, the stronger the violation of the Born-Markov approximations the larger the number of auxiliary states we must retain. Note that this hierarchy truncation still results in an exponential number of equations: if each hierarchy index can run from $0,1,\cdots,k_{\rm max} $ then in total we have $({k_{\rm max}+1})^M$ auxiliary states. This motivates the incorporation of MPS techniques which allow us to time-evolve many-body states of one-dimensional Hamiltonians without explicitly working with the full Hilbert space~\cite{SCHOLLWOCK201196}. As each hierarchy only couples locally with a system operator of site $n$ this allows us to efficiently write this problem as an MPS simply with an enlarged local Hilbert space consisting of the physical local dimension of the system, but now also an effective local dimension for the auxiliary state of that effective environment mode [see Fig.~\ref{NMSystem}(b)] modelled as a boson Hilbert space. This is particularly convenient as we can then apply standard MPS techniques for time-evolution~\cite{Paeckel:2019aa}. This does result in an MPS with a large local dimension but in the following sections we show that it can be used to make important quantitative predictions with practical numerical values for the size of the hierarchy dimension $k_{\rm max} $ and also the bond dimension of the MPS $D$ (see the supplemental material for a detailed error analysis). Note finally that providing $k_{\rm max} $ and $D$ are large enough, Eq.~(\ref{HOPS_MPS}) numerically converges to the \textit{exact} dynamics of the system (as well as of the environment via monitoring of the noises as we will discuss below), as it does not directly rely on any approximation (neither Born nor Markov).
\textit{Benchmarking.} We first consider the out-of-equilibrium dynamics of a Holstein model~\cite{Holstein:1959aa,Holstein:2000aa}. We use, \begin{equation}\label{Hol_Ham} \hat{H}_s = -J \sum_n \left( \hat{c}^{\dagger}_n \hat{c}_{n+1} + \hat{c}^{\dagger}_{n+1} \hat{c}_n \right), \end{equation} as the system Hamiltonian in Eq.~(\ref{HOPS_MPS}), where $J$ describes the tunnelling of the (spinless) electrons. Additionally, we use the number operator as our system-environment coupling operators $\hat{L}_n = \hat{n}_n = \hat{c}^{\dagger}_n \hat{c}_{n}$, and as mentioned we include dissipation on the phonons yielding the damped correlation functions, Eq.~(\ref{Ph_Corr}).
\begin{figure}
\caption{Dynamics in the dissipative Holstein model [Eq.~(\ref{Open_problem}) with the system Hamiltonian~(\ref{Hol_Ham}), $\hat{L}_n = \hat{n}_n$ and the environment correlation functions~(\ref{Ph_Corr})] upon beginning in an initial state $|1,0,1,0,\cdots \rangle$. (a-b) The evolution of the CDW correlations $O_{\rm CDW} (t) = (1/M) \sum_n (-1)^n \langle \hat{n}_n(t) \rangle$ for different coupling strengths $g$ ($g=0$ in black dotted) and phonon dispersion rates $\kappa = [ g,2g,\infty ]$ (dark to light blue). The Born-Markov limit ($\kappa \rightarrow \infty$) was calculated using a conventional quantum trajectory method~\cite{Daley:2014aa}) (c-d) Finite temperature analysis for different coupling strengths $g = [ 1,3,5 ] J$ (light to dark blue, $g=0$ in black dotted) and phonon dispersion rates $\kappa$. See Ref.~\cite{Hartmann:2017aa} on how to adapt the algorithm for finite temperature environments. In all cases we average the observables over $N_{\rm traj}=100$ trajectories and use $\omega = J$ and $M=20$ lattice sites. For our hybridized HOPS+MPS algorithm, we use the numerical parameters $k_{\rm max}=8$, $D=128$ and $Jdt=0.01$.}
\label{Fig_CDW}
\end{figure}
We begin with the initial state $|1,0,1,0,\cdots \rangle$ and time-dependently calculate a charge density wave (CDW) correlations $O_{\rm CDW} (t) = (1/M)\sum_n (-1)^n \langle \hat{n}_n(t) \rangle$. We plot this in Fig.~\ref{Fig_CDW}(a-b) for different coupling strengths $g$ and phonon dispersions $\kappa$. Comparing to the results obtained in Ref.~\cite{PhysRevB.101.035134}, which analyzes this system in the limit of dispersionless phonons ($\kappa \rightarrow 0$), we find the same qualitative behaviour, where for $g=J$ the dynamics are similar to the closed system ($g=0$) case where there are oscillations but the CDW \emph{melts} into a homogeneous steady state. Increasing the coupling strength to $g=5J$ we can see that the CDW melting is slowed for short times and the oscillations become completely damped.
\textit{Born-Markov limit.} We also compare our results to that of a conventional quantum state diffusion (QSD) equation valid in the Born-Markov limit~\cite{Daley:2014aa}. This is achieved by setting $k_{\rm max}=1$ and $\alpha_{n}(\tau) = \delta(\tau)$ (see Ref.~\cite{PhysRevLett.113.150403} and the supplemental material) which physically corresponds to the approximation that the phonon dispersion $\kappa$ goes to infinity. From Fig.~\ref{Fig_CDW} we see that for strong coupling ($g = 5J$) this model completely fails to predict the suppression of the CDW correlations at short times.
\textit{Finite temperature.} Within the framework of HOPS it is also possible to efficiently include finite temperature effects of the environment (see Ref.~\cite{Hartmann:2017aa}). In Fig.~\ref{Fig_CDW}(c-d) we plot the dependence on the CDW correlations upon increasing the initial temperature of the environment modes. We see that the suppression of the CDW melting is enhanced for increasing temperatures which is due to a non-zero population of phonons in the initial state, allowing for a greater effect on the short time dynamics. Including finite temperature effects in the Born-Markov QSD simply increases the effective system-environment coupling strength (see the supplemental material) which as seen from (a-b) predicts an increased decay of the CDW. Increasing the temperature of the phonon modes in this model therefore results in further deviations from the Born-Markov regime, in contrast to the more common cases where larger temperatures suppress non-Markovian features~\cite{RevModPhys.89.015001,doi:10.1142/1476}.
\textit{Correlation spreading.} We move on and consider the Hubbard-Holstein model describing two-species fermions coupled to phonon modes and now with an onsite interaction $U$. Explicitly our system Hamiltonian is given by \begin{equation}\label{Hubb_hol} \hat{H}_s = -J \sum_{n,\sigma} \left( \hat{c}^{\dagger}_{n,\sigma} \hat{c}_{n+1,\sigma} + \hat{c}^{\dagger}_{n+1,\sigma} \hat{c}_{n,\sigma} \right) + U \sum_n \hat{n}_{n,\uparrow} \hat{n}_{n,\downarrow}, \end{equation} where $\hat{n}_{n,\sigma} = \hat{c}^{\dagger}_{n,\sigma} \hat{c}_{n,\sigma}$ and our system-environment coupling operators are $\hat{L}_n = \hat{n}_{n,\uparrow} + \hat{n}_{n,\downarrow}$. As earlier, we go beyond the usual case and include phonon dissipation.
In Fig.~\ref{Fig_HuHolCorr} we begin in the initial product state $|\uparrow,\downarrow,\uparrow,\downarrow,\cdots \rangle$ and in (a) we analyze the fermionic pairing correlation functions, \begin{equation} \label{Corr_equ} P_m = \frac{1}{M-m} \sum_{\tilde{m}} \langle \hat{c}^{\dagger}_{\tilde{m},\uparrow} \hat{c}^{\dagger}_{\tilde{m},\downarrow} \hat{c}_{{\tilde{m}+m},\downarrow} \hat{c}_{{\tilde{m}+m},\uparrow} \rangle. \end{equation} For the case where there is no coupling to the phonons $g=0$ we observe a peak in these correlations which spreads out in time, and beyond this the correlations decay exponentially which is the usual light cone spreading of correlations~\cite{Lieb1972,PhysRevLett.97.050401,PhysRevB.79.155104}. Including coupling to the phonon modes with $g=J$ we see similar behaviour, although the dissipation damps the amplitude of this peak in time, gradually suppressing correlations in the steady state. For finite $\kappa$ (i.e., non-Markovian environment behaviour), we see a strong enhancement of the correlation length beyond the light cone at short times ($tJ\sim 0.5,1$) which is qualitatively different to the case of purely Markovian dissipation ($\kappa \rightarrow \infty$) where the correlation length is unaffected.
\begin{figure}
\caption{Dynamics in the dissipative Hubbard-Holstein model [Eq.~(\ref{Open_problem}) with the system Hamiltonian~(\ref{Hubb_hol}), $\hat{L}_n = \hat{n}_{n,\uparrow} + \hat{n}_{n,\downarrow}$ and the environment correlation functions~(\ref{Ph_Corr})], upon beginning in an initial CDW state $|\uparrow,\downarrow,\uparrow,\downarrow,\cdots \rangle$. (a) The pair correlation function [Eq.~(\ref{Corr_equ})], where we use $g=J$ ($g=0$ in black) and compare different phonon dispersion rates $\kappa = [ g/5,5g,\infty ]$ (dark to light blue). (b) Dynamics of the average phonon mode occupation, $\langle a^{\dagger} a \rangle_{\rm av} = (1/M) \sum_n \langle a^{\dagger}_n a_{n} \rangle = (1/M) \sum_n \left( |\tilde{z}_n^*(t)|^2 - 1 \right) $. (c) The real (blue) and imaginary (red) part of the phonon coherences $\langle a^{\dagger} \rangle_{\rm av} = (1/M) \sum_n \langle a^{\dagger}_n \rangle = (1/M) \sum_n \tilde{z}_n^*(t)$. For our hybridized HOPS+MPS algorithm, we use the parameters $k_{max}=6$, $D=300$ and $Jdt=0.01$ where we also have incorporated conserved quantum numbers into the MPS algorithm~\cite{itensor}. We average the observables over $N_{\rm traj}=100$ trajectories. In all cases we use $U=J$, $\omega = 2 J$ and $M=50$ sites.}
\label{Fig_HuHolCorr}
\end{figure}
These features of the non-Markovian dynamics can be understood by realising that the coupling to the phonons \textit{dresses} the electrons~\cite{PhysRevB.86.045110}, modifying the quasi-particle excitations and shifting the effective fermion-fermion interaction strength $U_{\rm eff} \rightarrow U - 2g^2/\omega$. Here there is a competition between a generated effective attractive interaction and then the dynamical generation of phonons in the environment, the presence of which can strongly suppress dynamics and correlation growth resulting in CDW order~\cite{PhysRevB.86.045110}. We can see the competition of these effects in Fig.~\ref{Fig_HuHolCorr}(a), which are made even more clear by analysing the phonon mode observables in Fig.~\ref{Fig_HuHolCorr}(b-c) which we can directly calculate from HOPS using the time-dependent colored noise term $\tilde{z}_n^*(t)$ in Eq.~(\ref{HOPS_MPS}) (see the supplemental material). We see that initially the phonon population is low and so the effective interaction between fermions dominates, enhancing the growth of pairing correlations, before the dynamical generation of phonons begins to dominate, suppressing correlations at later times, which for smaller $\kappa$ is larger due to an increased phonon population.
\textit{Discussion and outlook.} Our combination of the HOPS algorithm with MPS techniques opens up the ability to explore a wide range of new and interesting regimes that were previously only possible to simulate qualitatively and/or through invoking some strong approximations. By considering the dispersive Hubbard-Holstein model we demonstrated that we can simulate the exact dynamics of open many-body systems well into the non-Markovian and strong coupling regimes and we are able to quantitatively analyze the dynamical properties of long-range correlation functions. In particular, we found strong qualitative differences in the dynamics of fermionic pairing correlations between the non-Markovian and Markovian cases. This work can be generalised to describe microscopic dynamics in a range of experimental settings, such as impurities immersed in BECs~\cite{PhysRevLett.94.040404,PhysRevA.84.031602,PhysRevA.95.033610,PhysRevA.98.062106,PhysRevA.101.033612} or atoms in multi-mode cavities~\cite{PhysRevX.8.011002,Kollar:2017vn,PhysRevA.87.043817}.
Other non-Markovian techniques could be adapted in order to probe the features investigated in our work, for example TEMPO~\cite{Strathearn:2018aa,PhysRevResearch.2.013265,PhysRevLett.126.200401} or HEOM~\cite{Ishizaki:2005aa,Kato:2016aa}. It may similarly be possible to combine these methods with MPS, as we have done here with HOPS. The combination is particularly amenable to our case as it involves the evolution of a single 1D matrix product state to capture the strongly interacting open system.
Alternatively, explicitly retaining the phonon basis states would result in an equivalent simulation with Markovian dissipation, allowing for the solution within a standard Born-Markov QSD~\cite{RevModPhys.89.015001,Daley:2014aa}. However, we find that the number of phonon basis states required (the local dimension of the MPS) is generally larger than that required for the present HOPS algorithm. In addition, HOPS has two main additional advantages. Firstly it can simulate phonon modes initially at finite temperatures and then track the induced dynamics in real time as demonstrated here, whereas explicitly retaining the basis states in this case would further increase the complexity. But secondly, improvements and extensions to the MPS representation can be immediately implemented~\cite{10.21468/SciPostPhys.10.3.058,Stolpp:2021vc,PhysRevLett.80.2661,PhysRevB.101.035134,arxiv_mps_hops}, allowing us to generalise this approach and approximate the dynamics induced by, up to reasonable timescales, environments that have algebraically decaying correlations~\cite{PhysRevLett.113.150403,Hartmann:2017aa} such as those that arise from power law spectral densities~\cite{Caldeira:1983wy,RevModPhys.59.1,RevModPhys.89.015001}.
\textit{Acknowledgements.} We thank Walter Strunz, Valentin Link, Richard Hartmann, Adrian Kantian, Sebastian Paeckel and Peter Kirton for helpful discussions. Work at the University of Strathclyde was supported by the EPSRC Programme Grant DesOEQ (EP/P009565/1), by AFOSR grant number FA9550-18-1-0064, and by the European Union’s Horizon 2020 research and innovation program under grant agreement No.~817482 PASQuanS. F.D.\ acknowledges the Belgian F.R.S.-FNRS for financial support. Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region.
All data underpinning this publication are openly available from the University of Strathclyde KnowledgeBase at https://doi.org/10.15129/92dd009b-00c5-4e42-ae13-38c6b21df9fd
\appendix
\section{Supplemental material}
This supplementary material gathers numerical details and additional information about the results presented in the main text. In Sec.~I we provide more details on HOPS and how it is used in practice. In Sec.~II, we present a numerical benchmarking of the hybridized HOPS + MPS algorithm. In Sec.~III, we show how we generate the stochastic processes appearing in the HOPS Eqs.~(3) and (6) of the main text. In Sec.~IV, we present a simple Markovian master equation for the Hubbard system only. Finally, in Sec.~V, we discuss how to extract the time-dependent dynamics of the environment from the algorithm.
\section{Hierarchy of Pure States}
The hierarchy of pure states (HOPS)~\cite{PhysRevLett.113.150403,Hartmann:2017aa} is a newly developed numerical method for evaluating the non-Markovian quantum state diffusion equation derived nearly 20 years prior~\cite{Strunz:1996uk,DIOSI1997569}. The insight is to extend the computational basis by introducing a hierarchy of auxiliary states, which act to facilitate the \textit{memory} effects characteristic of non-Markovian environments, thus deriving a numerically exact equation of motion without needing to invoke weak-coupling approximations or relying on large separations of timescales between the system and environment. The method is thus very general and can in principle be applied to a wide range of problems. While the initial version of this approach is only valid for environments that consist of non-interacting bosons, however, an extension to these methods for fermionic environments has also been derived~\cite{Suess:2015aa}.
Successes of the method have already been demonstrated for the spin-boson model with a two-level atom coupled to an environment that has a power law spectral density at zero~\cite{PhysRevLett.113.150403} and finite temperatures~\cite{Hartmann:2017aa}. The method is less computationally expensive when applying it to model couplings to environments that have Lorentzian spectral densities, as considered in this work, strongly suggesting that it can have further great success upon modelling atom-only dynamics in cavity QED systems~\cite{PhysRevResearch.3.L032016,PhysRevA.99.033845}.
\subsection{Example: One Site, One Environment}
For clarity, we explicitly write down the HOPS equations first for the case where we have a single site coupled to a single damped phonon environment (i.e., case $M = 1$ in the main text). This case is particularly simple as our total hierarchy state can be modelled with a single bosonic mode, $|\vec{\mathbf{k}}\rangle = |k\rangle$. Let us also assume that the system is a single site that can either have no particles or a single spinless fermion such that the system Hilbert space is of local dimension $d = 2$. We can write the state then in the form \begin{equation}
|{\Psi} (t) \rangle = \sum_k \begin{pmatrix}
A_k(t)\\B_k(t)
\end{pmatrix} \otimes |k \rangle \end{equation} where we have introduced two sets of complex numbers, $ A_k(t)$ and $B_k(t)$ to represent an arbitrary system state.
For concreteness we will also choose $k_{\rm max} = 2$, such that the hierarchy state has a Hilbert space of size $k_{\rm max} + 1 = 3$. We can then write the initial state, which for the HOPS algorithm is always initialised with only the $k=0$ auxiliary state, as \begin{equation} \label{in_state}
|{\Psi} (0) \rangle = \begin{pmatrix}
A_0(0)\\B_0(0) \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}. \end{equation} We can see that this is simply a two-level system coupled to a single boson mode.
As in the main text, we consider a total Hamiltonian of the form \begin{equation}
\hat{H} = \hat{H}_s + \omega \hat{a}^\dagger \hat{a} + g\left(\hat{L} \hat{a}^\dagger + \hat{L}^\dagger \hat{a} \right), \end{equation}
where $\hat{L}$ acts on the system and $\hat{a}$ destroys an excitation in the physical picture of the environment. According to Eq.~(6) of the main text, the time evolution of the state $|{\Psi} (t) \rangle$ is given by, \begin{equation} \begin{split}\label{teeq}
\partial_t|{\Psi} (t) =& \left(-i \hat{H}_s + \tilde{z}^*(t) g \hat{L} - \left( \kappa + i \omega \right) \hat{K} \right. \\
& \left. + g\hat{L} \otimes \hat{K} \hat{b}^{\dagger} - g\left( \hat{L}^{\dagger} - \langle \hat{L}^{\dagger} \rangle_t \right) \otimes \hat{b}, \right)|{\Psi} (t), \end{split} \end{equation}
where the colored noise is defined by $\tilde{z}^*(t) = z^*(t) + g\int_0^t ds \alpha^*(t - s) \langle \hat{L}^{\dagger} \rangle_s$ with $\langle \hat{L}^{\dagger} \rangle_s = \langle {\psi}^{(0)}(s) | \hat{L}^{\dagger} | {\psi}^{(0)}(s) \rangle$ and $\mathcal{E}[z(t) z^*(t')] = \alpha(t-t')$, where $\alpha(t-t') = \langle \hat{a}(t) \hat{a}^{\dagger}(t') \rangle = e^{- \kappa |t-t'| - i \omega (t-t')}$ is the environment correlation function. The operators $\hat{b}$, $\hat{b}^\dagger$ and $\hat{K}$ act on the hierarchy Hilbert space. In the basis $\{ |0\rangle, |1\rangle, |2\rangle\}$, they have the matrix representations \begin{equation}\label{ops} \begin{split} \hat{K} = & \begin{pmatrix} 0~~0~~0 \\ 0~~1~~0 \\ 0~~0~~2 \end{pmatrix}, \\ \\ \hat{b} = & \begin{pmatrix} 0~~1~~0 \\ 0~~0~~1 \\ 0~~0~~0 \end{pmatrix}, \\ \\ \hat{b}^{\dagger} = & \begin{pmatrix} 0~~0~~0 \\ 1~~0~~0 \\ 0~~1~~0 \end{pmatrix}. \end{split} \end{equation}
While initially [Eq.~(\ref{in_state})] the system and hierarchy auxiliary states are uncorrelated, we can see from Eq.~(\ref{teeq})] that the terms $ \hat{L} \otimes \hat{K} \hat{b}^{\dagger}$ and $\hat{L}^{\dagger} \otimes \hat{b}$ can (and generally do) generate correlations and entanglement between them. Note that, these coupling operators are dissipative, decreasing and increasing the relative norm between the different states in the hierarchy $ P_k(t) = \langle \psi^{({{k}})}(t) |\psi^{({{k}})}(t)\rangle = |A_k(t)|^2 + |B_k(t)|^2$, but can still generate entanglement.
After the application of each numerical time-step,
we rescale all states $|\psi^{({{k}})}(t) \rangle$ by the square root of the norm of the physical system state $|\psi^{({{0}})}(t) \rangle = \langle 0| {\Psi} (t) \rangle$ by setting $|\psi^{({{k}})}(t)\rangle \rightarrow \frac{1}{\sqrt{P_0(t)}} |\psi^{({{k}})}(t)\rangle$. In doing so, we enforce the normalization condition that $P_0(t) = 1$ during time evolution while keeping the relative norms $P_{k}(t)/P_{k'}(t)$ of each auxiliary state invariant. Note that we calculate the expectation values of the observables at each time-step using the normalized physical system state via
\begin{equation}
O(t) = \frac{1}{P_0(t)} \langle \psi^{({{0}})}(t)| \hat{O} |\psi^{({{0}})}(t)\rangle, \end{equation} and that we must average them over many different realisations (trajectories) of the random numbers $z^*(t)$.
\subsection{Example: Two Sites, Two Environments}
Additionally, we consider how to generalise HOPS to larger systems and more environment modes by considering the case of a two-site system and two independent damped phonon environments, where as in the main text each environment couples to a single different system site with an interaction Hamiltonian, \begin{equation} \hat{H}_{\rm Int} = g\sum_{n=1}^M \Big( \hat{L}_n\hat{a}^{\dagger}_n + \hat{L}_n^{\dagger}\hat{a}_n \Big). \end{equation}
We can represent our total system-hierarchy state as, \begin{equation} \begin{split}
|{\Psi} (t) \rangle = \sum_{k_1,k_2} \begin{pmatrix}
A_{k_1,k_2} (t)\\B_{k_1,k_2} (t)\\C_{k_1,k_2} (t)\\D_{k_1,k_2} (t)
\end{pmatrix} \otimes |k_1 \rangle \otimes |k_2 \rangle, \\ \\ \end{split} \end{equation} where we have introduced now four sets of complex numbers, $ A_{k_1,k_2} (t)$, $B_{k_1,k_2} (t)$,$C_{k_1,k_2} (t)$ and $D_{k_1,k_2} (t)$ to represent an arbitrary system state which now has a Hilbert space of $d^2 = 4$. For the case where $k_{\rm max} = 2$ (for both hierarchies, but note that these do not necessarily have to have the same dimension) we initialise the state as, \begin{equation}
|{\Psi} (t) \rangle = \sum_{k_1,k_2} \begin{pmatrix}
A_{k_1,k_2} (t)\\B_{k_1,k_2} (t)\\C_{k_1,k_2} (t)\\D_{k_1,k_2} (t) \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}. \end{equation}
This time, our evolution equation for $|{\Psi} (t) \rangle$ contains two types of hierarchy operators acting on different sectors of the Hilbert space: $\hat{K}_1$, $\hat{b}_1$ and $\hat{b}^{\dagger}_1$, acting on the hierarchy dimension which models the first environment and $\hat{K}_2$, $\hat{b}_2$ and $\hat{b}^{\dagger}_2$ which act on the second hierarchy dimension responsible for modelling the effects of the second environment. Both sets of matrix representations are defined as in Eq.~(\ref{ops}).
\subsection{HOPS $+$ MPS for $M$ sites and $M$ Environments}
It can be seen that upon further increasing the size of the system and the number of environment modes that the Hilbert space grows exponentially thus motivating the incorporation of MPS techniques. Representing a 1D many-body system efficiently with these techniques is now quite standard, but it remains to be investigated on the best way to incorporate the hierarchy basis states. As a first demonstration here we simply incorporate each hierarchy dimension into a separate but different local dimension of the MPS. Due to the physical system that we are considering with an interaction Hamiltonian that couples an environment locally to a particular lattice site, this is a convenient representation.
Explicitly, the local dimension of the $m$th site is extended to include the basis states $| k_m \rangle $, \begin{widetext} \begin{equation} \begin{split}
| \Psi(t) \rangle & = \sum_{d_1,d_2,\cdots, d_M} \sum_{k_1,k_2,\cdots,k_M} C_{(d_1,d_2,\cdots, d_M),(k_1,k_2,\cdots,k_M)} |d_1,d_2,\cdots, d_M \rangle |k_1,k_2,\cdots,k_M \rangle \\
& = \sum_{d_1,k_1,d_2,k_2,\cdots, d_M,k_M} C_{d_1,k_1,d_2,k_2,\cdots, d_M,k_M} | d_1,k_1 \rangle | d_2,k_2 \rangle,\cdots,| d_M,k_M \rangle. \end{split} \end{equation} The coefficients $C_{d_1,k_1,d_2,k_2,\cdots, d_M,k_M}$ can then decomposed into a product of matrices~\cite{SCHOLLWOCK201196}, \begin{equation}
| \Psi(t) \rangle = \sum_{d_1,k_1,d_2,k_2,\cdots, d_M,k_M}A_{(d_1\times k_1)}^{1,D_1} A_{(d_2\times k_2)}^{D_1,D_2} \cdots A_{(d_M \times k_M)}^{D_{M-1},1} | d_1,k_1 \rangle | d_2,k_2 \rangle,\cdots,| d_M,k_M \rangle. \end{equation} \end{widetext} This is represented graphically in Fig.~1(b) in the main text. This representation, while most likely not being the most optimal, is particularly convenient as we can apply standard MPS techniques for time-evolution~\cite{Paeckel:2019aa} without modification, in our case we used the time-dependent variational principle (TDVP)~\cite{PhysRevLett.107.070601}.
\section{Numerical Benchmarking}
In this section we quantitatively benchmark the errors in the algorithm introduced by limiting the numerical precision. In all cases presented here, we calculate the errors induced in the Holstein model on the off-diagonal correlation functions,
\begin{equation}
\varepsilon = \frac{1}{M^2} \sum_{n,m} \left| \langle c^{\dagger}_n c_m \rangle - \langle c^{\dagger}_n c_m \rangle_{\rm exact} \right|, \end{equation} where by \textit{exact}, we mean the results predicted upon taking the limit that the bond dimension $D$ and the hierarchy depth $k_{\rm max}$ go to infinity.
The main numerical parameter for the HOPS algorithm is the depth of the hierarchy $k_{\rm max}$. Note that we use the same $k_{\rm max}$ for each hierarchy index in our many-body algorithm. We demonstrate the time-dependence of the errors in Fig.~\ref{Fig_NBM_Kmax}, where we can see that it is in fact possible to reach numerical precision while keeping the size of $k_{\rm max}$ relatively small, i.e., $k_\mathrm{max} <10$. We can see that for $g\sim J$ and smaller $\kappa$, i.e., slower decay of environment correlations and larger memory effects, that the errors grow rapidly, signifying the build up of non-Markovian effects. Intriguingly, for stronger coupling $g\sim5J $ we find the errors are strongly saturated indicating that perhaps strong coupling may actually be limiting the memory effects of the environment.
In our case we also have the additional consideration of the maximum allowed bond dimension of the MPS representation $D$. In Fig.~\ref{Fig_NBM_D} we plot the time-dependence of the errors (in the same observable) upon restricting this quantity, where again, we see that in principle we are able to obtain numerical precision for the short time dynamics after a global quench. For coupling strengths around $g\sim J $ we find similar behaviour as for closed system dynamics, where the errors grow approximately exponentially in time. Of course, this means that we are limited to short time dynamics after a global quench but this is also the case for conventional MPS simulations. In contrast, stronger coupling $g\sim5J $ we find a saturation of the errors, simply indicating that the strong coupling to the environment is suppressing the build up of correlations and entanglement in the dynamics.
\begin{figure}
\caption{Errors in the off-diagonal correlation functions in the Holstein model with $M=20$ lattice sites, upon limiting the maximum hierarchy depth of HOPS $k_{\rm max}$. Results taken with respect to $D=200$ and $k_{\rm max} = 8$. We have also incorporated conserved quantum numbers into the MPS algorithm~\cite{itensor}. These errors are averaged over 20 trajectories.}
\label{Fig_NBM_Kmax}
\end{figure}
\begin{figure}
\caption{Errors in the off-diagonal correlation functions in the Holstein model with $M=20$ lattice sites, upon limiting the MPS bond dimension $D$. Results taken with respect to $D=500$ and $k_{\rm max} = 8$. We have also incorporated conserved quantum numbers into the MPS algorithm~\cite{itensor}. These errors are averaged over 20 trajectories.}
\label{Fig_NBM_D}
\end{figure}
\section{Numerical generation of colored noise}
In this section we describe one way of numerically generating the random noise terms that reproduces the desired statistics of the complex gaussian colored noise source $z^*(t) $ that we use in the HOPS algorithm. The derivation follows what is presented in Ref.~\cite{Gaspard:1999wx,PhysRevA.71.023812}
First we must define a response function $R(t)$, \begin{equation} R(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} d\omega G(\omega) e^{-i \omega t}, \end{equation} where $G(\omega)$ is related to the correlation function through, \begin{equation} G(\omega) = \left( \int_{-\infty}^{\infty}dt \alpha(t) e^{i \omega t}\right)^{1/2}. \end{equation} Finally, we then calculate the colored noise $z(t) $ through, \begin{equation} z(t) = \int_{-\infty}^{\infty} ds R(s) \xi(t-s), \end{equation} where $\xi(\tau) = 1/\sqrt{2} \left( \xi'(\tau) + i \xi''(\tau) \right)$ and $\xi'(\tau)$ and $\xi''(\tau)$ are real independent gaussian white noise terms. The colored noise then has the desired properties, \begin{equation} \begin{split} &\mathcal{E}\left[ z(t) z^*(t) \right] = \alpha(t - t'),\\ &\mathcal{E}\left[ z(t) z(t) \right] = 0. \end{split} \end{equation}
\section{Lindblad Master equation description}
In this section,
we derive a simple Markovian master equation in the Lindblad form for the fermionic system only for an interaction Hamiltonian of the form considered in this letter, \begin{equation} \hat{H}_{\rm Int} = \sum_{n=1}^M \Big( \hat{L}_n\hat{a}^{\dagger}_n + \hat{L}_n^{\dagger}\hat{a}_n \Big). \end{equation}
We first invoke the Born approximation and write the total density matrix of the whole system made up of the fermions and the phonons as $\rho_\mathrm{tot}(t) \approx \rho(t) \otimes \rho_E$, where $\rho(t)$ and $\rho_E$ are respectively the fermionic system and phonon environment density matrices~\cite{Breur_book}. At second order in the interaction Hamiltonian, the master equation for the system density matrix $\rho(t)$ reads in interaction picture~\cite{Breur_book,MilWise_Book} \begin{equation} \label{Lindblad_SM}
\partial_t \rho(t) = - \int^{t}_0dt' {\rm Tr_E} \Big( [ \hat{H}_{\rm Int}(t),[ \hat{H}_{\rm Int}(t'),\rho(t)\otimes \rho_E ] ] \Big), \end{equation} where ${\rm Tr_E}(\cdot)$ denotes the trace over the environment degrees of freedom and where, \begin{equation} \begin{split} \hat{H}_{\rm Int}(t) & = e^{i(\hat{H}_s+\hat{H}_E)t}\hat{H}_{\rm Int}e^{-i(\hat{H}_s+\hat{H}_E)t} \\ & = \sum_{n=1}^M \hat{L}_n(t) \hat{a}^{\dagger}_n(t) + \hat{L}_n^{\dagger}(t) \hat{a}_n(t), \end{split} \end{equation} with $\hat{L}_n(t) = e^{i \hat{H}_s t} \hat{L}_n e^{-i \hat{H}_s t} $ and $\hat{a}_n(t) = e^{i \hat{H}_E t} \hat{a}_n e^{-i \hat{H}_E t}$. Note that in Eq.~(\ref{Lindblad_SM}), we performed the Markov approximation, which consists in replacing $\rho(t')$ by $\rho(t)$ under the integral.
Expanding the interaction Hamiltonians explicitly in Eq.~(\ref{Lindblad_SM}) leads to \begin{equation}
\begin{aligned}
\partial_t \rho(t)
= -g^2 \int_0^t dt' & \Big\{\langle \hat{a}_n(t) \hat{a}^\dagger_n(t')\rangle \hat{L}^\dagger_n(t) \hat{L}_n(t') \rho(t) \Big. \\
& + \langle \hat{a}_n^\dagger(t) \hat{a}_n(t')\rangle \hat{L}_n(t) \hat{L}_n^\dagger(t') \rho(t) \\
& - \langle \hat{a}_n^\dagger(t') \hat{a}_n(t)\rangle \hat{L}_n^\dagger(t) \rho(t) \hat{L}_n(t') \\
& - \langle \hat{a}_n(t') \hat{a}_n^\dagger(t)\rangle \hat{L}_n(t) \rho(t) \hat{L}_n^\dagger(t') \\
& - \langle \hat{a}_n^\dagger(t) \hat{a}_n(t')\rangle \hat{L}_n^\dagger(t') \rho(t) \hat{L}_n(t) \\
& - \langle \hat{a}_n(t) \hat{a}_n^\dagger(t')\rangle \hat{L}_n(t') \rho(t) \hat{L}_n^\dagger(t) \\
&+ \langle \hat{a}_n(t') \hat{a}^\dagger_n(t)\rangle \hat{L}^\dagger_n(t') \hat{L}_n(t) \rho(t) \\
& \Big. + \langle \hat{a}_n^\dagger(t') \hat{a}_n(t)\rangle \hat{L}_n(t') \hat{L}_n^\dagger(t) \rho(t) \Big\}. \\
\end{aligned} \end{equation} As we have a many-body system Hamiltonian $\hat{H}_s$, evaluating the term $\hat{L}_n(t) = e^{i \hat{H}_s t} \hat{L}_n e^{-i \hat{H}_s t} $ is not easy, as it requires to know its full energy spectrum. This is a general issue with applying open quantum systems to many-body Hamiltonians (but a major advantage of the HOPS algorithm considered in this work, as it does not require to calculate explicitly the system spectrum). Because of this, it is not possible to proceed further without having to introduce additional strong approximations on the behavior of $\hat{L}_n(t)$, if one wants to consider the environment correlation functions given in Eq.~(2) of the main text. For this reason, we assume here the limit of structureless phonon environments at thermal equilibrium, for which the correlation functions are given by \begin{equation} \begin{split} \langle \hat{a}_n(t) \hat{a}_n^{\dagger}(t') \rangle &= (\bar{n} + 1) \delta(t-t') \\ \langle \hat{a}_n^{\dagger}(t) \hat{a}_n(t') \rangle &= \bar{n} \delta(t-t'), \end{split} \end{equation} where $\bar{n} = \left(\exp[\omega/(k_B T)] - 1 \right)^{-1}$ is the Bose factor with $k_B$ the Boltzmann constant and $T$ the temperature. This gives directly rise to the Lindblad master equation, \begin{equation} \begin{split}
\partial_t \rho(t) &= -i[\hat{H}_s,\rho(t) ] \\
&+ g^2 (\bar{n} + 1 ) \sum_n \Big( \hat{L}_n \rho(t) \hat{L}^{\dagger}_n - \frac{1}{2}\left\{\hat{L}_n^{\dagger} \hat{L}_n , \rho(t)\right\} \Big) \\
&+ g^2 \bar{n} \sum_n \Big( \hat{L}_n^\dagger \rho(t) \hat{L}_n - \frac{1}{2}\left\{\hat{L}_n \hat{L}_n^\dagger , \rho(t)\right\} \Big),
\end{split} \end{equation} where $\{ \hat{A} , \hat{B} \} = \hat{A}\hat{B} + \hat{B}\hat{A}$ denotes the anti-commutator. It is a quantum trajectory approach equivalent to this expression that we have used in the main text to compare to the results of HOPS.
\section{Environment correlations}
Through features arising from the HOPS algorithm, it is also possible to calculate time-dependent observables within the environment~\cite{RevModPhys.89.015001}, thus enabling us to quantify the deviation away from the initial phonon population as well as capturing the build up of correlations between the system and the environment. This can be seen using the starting point of the derivation of NMQSD, namely the expansion of the total state $|\Psi_T \rangle$ of the system and the environment in a basis of Bargmann coherent states of the environment~\cite{Suess:2015aa}. For the model considered in this paper of exponentially decaying correlation functions, one would normally model the environment of each site as a bath of harmonic oscillators with a Lorentzian spectral density and define the Bargmann coherent states according to this bath. Since here we work within the pseudo-mode picture of such a bath, we consider an expansion directly into a basis of pseudo-mode states $|z\rangle = |z_1, \cdots, z_M\rangle$ which reads (in interaction picture with respect to the bath) \begin{equation}\label{expansion}
|\Psi_T \rangle = \int \frac{d^2z}{\pi} e^{-|z|^2} |\psi_z(t) \rangle \otimes |z\rangle, \end{equation}
with the shorthand $d^2z = d^2z_1 \cdots d^2z_M$ and $|z|^2 = \sum_n |z_n|^2$ and where $|\psi_z(t) \rangle = \langle z | \Psi_T \rangle $ is the system state whose the dynamics is governed by Eq.~(3) of the main text, where the subscript $z$ added here makes it clear that it is relative to a given environment state. The pseudo-mode states $|z_n\rangle$ are defined through $\hat{a}_n(t) |z_n\rangle = z_n(t) |z_n\rangle$, which define stochastic processes $z_n(t)$ whose ensemble average \begin{equation}
\mathcal{E}\left[ \dots \right] = \int \frac{d^2z}{\pi} e^{-|z|^2} \left[ \dots \right] \end{equation} gives \begin{equation}
\mathcal{E}[ z_n(t) z_{n'}^*(t') ] = \langle \hat{a}_n^\dagger(t) \hat{a}_n'(t') \rangle = \delta_{n,n'} \alpha_n(t-t'). \end{equation}
Hence, any observable such as $\hat{a}^{\dagger}_n \hat{a}_n \hat{S}$ where $\hat{S}$ is an arbitrary system operator can be calculated via, \begin{equation} \begin{split}
\langle \hat{a}^{\dagger}_n &\hat{a}_n \hat{S} \rangle = {\rm Tr}\left[ \hat{a}_n^{\dagger}(t)\hat{a}_n(t) \hat{S} |\Psi_T \rangle \langle \Psi_T | \right] \\
& = {\rm Tr}\left[ \hat{a}_n^{\dagger}(t) \hat{S} |\Psi_T \rangle \langle \Psi_T | \hat{a}_n(t) \right] - \langle \hat{S} \rangle \\
& = \int \frac{d^2z}{\pi} e^{-|z|^2} \Big( \langle \psi_z(t) | \otimes \langle z | \hat{a}_n^{\dagger}(t) \hat{S} |\Psi_T \rangle \langle \Psi_T | \hat{a}_n(t) \\
&~~~~~~~~~~~~~~| \psi_z(t) \rangle \otimes | z \rangle \Big) - \langle \hat{S} \rangle\\
&= \int \frac{d^2z}{\pi} e^{-|z|^2} \langle \psi_z(t) |\hat{S} |\psi_z(t) \rangle |z_n(t)|^2 - \langle \hat{S} \rangle \\
&= \mathcal{E} \left[ \langle \psi_z(t) |\hat{S} |\psi_z(t) \rangle |z_n(t)|^2 - \langle \hat{S} \rangle \right] \end{split} \end{equation}
For $\hat{S}$ equal to the identity, this shows that the population of the pseudo-modes are related to the squared absolute values of the noise terms. One can thus track their time-dependent dynamics via monitoring of the noises used to generate the dynamical evolution of the system state. Any higher-order environment correlation functions can be obtained in a similar fashion.
Note finally that since the non-linear version of HOPS has better convergence properties~\cite{Suess:2015aa, Hartmann:2017aa}, we used the expressions above with the system state of the non-linear HOPS Eq.~(6) of the main text with the transformed noises $\tilde{z}^*_n(t) = z^*_n(t) + \int_0^t ds \alpha^*_n(t - s) \langle \hat{L}_n^{\dagger} \rangle_s$ with $\langle \hat{L}^{\dagger}_n \rangle_s = \langle {\psi}^{(0)}(s) | \hat{L}^{\dagger}_n | {\psi}^{(0)}(s) \rangle$ instead of $z_n(t)$.
\end{document} | arXiv |
Webinar in PDEA
Webinar in NT
French-Korean IRL in Mathematics
International Research Laboratory
Webinar in Number Theory
YoungJu Choie (POSTECH), Bo-Hae Im (KAIST),
Laurent Berger (ENS Lyon)
Every 1st and 3rd Monday of every month: a 50 mn talk
Winter French time: 9:00-10:00, Korean time: 17:00-18:00
Zoom link: ask the organizers
The webinar will resume on Monday October 17 2022
Forthcoming speakers
2022/12/12: Professor Jungyun Lee (Kangwon University, Korea)
Title: The mean value of the class numbers of cubic function fields
Abstract: We compute the mean value of $|L(s,\chi)|^2$ evaluated at $s=1$ when chi goes through the primitive cubic Dirichlet characters of $A:=F_q[T]$, where $F_q$ is a finite field with $q$ elements and $q \equiv 1 \ \text{mod 3}$. Furthermore, we find the mean value of the class numbers for the cubic function fields $K_m=k(\sqrt[3]{m})$, where $k:= F_q(T)$ is the rational function field and $m$ in $A$ is a cube-free polynomial. (This is a joint work with Yoonjin Lee and Jinjoo Yoo.)
2022/12/05: Anthony Poëls (Université Claude Bernard Lyon 1, France )
PDF Title: Rational approximation to real points on quadratic hypersurfaces
Abstract: This is a joint work with Damien Roy. Let $\mathbb{Z}$ be a quadratic hypersurface of $\mathbb{R}^n$ defined over $\mathbb{Q}$ (such as the unit sphere). We compute the largest exponent of uniform rational approximation of the points belonging to $\mathbb{Z}$ whose coordinates together with 1 are linearly independent over $\mathbb{Q}$. We show that it depends only on $n$ and on the Witt index (over $\mathbb{Q}$) of the quadratic form defining $\mathbb{Z}$. This completes a recent work of Kleinbock and Moshchevitin.
2022/11/21: Professor Jaehyun Cho (UNIST, Korea)
Title: The average residue of the Dedekind zeta function
Abstract: We find the explicit formula for the average residue of the Dedekind zeta functions over all non-Galois cubic fields. The main tool is a recent result of Bhargava, Taniguchi, and Thorne's on improving the error term in counting cubic fields.
2022/11/07: François Ballaÿ (Université de Caen Normandie )
PDF Title: Positivity in Arakelov geometry and arithmetic Okounkov bodies
Abstract: Arakelov theory is a powerful approach to Diophantine geometry that develops arithmetic analogues of tools from algebraic geometry to tackle problems in number theory. It permits to study the arithmetico-geometric properties of a projective variety over a number field by looking at its adelic line bundles, which are usual line bundles equipped with a suitable collection of metrics. Since the seminal work of Zhang on arithmetic ampleness, several notions of positivity for adelic line bundles have been introduced and studied in analogy with the algebro-geometric setting (nefness, bigness, pseudo-effectiveness...). In this talk, I will present these notions and emphasize their connection with the study of height functions in Diophantine geometry. I will then describe how these positivity properties can be studied through convex analysis, thanks to the theory of arithmetic Okounkov bodies introduced by Boucksom and Chen
2022/10/17: Professor Joachim Koenig (Korea National University of Education)
PDF Title: On the arithmetic-geometric complexity of the Grunwald problem
Abstract: The Grunwald problem for a group $G$ over a number field $k$ asks whether, given Galois extensions of $k_p$ of Galois group embedding into $G$ at finitely many completions $k_p$ of $k$ (possibly away from some finite set of primes depending only on $G$ and $k$), there always exists a $G$-extension of $k$ approximating all these local extensions. This question grew naturally out of the Grunwald-Wang theorem, which deals with the case of abelian groups. Following more general concepts of arithmetic-geometric complexity in inverse Galois theory, we develop a notion of complexity of Grunwald problems by looking for Galois covers of varieties which encapsulate solutions to arbitrary Grunwald problems (for a given group). In particular, we determine the groups $G$ for which solutions to arbitrary Grunwald problems may be obtained via specialization of a $G$-cover of {\it curves}. Joint with D. Neftin.
2022/06/27: Baptiste Peaucelle (University of Clermont-Ferrand)
Title: Exceptional images of modular Galois representations
Abstract: Given a modular form $f$ and a prime ideal $\lambda$ in the coefficient field of $f$, one can attach a residual Galois representation of dimension 2 with values in the residue field of $\lambda$. A theorem of Ribet states that this representation has small image for a finite number of prime ideals $\lambda$. In this talk, I will explain how one can bound explicitly these exceptional ideals, and how to compute them for some types of small image.
2022/06/13: Prof. Yeongseong Jo (The University of Maine)
PDF Title: Rankin-Selberg integrals in positive characteristic and its connection to Langlands functoriality
Abstract: The prominent Langlands functoriality conjecture predicts deep relationships among representations on different groups. One of the well-understood cases is a local functorial transfer of irreducible generic supercuspidal representations of ${\rm SO}_{2r+1}(F)$ to irreducible supercuspidal ones of ${\rm GL}_{2r}(F)$ over $p$-adic fields $F$. This functorial lift is defined by Lomel\'{\i} over non-archimedean local fields $F$ of positive characteristic, but it is rarely studied. Following the spirit of Cogdell and Piatetski-Shapiro, the purpose of this talk is to take one more step further to investigate the transfer thoroughly. We first consider the image of the map. Somewhat surprisingly, this is related to poles of local exterior square $L$-functions via integral representations due to Jacquet and Shalika. We then discuss whether the map is injective. It turns out that the problem is relevant to what is known as the local converse theorem for ${\rm SO}_{2r+1}(F)$.
2022/05/23: Thomas Lanard (University of Vienna)
PDF Title: Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$
Abstract: In this talk, I will talk about the category of depth zero representations of a $p$-adic group with coefficients in $\overline{\mathbb{Z}}[\frac{1}{p}]$. When the group $\mathbf{G}$ is quasi-split and tamely ramified, the depth zero category over $\overline{\mathbb{Z}}[\frac{1}{p}]$ is indecomposable. In general, for a quasi-split group, we will see that the blocks (indecomposable summands) of this category are in natural bijection with the connected components of the space of tamely ramified Langlands parameters. In the last part, I will explain some potential applications to the Fargues-Scholze and Genestier-Lafforgue semisimple local Langlands correspondences. This is joint work with Jean-François Dat.
2022/05/02: Prof. Seungki Kim (University of Cincinnati)
PDF Title: Adelic Rogers integral formula
Abstract: The Rogers integral formula, a natural generalization of the well-known Siegel integral formula, first appeared in the 1950's as an essential tool in the geometry of numbers. Very recently, there has been a surprising resurgence of interest in the formula, thanks in much part to its usefulness in homogeneous dynamics, and a number of variants and extensions have been proposed. I will introduce the audience to the relevant literature, in particular the recently proved formula over an adele of a number field.
2022/04/04: Valentin Hernandez (Université Paris-Sud, Orsay)
Title: The Infinite Fern in higher dimensions
Abstract: In general, deformations spaces of residual Galois representation are quite mysterious objects. It is natural to ask if there is at least enough modular points in their generic fiber X. A related question is the density of the p-adic modular forms, which form a fractal-like object called the Infinite Fern. In dimension 2, in most cases Gouvea and Mazur proved that this infinite fern is Zariski dense in X. In higher dimension we look at \emph{polarized} Galois representation, and the analogous question becomes much more complicated. Chenevier explained a strategy by looking for \emph{good} (called generic) points in Eigenvarieties, studied the analogous local (p-adic) question and solved the case of dimension 3. Recently Breuil-Hellmann-Schraen studied the local Infinite Fern at well behaved crystalline points, and Hellmann-Margerin-Schraen, under strong Taylor-Wiles hypothesis, managed to prove the density of the (global) Infinite Fern (in a union of connected components) in all dimensions using the \emph{patched} Eigenvariety. In this talk I would like to explain how to only use the local geometric input to deduce the analogous density result without using the Taylor-Wiles hypothesis, but using another kind of \emph{good} points as in Chenevier's strategy. This is a joint work with Benjamin Schraen.
2022/03/21: Junho Peter Whang (Seoul National University)
PDF Title: Decidable problems on integral SL2-characters
Abstract: Classical topics in the arithmetic study of quadratic forms include their reduction theory and representation problem. In this talk, we discuss their nonlinear analogues for SL2-characters of surface groups. First, we prove that the set of integral SL2-characters of a surface group with prescribed invariants can be effectively determined and finitely generated, under mapping class group action and related dynamics. Second, we prove that the set of values of an integral SL2-character of a finitely generated group is a decidable subset of the integers.
2022/03/07: Silvain Rideau-Kikuchi ( Institut de Mathématiques de Jussieu-Paris Rive Gauche)
Title: H-minimality (with R. Cluckers, I. Halupczok)
Abstract: The development and numerous applications of strong minimality and later o-minimality has given serious credit to the general model theoretic idea that imposing strong restrictions on the complexity of arity one sets in a structure can lead to a rich tame geometry in all dimensions. O-minimality (in an ordered field), for example, requires that subsets of the affine line are finite unions of points and intervals.
In this talk, I will present a new minimality notion (h-minimality), geared towards henselian valued fields of characteristic zero, generalising previously considered notions of minimality for valued fields (C,V,P ...) that does not, contrary to previously defined notions, restrict the possible residue fields and value groups. By analogy with o-minimality, this notion requires that definable sets of of the affine line are controlled by a finite number of points. Contrary to o-minimality though, one has to take special care of how this finite set is defined, leading us to a whole family of notions of h-minimality. I will then describe consequences of h-minimality, among which the jacobian property that plays a central role in the development of motivic integration, but also various higher degree and arity analogs.
2022/02/21: Dr Jun-Yong Park (MPIM, Bonn)
PDF Title: Arithmetic Moduli of Elliptic Surfaces
Abstract: By considering the arithmetic geometry of rational orbi-curves on modular curve $\overline{\mathcal{M}}_{1,1}$ where $\overline{\mathcal{M}}_{1,1}$ is the Deligne--Mumford stack of stable elliptic curves, we formulate the moduli stack of minimal elliptic fibrations over $\mathbb{P}^{1}$, also known as minimal elliptic surfaces with section over any base field $K$ with $\mathrm{char}(K) \neq 2,3$. Inspired by the classical work of [Tate] which allows us to determine the Kodaira--N\'eron type of fibers over global fields, we establish Tate's correspondence between the moduli stacks $\mathrm{Rat}_{n}^{\gamma}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of quasimaps with vanishing constraints $\gamma$ and $\mathrm{Hom}^{\Gamma}_n(\mathcal{C}, \overline{\mathcal{M}}_{1,1})$ of twisted maps with cyclotomic twistings $\Gamma$. Afterward, we acquire the exact arithmetic invariants of the moduli for each Kodaira--N\'eron types which naturally renders new sharp enumerations with a main leading term of order $\mathcal{B}^{\frac{5}{6}}$ and secondary & tertiary order terms $\mathcal{B}^{\frac{1}{2}} ~\&~ \mathcal{B}^{\frac{1}{3}}$ on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting elliptic curves over $\mathbb{P}_{\mathbb{F}_q}^{1}$ with additive reductions ordered by bounded height of discriminant $\Delta$. The emergence of non-constant lower order terms are in stark contrast with counting the semistable (i.e., strictly multiplicative reductions) elliptic curves. In the end, we formulate an analogous heuristic on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for counting elliptic curves over $\mathbb{Q}$ through the global fields analogy. This is a joint work with Dori Bejleri (Harvard) and Matthew Satriano (Waterloo).
2022/02/07: Gautier Ponsinet (Post doctoral at the Università degli Studi di Genova)
PDF Title: Universal norms of p-adic Galois representations
Abstract: In 1996, Coates and Greenberg observed that perfectoid fields appear naturally in Iwasawa theory. In particular, they have computed the module of universal norms associated with an abelian variety in a perfectoid field extension. A precise description of this module is essential in Iwasawa theory, notably to study Selmer groups over infinite algebraic field extensions. In this talk, I will explain how to use properties of the Fargues-Fontaine curve to generalise their results to p-adic representations.
2021/12/20: Professor Kwangho Choiy ( Southern Illinois University)
PDF Title: Invariants in restriction of admissible representations of $p$-adic groups
Abstract: The local Langlands correspondence, LLC, of a $p$-adic group over complex vector spaces has been proved for several cases over decades. One of interesting approaches to them is the restriction method which was initiated for $SL(2)$ and its inner form. It proposes in line with the functoriality principle that the LLC of one group can be achieved from the LLC of the other group sharing the same derived group. In this context, we shall explain how the method is extended to some other cases of LLC's, the multiplicity formula in restriction, and the transfer of the reducibility of parabolic induction.
2021/11/22: Lucile Devin (Université du Littoral)
PDF Title: Chebyshev's bias and sums of two squares
Abstract: Studying the secondary terms of the Prime Number Theorem in Arithmetic Progressions, Chebyshev claimed that there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4. We will explain and qualify this claim following the framework of Rubinstein and Sarnak. Then we will see how this framework can be adapted to other questions on the distribution of prime numbers. This will be illustrated by a new Chebyshev-like claim : "for more than half of the prime numbers that can be written as a sum of two squares, the odd square is the square sof a positive integer congruent to 1 mod 4.
2021/11/08: Wansu Kim (KAIST)
PDF Title: Equivariant BSD conjecture over global function fields
Abstract: Under a certain finiteness assumption of Tate-Shafarevich groups, Kato and Trihan showed the BSD conjecture for abelian varieties over global function fields of positive characteristic. We explain how to generalise this to semi-stable abelian varieties ``twisted by Artin character'' over global function field (under some additional technical assumptions), and discuss further speculations for generalisations if time permits. This is a joint work with David Burns and Mahesh Kakde.
2021/10/18: Richard Griffon (University Clermont-Auvergne)
PDF Title: Isogenies of elliptic curves over function fields
Abstract: I will report on a recent work, joint with Fabien Pazuki, in which we study elliptic curves over function fields and the isogenies between them. More specifically, we prove analogues in the function field setting of two famous theorems about isogenous elliptic curves over number fields. The first of these describes the variation of the Weil height of the j-invariant of elliptic curves within an isogeny class. Our second main result is an ''isogeny estimate'' in the spirit of theorems by Masser-Wüstholz and by Gaudron-Rémond. After stating our results and giving quick sketches of their proof, I will, time permitting, mention a few Diophantine applications.
2021/10/04: Dr. Seoyoung Kim (Queen's University)
PDF Title: On the generalized Diophantine m-tuples
Abstract: For non-zero integers $n$ and $k\geq2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $\{a_1,a_2,\ldots, a_m\}$ such that $a_ia_j + n$ is a $k$-th power for any distinct i and j. Define by $M_k(n)$ the supremum of the size of the set which has property $D_k(n)$. In this paper, we study upper bounds on $M_k(n)$, as we vary $n$ over positive integers. In particular, we show that for $k\geq 3$, $M_k(n)$ is $O(\log n)$ and further assuming the Paley graph conjecture, $M_k(n)$ is $O((\log n)^{\epsilon})$. The problem for $k=2$ was studied by a long list of authors that goes back to Diophantus who studied the quadruple $\{1,33,68,105\}$ with property $D(256)$. This is a joint work with A. Dixit and M. R. Murty.
2021/06/21: Vlerë Mehmeti (Université Paris-Saclay, France)
PDF Title: Non-Archimedean analytic curves and the local-global principle
Abstract: In 2009, a new technique, called algebraic patching, was introduced in the study of local-global principles. Under different forms, patching had in the past been used for the study of the inverse Galois problem. In this talk, I will present an extension of this technique to non-Archimedean analytic curves. As an application, we will see various local-global principles for function fields of curves, ranging from geometric to more classical forms. These results generalize those of the previous literature and are applicable to quadratic forms. We will start by a brief introduction of the framework of non-Archimedean analytic curves and will conclude by a presentation of a first step towards such results in higher dimensions.
2021/06/07: Hae-Sang Sun (UNIST, Korea)
PDF Title: Cyclotomic Hecke L-values of a totally real field
Abstract: It is known that any Fourier coefficient of a newform of weight 2 can be expressed as a polynomial with rational coefficients, of a single algebraic critical value of the corresponding L-function twisted by a Dirichlet character of $p$-power conductor for a rational prime $p$. In the talk, I will discuss a version of this result in terms of Hecke L-function over a totally real field, twisted by Hecke characters of $p$-power conductors. The discussion involves new technical challenges that arise from the presence of the unit group, which are (1) counting lattice points in a cone that $p$-adically close to units and (2) estimating an exponential sum over the unit group. This is joint work with Byungheup Jun and Jungyun Lee.
2021/05/17: Riccardo Pengo (ENS Lyon, France)
PDF Title: Entanglement in the family of division fields of a CM elliptic curve
Abstract: Division fields associated to an algebraic group defined over a number field, which are the extensions generated by its torsion points, have been the subject of a great amount of research, at least since the times of Kronecker and Weber. For elliptic curves without complex multiplication, Serre's open image theorem shows that the division fields associated to torsion points whose order is a prime power are "as big as possible" and pairwise linearly disjoint, if one removes a finite set of primes. Explicit versions of this result have recently been featured in the work of Campagna-Stevenhagen and Lombardo-Tronto. In this talk, based on joint work with Francesco Campagna (arXiv:2006.00883), I will present an analogue of these results for elliptic curves with complex multiplication. Moreover, I will present a necessary condition to have entanglement in the family of division fields, which is always satisfied for elliptic curves defined over the rationals. In this last case, I will describe in detail the entanglement in the family of division fields.
2021/05/03: Chan-Ho Kim (KIAS, Korea)
PDF Title: On the Fitting ideals of Selmer groups of modular forms
Abstract: In 1980's, Mazur and Tate studied ``Iwasawa theory for elliptic curves over finite abelian extensions" and formulated various related conjectures. One of their conjectures says that the analytically defined Mazur-Tate element lies in the Fitting ideal of the dual Selmer group of an elliptic curve. We discuss some cases of the conjecture for modular forms of higher weight. | CommonCrawl |
Entropy Measures of Distance Matrix
Bünyamin Şahin, Abdulgani Şahi̇n
Subject: Keywords: Distance; Wiener Index; Distance Matrix; Entropy Measure
Online: 8 November 2021 (13:33:09 CET)
Bonchev and Trinajstic defined two distance based entropy measures to measure the molecular branching of molecular graphs in 1977 [Information theory, distance matrix, and molecular branching, J. Chem. Phys., 38 (1977), 4517–4533]. In this paper we use these entropy measures which are based on distance matrices of graphs. The first one is based on distribution of distances in distance matrix and the second one is based on distribution of distances in upper triangular submatrix. We obtain the two entropy measures of paths, stars, complete graphs, cycles and complete bipartite graphs. Finally we obtain the minimal trees with respect to these entropy measures with fixed diameter.
Bounds for the Minimum Distance and Covering Radius of Orthogonal Arrays via Their Distance Distributions
Silvia Boumova, Peter Boyvalenkov, Maya Stoyanova
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Orthogonal Arrays; Distance distributions; Minimum distance; Covering radius
We propose two methods for obtaining estimations on the minimum distance and covering radius of orthogonal arrays. Both methods are based on knowledge about the (feasible) sets of distance distributions of orthogonal arrays with given length, cardinality, factors and strength. New bounds are presented either in analytic form and as products of an ongoing project for computation and investigation of the possible distance distributions of orthogonal arrays with parameters in doable ranges.
The Terminology of Survival Modeling: An Insight and Alternative Modeling of Student Retention
Dewi Juliah Ratnaningsih, Asep Saefuddin, Anang Kurnia, I Wayan Mangku
Subject: Social Sciences, Accounting Keywords: distance education; open and distance education; student retention; survival analysis
Online: 12 October 2020 (13:22:55 CEST)
Student retention is one indicator of accountability in the implementation of educational programs. Achievement of student retention rates indicates the performance of the quality objectives of an institution or college. To get an accurate picture of the factors related to retention, we need to do modeling. The retention variable is the time response variable measured in semester units. One of the statistical analyzes that can be used to analyze response data in time is survival analysis. The selection of an accurate analytical method in modeling will produce valid conclusions and impact making policies that are right and on target. This paper presents alternative modeling of student retention in distance education using survival analysis. The method used is a literature review. This paper also briefly describes distance education, open and distance education, distance education students' characteristics, distance education student retention, and survival models for modeling student retention in distance education.
DEBoost: A Python Library for Weighted Distance Ensembling in Machine Learning
Wei Hao Khoong
Subject: Mathematics & Computer Science, Other Keywords: ensemble learning; machine learning; Python; spatial distance; statistical distance; weighted ensemble
In this paper, we introduce deboost, a Python library devoted to weighted distance ensembling of predictions for regression and classification tasks. Its backbone resides on the scikit-learn library for default models and data preprocessing functions. It offers flexible choices of models for the ensemble as long as they contain the predict method, like the models available from scikit-learn. deboost is released under the MIT open-source license and can be downloaded from the Python Package Index (PyPI) at https://pypi.org/project/deboost. The source scripts are also available on a GitHub repository at https://github.com/weihao94/DEBoost.
Iterative Positioning Algorithm of the Target Node Based on Distance Correction in WSN
Jing Chen, Shixin Wang, Mingsan Ouyang, Yudi Chen, Yuting Xuan
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: iterative positioning algorithm; distance correction; RSSI; noise impact factor; distance deviation coefficient
The node position information is critical in the wireless sensor network (WSN). However, the existing positioning algorithms commonly have low positioning accuracy because of noise interferences in communication. To solve this problem, this paper presents an iterative positioning model based on distance correction to improve the positioning accuracy of the target node in WSN. First, the log-distance distribution model of received signal strength indication (RSSI) ranging is built and the noise impact factor is derived based on the model. Second, the initial position coordinates of the target node are obtained based on the triangle centroid localization algorithm, thereby calculating the distance deviation coefficient under the influence of noise. Then, the ratio of the distance measured by the log-normal distribution model to the median distance deviation coefficient is taken as the new distance between the anchor node and the target node. Based on the new distance, the triangular centroid positioning algorithm is used again to calculate the target node coordinates. Finally, the iterative positioning model is constructed, and the distance deviation coefficient is updated repeatedly to update the positioning result until the set number of iterations is reached. Experiment results show that the proposed iterative positioning model can improve positioning accuracy effectively.
Investigating Benefits and Barriers of Distance Education during Coronavirus Pandemic
Khalid Abdullah Alotaibi
Subject: Arts & Humanities, Other Keywords: Benefits; barriers; distance education; coronavirus
Online: 30 September 2021 (13:57:47 CEST)
This study aims to identify benefits and barriers to distance education, particularly from the perspective of teachers in Saudi Arabia. As the applied data collection tool, a questionnaire was distributed to the general education teachers in three districts. The sample size of the study was 1076 teachers. The results revealed that despite several benefits gained from distance learning, there are also some barriers. Teachers found that the most important advantage in distance learning is the acquisition of technical skills during the online teaching processes, they learn more and use digital education platforms, they have sufficient time to prepare the scientific content, they were able to provide adequate technical solutions for their courses, and they have the opportunity to use multiple media to deliver their courses. With the introduction of distance learning, teachers have explored new ways to deliver course contents to students. It has fostered better ways to provide more interactive real-time and on-demand teaching and learning using modern technology, thus, helping teachers become familiar with the use of electronic resources. It seems that teachers invest in technical methods to enhance students' performance. Also, teachers reported some obstacles that they face during remote teaching. Most of these problems are connection problems applied with devices and the internet, lack of students' motivation to learn in distance, problems associated with urban learners.
A Non-Artificial Setting Method for Fault Feeder Detection Systems Based on Data Fusion Used in Resonant Grounding Systems
Lixing Zhou, Junchen Peng, Zeyu Xu, Zhanguo Xia, Tao Zhou
Subject: Engineering, Electrical & Electronic Engineering Keywords: fault line detection; data fusion; non-artificial setting; sound distance; fault distance; resonant grounding system
Fault line detection timely and accurately when single-phase-to-earth fault occurs in resonant grounding system is still a focus of research. This paper presents a new approach for fault detection based on data fusion and it has non-artificial setting. Firstly, the fault criterion for interphase difference energy ratio and time-frequency correlation coefficient of each line is proposed. Subsequently, the paper establish a coordinate system with the interphase difference energy ratio as X axis and the time-frequency correlation coefficient as Y axis, and it uses the Euclidean distance algorithm to get the characteristic distance of each line by fusing two-dimensional information. Finally, comparing the sound distance and the fault distance of each line to discriminate the fault line. Electromagnetic Transients Program (EMTP) simulation results and adaptability analysis have confirmed the effectiveness and reliability of the proposed scheme.
From Text to E-text: Perceptions of Medical, Dental and Allied Students About E-learning
Ayesha Fahim, Sadia Rana, Saira Atif, Sakeenabi Basha, Irsam Haider, Mohammad Khursheed Alam, Anil Kumar Nagarajappa
Subject: Medicine & Pharmacology, Dentistry Keywords: Undergraduate; Medical; Online; Distance Education; Perception
In 2020, students of Pakistan had to adapt to the online environment for the very first time. This study aims to analyze the perceptions of medical, dental, and allied health students about online education in Pakistan. A descriptive, cross-sectional study was done to assess the level of acceptance of undergraduate students. A pre-validated questionnaire regarding demographics, past-experience of e-learning, advantages disadvantages of e-learning, and general perception of students towards e-learning was distributed. Descriptive statistics were computed for demographics, Mann-Witney-U test was used to compare the differences of perceptions between pre-clinical year and clinical years students. Kruskal-Wallis test was applied to compare the results of three specialties of students. Chi-square was used to compare overall category-wise positive and negative responses of students. 1200 students participated in the study. The major advantage identified by all students was the 'comfortable environment' in which they studied online. The major disadvantage selected by preclinical year students was 'anxiety due to social isolation' and that chosen by clinical year students was 'lack of patient interaction'. Overall, 72% of students had a negative perception of e-learning. Student-teacher training, student counselling sessions, and innovative techniques need to be introduced to enhance student engagement and reduce pandemic stress.
Using Microcomputers in an Online Introduction to Horticulture Class
Stephanie Elaine Burnett
Subject: Life Sciences, Other Keywords: Raspberry Pi; undergraduate education; distance learning
Online courses in horticulture increase the breadth of students who may be able to enroll. However, it is challenging to create hands-on learning experiences in online classes that are valuable for student learning. In an online introduction to horticulture class at the University of Maine, we created a hands-on project that is appropriate for students to work on independently at home. Students built an environmental monitoring system using a relatively inexpensive Raspberry Pi microcomputer and sensors for monitoring environmental factors that impact plant growth with a particular focus on monitoring temperature and humidity. They monitored the growing environment in their homes while growing house plants and used the information from their environmental monitoring system to determine whether their home environment was suitable for growing plants. Students were asked to use a pre-existing computer program in the Python language to monitor the environment. They also learned about how components of the code function and changed some simple parts of the code. A majority of students working on this project felt moderately confident, somewhat more confident, or very confident about their ability to use a Raspberry Pi microcomputer in the future. This project provides students with valuable hands-on experience in building environmental monitoring systems and provides them with a deeper understanding of the impact of the environment on plant growth.
Resistance Distance and Kirchhoff Index of Graphs with Pockets
Qun Liu, Jiabao Liu
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Kirchhoff index; resistance distance; generalized inverse.
Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1,Vk= {v1,v2,··· ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m ≥ 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek= {e1,e2,···ek} is a subset of the edge set of F and Huvis a simple graph of order m ≥ 3, uv is a specified edge of Huvsuch that Huv− u is isomorphic to Huv− v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.
Multiple Pictures of a Perspective Scene Reveal the Principles of Picture Perception
Casper J. Erkelens
Subject: Behavioral Sciences, Cognitive & Experimental Psychology Keywords: picture perception; pictorial distance; angular size
A picture is a powerful and convenient medium for inducing the illusion that one perceives a real three-dimensional scene. The relative invariance of picture perception across viewing positions has aroused the interest of painters, photographers and visual scientists. Many studies have been devoted to perceptual invariance when pictures are viewed from oblique directions. Invariance across viewing distances has received less attention. This study presents a computational analysis of pictures of perspective scenes taken from different distances between camera and physical objects. Distances and directions of pictorial objects were computed as function of viewing distance to the picture and compared with distances and directions of the physical objects as function of camera position. The computations show that pictorial distance and direction are determined by angular size of the depicted objects. Pictorial distance and direction are independent of camera position, focal length of the lens, and picture size. Ratios of pictorial distances, directions and sizes are constant as function of viewing distance. The constant ratios are proposed as the reason for invariance of picture perception over a range of viewing distances. Reanalysis of distance judgments obtained from the literature shows that perspective space, previously proposed as the model for visual space, is also a good model for pictorial space. The geometry of pictorial space contradicts some conceptions about picture perception.
Calculating Hamming Distance with the IBM Q Experience
José Manuel Bravo
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: quantum algorithm; Hamming weight; Hamming distance
In this brief paper a quantum algorithm to calculate the Hamming distance of two binary strings of equal length (or messages in theory information) is presented. The algorithm calculates the Hamming weight of two binary strings in one query of an oracle. To calculate the hamming distance of these two strings we only have to calculate the Hamming weight of the xor operation of both strings. To test the algorithms the quantum computer prototype that IBM has given open access to on the cloud has been used to test the results.
Differences in Household size, Employment Status and Ability to pay for the service, are Associated with Distance Travelled for Inpatient Care in Kenya
Ngugi Mwenda, Ruth Nduati, Mathew Kosgey, Gregory Kerich
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: distance; inpatient care; SDG's; Kenya; Tweedie; clustered
Background: Distance to a health facility for inpatient care in developing countries has been a huge hindrance towards the achievement of the Sustainable Development Goal three. The United Nation encourages countries to research on access to inpatient care, so as to form health policies based on data. Methods: Data on four hundred and eighty-one participants of all ages from forty-seven counties in Kenya who sought inpatient care in Kenya in 2018 were analyzed. Distance to a health facility was captured as a continuous variable and was self-reported by the respondent. The response exhibited a discrete mass at zero and continuous characteristic, therefore a Tweedie distribution was adopted for modelling. Due to the correlation nature of clustered data, we embraced the Generalized Estimating Equations approach with an exchangeable correlation. Since no standard software was available to analyze this problem, we developed an R functions. We assessed the best model fit using the QICu and criteria, in which the lowest value for the former and the highest for the later are preferred.Findings: Differences in employment, ability to pay for the service and household size are associated with the distance covered to access government facilities. Interpretation: Poor people tend to have large households and are more likely to live in rural areas and slums, thus are forced to travel for long distance to access inpatient care. Compared to unemployed, the employed could have better socio-economic status and possibly live within reach of the inpatient health facilities, therefore travel less distances to access. Longer distances are associated with high payments, signifying some form of specialized treatment care due to the complexity of the medical cases, that are expensive to treat.
A Novel Method to Calculate Objects' Apparent Size
Ermanno Lo Cascio
Subject: Keywords: apparent size; angular diameter distance; apparent diameter
The angular diameter is the angle subtended by a generic object – an apple or a star – to the eye of an observer, and it describes how large the object appears from a given viewpoint. The angular diameter represents a powerful tool for distance calculations starting from a directly measurable information and it finds application in several contexts varying from cosmography to architecture. In this article, the author proposes a novel equation to calculate the apparent diameter of whatever object. This equation defines the relationship between the object's apparent diameter with respect to the travelled distance starting from the initial distance R0 at which the observed object is located. Based on the preliminary tests conducted, the model seems to faithfully portray this relation with respect to measured values, also at the astronomical scale, thus considering the Earth-Moon distance, where, the absolute error detected is about 0.56%. Tests highlight also a dependency between the results accuracy and the measurement conditions suggesting a high level of sensibility linked to the initial magnification effect produced by the retina or the artificial lens employed.
Allocation of Tutors and Study Centers in Distance Learning Using Geospatial Technologies
Shahid Nawaz Khan, Kamran Mir, Ali Tahir, Arshad Awan, Zaib un Nisa, Syeda Aareeba Gillani
Subject: Earth Sciences, Geoinformatics Keywords: geospatial technologies; distance learning; resource allocation; AIOU
Allama Iqbal Open University (AIOU) is the largest distance learning institute of Pakistan and providing education to 1.4 million students. This is fairly a large setup across the country where students are geographically distributed. Currently the system works on a manual approach which is not efficient. Allocation of tutors and study centers to students plays a key role in distance learning for a better learning environment. Assigning tutors and study centers to distance learning students is a challenging task when there is huge geographical spread. The utilization of geospatial technologies in open and distance learning can fix allocation problems. This research analyzes the real data of twin cities Islamabad and Rawalpindi. The results show that the geospatial technologies can be used for efficient and proper resource utilization and allocation, which in turn can save the time and money. The overall idea fits into improved distance learning framework and related analytics.
Mathematical Incorrectnes of So Called Higuchi'S Fractal Dimension
Dalibor Martišek
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: Higuchi's method; Higuchi's fractal dimension; distance; metric space
So called Higuchi's method of fractal dimension estimation is widely used and the term Higuchi's fractal dimension even occurs in many publications. This paper deals with this method from mathematical point of view. Terms distance and dimension and its basic properties are explained and Higuchi's dimension according the original source is defined. Definition of Higuchi's dimension was comparated with mathematical definition of the distance and dimension. It is showed, that the definition of the Higuchi's dimension does not satisfy axioms of distance and dimension. So called Higuchi's method and Higuchi's dimension are mathematically incorrect. Therefore, all results achieved by this method are scientifically unreliable.
Perception and Preference towards Online Education in Nepali Academic Setting
Manish Thapa
Subject: Social Sciences, Accounting Keywords: Distance; Learning; Academic; Education; Students; Teaching-Learning; Modality
Online: 25 January 2021 (10:59:30 CET)
Education setting evolved from historical open learning system to traditional classroom set-up to distance learning modality. Teaching-Learning practice is transformed with an evolution of teaching-learning materials. With technological advancement in progressive manner and it's increasing use in academic setting, distance learning has been the on-demand and on-debate topic in current educational discourse. Comparatively fresh topic in Nepali academic setting, this paper intended to analyze the perception of Nepali students towards online modality in Nepali academic setting. This paper further analyzed the student's preference towards distance learning in current Nepali academic setting. Research findings were analyzed based on data collected through literature review, interview with students and professor and quantitative data collection through use of google form. Study identified opportunities as revenue generation; continuation of academic career from any part of country; increase learning outcome among jobholders. Study identified challenges as unequal access and quality of internet facilities; affordability of laptops/computers; limited interaction; and frequent disturbances. Seeing the better prospects, study strongly supported the need of shift in academic shift from traditional setting to non-traditional setting in Nepali context to meet the global needs of competitive and quality education.
Quantum Minimum Distance Classifier
Enrica Santucci
Subject: Physical Sciences, Mathematical Physics Keywords: quantum formalism applications; minimum distance classification; rescaling parameter
We propose a quantum version of the well known minimum distance classification model called "Nearest Mean Classifier" (NMC). In this regard, we presented our first results in two previous works. In [34] a quantum counterpart of the NMC for two-dimensional problems was introduced, named "Quantum Nearest Mean Classifier" (QNMC), together with a possible generalization to arbitrary dimensions. In [33] we studied the n-dimensional problem into detail and we showed a new encoding for arbitrary n-feature vectors into density operators. In the present paper, another promising encoding of n-dimensional patterns into density operators is considered, suggested by recent debates on quantum machine learning. Further, we observe a significant property concerning the non-invariance by feature rescaling of our quantum classifier. This fact, which represents a meaningful difference between the NMC and the respective quantum version, allows to introduce a free parameter whose variation provides, in some cases, better classification results for the QNMC. The experimental section is devoted to: i) compare the NMC and QNMC performance on different datasets; ii) study the effects of the non-invariance under uniform rescaling for the QNMC.
A Fast K-prototypes Algorithm Using Partial Distance Computation
Byoungwook KIM
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: clustering algorithm; k-prototypes algorithm, partial distance computation
The k-means is one of the most popular and widely used clustering algorithm, however, it is limited to only numeric data. The k-prototypes algorithm is one of the famous algorithms for dealing with both numeric and categorical data. However, there have been no studies to accelerate k-prototypes algorithm. In this paper, we propose a new fast k-prototypes algorithm that gives the same answer as original k-prototypes. The proposed algorithm avoids distance computations using partial distance computation. Our k-prototypes algorithm finds minimum distance without distance computations of all attributes between an object and a cluster center, which allows it to reduce time complexity. A partial distance computation uses a fact that a value of the maximum difference between two categorical attributes is 1 during distance computations. If data objects have m categorical attributes, maximum difference of categorical attributes between an object and a cluster center is m. Our algorithm first computes distance with only numeric attributes. If a difference of the minimum distance and the second smallest with numeric attributes is higher than m, we can find minimum distance between an object and a cluster center without distance computations of categorical attributes. The experimental shows proposed k-prototypes algorithm improves computational performance than original k-prototypes algorithm in our dataset.
A Literature Review on Intelligent Services Applied to Distance Learning
Lidia Martins da Silva, Lucas P. S. Dias, Sandro Rigo, Jorge L. V. Barbosa, Daiana R. F. Leithardt, Valderi Reis Quietinho Leithardt
Subject: Arts & Humanities, Other Keywords: distance learning; intelligent services; literature review; virtual learning environments.
Distance learning has assumed a relevant role in the Educational scenario. The use of Virtual Learning Environments contributes to obtain a substantial amount of educational data. In this sense, the analyzed data generate knowledge used by institutions to assist managers and professors in strategic planning and teaching. The discovery of students' behaviors enables a wide variety of intelligent services for assisting in the learning process. This article presents a literature review in order to identify the intelligent services applied in distance learning. The research covers the period from January 2010 to May 2021. The initial search found 1,316 articles, among which 51 were selected for further studies. Considering the selected articles, 33% (17/51) focus on learning systems, 35% (18/51) propose recommendation systems, 26% (13/51) approach predictive systems or models, and 6% (3/51) use assessment tools. This review allowed to observe that the principal services offered are recommendation systems and learning systems. In these services, the analysis of student profiles stands out to identify patterns of behavior, detect low performance and identify probabilities of dropouts from courses.
Performance Evaluation of a Two-Parameters Monthly Rainfall-Runoff Model in The Southern Basin of Thailand
Pakorn Ditthakit, Sirimon Pinthong, Nureehan Salaeh, Fadilah Binnui, Laksanara Khwanchum
Subject: Engineering, Automotive Engineering Keywords: GR2M; Inverse Distance Weighting; Rainfall-Runoff Model; Sensitivity Analysis
Accurate monthly runoff estimation is fundamental in water resources management, planning, and development, resulting in preventing and reducing water-related problems, such as flooding and drought. This article evaluates the performance of the monthly hydrological rainfall-runoff model, GR2M model, in Thailand's southern basins. The GR2M model requires only two parameters, and no prior research has been reported on its application in this region. The 37 runoff stations, which are distributively located in three sub-watersheds of Thailand's southern region, namely; Thale Sap Songkhla, Peninsular-East Coast, and Peninsular-West Coast, were selected as study cases. The available monthly hydrological data of runoff, rainfall, air temperature from the Royal Irrigation Department (RID) and the Thai Meteorological Department (TMD) were collected and analyzed. Thornthwaite method was utilized for the determination of evapotranspiration. The model's performance was conducted using three statistical indices: Nash-Sutcliffe Efficiency (NSE), Correlation Coefficient (r), and Overall Index (OPI). The model's calibration results for 37 runoff stations gave the average of NSE, r, and OPI of 0.637, 0.825, and 0.757, and those values for verification of 0.465, 0.750, and 0.639, respectively. It indicated a model's acceptable performance and could apply the GR2M model for determining monthly runoff variation in this region. The spatial distribution of X1 and X2 values was conducted by using IDW method. It was susceptible to the X1 value and X2 value of approximately more than 0.90 gave the higher model's performance.
Mechanisms of Enhancer-Promoter Interactions in Higher Eukaryotes
Pavel Georgiev, Olga Kyrchanova
Subject: Life Sciences, Biochemistry Keywords: C2H2 proteins; CTCF; LDB1; chromatin insulator; long-distance interactions
Online: 8 December 2020 (08:29:36 CET)
In higher eukaryotes, enhancers determine the activation of developmental gene transcription in specific cell types and stages of embryogenesis. Enhancers transform the signals produced by various transcription factors within a given cell, activating the transcription of the targeted genes. Often, developmental genes can be associated with dozens of enhancers, some of which are located at large distances from the promoters that they regulate. Currently, the mechanisms that underly the specific distance interactions between enhancers and promoters remain unknown. This review describes the properties and activities of enhancers and discusses the mechanisms of distance interactions and potential proteins involved in this process.
WebGIS and Geospatial Technologies for Landscape Education on Personalized Learning Contexts
María Luisa de Lázaro Torres, Rafael de Miguel González, Francisco José Morales Yago
Subject: Earth Sciences, Geoinformatics Keywords: WebGIS; Landscape; heritage; personalized learning; the cloud; distance learning
The value of landscape, as part of collective heritage, can be acquired by GIS due to the multilayer approach of the spatial configuration. Proficiency in geospatial technologies in order to collect, process, analyze, interpret, visualize and communicate geographic information is being increased by undergraduate and graduate students, but in particular by those who are training to become geography teachers at secondary education. This training can be carried out through personalized learning and distance learning methodology. Personalized GIS education aims to integrate students and enhance their understanding of landscape. Some teaching experiences are shown whereby opportunities offered by WebGIS will be described, through quantitative tools and techniques that will allow this modality of learning and improve its effectiveness. Results of this research show that students, through geospatial technologies, learn landscape as a diversity of elements but also the complexity of physical and human factors involved. Several conclusions will be highlighted: i) the contribution of geospatial training to education for sustainable development; ii) spatial analysis as a mean of skills acquisition about measures for landscape conservation; iii) expanding and applying acquired knowledge to other geographic spaces and different landscapes.
Quality of Research in Residents of Medical Specialties after a Standardized Digital Training Program with Rubrics
Valeria Jimenez Baez, Maria Erika Gutierrez De la Cruz, Luis Sandoval Jurado, Luis Roberto Martínez Castro, Francisco Javier Alcocer Nuñez
Subject: Medicine & Pharmacology, Dentistry Keywords: Scoring; Rubrics; Health; Personnel; Program Evaluation; Distance Education; Residency Education; Speciality
Introduction: In the medical area, teaching is essential since it must offer the appropriate instruments to demonstrate that graduates have acquired the necessary skills. Objective: Evaluate the quality of research in residents of medical specialties after a standardized digital training program with rubrics. Methods: An observational, prospective research study in resident physicians of seven medical specialties first-year of an introductory program to methodology. It is integrated with the result variable through the quality of the final product and the quality variable will be measured with an ad hoc questionnaire validated by the Delphi method with a consistency level of 3-3. The data will be integrated into a base of the SPSS system and determined with the Chi-square test considering a minimum significance of 0.05. Results: 85 first-year medical residents (n=85) enrolled in the Research Seminar. The mean age was 31.34 years (± 3.96). About gender Male 38±31.13 Female 31.51±3.83. The global final grade was 80.61 (± 9.59) and the global sat-isfaction of the course was referred to as good by 62.2%. We observed a positive relationship between the scope of evaluation and the level of satisfaction. Conclusion: The research seminar implementation in a b-learning mode in response to the educational needs in medical residents for the field of health education showed a relationship between higher qualification, higher satisfaction, as well as determining that the comprehensive evaluation through the use of rubrics standardized allowed to delimit the deficiencies and strengths for timely feedback influencing the process of acquiring skills and the quality of the final product.
psi-Hilfer Fractional Approximations of Csiszar's f-Divergence
George Anastassiou
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Csiszar's discrimination; Csiszar's distance; fractional calculus; psi-Hilfer fractional derivative
Online: 4 March 2021 (14:19:06 CET)
Here are given tight probabilistic inequalities that provide nearly best estimates for the Csiszar's f-divergence. These use the right and left psi -Hilfer fractional derivatives of the directing function f. Csiszar's f- divergence or the so called Csiszar's discrimination is used as a measure of dependence between two random variables which is a very essential aspect of stochastics, we apply our results there. The Csiszar's discrimination is the most important and general measure for the comparison between two probability measures. We give also other applications.
Working Paper ARTICLE
A Stepwise GIS Approach for the Delineation of River Valley Bottom within Drainage Basins Using a Cost Distance Accumulation Analysis
Gasper L. Sechu, Bertel Nilsson, Bo V. Iversen, Mette B. Greve, Christen D. Børgesen, Mogens H. Greve
Subject: Earth Sciences, Atmospheric Science Keywords: river valley bottom; GIS; cost distance accumulation; groundwater dependent ecosystems
River valley bottoms have hydrological, geomorphological, and ecological importance and are buffers for protecting the river from upland nutrient loading coming from agriculture and other sources. They are relatively flat, low-lying areas of the terrain that are adjacent to the river and bound by increasing slopes at the transition to the uplands. These areas have under natural conditions, a groundwater table close to the soil surface. The objective of this paper is to present a stepwise GIS approach for the delineation of river valley bottom within drainage basins and use it to perform a national delineation. We developed a tool that applies a concept called cost distance accumulation with spatial data inputs consisting a river network and slope derived from a digital elevation model. We then used wetlands adjacent to rivers as a guide finding the river valley bottom boundary from the cost distance accumulation. We present results from our tool for the whole country of Denmark carrying out a validation within three selected areas. The results reveal that the tool visually performs well and delineates both confined and unconfined river valleys within the same drainage basin. We use the most common forms of wetlands (meadow and marsh) in Denmark's river valleys known as Groundwater Dependent Ecosystems (GDE) to validate our river valley bottom delineated areas. Our delineation picks about half to two-thirds of these GDE. However, we expected this since farmers have reclaimed Denmark's low-lying areas during the last 200 years before the first map of GDE was created. Our tool can be used as a management tool, since it can delineate an area that has been the focus of management actions to protect waterways from upland nutrient pollution.
Assessing the Impact of Different Levels of Interactivity on the Effectiveness of Self-Learning
Nisala Kalupahana
Subject: Social Sciences, Education Studies Keywords: interactive learning environments; distance education and telelearning; human-computer interface
Online: 8 January 2019 (09:42:35 CET)
As education becomes more and more important in creating improved societies, many people who do not have access to it are falling behind. To help them catch up, many people, especially those in rural areas and developing countries, are turning to different methods of self-learning, especially those that utilize cheap technology and use interactive methods to teach. Our empirical study tests the effectiveness of an e-learning system that utilizes newer, less tested forms of interactivity and could potentially be used in these areas as a self-learning system and compares it to non-interactive video and textbook self-learning in two different topics. The results of the experiment showed that the increased interactivity provided by the e-learning system achieved significantly better learning performance over both non-interactive video and textbook self-learning. It was also found that students who learned through non-interactive video performed significantly better than those who used textbooks for self-learning.
Lost in Optimization of Water Distribution Systems: Better Call Bayes
Antonio Candelieri, Andrea Ponti, Ilaria Giordani, Francesco Archetti
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: Pump scheduling optimization; Bayesian optimization; Optimal sensor placement; Wasserstein distance; Robustness
The main goal of this paper is to show that Bayesian optimization could be regarded as a general framework for the data driven modelling and solution of problems arising in water distribution systems. Hydraulic simulation, both scenario based, and Monte Carlo is a key tool in modelling in water distribution systems. The related optimization problems fall in a simulation/optimization framework in which objectives and constraints are often black-box. Bayesian Optimization (BO) is characterized by a surrogate model, usually a Gaussian process, but also a random forest and increasingly neural networks and an acquisition function which drives the search for new evaluation points. These modelling options make BO nonparametric, robust, flexible and sample efficient particularly suitable for simulation/optimization problems. A defining characteristic of BO is its versatility and flexibility, given for instance by different probabilistic models, in particular different kernels, different acquisition functions. These characteristics of the Bayesian optimization approach are exemplified by the two problems: cost/energy optimization in pump scheduling and optimal sensor placement for early detection on contaminant intrusion. Different surrogate models have been used both in explicit and implicit control schemes. Showing that BO can drive the process of learning control rules directly from operational data. BO can also be extended to multi-objective optimization. Two algorithms have been proposed for multi-objective detection problem using two different acquisition functions.
The Multi-Attributive Border Approximation Area Comparison (Mabac) Method for Decision Making under Interval-Valued Fermatean Fuzzy Environment for Green Supplier Selection
Manoj Mathew, Ripon K. Chakrabortty, Michael J. Ryan, Muhammad Fazal Ljaz, Syed Abdul Rehman Khan
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: Interval-valued Fermatean fuzzy; MABAC; Green supplier selection; MCDM; Hamming distance
In the era of sustainable development, green supplier selection has become a key component of supply chain management, as it considers criteria such as carbon footprint, water usage, energy usage and recycling capacity. Since the green supplier selection problem involves subjective criteria and uncertainty in preferences, it is well suited to using multi-criteria decision-making methods (MCDM). Although few researchers have investigated MCDM for green supplier selection under uncertainty using intuitionistic and Pythagorean fuzzy sets. Still, the newly developed fermatean fuzzy can handle greater data uncertainty than intuitionistic and Pythagorean fuzzy sets. Since interval extension of fuzzy theory provides more accurate modeling, in this paper, we propose an interval extension of fermatean fuzzy set and discuss its fundamental set operations, arithmetic operations and related properties. We propose the Hamming distance function and the score function of the interval-valued fermatean fuzzy numbers. The recently developed Multi-attribute Border Approximation Area Comparison (MABAC) method is also considered due to its stable and simple computation compared to other conventional MCDM methods. Therefore, an interval-valued fermatean fuzzy MABAC method is proposed and solved to select a green supplier based on multiple criteria. The proposed method's comparative analysis and sensitivity analysis validate the proposed method's effectiveness and robustness.
Ultrasound Measurement of Tumor-Free Distance from the Serosal Surface as the Alternative to Measuring the Depth of Myometrial Invasion in Predicting Lymph Node Metastases in Endometrial Cancer
Marcin Liro, Marcin Śniadecki, Ewa Wycinka, Szymon Wojtylak, Michał Brzeziński, Agata Stańczak, Dariusz Grzegorz Wydra
Subject: Medicine & Pharmacology, Allergology Keywords: ultrasound; endometrial cancer; lymph nodes metastasis; myometrial invasion; tumor-free distance
Background: Ultrasonography's usefulness in endometrial cancer (EC) diagnosis consists of its staging and predictive roles. Ultrasound-measured tumor-free distance from the tumor to the uterine serosa (uTFD) is a promising marker for this variable. The aim of the study was to determine the usefulness of this biomarker in locoregional staging, and thus in the prediction of lymph node metastasis (LNM). Methods: We conducted a single-institutional, prospective study on 116 consecutive patients with EC who underwent 2D transvaginal ultrasound examination. The uTFD marker was compared with the depth of ultrasound-measured myometrial invasion (uMI). Univariate and multivariate logit models were evaluated to assess the predictive power of the uTFD and uMI in regard to LNM. The reference standard was a final histopathology result. Survival was assessed by the Kaplan-Meyer method. Results: LNM was found in 17% of the patients (20/116). In the univariate analysis, uMI and uTFD were significant predictors of LNM. Accuracy was 70.7%, and NPV was 92.68% (OR 4.746, 95% CI 1.710-13.174) for uMI (p = 0.002), and 63.8%, and 89.02% (OR 0.842, 95% CI 0.736 – 0.963), respectively, for uTFD (p = 0.01). The cut-off value for uTFD in the prediction of LNM was 5.2 mm. The absence of LNM was associated more with biomarker values uMI <1/2 and uTFD >=5.2 mm than with the presence of metastases with uMI >1/2 and uTFD values <5.2 mm. In the multivariate analysis, the accuracy of the uMI-uTFD model was 74%, and NPV was 90.24% (p = NS). Neither uMI nor uTFD are surrogates for overall and recurrence-free survivals in endometrial cancer. Conclusions: Both uMI and uTFD, either alone or in combination, are valuable tools for gaining additional preoperative information on expected lymph node status. Negative lymph nodes status is better described by ultrasound biomarkers than a positive status. It is easier to use uTFD measurement as a biomarker of EC invasion than uMI, and the former still maintains a similar predictive value for lymph node metastases to the latter at diagnosis.
Preprint DATA DESCRIPTOR | doi:10.20944/preprints202106.0368.v1
Mash Sketched Reference Dataset for Genome-Based Taxonomy and Comparative Genomics
Ayixon Sánchez-Reyes, Maikel Gilberto Fernández-López
Subject: Life Sciences, Biochemistry Keywords: Microbial Mash database, Mash distance, Genome containment, Type material, Microbial taxonomy
The analysis of curated genomic, metagenomic, and proteomic data are of paramount importance in the fields of biology, medicine, education, and bioinformatics. Although this type of data is usually hosted in raw form in free international repositories, its access requires plenty of computing, storage, and processing capacities for the domestic user. The purpose of the study is to offer a comprehensive set of genomic and proteomic reference data, in an accessible and easy-to-use form to the scientific community. A representative type material set of genomes, proteomes and metagenomes were directly downloaded from the site: https://www.ncbi.nlm.nih.gov/assembly/ and from Genome Taxonomy Database, associated with the major groups of Bacteria, Archaea, Virus, and Fungi. Sketched databases were subsequently created and stored on handy raw reduced representations, by using Mash software. Our dataset contains near to 100 GB of space disk reduced to 585.78 MB and represents 87,476 genomics/proteomic records from eight informative contexts, which have been prefiltered to make them accessible, usable, and user-friendly with computational resources. Potential uses of this dataset include but are not limited to, microbial species delimitation, estimation of genomic distances, genomic novelties, paired comparisons between proteomes, genomes, and metagenomes.
Principal Determinants of Aquatic Macrophyte Communities in Least-Impacted Small Shallow Lakes in France
Frédéric Labat, Gabrielle Thiébaut, Christophe Piscart
Subject: Earth Sciences, Atmospheric Science Keywords: wetlands; ponds; alkalinity; geology; distance from source; connectivity; climate; altitude; hydroperiod
Small Shallow Lakes (SSL) support exceptionally high and original biodiversity, providing numerous ecosystem services. Their small size makes them especially sensitive to anthropic activities, that causes a shift to dysfunctional turbid states and induces loss of services and biodiversity. In this study we investigated the relationships between environmental factors and macrophyte communities. Macrophytes play a crucial role in maintaining functional clear states. Better understanding factors determining the composition and richness of aquatic plant communities in least-impacted conditions may be useful to protect them. We inventoried macrophyte communities and collected chemical, climatic and morphological data from 89 least-impacted SSL widely distributed in France. SSL were sampled across four climatic ecoregions, various geologies and elevations. Hierarchical cluster analysis showed a clear separation of four macrophyte assemblages strongly associated with mineralisation. Determinant factors identified by db-RDA analysis are, in order of importance, geology, distance from source (DIS, a proxy for connectivity with river hydrosystems), surface area, climate and hydroperiod (water permanency). Surprisingly, at country-wide scale, climate and hydroperiod filter macrophyte composition weakly. Geology and DIS are the major determinants of community composition, whereas surface area determines floristic richness. DIS is identified as determinant in freshwater lentic ecosystems for the first time.
A Generalized 90° Out-of-Phase Wilkinson Power Divider for Dual Port UHF CP RFID Antennae with variable Port Distance
W Akash Sovis, Manilka Jayasooriya
Subject: Engineering, Electrical & Electronic Engineering Keywords: Wilkinson power divider; RFID; dual port feeder; port distance; circular polarization
In this paper a microstrip Wilkinson power divider with a 90° phase delay at one output port is proposed to obtain circular polarization to feed a dual port RFID antenna. The 90° phase delay was obtained by embedding an extra quarter wavelength at one port of the Wilkinson power divider. The feeding circuit is then mounted on the ground plane of the microstrip antenna feeding the radiating patch directly through the ground plane and dielectric layer thus reducing any fringing effect and resulting a mechanically compact unit. The proposed feeding method offers better expectation of antenna performance with minimal attenuation and coupling losses. The design process generalizes geometric pa- rameters of the Wilkinson power divider for variable port distances. The paper considers both UK and US RFID center frequencies, 870 MHz and 915 MHz respectively. Numeri- cally computed values for geometric design parameters for both frequencies are tabled as future design tools for port distances varying from 18 mm up to 34 mm at 870 MHz and 17 mm up to 32 mm at 915 MHz. Simulation results indicate a return loss (S11) of -20 dB and -26 dB at 870 MHz and 915 MHz operational frequencies respectively at 270° angled quarter wavelength.
Interpolation of Small Datasets in the Sandstone Hydrocarbon Reservoirs, Case Study from the Sava Depression, Croatia
Tomislav Malvić, Josip Ivšinović, Josipa Velić, Rajna Rajić
Subject: Earth Sciences, Geology Keywords: interpolation; permeability; injected water; inverse distance weighting; sava depression; miocene; Croatia
Online: 28 February 2019 (06:54:38 CET)
Here are analysed data taken in two hydrocarbon fields ("A" and "B"), located in the western part of Sava Depression (North Croatia). They are in the secondary phase of production. The selected reservoirs "L" (in the "A" Field) and "K" ("B") are of the Lower Pontian (Upper Miocene) age and belong to Kloštar-Ivanić Formation. Due to strong tectonics, there are numerous tectonic block, relatively rarely sampled with well and laboratory tests. Here are selected two variables for interpolation - reservoirs permeabilities and the injected volumes of field water. The following interpolation methods are described, compared and applied: Nearest Neighbourhood, Natural Neighbour (the first time in the Sava Depression) and Inverse Distance Weighting. The last one has been proven as the most appropriate for datasets with size lower than 20 points.
Cross-Linguistic Influence in Chinese Learners of Two Foreign Languages While Studying Abroad in Spain
Wenxiao Zhao
Subject: Arts & Humanities, Linguistics Keywords: cross-linguistic influence (CLI); L3 Spanish; borrowing; functional transfer; language distance
Studies in the area of cross-linguistic influence (CLI) have attracted the focus of multi-linguistic learners. However, little research on CLI deals with Asian learners, particularly Chinese-speaker with knowledge of two or more foreign languages. The present study explores CLI in L1 Chinese learners with both English (L2) and Spanish (L3) as foreign languages who are studying in Madrid, a Spanish-speaking community; their studies coincided with data collection. English learners were instructed to speak for analysis purposes, with the following aims: (i) to observe the most frequent category (functional transfer, code-switching, borrowing and coinage) in the CLI instances; (ii) to determine the source language of CLI; (iii) to investigate whether CLI factors, including language distance, L2 status, proficiency and recency of use, intervene in the appearance of CLI instances in the participants. Data was gathered from 16 female Chinese students at Complutense University of Madrid (UCM). These were master students aged 22 to 26, who visited Spain for more than 5 months when they participated in the present study. The instrument used was an English semi-structured interview. Results primarily reveal that (a) borrowing is the most prevalent category, accounting for 70% of the CLI instances; (b) Spanish is the main source language of CLI while Chinese plays a functional role in the transfer process; (c) language distance proves to be the strongest predictor of CLI.
Relationship between City Size, Coastal Land Use and Summer Daytime Air Temperature Rise with Distance from Coast
Hideki Takebayashi, Takahiro Tanaka, Masakazu Moriyama, Hironori Watanabe, Hiroshi Miyazaki, Kosuke Kittaka
Subject: Engineering, Other Keywords: distance from coast; air temperature; land use; city size; Japan; Germany
The relationship between city size, coastal land use and air temperature rise with distance from coast during summer day is analyzed using the meso-scale Weather Research & Forecasting (WRF) model in five coastal cities in Japan with different sizes and coastal land use (Tokyo, Osaka, Nagoya, Hiroshima and Sendai) and inland cities in Germany (Berlin, Essen and Karlsruhe). Air temperature increased as distance from the coast increased, reached its maximum, and then decreased slightly. In Nagoya and Sendai, the number of urban land use in coastal areas is less than the other three cities, where air temperature is a little lower. As a result, air temperature difference between coastal and inland urban area is small and the curve of air temperature rise is smaller than those in Tokyo and Osaka. In Sendai, air temperature in the inland urban area is the same as in the other cities, but air temperature in the coastal urban area is a little lower than the other cities, due to about one degree lower sea surface temperature influenced by the latitude. In three German cities, the urban boundary layer may not develop sufficiently because the fetch distance is not enough.
Perspectives from Montiaceae (Portulacineae) Evolution. I. Phylogeny and Phylogeography
Mark A. Hershkovitz
Subject: Biology, Plant Sciences Keywords: Montiaceae; phylogeny; phylogeography; long-distance dispersal; idiosyncrasy; Principal of Evolutionary Idiosyncraticity
Montiaceae comprise a clade of at least 270 species plus about 20 accepted subspecific taxa, primarily of western America and Australia. The present paper is the first of a two-part work that seeks to evaluate evolutionary theory via metadata analysis of Montiaceae. In particular, it uses metadata analysis to evaluate the theory in theory-laden methods that have been applied in evolutionary analyses of Montiaceae. This part focuses on phylogeny and phylogeography. The second part focuses on phenotypic and ecological diversification. An emergent theme in this paper is the degree to which historical idiosyncrasy during Montiaceae evolution misleads quantitative methods of evolutionary reconstruction and phylogeographic interpretation. This suggests that idiosyncraticity itself is a fundamental property of evolution. The second part of this work elaborates this notion as the Principle of Evolutionary Idiosyncraticity. The present part describes idiosyncraticity in molecular phylogenetic and phylogeographic data and uses this notion to refine ideas on Montiaceae evolution. Phylogenetic metadata conflicts and conflicting phylogeographic interpretations are discussed. I conclude that, owing to PEI, quantitative methods of evolutionary analysis cannot be globally accurate, though they are useful heuristically. In contrast, classical narrative analysis is robust in the face of PEI.
Variational Mode Decomposition Denoising Combined with the Euclidean Distance for Diesel Engine Vibration Signal
Gang Ren, Jide Jia, Xiangyu Jia, Jiajia Han
Subject: Engineering, Automotive Engineering Keywords: variational mode decomposition; Euclidean Distance; diesel engine; vibration signal; denoising algorithm
Variational mode decomposition (VMD) is a recently introduced adaptive signal decomposition algorithm with a solid theoretical foundation and good noise robustness compared with empirical mode decomposition (EMD). There is a lot of background noise in the vibration signal of diesel engine. To solve the problem, a denoising algorithm based on VMD and Euclidean Distance is proposed. Firstly, a multi-component, non-Gauss, and noisy simulation signal is established, and decomposed into a given number K of band-limited intrinsic mode functions by VMD. Then the Euclidean distance between the probability density function of each mode and that of the simulation signal are calculated. The signal is reconstructed using the relevant modes, which are selected on the basis of noticeable similarities between the probability density function of the simulation signal and that of each mode. Finally, the vibration signals of diesel engine connecting rod bearing faults are analyzed by the proposed method. The results show that compared with other denoising algorithms, the proposed method has better denoising effect, and the fault characteristics of vibration signals of diesel engine connecting rod bearings can be effectively enhanced.
Why the Many-Worlds Interpretation?
Lev Vaidman
Subject: Physical Sciences, General & Theoretical Physics Keywords: many-worlds interpretation; interpretations of quantum mechanics; determinism; action at a distance
A brief (subjective) description of the state of the art of the many-worlds interpretation of quantum mechanics (MWI) is presented. It is argued that the MWI is the only interpretation which removes action at a distance and randomness from quantum theory. Limitations of the MWI regarding questions of probability which can be legitimately asked are specified. The ontological picture of the MWI as a theory of the universal wave function decomposed in a superposition of world wave functions, the important part of which are defined in three-dimensional space, is viewed from the point of view of our particular branch. Some speculations about misconceptions which apparently prevent the MWI to be in the consensus are mentioned.
Estimating Interpersonal Distance and Crowd Density with a Single Edge Camera
Alem Fitwi, Yu Chen, Han Sun, Robert Harrod
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: Area Estimation, Crowd Management, COVID-19, Edge Camera, Interpersonal Distance, Social Distancing.
Online: 1 October 2021 (15:37:26 CEST)
For public safety and physical security, currently more than a billion closed-circuit television (CCTV) cameras are deployed around the world. Proliferation of artificial intelligence (AI) and machine learning (ML) technologies has gained significant applications including crowd surveillance. The state-of-the-art distance and area estimation algorithms either need multiple cameras or a reference scale as a ground truth. It is an open question to obtain an estimation using a single camera without a scale reference. In this paper, we propose a novel solution called E-SEC, which estimates interpersonal distance between a pair of dynamic human objects, area occupied by a dynamic crowd, and density using a single edge camera. The E-SEC framework comprises edge CCTV cameras responsible for capture a crowd on video frames leveraging a customized YOLOv3 model for human detection. E-SEC contributes an interpersonal distance estimation algorithm vital for monitoring the social distancing of a crowd, and an area estimation algorithm for dynamically determining an area occupied by a crowd with changing size and position. A unified output module generates the crowd size, interpersonal distances, social distancing violations, area, and density per every frame. Experimental results validate the accuracy and efficiency of E-SEC with a range of different video datasets.
Application of Modified Shepard's Method (MSM) case study with interpolation of Neogene reservoirs variables in the Northern Croatia
Tomislav Malvić, Josip Ivšinović, Josipa Velić, Jasenka Sremac, Uroš Barudžija
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: Modified Shepard's Method (MSM); Inverse Distance Weighting (IDW); sandstone; neogene; Northern Croatia
Interpolation is procedure that depends on spatial and/or statistical properties of analysed variable(s). It is special challenging task for data that included low number of samples, like dataset with less than 20 data. This problem is especially emphasized in the subsurface geological mapping, i.e. in the cases where data are taken solely from wells. Successful solutions of such mapping problems ask for knowledge about interpolation methods designed primarily for small datasets and dataset itself. Here are compared two methods, namely Inverse Distance Weighting and Modified Shepard's Method, applied for three variables (porosity, permeability, thickness) measured in the Neogene sandstone hydrocarbon reservoirs (Northern Croatia). The results showed that pure cross-validation is not enough condition for appropriate map selection, but also geometrical features need to be considered, for datasets with less than 20 points.
On the Distributional Characterization of Graph Models of Water Distribution Networks in Wasserstein Spaces
Antonio Candelieri, Andrea Ponti, Francesco Archetti
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: multi-objective; evolutionary algorithms; Pareto optimality; Wasserstein distance; network vulnerability; resilience; sensor placement.
This paper is focused on two topics very relevant in water distribution networks (WDNs): vulnerability assessment and the optimal placement of water quality sensors. The main novelty element of this paper is to represent the data of the problem, in this case all objects in a graph underlying a water distribution network, as discrete probability distributions. For vulnerability (and the related issue of re-silience) the metrics from network theory, widely studied and largely adopted in the water research community, reflect connectivity expressed as closeness centrality or, betweenness centrality based on the average values of shortest paths between all pairs of nodes. Also network efficiency and the related vulnerability measures are related to average of inverse distances. In this paper we propose a different approach based on the discrete probability distribution, for each node, of the node-to-node distances. For the optimal sensor placement, the elements to be represented as dis-crete probability distributions are sub-graphs given by the locations of water quality sensors. The objective functions, detection time and its variance as a proxy of risk, are accordingly represented as a discrete e probability distribution over contamination events. This problem is usually dealt with by EA algorithm. We'll show that a probabilistic distance, specifically the Wasserstein (WST) distance, can naturally allow an effective formulation of genetic operators. Usually, each node is associated to a scalar real number, in the optimal sensor placement considered in the literature, average detection time, but in many applications, node labels are more naturally expressed as histograms or probability distributions: the water demand at each node is naturally seen as a histogram over the 24 hours cycle. The main aim of this paper is twofold: first to show how different problems in WDNs can take advantage of the representational flexibility inherent in WST spaces. Second how this flexibility translates into computational procedures.
IoT application for vehicles identification using the Optical Fiber Sensors and Wireless Sensor Network
Hacen Khlaifi, Amira Zrelli, Tahar Ezzedine
Subject: Engineering, Other Keywords: Wireless Sensors Networks; Fiber Bragg Grating; Pressure; Speed; Wheelbase distance; Weight; Vehicle; Identification.
Due to the renewed variation in government and political systems inside and outside countries, and with the high tariffs at borders, the latter have become an outlet for terrorism and smugglers. Therefore, each country seeks to develop its own protection system, and the technologies used in these systems vary according to the severity and the importance of the installations to be protected, it is found that some of them are expensive and unnecessary, but other have good and variable levels of efficiency. Consequently, the idea of designing a surveillance system that can monitor and control access becomes indispensable. In the same context, this work is of crucial strategic and geopolitical importance. It combines pre-existing alarm and monitoring methods and revolutionary Internet of Things (IoT) application products, of which Wireless Sensor Networks (WSN) and Optical Fiber Sensors (OFS) are part of this application. This article presents the distribution of wireless radar nodes accompanying with a Bragg fiber sensor to identify each rolling intruder incoming the zone to be monitored, from the determination of its speed, weight and wheelbase distance.
MONG: An extension to Galaxy Clusters
Louise Rebecca, Arun Kenath, C Sivaram
Subject: Physical Sciences, Astronomy & Astrophysics Keywords: dark matter; dark energy; modified Newtonian gravity; flat rotation curve; velocity-distance curve
The presence of dark matter, though well established by indirect evidence is yet to be observed directly. Various dark matter detection experiments running for several years have yielded no positive results so far. In view of these negative results, we had earlier proposed alternate models by postulating a minimum gravitational field strength (minimum curvature) and also a minimum acceleration. These postulates led to the modified Newtonian dynamics and modified Newtonian gravity (MONG). The observed flat rotation curves of galaxies had also been accounted for through these postulates. Here we extend these postulates to galaxy clusters and model the dynamical velocity-distance curves for such large-scale structures. The velocity-distance curve of the Virgo cluster, plotted with this model is found to be in accordance with that observed.
MOOCs at the Crossroads: A Literature Review and Reflection Drawing upon Discourse Analysis
Laila Mohebi, Marcelo F. Ponce
Subject: Social Sciences, Education Studies Keywords: MOOCs; distance education; self-directed learning; self-defined learning pathways; 21st century abilities
This study is a synthesis of 159 articles that were selected for their relevance to comprehend key aspects of the Massive Open Online Courses (MOOCs) phenomenon, from a discourse analysis perspective. Since 2011, MOOCs are expanding worldwide so that the number of subscribers outpointed 101 million at the end of 2018. This paper explores the question whether the MOOCs are the embodiment of the global one-world classroom or whether, instead, they represent a low-cost alternative tailored to a segment that doesn´t have enough time or resources to attend a brick-and-mortar college. In addition, the review tackles the link between motivation and low completion rates. Finally, we discuss the need to devise better methods to assess the pedagogical value of MOOCs.
A Machine Learning Approach to the Residential Relocation Distance of Households Living in the Seoul Metropolitan Region
Changhyo Yi, Kijung Kim
Subject: Social Sciences, Other Keywords: residential relocation distance; residential movement; machine learning; decision tree regression; Seoul metropolitan region
This study aimed to ascertain the applicability of a machine learning approach to the description of residential mobility patterns of households in the Seoul metropolitan region (SMR). The spatial range and temporal scope of the empirical study were set to 2015 to review the most recent residential mobility patterns in the SMR. The analysis data used in this study involve the microdata of Internal Migration Statistics provided by the Microdata Integrated Service of Statistics Korea. We analysed the residential relocation distance of households in the SMR by using machine learning techniques such as ordinary least squares regression and decision tree regression. The results of this study showed that a decision tree model can be more advantageous than ordinary least squares regression in terms of the explanatory power and estimation of moving distance. A large number of residential movements are mainly related to the accessibility to employment markets and some household characteristics. The shortest movements occur when households with two or more members move into densely populated districts. In contrast, job-based residential movements have relatively longer distance. Furthermore, we derived knowledge on residential relocation distance, which can provide significant information on the urban management of metropolitan residential districts and the construction of reasonable housing policies.
Total Factor Energy Efficiency of China's Industrial Sector: A Stochastic Frontier Analysis
Xiaobo Shen, Boqiang Lin
Subject: Social Sciences, Economics Keywords: malmquist productivity index; total factor energy efficiency; stochastic input distance function; China's industry
Based on stochastic frontier analysis and translog input distance function, this paper examines the total factor energy efficiency of China's industry using input-output data of 30 sub-industries from 2002 to 2014, and decomposes the changes in estimated total factor energy efficiency into the effects of technical change, technical efficiency change, scale efficiency change and input-mix effect. The results show that during this period the total factor energy efficiency in China's industry grows annually at a rate of 3.63%, technical change, technical efficiency change and input-mix effect contribute positively to the change in total factor energy efficiency, while scale efficiency change contributes negatively to it.
How Many Reindeer? UAV Surveys as an Alternative to Helicopter or Ground Surveys for Estimating Population Abundance in Open Landscapes
Ingrid Marie Garfelt Paulsen, Åshild Ønvik Pedersen, Richard Hann, Marie-Anne Blanchet, Isabell Eischeid, Charlotte Van Hazendonk, Virve Tuulia Ravolainen, Audun Stien, Mathilde Le Moullec
Subject: Biology, Ecology Keywords: Aaerial survey; animal detection; distance sampling; helicopter; monitoring; strip transect; Svalbard; total count; ungulate
Conservation of wildlife depends on precise and unbiased knowledge on the abundance and distribution of species. It is challenging to choose appropriate methods to obtain a sufficiently high detectability and spatial coverage matching the species characteristics and spatiotemporal use of the landscape. In remote regions, such as in the Arctic, monitoring efforts are often resource-intensive and there is a need for cheap and precise alternative methods. Here, we compare an uncrewed aerial vehicle (UAV; quadcopter) pilot-survey of the non-gregarious Svalbard reindeer to traditional population abundance surveys from ground and helicopter to investigate whether UAVs can be an efficient alternative technology. We found that the UAV survey underestimated reindeer abundance compared to the traditional abundance surveys when used at management relevant spatial scales. Observer variation in reindeer detection on UAV imagery was influenced by the RGB greenness index and mean blue channel. In future studies, we suggest to test long-range fixed-wing UAVs to increase the sample size of reindeer and area coverage and incorporate detection probability in animal density models from UAV imagery. In addition, we encourage focus on more efficient post-processing techniques, including automatic animal object identification with machine learning and analytical methods that account for uncertainties.
About the Ultimate Efficiency of Interference Binary Codes
Artem Sergeevich Adzhemov, Nicolay Yurievich Albov
Subject: Engineering, Electrical & Electronic Engineering Keywords: Code construction; minimum code distance; noise immunity; coding efficiency; theoretically achievable boundary; correction codes
The digital representation of various signals allows, at the subsequent stages of their transmission, to apply correction codes that provide protection against possible errors arising from the action of interference in the communication channel. At the same time, it is important that, with the required correcting ability, these codes have the maximum possible speed. The article presents the results of calculations for linear codes, showing their really achievable limiting capabilities.
Genetic Diversity and Population Structure in a Regional Collection of Kersting's Groundnut (Macrotyloma geocarpum (Harms) Maréchal & Baudet)
Konoutan M. Kafoutchoni, Eric E. Agoyi, Symphorien Agbahoungba, Achille E. Assogbadjo, Clément Agbangla
Subject: Life Sciences, Genetics Keywords: orphan crop; genotyping-by-sequencing; inbreeding; pre-breeding; population genetics; DArTseq; isolation by distance
Kersting's groundnut is an important source of protein and essential nutrients that contribute to food security in West Africa. However, the crop is still underexploited by the populations and under-researched by the scientific community. This study aimed to investigate the genetic diversity and population structure of 217 Kersting's groundnut accessions from five origins using 886 DArTseq markers. Gene diversity was low and ranged from 0.049 to 0.064. The number of private alleles greatly varied among populations (42–192) and morphotypes (40–339). Moderate to very high levels of selfing and inbreeding were observed among populations (s=56–85%, FIS=0.389–0.736) and morphotypes (s=57–82%, FIS=0.400–0.691). Moreover, little to very high genetic differentiations were observed among populations (0.006≤FIS≤0.371) and morphotypes (0.029≤FIS≤0.307). Analysis of molecular variance partitioned 38.5% of the genetic variation among and 48.7% within populations (P<0.001). Significant isolations by distance were detected between populations (R2=0.612, P=0.011) and accessions (R2=0.499, P<0.001). Discriminant analysis of principal components and neighbour joining consistently distinguished eight distinct clusters. These data provide a global picture of the existing genetic diversity for Kersting's groundnut and will guide the choice of breeding strategies to increase production.
Optimal Rescue Ship Locations Using Image Processing and Clustering
Cho-Young Jung, Sang-Lok Yoo
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: clustering-based optimization; location optimization; flood-filling algorithm; marine accident; rescue ship; shortest distance
Currently, maritime traffic is increasing with economic growth in several regions worldwide. However, this growth in maritime traffic has led to increased risk of marine accidents. These accidents have a higher probability of occurring in regions where geographical features, such as islands, are present. Further, the positioning of rescue ships in a particular ocean region with a high level of maritime activity is critical for rescue operations. This paper proposes a method for determining an optimal set of locations for stationing rescue ships in an ocean region with numerous accident sites in the Wando islands of South Korea. The computational challenge in this problem is identified as the positioning of numerous islands of varying sizes located in the region. Thus, the proposed method combines a clustering-based optimization method and an image processing approach that incorporates flood filling to calculate the shortest distance between two points in the ocean that detours around the islands. Experimental results indicate that the proposed method reduces the distance from rescue ships and each accident site by 5.0 km compared to the original rescue ship locations. Thus, rescue time is reduced.
An Effective FCM Approach of Similarity and Dissimilarity Measures with Alpha-Cut
Sayan Mukhopadhaya, Anil Kumar, Alfred Stein
Subject: Earth Sciences, Geoinformatics Keywords: Fuzzy c-Means (FCM) Classifier, Similarity and Dissimilarity measures, Distance, Fuzzy Error Matrix (FERM)
In this study, the fuzzy c- means classifier has been studied with nine other similarity and dissimilarity measures: Manhattan distance, chessboard distance, Bray-Curtis distance, Canberra, Cosine distance, correlation distance, mean absolute difference, median absolute difference and normalised squared Euclidean distance. Both single and composite modes were used with a varying weight constant (m) and also at different α-cuts. The two best single norms obtained were combined to study the effect of composite norms on the datasets used. An image to image accuracy check was conducted to assess the accuracy of the classified images. Fuzzy Error Matrix (FERM) was applied to measure the accuracy assessment outcomes for a Landsat-8 dataset with respect to the Formosat-2 dataset. To conclude FCM classifier with Cosine norm performed better than the conventional Euclidean norm. But, due to the incapability of the FCM classifier to handle noise properly, the classification accuracy was around 75%.
MADM Strategy Based on Some Similarity Measures in Interval Bipolar neuTrosophic Set Environment
Surapati Pramanik, Partha Pratim Dey, Florentin Smarandache
Subject: Mathematics & Computer Science, Applied Mathematics Keywords: neutrosophic set; interval bipolar neutrosophic set; multi-attribute decision making; distance measures; similarity measures
The paper investigates some similarity measures in interval bipolar neutrosophic environment for multi-attribute decision making problems. At first, we define Hamming and Euclidean distances measures between interval bipolar neutrosophic sets and establish their basic properties. We also propose two similarity measures based on the Hamming and Euclidean distance functions. Using maximum and minimum operators, we define new similarity measures and prove their basic properties. Using the proposed similarity measures, we propose a novel multi attribute decision making strategy in interval bipolar neutrosophic set environment. Lastly, we solve an illustrative example of multi attribute decision making and present comparison analysis to show the feasibility, applicability and effectiveness of the proposed strategy.
The Source of the Symbolic Numerical Distance and Size Effects
Attila Krajcsi, Gábor Lengyel, Petia Kojouharova
Subject: Behavioral Sciences, Cognitive & Experimental Psychology Keywords: numerical cognition; numerical distance effect; numerical size effect; analogue number system; discrete semantic system
Human number understanding is thought to rely on the analogue number system (ANS), working according to Weber's law. We propose an alternative account, suggesting that symbolic mathematical knowledge is based on a discrete semantic system (DSS), a representation that stores values in a semantic network, similar to the mental lexicon or to a conceptual network. Here, focusing on the phenomena of numerical distance and size effects in comparison tasks, first we discuss how a DSS model could explain these numerical effects. Second, we demonstrate that DSS model can give quantitatively as appropriate a description of the effects as the ANS model. Finally, we show that symbolic numerical size effect is mainly influenced by the frequency of the symbols, and not by the ratios of their values. This last result suggests that numerical distance and size effects cannot be caused by the ANS, while the DSS model might be the alternative approach that can explain the frequency-based size effect.
A Practical Approach for Determining Multi-Dimensional Spatial Rainfall Scaling Relations Using High Resolution Time Height Doppler Data from a Single Mobile Vertical Pointing Radar
A.R. Jameson
Subject: Earth Sciences, Atmospheric Science Keywords: Time-height rainfall rate profiles from MRR radars; Advection correction for conversion to height-distance profiles, Computing radial power spectra using height-distance profiles; Using derived radial power spectra for downscaling and upscaling
Rescaling of rainfall requires measurements of rainfall rates over many dimensions. This paper develops one approach using 10 m vertical spatial observations of the Doppler spectra of falling rain every 10 seconds over intervals varying from 15 up to 41minutes in two different locations and in two different years using two different Micro-Rain Radars (MRR). The transformation of the temporal domain into spatial observations uses the Taylor 'frozen' turbulence hypothesis to estimate an average advection speed over an entire observation interval. Thus, when no other advection estimates are possible, this paper offers a new approach for estimating the appropriate frozen turbulence advection speed by minimizing power spectral differences between the ensemble of purely spatial radial power spectra observed at all times in the vertical and those using the ensemble of temporal spectra at all heights to yield statistically reliable scaling relations. Thus, it is likely that, MRR and other vertically pointing Doppler radars may often help to obviate the need for expensive and immobile large networks of instruments in order to determine such scaling relations, but not the need of those radars for surveillance.
Geostatistical Modeling and Heterogeneity Analysis of Tumor Molecular Landscape
Morteza Hajihosseini, Payam Amini, Dan Voicu, Irina Dinu, Saumyadipta Pyne
Subject: Medicine & Pharmacology, Oncology & Oncogenics Keywords: spatial single-cell analysis; intratumor heterogeneity; kriging; spatial entropy; Was-serstein distance; cancer; RNA-seq
Intratumor heterogeneity (ITH) is associated with therapeutic resistance and poor prognosis in cancer patients, and attributed to genetic, epigenetic, and microenvironmental factors. We developed a new computational platform, GATHER, for geostatistical modeling of single cell RNA-seq data to synthesize high-resolution and continuous gene expression landscapes of a given tumor sample. Such landscapes allow GATHER to map the enriched regions of pathways of interest in the tumor space and identify genes that have spatial differential expressions at locations representing specific phenotypic contexts using measures based on optimal transport. GATHER provides new applications of spatial entropy measures for quantification and objective characterization of ITH. It includes new tools for insightful visualization of spatial transcriptomic phenomena. We illustrate the capabilities of GATHER using real data from breast cancer tumor to study hallmarks of cancer in the phenotypic contexts defined by cancer associated fibroblasts.
Emergency Online Learning in Low-Resource Contexts: Student Perceptions of Effective Engagement Strategies
Victoria Abou-Khalil, Samar Helou, Eliane Khalifé, MeiRong Alice Chen, Rwitajit Majumdar, Hiroaki Ogata
Subject: Social Sciences, Education Studies Keywords: online learning; emergency; low-resource; engagement; distance learning; student perception; survey; COVID-19; Moore framework
The COVID-19 pandemic forced the transition to emergency online learning without prior preparation or guidelines. This transition has been particularly challenging in developing countries and low-resource contexts and hindered student engagement. We aim to identify the engagement strategies which students, engaging in emergency online learning in low-resource contexts, perceive to be effective. We conducted a sequential mixed-methods study based on Moore's interaction framework for distance education. First, we conducted a literature review and interviewed ten teachers and ten students to identify a list of engagement strategies. Then, we designed a questionnaire that examines student perceptions of these strategies. We administered the questionnaire to 313 students engaging in emergency online learning in low-resource contexts. Our analysis results showed that student-content engagement strategies, e.g. screen sharing, summaries, and class recordings, are perceived as the most effective, closely followed by student-teacher strategies, e.g. Q&A sessions and reminders. Student-student strategies, e.g. group chat and collaborative work, are perceived as the least effective. The perceived effectiveness of engagement strategies depends on the context and the students' characteristics, e.g. gender, major, and technology access. To support instructors, instructional designers, and researchers we propose a ten-level guide for engaging students during emergency online classes in low-resource contexts.
Long-Distance Water Transport of Land Plants Using the Thermodynamic Sorption Principle
Karlheinz Hahn
Subject: Biology, Physiology Keywords: Plant water transport, plant long-distance water transport, sorption hypothesis, cohesion theory, cohesion-tension theory
In the case of vascular plants the process of water loss by leafs and water absorption by the root is well known. There is agreement on the passive nature of long-distance moisture movement in the dead cells of the xylem; however, controversy exists focusing on the long-distance water transport principle. Hales (1726) founded a view of bulk flow based on water suction after experiments with cut twigs. The previous doctrine of long-distance water transport within vessel elements and tracheid of the xylem of intact plants – the relevant cohesion theory in text books – was developed mainly by Boehm (1893), Renner (1911) and Dixon (1914) with plant artefacts. Water movement according to this theory is based on an assumed hydrodynamic bulk fluid flow in xylem in continuous water columns (free of water vapour space), under tension, according to the law of Poiseuille (see e.g. Dixon 1914). Physically hydrodynamics is part of fluid mechanics, as a result Poiseuille's law is usually valid only for hydrodynamic bulk flow in ideal capillaries (Sutera & Skalak, 1993). Besides the basic requirement for transport, according to cohesion theory, the existence of ideal capillaries is not compatible with either: "Because vessel elements and tracheid do not stand as ideal capillaries. …" (Bresinsky et al. 2008, translated from German). Unlike ideal capillaries, the walls of vessel elements and tracheid interact with the transported water. These walls are able to function as a source or as a sink for the transported water because of interaction with the cell walls. With the interaction, vessel elements and tracheid, part of the xylem, can shrink and swell, unlike ideal capillaries. Because the xylem (in woody plants part of the wood) is inconsistent with the basic law of fluid flow, the equation of mass balance (Zimmermann et al. 2004) and cohesion theory are not strictly followed.Many plant physiologists view the cohesion theory as appropriate, however, this theory remains controversial, i.e. by Eisenhut (1988), Laschimke (1990) and Hahn (1993). Nultsch (1996) gives doubts referring to the present doctrine of plant water transport. Zimmermann et al. (2004) reject the cohesion theory and conclude: "... that the arguments of the proponents of the Cohesion Theory are completely misleading" (Zimmermann et al. 2004). Hence cohesion theory can be treated as inapplicable and the question arises: how does water transport in fact function? In the following, it is gone into in more detail. A sorption hypothesis of actual water transport, based on empirical fact, shall be addressed in this paper.
Experimental Study on Improvement of Performance by Wave Form at Cathode Channels in PEM Fuel Cell
Sun-Joon Byun, Zhen Huan Wang, Jun Son, Young-Chul Kwon, Dong-Kurl Kwak
Subject: Engineering, Energy & Fuel Technology Keywords: Wave form; PEMFC; Cathode channel; Gas diffusion layer (GDL); Adhesive distance (AD); Expansion ratio (ER)
We propose a wave-like design on the surface of cathode channels (wave form cathode channels) to improve oxidant delivery to gas diffusion layers (GDLs) [1-2]. We performed experiments using PEMFCs combined with wave form surface design on cathodes. We varied the factors of the distance between wave-bumps (the Adhesive distance, AD), and the size of the wave-bumps (the Expansion ratio, ER). The ADs are 3, 4, and 5 times the size of the half-circle bump's radius, and the ERs are 1/1.5, 1/2, and 1/3 times the channel's height. We evaluated the performances of the fuel cells, and compared the current-voltage (I-V) relations. For comparison, we prepared PEMFCs with conventional flat-surfaced oxygen channels. Our aim in this work is to identify fuel cell operation by modifying the surface design of channels, and ultimately to find the optimal design of cathode channels that will maximize fuel cell performance.
Not Hydraulic but an Adsorption Water Transport occurs in the Xylem of Land Plants
Subject: Biology, Forestry Keywords: Adsorption water movement; adsorption hypothesis; plant long-distance water transport; thermodynamic water movement; not-hydraulic movement
Ad- and desorption forces move water in living xylem/wood from the root to the leaf thermodynamically. The doctrine of plant water transport, the so-called cohesion- or cohesion-tension theory, postulates however that the process is physically based on a hydraulic fluid flow with negative pressure in water conducting tubes originating from the leaves. Lower pressure (suction) driven volume flow is physically a branch of mechanics. Moisture absorbed from the soil via the root is thought to be pulled up the stem by the leaves in continuous and tensioned threads of water. It is assumed that the hydraulic Hagen/Poiseuille flow law, derived for tubes, applies in the xylem. In a textbook of botany you can find the opinion: "Just as the pipes of a water pipe supply necessary water to each household, leaf nerves supply water and nutrient salts to each individual cell." (Translated from German). Many plant physiologists consider this hydraulic principle to be correct, but it does not remain unchallenged. Doubts are repeatedly expressed. The question arises: How does water transport actually take place? It is shown how the diffusive/adsorption transport principle works. The partial dehydration (desorption) of the plant, driven by the diffusive process of transpiration, forms a combined concentration and adsorption-site gradient for water in the xylem matrix. Especially with open stomata the lowest moisture concentration and the highest number of adsorption-sites for water (sites with free van der Waals forces), can be found in the mesophyll cell walls at the liquid/vapor boundary in the leaf. The water taken up by the root moves spontaneously in the direction of this boundary and can thus partially or completely compensate for the existing concentration- and adsorption-site- differences for water. Thus, a thermodynamic overlapping diffusive/adsorptive movement of moisture along the stationary xylem/wood takes place. After the introduction and a review of some controversies with cohesion theory, the physiology of the processes associated with long-distance water displacement is mentioned below. A thermodynamic adsorption hypothesis of the natural water transport in plants, based on known facts, is presented.
Moving Object Detection Based on a Combination of Kalman Filter and Median Filtering
Diana Kalita, Pavel Lyakhov
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: Kalman filter; median filter; impulse noise; estimate prediction; object distance determination; lidar; value calibration; point cloud.
The task of determining the distance from one object to another is one of the important tasks solved in robotics systems. Conventional algorithms rely on an iterative process of predicting distance estimates, which results in an increased computational burden. Algorithms used in robotic systems should require minimal time costs, as well as be resistant to the presence of noise. To solve these problems, the paper proposes an algorithm for Kalman combination filtering with a Goldschmidt divisor and a median filter. Software simulation showed an increase in the accuracy of predicting the estimate of the developed algorithm in comparison with the traditional filtering algorithm, as well as an increase in the speed of the algorithm. The results obtained can be effectively applied in various computer vision systems.
A Year of Online Classes Amid COVID-19 Pandemic: Advantages, Problems, and Suggestions of Economics Students at a Bangladeshi Public University
Musharrat Shabnam Shuchi, Sayeda Chandra Tabassum, MMK Toufique
Subject: Keywords: COVID-19; online learning; pandemic; online education; Bangladesh; students' perceptions; higher education; distance learning; online classes
Though there have been works highlighting the advantages and disadvantages of online learning, no study focused on university-level economics students. None of the studies explored students' opinions about improving the quality and effectiveness of online classes. Many used questionable samples, closed-ended questions, and all those researches were carried out at the beginning of online classes. In this paper, we overcome these limitations of earlier studies. Using a convenience sampling technique and open-ended questions, we collect data from 154 university-level economics students after being exposed to the online class for a year. Some advantages of online classes are: students can do classes from home without being exposed to health risks, easily accessible, flexible class schedule, students remained connected with the study, it saves costs, reduce the likelihood of semester loss, easy to understand, less stressful, and learning new technologies. Major problems from students' perspectives include network problems, difficulties in understanding the topic, unsuitable for mathematical courses, concentration problem, class not interactive, financial constraint, adverse health impacts, device issues, power outages, unfamiliarity with digital technology, internet problem, and unfixed class-schedule. Disadvantages outnumbered advantages. Students made several suggestions to improve the quality and effectiveness of online classes. Some of the vital suggestions are: using state-of-the-art digital tools, recording and uploading lectures, resolving internet issues, holding classes regularly, higher efforts to make the topics easier, resolving network issues, lowering class duration, institutional support, implementing a fixed class schedule, and introducing online evaluation system.
Edge-based Color Image Segmentation using Particle Motion on Vector Fields Derived from Local Color Distance Images
Wutthichai Phornphatcharaphong, Nawapak Eua-Anant
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: Color images segmentation; Particle Motion; Local Color Distance Images; Normal compressive vector field; Edge vector field
This paper presents an Edge-based color image segmentation derived from the method of Particle Motion in a Vector Image Fields (PMVIF) that could previously be applied only to monochrome images. Instead of using an edge vector field derived from a gradient vector field and a normal compressive vector field derived from a Laplacian-gradient vector field, two novel orthogonal vector fields, directly computed from a color image, one parallel and another orthogonal to the edges, were used in the model to force a particle to move along the object edges. The normal compressive vector field is derived from the center-to-centroid vectors of local color distance images. Next, the edge vector field is derived by taking the normal compressive vector field, multiplied by differences of auxiliary image pixels to obtain a vector field analogous to a Hamiltonian gradient vector field. Using the PASCAL Visual Object Classes Challenge 2012 (VOC2012) and the Berkeley Segmentation Data Set and Benchmarks 500 (BSDS500), the benchmark score of the proposed method is provided in comparison with those of the traditional PMVIF, Watershed, SLIC, K-means, Mean shift, and JSEG. The proposed method yields better RI, GCE, NVI, BDE, Dice coefficients, faster computation time, and noise resistance.
Graphs and Binary Systems
Hee Sik Kim, J. Neggers, Sun Shin Ahn
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: binary system(groupoid); minimum (mutual) covering set; (mutual) shortest distance; (di)frame graph; d/BCK-algebra
In this paper, we observe that if X is a set and (Bin(X), □) is the semigroup of binary systems on X, then its center ZBin(X) consists of the locally-zero-semigroups and that these can be modeled as (simple) graphs and conversely. Using this device we show that we may obtain many results of interest concerning groupoids by reinterpreting graph theoretical properties and at the same time results on graphs G may be obtained by considering them as elements of centers of the semigroups of binary systems (Bin(X), □) where X = V(G), the vertex set of G.
Normal and Abnormal Human Face Detection Based on DCT and FFT Techniques - A Proposed Method
Samir Bandyopadhyay, Shawni Dutta, Vishal Goyal, Payal Bose
Subject: Keywords: Face Detection; Euclidean Distance; Fast Fourier Transformation; Discrete Cosine Transformation; Facial Parts Detection; Frequency domain; Spatial domain
In today's world face detection is the most important task. Due to the chromosomes disorder sometimes a human face suffers from different abnormalities. For example, one eye is bigger than the other, cliff face, different chin-length, variation of nose length, length or width of lips are different, etc. For computer vision currently this is a challenging task to detect normal and abnormal face and facial parts from an input image. In this research paper a method is proposed that can detect normal or abnormal faces from a frontal input image. This method used Fast Fourier Transformation (FFT) and Discrete Cosine Transformation of frequency domain and spatial domain analysis to detect those faces.
Estimating Lower Limb Kinematics using a Lie Group Constrained Extended Kalman Filter with a Reduced Wearable IMU Count and Distance Measurements
Luke Wicent F. Sy, Nigel H. Lovell, Stephen J. Redmond
Subject: Engineering, Automotive Engineering Keywords: Lie group; Constrained extended Kalman filter; Gait analysis; Motion capture; Pose estimation; Wearable devices; IMU; Distance measurement
Tracking the kinematics of human movement usually requires the use of equipment that constrains the user within a room (e.g., optical motion capture systems), or requires the use of a conspicuous body-worn measurement system (e.g., inertial measurement units (IMUs) attached to each body segment). This paper presents a novel Lie group constrained extended Kalman filter to estimate lower limb kinematics using IMU and inter-IMU distance measurements in a reduced sensor count configuration. The algorithm iterates through the prediction (kinematic equations), measurement (pelvis height assumption/inter-IMU distance measurements, zero velocity update for feet/ankles, flat-floor assumption for feet/ankles, and covariance limiter), and constraint update (formulation of hinged knee joints and ball-and-socket hip joints). The knee and hip joint angle root-mean-square errors in the sagittal plane for straight walking were 7.6±2.6∘ and 6.6±2.7∘, respectively, while the correlation coefficients were 0.95±0.03 and 0.87±0.16, respectively. Furthermore, experiments using simulated inter-IMU distance measurements show that performance improved substantially for dynamic movements, even at large noise levels (σ=0.2 m). However, further validation is recommended with actual distance measurement sensors, such as ultra-wideband ranging sensors.
What is the Real Shape of the Hubble Diagram, z = H0*D or z+1= eH0*t? Analysis of the SN1a Supernovae and Gamma Ray Burst Redshift/Magnitude Data including the High Redshift Range up to z = 8.1
Laszlo Marosi
Subject: Keywords: galaxies; distances and redshift; high redshift; stars; Gamma ray bursts; individual; supernovae; individual; cosmology; distance scale; observations
Analyses of the Hubble diagrams are presented for SN1a supernovae and gamma ray bursts in the redshift ranges z = 0.01–1.3 and 0.034–8.1, respectively. Data are presented on the typical z/μ scale and also on the less common yet increasingly sensitive photon flight time t/(z+1) scale. The primary conclusion is that on the basis of the presently accessible data in the redshift range z = 0.01–8.1, the slope of the Hubble diagram is, or is extremely close to, exponential.
Working Paper SHORT NOTE
Comments on "On a Continuum Model for Avalanche Flow and Its Simplified Variants"' by S. S. Grigorian and A. V. Ostroumov
Dieter Issler
Subject: Keywords: Snow avalanches; mathematical models; snow entrainment; Voellmy and Grigorian friction laws; hydraulic models; runout distance; analytic solutions
Online: 6 February 2020 (09:11:48 CET)
This note first summarizes the history of the manuscript "On a Continuum Model for Avalanche Flow and Its Simplified Variants" by Grigorian and Ostroumov―published in the Special Issue on snow avalanche dynamics of Geosciences―since the early 1990s and explains the guiding principles in editing it for publication. The changes are then detailed and some explanatory notes given for the benefit of readers who are not familiar with the early Russian work on snow avalanche dynamics. Finally, the editor's personal views as to why he still considers this paper of relevance for avalanche dynamics research today are presented in brief essays on key aspects of the paper, namely the role of simple and complex models in avalanche research and mitigation work, the status and possible applications of Grigorian's stress-limited friction law, and non-monotonicity of the dynamics of the Grigorian–Ostroumov model in the friction coefficient. A comparison of the erosion model proposed by those authors with two other models suggests to enhance it with an additional equation for the balance of tangential momentum across the shock front. A preliminary analysis indicates that continuous scouring entrainment is possible only in a restricted parameter range and that there is a second erosion regime with delayed entrainment.
Analytical Study of Colour Spaces for Plant Pixel Detection
Pankaj Kumar, Stanley J. Miklavcic
Subject: Mathematics & Computer Science, Artificial Intelligence & Robotics Keywords: Plant phenotyping, Plant pixel classification, Colour space, , Gaussian mixture model, Earth mover distance, Variance ratio, Plant segmentation.
Segmentation of a region of interest is an important pre-processing step for many colour image analysis techniques. Similarly segmentation of plant in digital images is an important preprocessing step in phenotying plants by image analysis. In this paper we present an analytical study to statistically determine the suitability of colour space representation of an image to best detect plant pixels and separate them from background pixels. Our hypothesis is that the colour space representation in which the separation of the distributions representing plant pixels and background pixels is maximized would be the best for detection of plant pixels. The two classes of pixels are modelled as a Gaussian mixture model (GMM). In our GM modelling we don't make any prior assumption about the number of Gaussians in the model. Rather a constant bandwidth mean-shift filter is used to cluster the data and the number of clusters and hence the number of Gaussians is automatically determined. Here we have analysed following representative colour spaces like $RGB$, $rgb$, $HSV$, $Ycbcr$ and $CIE-Lab$. This is because these colour spaces represent several other similar colour spaces and also an exhaustive study of all the colour space will be too voluminous. We also analyse the colour space feature from the two-class variance ratio perspective and compare the results of our hypothesis with this metric. The dataset for this empirical study consist of 378 digital images of plants and their manual segmentation. Dataset consist of various species of plants (arabidopsi, tobacco, wheat, rye grass etc.) imaged under different lighting conditions, indoor and outdoor, controlled and uncontrolled background. In results we obtain better segmentation of the plants in $HSV$ colour space, which is supported by its Earth mover distance (EMD) on the GMM distribution of plant and background pixels.
The Ecology of Plant Interactions: A Giant with Feet of Clay
Ciro Cabal, Fernando Valladares, Ricardo Martinez-Garcia
Subject: Biology, Ecology Keywords: plant-plant interactions; stress gradient hypothesis; functional trait ecology; inter-plant distance; individual-based models; consumer-resource models
Ecologists use the net biotic interactions among plants to predict fundamental ecosystem features. Following this approach, ecologists have built a giant body of theory founded on observational evidence. However, due to the limitations that a phenomenological approach raises both in empirical and theoretical studies, an increasing number of scientists claim the need for a mechanistic understanding of plant interaction outcomes, and a few studies have taken such a mechanistic approach. In this synthesis, we propose a modeling framework to study the plant interaction mechanistically. We first establish a conceptual ground to frame plant-plant interactions, and then, we propose to formalize this research line theoretically developing a family of individual-based, spatially-explicit models in which biotic interactions are an emergent property mediated by the interaction between plants' functional traits and the environment. These models allow researchers to evaluate the strength and sign of biotic interactions under different environmental scenarios and thus constitute a powerful tool to investigate the mechanisms underlying facilitation, species coexistence, or the formation of vegetation spatial patterns.
Prerequisites for Shallow-Transfer Machine Translation of Mordvin Languages: Language Documentation with a Purpose
Jack Rueter, Mika Hämäläinen
Subject: Mathematics & Computer Science, Other Keywords: Erzya, Moksha, Uralic, Shallow-transfer machine translation, Measurable language research, Measurable language distance, Finite-State Morphology, Universal Dependencies
This paper presents the current lexical, morphological, syntactic and rule-based machine translation work for Erzya and Moksha that can and should be used in the development of a roadmap for Mordvin linguistic research. We seek to illustrate and outline initial problem types to be encountered in the construction of an Apertium-based shallow-transfer machine translation system for the Mordvin language forms. We indicate reference points within Mordvin Studies and other parts of Uralic studies, as a point of departure for outlining a linguistic studies with a means for measuring its own progress and developing a roadmap for further studies.
Brain Network Modeling Based on Mutual Information and Graph Theory for Predicting the Connection Mechanism in the Development of Alzheimer's Disease
Si Shuaizong, Wang Bin, Liu Xiao, Yu Chong, Ding Chao, Zhao Hai
Subject: Physical Sciences, Applied Physics Keywords: Alzheimer's disease; graph theory; mutual information; network model; connection mechanism; functional magnetic resonance imaging; topological structures; anatomical distance
Abnormal connections in brain networks of healthy people always bring the problems of cognitive impairments and degeneration of specific brain circuits, which may finally result in Alzheimer's disease (AD). Exploring the development of the brain from normal controls (NC) to AD is an essential part of human research. Although connections changes have been found in the development, the connection mechanism that drives these changes remain incompletely understood. The purpose of this study is to explore the connection changes in brain networks in the process from NC to AD, and uncover the underlying connection mechanism that shapes the topologies of AD brain networks. In particular, we propose a model named MINM from the perspective of topology-based mutual information to achieve our aim. MINM concerns the question of estimating the connection probability between two cortical regions with the consideration of both the mutual information of their observed network topologies and their Euclidean distance in anatomical space. In addition, MINM considers establishing and deleting connections, simultaneously, during the networks modeling from the stage of NC to AD. Experiment results show that MINM is sufficient to capture an impressive range of topological properties of real brain networks such as characteristic path length, network efficiency, and transitivity, and it also provides an excellent fit to the real brain networks in degree distribution compared to experiential models. Thus, we anticipate that MINM may explain the connection mechanism for the formation of the brain network organization in AD patients.
Surprising Drought Tolerance of Fir (Abies) Species between Past Climatic Adaptation and Future Projections Reveals New Chances for Adaptive Forest Management
Csaba Mátyás, Jaroslav Dostál, František Beran, Jiří Čáp, Martin Fulín, Monika Vejpustková, Gregor Božič, Pál Balázs, Josef Frýdl
Subject: Life Sciences, Biochemistry Keywords: common garden; climate change; silver fir; grand fir; Balkan firs; drought stress; provenance test; resilience; climate transfer distance; adaptation
Research Highlights: Data of advanced-age provenance tests were reanalyzed applying a new approach, to directly estimate the growth of populations at their original sites under individually generated future climates. The results reveal surprisingly high resilience potential of fir species. Background and Objectives: The growth and survival of silver fir under future climatic scenarios is insufficiently investigated at the xeric limits. The selective signature of past climate determining the current and projected growth was investigated to analyze the prospects of adaptive silviculture and assisted transfer of silver fir populations, and of the introduction of non-autochthonous species. Materials and Methods: Hargreaves' climatic moisture deficit was selected to model height responses of adult populations. Climatic transfer distance was used to assess the relative drought stress of populations at the test site, relating these to the past conditions to which the populations had adapted. ClimateEU and ClimateWNA pathway RCP8.5 data served to determine individually past, current, and future moisture deficit conditions. Beside silver fir, other fir species from South Europe and the American Northwest were also tested. Results: Drought tolerance profiles explained the responses of transferred provenances and predicted their future performance and survival. Silver fir displayed significant within-species differentiation regarding drought stress response. Applying the assumed drought tolerance limit of 100mm relative moisture deficit, most of the tested silver fir populations seem to survive their projected climate at their origin until the end of the century. Survival is likely also for transferred Balkan fir species and for grand fir populations, but not for the Mediterranean species. Conclusions: The projections are less dramatic than provided by usual field assessments. Some results contradict generally accepted concepts. The method fills the existing gap between experimentally determined adaptive response and the predictions needed for management decisions. It also underscores the unique potential of provenance tests.
DeepSOCIAL: Social Distancing Monitoring and Infection Risk Assessment in COVID-19 Pandemic
Mahdi Rezaei, Mohsen Azarmi
Subject: Mathematics & Computer Science, Artificial Intelligence & Robotics Keywords: Social Distancing; COVID-19; Human Detection and Tracking; Distance Estimation, Deep Convolutional Neural Networks; Crowd Monitoring, Inverse Perspective Mapping
Social distancing is a recommended solution by the World Health Organisation (WHO) to minimise the spread of COVID-19 in public places. The majority of governments and national health authorities have set the 2-meter physical distancing as a mandatory safety measure in shopping centres, schools and other covered areas. In this research, we develop a Deep Neural Network-based Model for automated people detection, tracking, and inter-people distances estimation in the crowd, using common CCTV security cameras. The proposed DNN model along with an inverse perspective mapping technique leads to a very accurate people detection and social distancing monitoring in challenging conditions, including people occlusion, partial visibility, and lighting variations. We also provide an online infection risk assessment scheme by statistical analysis of the Spatio-temporal data from the moving trajectories and the rate of social distancing violations. We identify high-risk zones with the highest possibility of virus spread and infection. This may help authorities to redesign the layout of a public place or to take precaution actions to mitigate high-risk zones. The efficiency of the proposed methodology is evaluated on the Oxford Town Centre dataset, with superior performance in terms of accuracy and speed compared to three state-of-the-art methods.
Gravitation: Immediate Action at a Distance or Close Up Events?
Harmen Henricus Hollestelle
Subject: Physical Sciences, General & Theoretical Physics Keywords: Gravitation, General relativity, Immediate action at a distance, Spirituality, Tactile interaction, Dark energy, Dark matter, Cosmology, Wave function reduction
Is immediate action at a distance, like gravitational attraction, imaginable using the contrary concept of close up, tactile, events? Tactile events, defined with the term 'tap-tapping' as a blind man does, described in a two-way spiritual interaction theory, are implemented in physics to understand gravitation from this respect. The quantum mechanical wave function reduction during measurements receives a new approach. Formulated is a new proof for Einstein's Equivalence Principle, extending it beyond locality, and a sketch of how tactile interaction could explain dark energy and an accelerated expansion of the universe. Dark energy and dark matter are examples starting from which to discuss properties of matter and space and gravitation as immediate tactile action rather than mediated action such as electromagnetism.
Building 2D Model of Compound Eye Vision for Machine Learning
Leonid B. Sokolinsky, Artem E. Starkov
Subject: Mathematics & Computer Science, Artificial Intelligence & Robotics Keywords: robot vision; compound eye; two-dimensional model; distance measurement; azimuth measurement; deep learning; training data set generation; deep neural network
This paper presents a two-dimensional mathematical model of compound eye vision. Such a model is useful for solving navigation issues for autonomous mobile robots on the ground plane. The model is inspired by the insect compound eye that consists of ommatidia, which are tiny independent photoreception units, each of which combines a cornea, lens, and rhabdom. The model describes the planar binocular compound eye vision, focusing on measuring distance and azimuth to a circular feature with an arbitrary size. The model provides a necessary and sufficient condition for the visibility of a circular feature by each ommatidium. On this basis, an algorithm is built for generating a training data set to create two deep neural networks (DNN): the first detects the distance, and the second detects the azimuth to a circular feature. The hyperparameter tuning and the configurations of both networks are described. Experimental results showed that the proposed method could effectively and accurately detect the distance and azimuth to objects.
On the Role of Cosmic Mass in Understanding the Relationships among Galactic Dark Matter, Visible Matter and Flat Rotation Speeds
U.V.S. Seshavatharam, S. Lakshminarayana
Subject: Physical Sciences, Astronomy & Astrophysics Keywords: Planck mass; Mach's principle; distance cosmic mass; galactic visible mass; galactic dark mass; galactic flat rotation speeds; time dependent reference mass;
With reference to our recently proposed Planck Scale White Hole Cosmology (PS-WHC) or Flat Space Cosmology (PS-FSC), we make an attempt to quantify galactic dark matter and flat rotation speeds in terms of galactic visible matter and cosmic mass. Considering recently observed dwarf galaxies having very little dark matter and assuming a time dependent reference mass unit of $M_X\cong \left(\mbox{3.0 to 4.0}\right)\times 10^{38}$ kg, we suggest an empirical relation for galactic dark matter $M_d$ via galactic visible mass $M_v$ as,$M_d \cong \frac{M_v^{3/2}}{M_X^{1/2}}$. This relation helps in fitting flat rotation speeds starting from 8 km/sec (for Segue 2) to 500 km/sec (for UGC12591). Modifying MOND's galactic flat rotation speed relation with Hubble mass $M_0\cong \left(\frac{c^3}{2GH_0}\right)$ of the universe, ratio of galactic flat rotation speed $V_G$ to speed of light $c$ can be shown to be approximately $\frac{V_G}{c} \cong 0.5 \left(\frac{M_v}{M_0}\right)^{1/4}$. Considering the sum of galactic dark matter and visible matter, ratio of galactic flat rotation speed to speed of light can be shown to be approximately $\frac{V_G}{c}\cong 0.25 \left(\frac{M_v+M_d}{M_0}\right)^{1/4}$. With further study, dark matter's nature, effect and distribution can be understood in terms of visible matter's extended gravity and extended theories of gravity can be understood with 'distance cosmic mass' rather than the empirical 'minimum acceleration'.
Badging for Sustainable Development: Applying EdTech Micro-Credentials for Advancing SDGs amongst Mountain and Pastoralist Societies
Randall Gwin, Marc Foggin
Subject: Social Sciences, Education Studies Keywords: education; skills development; online distance learning; credentials; open badges; blockchain; sustainable livelihoods; sustainable mountain development; traditional knowledge; culture; Kyrgyzstan; Central Asia
Online: 27 March 2020 (03:07:11 CET)
Mountain and pastoralist societies around the world have for centuries sustained their livelihoods and cultures by accumulating specialist knowledge about their local and regional socio-ecological environments. Developing traditional knowledge and customary practices takes time, sometimes spanning across generations. As macro-level changes to social and natural environment are now taking place, such as globalization and climate change, local communities could potentially also benefit from complementary, suitably adapted educational opportunities for sustainable development. However, access to education has often required moving to urban centres, which can weaken community structures and cohesion, and could also foster increased dependence on external specialists, providers or decision-makers. Careful introduction of emerging Educational Technologies could alleviate and possibly reverse such trends as mobile Internet access spreads to remote areas. This paper examines the role of education in sustainable development and specifically explores the potential for two educational innovations, open badges and blockchain, to provide a new construct for transformation in sustainable development amongst mountain and pastoralist societies. These technologies could not only facilitate education through online distance learning, but also allow geographically remote populations to highlight the value of their traditional knowledge and to engage more comprehensively in their changing worlds.
Estimation of Temperature Recovery Distance and the Influence of Heat Pump Discharge on Fluvial Ecosystems
Jaewon Jung, Jisu Nam, Jungwook Kim, Young Hye Bae, Hung Soo Kim
Subject: Engineering, Energy & Fuel Technology Keywords: hydrothermal energy; river-water heat pump; water temperature recovery distance; heat transfer equation; Environmental Fluid Dynamic Code (EFDC); Han river basin
Temperature differences between the atmosphere and river water allow rivers to be used as a hydrothermal energy source. The river-water heat pump system is a relatively non-invasive renewable energy source; however, effluent discharged from the heat pump can cause downstream temperature changes which may impact sensitive fluvial ecosystems. In this study, the water temperature recovery distance of the effluent was estimated for a river section in the Han River Basin, Korea, using the heat transfer equation and the Environmental Fluid Dynamic Code (EFDC) model. Results showed that, compared to the EFDC model, the heat transfer equation tended to overestimate the water temperature recovery distance due to its simplified assumptions. The water temperature recovery distance could also be used as an objective indicator to decide the reuse of downstream river water. Furthermore, as the river system was found to support an endangered fish species that is sensitive to water environment changes, care should be taken to exclude the habitats of protected species affected by water temperatures within water temperature recovery distance.
Weighing Cosmological Models with SNe Ia and GRB Redshift Data
Rajendra P. Gupta
Subject: Physical Sciences, Astronomy & Astrophysics Keywords: galaxies; supernovae; GRB; distances and redshifts; cosmic microwave background radiation; distance scale; cosmology theory; cosmological constant; Hubble constant; general relativity; TMT
Many models have been proposed to explain the intergalactic redshift using different observational data and different criteria for the goodness-of-fit of a model to the data. The purpose of this paper is to examine several suggested models using the same supernovae Ia data and gamma-ray burst (GRB) data with the same goodness-of-fit criterion and weigh them against the standard Λ CDM model. We have used the redshift – distance modulus ( z−μ ) data for 580 supernovae Ia with 0.015≤z≤1.414 to determine the parameters for each model, and then use these model parameter to see how each model fits the sole SNe Ia data at z=1.914 and the GRB data up to z=8.1 . For the goodness-of-fit criterion, we have used the chi-square probability determined from the weighted least square sum through non-linear regression fit to the data relative to the values predicted by each model. We find that the standard ΛCDM model gives the highest chi-square probability in all cases albeit with a rather small margin over the next best model - the recently introduced nonadiabatic Einstein de Sitter model. We have made ( z−μ ) projections up to z=1096 for the best four models. The best two models differ in μ only by 0.328 at z=1096 , a tiny fraction of the measurement errors that are in the high redshift datasets.
A Possible Explanation for the Twin Paradox and Action at a Distance—The Relative Independence of Space and the Absoluteness of Simultaneity
Sheng Qin
Subject: Physical Sciences, General & Theoretical Physics Keywords: the absoluteness of simultaneity; the relative independence of space; special relativity; the problem of measuring; the action at a distance; cosmic inflation
This paper is mainly based on a stricter premise of the twin paradox and the assumption of inertial frame, discusses the properties of time and space under the premise of complete symmetry, and draws an interesting conclusion: the simultaneity of different reference frames is possible realized, and the space is relatively independent. And based on this, the twin paradox, cosmic inflation, ultra-distance action of quantum entanglement, microscopic space motion of particles, measurement problems and other phenomena are tentatively explained from a new angle. This interpretation is exploratory and new. At the same time, the author also proposes an experimental way to test the relative independence of space.At the same time, this paper attempts to strictly prove that Einstein's definition of simultaneity and spatial absoluteness in special relativity may be problematic.
Possible Unification of Quantum Mechanics and General Relativity Theory Based on the Three-Dimensional Quantized Spaces
Jae-Kwang Hwang
Subject: Physical Sciences, Particle & Field Physics Keywords: Quantum mechanics; General and special relativity theories; Origins of the energy and mass; Space-time volume and distance; Origin of quantum wave function; Three-dimensional quantized spaces; Quantum entanglement; Space-time curvatures and quantum metrics
Three-dimensional quantized space model is newly introduced. Quantum mechanics and relativity theory are explained in terms of the warped three-dimensional quantized spaces with the quantum time width (Dt=tq). The energy is newly defined as the 4-dimensional space-time volume of E = cDtDV in the present work. It is shown that the wave function of the quantum mechanics is closely related to the warped quantized space shape with the space time-volume. The quantum entanglement and quantum wave function collapse are explained additionally. The special relativity theory is separated into the energy transition associated with the space-time shape transition of the matter and the momentum transition associated with the space-time location transition. Then, the quantum mechanics and the general relativity theory are about the 4-dimensional space-time volume and the 4-dimensional space-time distance, respectively. | CommonCrawl |
Diagnosing hospital bacteraemia in the framework of predictive, preventive and personalised medicine using electronic health records and machine learning classifiers
Oscar Garnica ORCID: orcid.org/0000-0001-5064-25871,
Diego Gómez2,
Víctor Ramos2,
J. Ignacio Hidalgo ORCID: orcid.org/0000-0002-3046-63681 &
José M. Ruiz-Giardín ORCID: orcid.org/0000-0001-9459-73863
EPMA Journal volume 12, pages 365–381 (2021)Cite this article
The bacteraemia prediction is relevant because sepsis is one of the most important causes of morbidity and mortality. Bacteraemia prognosis primarily depends on a rapid diagnosis. The bacteraemia prediction would shorten up to 6 days the diagnosis, and, in conjunction with individual patient variables, should be considered to start the early administration of personalised antibiotic treatment and medical services, the election of specific diagnostic techniques and the determination of additional treatments, such as surgery, that would prevent subsequent complications. Machine learning techniques could help physicians make these informed decisions by predicting bacteraemia using the data already available in electronic hospital records.
This study presents the application of machine learning techniques to these records to predict the blood culture's outcome, which would reduce the lag in starting a personalised antibiotic treatment and the medical costs associated with erroneous treatments due to conservative assumptions about blood culture outcomes.
Six supervised classifiers were created using three machine learning techniques, Support Vector Machine, Random Forest and K-Nearest Neighbours, on the electronic health records of hospital patients. The best approach to handle missing data was chosen and, for each machine learning technique, two classification models were created: the first uses the features known at the time of blood extraction, whereas the second uses four extra features revealed during the blood culture.
The six classifiers were trained and tested using a dataset of 4357 patients with 117 features per patient. The models obtain predictions that, for the best case, are up to a state-of-the-art accuracy of 85.9%, a sensitivity of 87.4% and an AUC of 0.93.
Our results provide cutting-edge metrics of interest in predictive medical models with values that exceed the medical practice threshold and previous results in the literature using classical modelling techniques in specific types of bacteraemia. Additionally, the consistency of results is reasserted because the three classifiers' importance ranking shows similar features that coincide with those that physicians use in their manual heuristics. Therefore, the efficacy of these machine learning techniques confirms their viability to assist in the aims of predictive and personalised medicine once the disease presents bacteraemia-compatible symptoms and to assist in improving the healthcare economy.
The paradigm shift from reactive to predictive, preventive and personalised medicine
Current best healthcare practices promote the assumption of a predictive medicine tailored to the patient under the Predictive, Preventive and Personalised Medicine (PPPM/3PM) paradigm that is based on, among others, the capacity to predict disease development and influence decisions about lifestyle choices or to customise the medical practice to the patient [1]. Many of these diseases can be accompanied by severe complications. Hence, applying machine learning techniques on the available patient's data in the electronic hospital records to predict the presence of complications is an example of practical multidisciplinary implementation of PPPM/3PM strategies to improve healthcare.
One of these complications that result in increased morbidity and mortality [2] is bacteraemia. The related in-hospital case-fatality rate in bacteraemia is 12% in some reports [3]. Sepsis is one of the most important causes of morbidity and mortality. It is estimated at 19 million cases, and up to 5 million sepsis-related deaths annually [4].
Machine learning (ML) techniques will contribute an important added value to the three pillars of 3P medicine. Thus, the prediction of this kind of infection is useful either (i) to prevent it or (ii) to decrease its morbidity and mortality by starting an early, appropriate and specific antibiotic treatment. It is recommended that antibiotic treatment be promptly administered whenever there is a suspected serious bacterial infection [5, 6] and, if possible, after blood cultures have been taken. The diagnosis can take up to 6 days using blood cultures which introduces a significant lag in the antibiotic treatment. The individual prediction of bacteraemia would reduce this diagnosis lag enabling the early administration, up to 6 days earlier, of a personalised antibiotic treatment that would significantly reduce the bacteraemia complications.
Additionally, ML techniques can also provide an important added value to the targeted prevention of bacteraemia by identifying patients with bacteraemia and their specific bacteraemia's source earlier. The bacteraemia's source determines (i) the specific and most appropriate antibiotic treatment, (ii) the specific diagnostic techniques to search the reasons for the bacteraemia source, and (iii) it helps determine additional treatments that sometimes must be combined with the antibiotic treatment, for example, surgery [7]. In this sense, preventative methods have been shown to be successful, for example, methods such as vaccination or the Michigan-keystone project to reduce central-line related bloodstream infections in children [8].
A personalised and specific antibiotic treatment follows the prediction of bacteraemia and its source. Personalised treatment means that each patient, with its own bacteraemia's focus and clinical situation (i.e. type of bacterial infection, source of infection, hemodynamic situation, temperature, laboratory markers, age, vaccination coverage, exposure to invasive procedures, if the patient has received antibiotics before, if he has suffered previous hospital incomes, or if a multiresistant microorganism has colonised him), needs a specific antibiotic treatment. All these factors determine the kind of antibiotic that the patient should receive [9, 10] which is intimately related to the morbidity and mortality of the patient.
ML techniques can consider all the previous variables to predict bacteraemia, prevent its complications and help personalise the treatments.
Bacteraemia
Bacteraemia is the presence of bacteria in the bloodstream [11]. In healthy patients, the blood does not contain bacteria, so its presence is associated with infections that can impact the patient's life.
The most typical origin for bacteraemia is an infection, restricted to a specific location in the body, that favours the bacteria's movement into the blood. The most frequent bacteraemia-producing infections are urinary (prostatitis or pyelonephritis), respiratory (pneumonia), vascular (infected catheters), digestive (cholecystitis or cholangitis), skin and soft tissues (cellulitis or myositis), or bones (osteomyelitis). When the origin is unknown, it is referred to as primary or idiopathic bacteraemia. Some medical procedures can also favour bacteria's passage into the blood in previously healthy patients, from sites usually colonised by bacteria, such as urinary catheters in the bladder or endoscopies of the digestive tract (colonoscopies). Likewise, certain habits such as intravenous drug use can favour the passage of bacteria from the skin to the blood [12].
The bacteria in the blood can spread the infection to other places in the body, producing endocarditis, arthritis, osteomyelitis, meningitis, or brain abscesses, among others. In [13], the authors describe the connection between the type of bacteraemia microorganism and the site of acquisition with associated mortality. They show that the mortality associated with bacteraemia ranges from 11 to 37% depending on the place and type of microorganism. There is a high mortality rate associated with bacteraemias [14], and blood cultures are the gold standard for testing for the diagnosis of bloodstream infections. Due to the high morbidity and mortality associated with bacteraemia, it is mandatory to initiate effective antibiotic treatment as soon as possible to reduce the death rate [15].
Therefore, as presented above, bacteraemia can be either the origin or the complication of diseases on which the PPPM/3PM [16] and personalised medicine [17] are focused on, and the very same principles that guide PPPM can be used to predict the complications' development and to customise their medical practice.
Deficits in the current treatment of bacteraemia
The means of detecting bacteraemia is via blood cultures [18, 19] in vials that contain growth media of two types: aerobic and anaerobic. To this aim, an amount of the patient's blood—from 20 to 40ml—is drawn and introduced into the vials. Then the vials are placed within a system that maintains the optimal environmental conditions (temperature, humidity, light) for the microorganism's growth. The microorganism's growth produces CO2, and the system detects its production. This process can take between hours and 5 days. If the system does not detect CO2 production during this time frame, it reports a negative culture (no bacteraemia), whereas if it does detect CO2 production, then it reports a positive culture. Nevertheless, a positive culture does not always imply bacteraemia. Therefore, it is also important to determine if this growth is a true bacteraemia or a contaminant (negative bacteraemia). If a positive culture appears, then the identification of the microorganism, the bacteria species that have grown in the vials, begins. The complete process of identifying the microorganism can take up to another 2 to 3 days. In many cases, the species identified came from the skin or was introduced in the blood sample either during blood extraction or during the culture. In such a case, the culture is contaminated and considered to have no bacteraemia. Finally, only those analyses in which the bacteria species comes from an infection are declared to be bacteraemia.
The prediction of true bacteraemia has two important moments. The first one is when the physician decides to extract blood from the patient for the blood culture. The second one is the moment (hours or days after the blood extraction) when some blood cultures are positive. From this second moment to the definitive identification of the microorganism can take 2 or 3 days. Among these positive blood cultures (i.e. the system detects CO2), some will be contaminants (considered to be negative bacteraemia), and others will be true cultures (considered to be true bacteraemia). The type of blood culture (aerobic or anaerobic blood cultures) and the time lapse to detect growth could be important to predict if the growth is true or not in this second period, before the definitive identification of the microorganism.
The deficits in the current treatment of bacteraemia begin at the moment that it is decided to obtain blood cultures. Blood cultures should not be obtained indiscriminately because this increases the number of contaminated blood cultures, leading to unnecessary antibiotic therapy and increasing economic costs. There are different situations in which blood cultures should be obtained, such as severe sepsis, suspected infection with organ dysfunction, high blood lactate levels, or infectious processes associated with bacteraemia (for example, pyelonephritis, cholangitis, severe pneumonia, meningitis, suspected endocarditis, or endovascular infections). Also, bacteraemia should be suspected in patients with fever and at least one other sign or symptom of infection in the absence of a known alternative diagnosis.
For the physician, it is important to predict bacteraemia before deciding to obtain blood cultures. Unfortunately, physicians are not good at predicting which patients have bacteraemia [20]. The result of this poor prediction of bacteraemia is a low rate of true positive blood cultures; [21] reports rates between 5 and 10% and [22] reports values as low as 3.6% per analysis.
The second point regarding deficits in the current treatment of bacteraemia is the interpretation of positive blood cultures. There are organisms that should never be considered contaminants when identified in blood cultures, such as gram-negative roads, Staphylococcus aureus, or Candida spp. On the other hand, organisms such as coagulase-negative Staphylococcus spp. and Corynebacterium sp. are usually common skin contaminants, and if they are obtained in blood cultures, they usually do not need antibiotic treatment. However, sometimes this last group, usually contaminants, could produce bacteraemia mostly related to catheters or prosthetic valves.
The items explained above are related to the decision regarding antibiotic treatment and how long a patient should be treated. Therefore, predictive models of bacteraemia could help the physician make the appropriate decision regarding these points. Thus, in this sense, PPPM/3PM has a very important point of intervention in suspected bacteraemia and its treatment.
Clinical, economic and structural consequences
The usefulness of blood cultures in predicting bacteraemia is low, with a range between 4.1 and 7% [21, 23]. Compared to the true positive rate, false positive results due to contamination are in a similar or a higher range, varying between 0.6 and over 8% [24]. These problems of blood culture analysis also have an important economic impact, with a 20% increase of total hospital costs for patients with false positive blood cultures [25, 26]. Economic analyses estimate the costs related to a single false positive blood culture can be between $6878 and $7502 per case [24, 27]. In 2012, the American Board of Internal Medicine introduced the Choosing Wisely campaign, which aimed to reduce medical waste and the overuse of blood cultures by setting clear guidelines for the use of blood cultures. Studies assessing risk factors for bacteraemia have led to the development of multiple stratification systems without consensus [28].
Specialised prediction models can help make clinical decisions. The goal is to provide patient risk stratification to support tailored clinical decision-making. Clinical prediction models use variables selected because they are thought to be associated (either negatively or positively) with the outcome of interest [29]. On the other hand, risk prediction models can be used to estimate the probability of either having (diagnostic model) or developing a particular disease or outcome (prognostic model) [30].
Regarding prediction models for bacteraemia, a physician's suspicion of bacteraemia lacks sensitivity, specificity, or predictive values to be clinically useful. Some examples of clinical prediction models have been developed with bacteraemia related to pneumonia [31, 32], skin infections [33], and community-acquired bacteraemias [34]. Unlike ours, they all are focused on specific infections, which applies to any source of intra- or extra-hospital bacteraemia. In addition, none of them uses ML techniques, but rather methodologies ranging from multivariable analysis to identify significant predictors for bacteraemia [31], stepwise logistic regression, or multiple mutually exclusive stepwise logistic regression.
To the best of our knowledge, there is no application of ML techniques to create diagnostic bacteraemia models. Nevertheless, ML has had a successful history in biomedicine with applications in almost all the facets of medicine [35]: neural networks for breast cancer diagnosis [36], bladder cancer [37] or colorectal cancer [38], ensemble classifiers in bioinformatics [39], deep residual networks for carcinoma subtype identification [40], Tree-Lasso logistic regression [41], Bayesian Networks [42] for the prediction of the causative pathogen in children with osteomyelitis or decision trees [43] to cite just a few recent examples. Regarding classifiers, recently they have been used for cancer diagnosis using K-Nearest Neighbours (KNN) [44], drug identification using Support Vector Machine (SVM) [45] or predicting risk of disease using Random Forest (RF) [46], again to cite some illustrative examples in a myriad of papers.
Working hypothesis
For the aforementioned reasons, it would be interesting to predict which patients suffer from this pathology before deciding on blood sample extraction, and if the physician has decided to obtain blood cultures, it would be of interest to predict which patients will suffer true bacteraemia without waiting for up to 6 days for the definitive results. There are no useful clinical, analytical or epidemiological studies that allow physicians to predict bacteraemia at the patient's initial assessment.
Hence, our work's main objective is to implement ML techniques on a set of patient data from electronic hospital records to predict the appearance of bacteraemia, thus eliminating the wait for the results of blood cultures and anticipating the application of therapeutic treatments. Three ML techniques have been used: SVM, RF and KNN. The potential of these models in terms of PPPM/3PM is that used in conjunction with clinical judgement, they can be useful in the decision-making process regarding blood culture collection, clinical monitoring and empirical antimicrobial therapy. This work could provide two benefits: first, the possibility of starting the personalised patient's treatment earlier; second, the number of blood cultures would be reduced since they would only be prescribed in cases where the techniques' predictions did not have high reliability.
The rest of the paper is structured as follows. Section "Materials and methods" is devoted to introducing the material and methods of this study. Next, Section "Data analysis" presents the data analysis, Section "Discussion of the results" discusses the findings, and, finally, Section "Conclusions and recommendations in the framework of 3P medicine" summarises the conclusions and presents the recommendations in the framework of 3P medicine.
Subject database
The database is provided by the Hospital Universitario de Fuenlabrada, Madrid, Spain, a 350-bed hospital with the following services: general surgery, urology, orthopaedic surgery, gynaecology and obstetrics, paediatrics, intensive care units (ICUs), haematology-oncology, internal medicine and cardiology. The database was gathered from 2005 to 2015, and it consists of 4357 anonymous patient records, a.k.a. instances, containing 117 features per patient, 49.3% female with age 65.1 ± 19.7, and 56.1% male with age 62.7 ± 20.2. Each instance contains demographic and medical data (medical history, clinical analysis, comorbidities, etc.) and the result of the blood culture, the feature to be predicted, which can take one of two values: bacteraemia and no bacteraemia. The database contains 2123 bacteraemia (51.3%), which includes aerobic, strict anaerobic and facultative anaerobic bacteria, and 2234 no bacteraemia (48.7%), including 1844 contaminations.The final classification of true bacteraemia was done in prospective time by an infectious disease physician, using all the previous data, including microbiological, clinical and analytical data.
Forty-seven out of the 117 features were discarded from the database because they are derived from other features, irrelevant to the study, or useful after the blood culture was identified.
Two datasets were created from the database. The first dataset, called pre_culture, only uses the features known previously to the blood culture, i.e. the ML techniques only use the 65 features available previous to the culture to predict the bacteraemia, having discarded the features that hold the suspected source of infection. The second dataset, called mid-culture, uses the data available when the concentration of CO2 starts rising. Note that, as stated in "Introduction", an increase of CO2 could be either due to a true bacteraemia or a contamination of the blood sample during extraction, so the increase of CO2 does not necessarily mean bacteraemia. In this sense, contamination has the same value as no bacteraemia. The number of features in this dataset is 69: the 65 features in pre-culture plus four new ones: the time to CO2 detection, the type of media with bacterial growth, either aerobic or anaerobic and the first vial where the growth is detected (see "Appendix A: Features in the study" for an enumeration of the features under study).
Data preprocessing
Categorical features
Both datasets contain a set of patient instances, \(\mathcal {P}_{i}\), so that every instance comprises the medical (microbiological, clinical and analytical) and demographic data of one patient. \(\mathcal {P}_{i}\) is the concatenation of a feature vector, fi, and the classification—predicted—variable, yi, that is \(\mathcal {P}_{i} = (\mathbf {f}_{i},y_{i})\). fi defined on a feature space, \(\mathbb {F}\), of dimension L, \(\mathbb {F} = F^{1} \times F^{2} \times {\ldots } \times F^{L}\) so that each Fi is the set of values of a medical or demographic feature of the patient, i.e. age, fever, comorbidities, etc., and yi ∈{− 1,1} is the result of the blood culture, either '1' when the patient has bacteraemia or '− 1' when he or she does not. Therefore, \(\mathbf {f}_{i} =\left ({f_{i}^{1}} \in F^{1}, {f_{i}^{2}} \in F^{2}, \ldots , {f_{i}^{L}} \in F^{L} \right )\) and the datasets are \( \left \{ \mathcal {P}_{i} = (\mathbf {f}_{i},y_{i}) \mid \mathbf {f}_{i} \in \mathbb {F}, y_{i} \in \left \{ -1,1 \right \} \right \}\).
SVM and KNN require a definition of distance on \(\mathbb {F}\). This requirement imposes the categorical features to be translated into numerical values. However, the mapping of categorical values onto numerical ones without detailed supervision will bias the ML algorithm because the numerical translation will define proximity relationships that are not present in the categorical feature. The most used codification to avoid these problems is the one-hot encoder. It loops through the dataset and separates each feature of a given categorical type into subcategories; that is, for each category in a feature, the technique generates a new feature with only two values: true or false. Consequently, this technique defines a new feature space, \(\mathbb {F}'\) with a number of features \(L^{\prime }\). On \(\mathbb {F}'\), the distance metric, \(d: \mathbb {F}' \times \mathbb {F}' \xrightarrow {} \mathbb {R}\), can be defined now. The Euclidean distance, given by Eq. (1), was chosen.
$$ d(\mathcal{P}_{i}, \mathcal{P}_{j}) = \sqrt{{\sum}_{d=1}^{L^{\prime}}{({f_{i}^{d}} - {f_{j}^{d}})^{2}}} $$
Missing data
The method to handle missing data depends on the nature of the data missingness. Three categories have been defined to classify missingness [47]: (i) missing completely at random (MCAR) in which the missingness is random, unrelated to the outcomes and does not contain valid information for analysis; (ii) missing at random (MAR) when the missingness depends on the outcomes observed; and (iii) missing not at random (MNAR) when missingness depends on unobserved measurements.
To check the missingness of the data, we define, one feature at a time, two classes, missing and non-missing data, a RF classifier is built upon this feature, and we evaluate if the missing data provides a good classification using the RF classifier [48]. If RF accuracy is high for this feature, a MAR behaviour is concluded for the feature and discard it from the dataset.
Three different approaches are evaluated to handle the high number of missing data [49]. The complete case data approach removes the instances with missing data to obtain a new dataset without misses; that is, all instances have valid data in all features. This approach presents two handicaps: (i) its usage would not allow a new instance with missing data to be evaluated once the ML model is trained and tested, and (ii) it significantly reduces the dataset.
An alternative approach that attempts to keep a large ratio of complete instances in the dataset is also evaluated [50]. This method ranks the features in decreasing order in the percentage of missing data and then iteratively removes the features following the ranking order. In each iteration, the number of complete instances is calculated and the total quantity of data in the complete instances, i.e. the number of complete instances times the number of instances. As the number of features decreases, the total amount of non-missing data in the complete instances increases to a maximum, beyond which the quantity of non-missing data in complete instances decreases. This maximum determines the number of features that most contribute to complete case instances, and it is the best option.
Both previously mentioned methods operate under the MCAR supposition, a supposition that we will prove to be false for one feature.
Thirdly, the separate class method [48] is evaluated to handle missing data. The separate class method defines a new category to represent the missing data of a feature so that each feature has its own category to represent its misses. In the case of numeric type features, the missing data receive a value that is outside the range of the feature's values. In this way, the required separation between the missing data and the correct values is created.
Each approach creates a different dataset size with a different number of patient samples and a different number of features per patient. Hence, our comparison selects the best approach in terms of the best training of the ML model. That is the approach that has the best trade-off between the number of samples and the features so that the RF provides the most accurate prediction.
Renormalisation
We renormalise the numerical features so that every feature's different values are separated based on the same scale, which is especially relevant for those techniques such as SVM or KNN that use the notion of distance in a metric space. Hence, all numerical data are rescaled to values in [0,1]. This renormalisation is also applied on the separate classes associated to the missing data, and we assign them the value − 0.5 since there are no negative values in any dataset.
Machine learning techniques
Three supervised ML classifiers are used: SVM, RF and KNN. We devote the next three sections to briefly presenting the ML techniques.
Support vector machine
SVM is a supervised ML technique [51, 52]. In binary classification problems over a dataset of instances of dimension L + 1, this technique finds an L-dimensional hyperplane that separates the two different classes, maximising the distance of the closest instances in the dataset -called support vectors- to the hyperplane. The distance from the support vectors to the hyperplane is called margin. In other words, SVM finds the hyperplane that maximises the margin of the support vectors. So, as stated above, it requires a definition of distance on the dataset's features to evaluate the separation between the instances and the hyperplane. The hyperplane is defined by its normal vector, w, and the hyperplane equation is wT ⋅ x + b = 0 with wT being the transpose of the normal vector and \(\frac {b}{\left \lVert w \right \rVert } \) the offset of the hyperplane from the origin. Equation 2 defines the optimisation problem.
$$ \begin{aligned} \min \quad & \left\lVert w \right\rVert \quad \textrm{subject to} \quad y_{i} \cdot (w^{T} \cdot x_{i} + b) \geq 1 \end{aligned} $$
There are two types of SVM classifiers: linear and nonlinear. In the former, SVM operates on the raw data to find the hyperplane under the supposition that the data are linearly separable, whereas the latter transforms the original instances by adding extra similarity features to try to create a linearly separable dataset under the supposition that the original one was not. The most used similarity function is the Gaussian Radial Basis Function [53]:
$$ \phi(x_{i},p) = e^{-\gamma \cdot \left\lVert x_{i} - p \right\rVert^{2}} $$
where the set of points p determines the landscape used to calculate the new features, and γ ∈ [0,1] is a regularisation hyperparameter used to control the over- and underfitting of the SVM model.
There are also two types of SVM models depending on whether a few instances of one class are allowed to be located within the margin region or even in the region assigned to the other class. If no instance of one class can be within the margin region or the region assigned to the other class, then a hard margin classification is defined. In any other case, it is a soft margin classification. The soft margin classification allows the misclassification of some instances but provides higher margins in the classification whereas hard margin classification typically provides a clean but narrower margin. In the former case, the SVM has better generalisation capabilities, that is, lower overfitting. SVM implementations provide a hyperparameter to control the softness of the margin, C. The higher the C, the stricter the classification.
RF is a supervised ML technique used in both classification and regression [54]. In classification problems, it creates multiple decision trees, each one providing its classification output, and combines the results of all the trees using an aggregation function to provide the classification of the given instance. The potential of this technique is based on the aggregation of weak learners in order to provide high-accuracy predictions. Nevertheless, high accuracy requires the technique to satisfy certain requirements, the first of which is the independence of the individual trees.
In this work, (i) the trees are binary and provide output that can take one of two values, {− 1,1}; (ii) the RF prediction is an aggregation function, i.e. the majority vote, of individual tree predictions; and (iii) independence is achieved by using different subsets of instances to train every individual tree. The sampling of the subsets can be performed using two different schemas: sampling with replacement, called bagging, or without replacement, called pasting. Thus, each individual tree has a larger bias than if it were trained using the complete training set, but the aggregation of trees provides a lower bias-aggregated classification.
The form of a single classification tree is determined by the order in which the features are used to create that tree; that is, in the same set of instances, a different order in the selection of the features used to create the tree generates different trees. One of the most used algorithms to train decision trees is the classification and regression tree (CART). CART splits the training subset into two subsets using a single feature and a threshold for such feature, searching for the tuple feature/threshold that provides the purest subsets. Equation 4 presents the fitness metric used by CART to measure the purity of a node's classification where m is the total number of instances being classified in the node, mleft and mright are the numbers of instances in the left and right splits, respectively, and G is the metric that measures the impurity of the splits. The lower the value of J, the purer the classification.
$$ J = \frac{m_{\text{left}}}{m} \cdot G_{\text{left}} + \frac{m_{\text{right}}}{m} \cdot G_{\text{right}} $$
Two impurity metrics are commonly used [55]: the Gini impurity, Eq. 5, and the entropy-based impurity, equation 6.
$$ \begin{array}{@{}rcl@{}} G & =& 1- {\sum}_{c=1}^{2} {p_{c}^{2}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} E & =& - {\sum}_{c=1}^{2} p_{c} \log(p_{c}) \end{array} $$
where pc is the ratio of instances of class c in the set of instances in the node. Each node only has instances of two classes: bacteraemia or no bacteraemia. For that reason, the sum upper limit is 2.
Finally, the decision tree can be regularised with the following hyperparameter [56]: the maximum depth of the trees, the minimum number of samples in a node to be split, the minimum number of samples of a leaf node, the maximum number of leaf nodes and the maximum number of features to be tested in order to split a node.
K-Nearest neighbours
We use the supervised flavour of this simple nonparametric ML technique to classify the binary-class instances [57]. Given a new feature vector, fl, it assigns its class, yl, by finding the k of nearest instances in the dataset feature space and combining their classifications (i.e. averaging or voting). So, like SVM, this technique requires the definition of distance in Eq. 1. However, this technique does not need a training phase, and it achieves a very high capacity: the larger the training set, the higher the capacity.
The selection of the value for k should follow these rules: (i) the value should be a prime number to avoid ties; (ii) it should be less than the total number of reference instances in an instance class; and (iii) its value should be large enough to avoid false classification caused by outliers. The actual value of k is found using a grid search on a range of reasonable values. The technique returns the majority of the k nearest neighbours that share the same class. The fine-tuning of this hyperparameter requires it to be searched for using a heuristic.
In our experiments, the 10-fold cross-validation approach is followed so that the dataset is divided into ten subsets and each subset is used as a validation set whereas the remaining nine subsets are used for training a model. This procedure is repeated for every subset, so ten models are obtained. The performance of the ML technique is measured as the average performance of the ten models obtained with different training sets and validated on different sets.
The analysis was performed in Python 3.7 using sklearn 0.23 for model inference and ELI5 0.10.1 for the permutation importance method.
Data bias
First, we study any bias in the distribution of values in the datasets. As stated at the beginning of this section, datasets contain a balanced percentage of values in the predicted variable: bacteraemia (51.3%) and no bacteraemia (48.7%); the latter includes both actual negative bacteraemias and contaminated cultures.
Similarly, we check whether missing data in \(\mathbb {F}\) are correlated with the predicted variable. That is, if the MCAR assumption holds for the data. Figure 1 presents the classification accuracy for all the features, one feature at a time, in the dataset.
Accuracy of the individual features when only two classes (missing and non-missing) are used to predict bacteraemia
The missing class of the suspected source (the peak in the histogram with an accuracy of 82.6%) is a good predictor of no bacteraemia. In contrast, the remaining features have a slight bias in the prediction. The ratio of missing data for this feature is around 40%, as Fig. 2 illustrates. The feature's importance, with such a high ratio of missing data, is suspicious and indicates a correlation between the missing-data class and the variable predicted. Hence, 72.4% of the instances with a suspected source, either 'unknown' or any organ in the body, are bacteraemia. On the contrary, only 7.2% of the missing suspected sources are bacteraemia.
Percentage of missing values for all the features in \(\mathbb {F}\). The features are sorted on x-axis as in Table 5. The annotations in the graph mark the inflection points, and they facilitate cross-searching in Table 5
These figures state a missing at random (MAR) [47] behaviour for this feature. During database generation, the physician, who is typically good at predicting the focus of infection but not so good at predicting which of them are accompanied by bacteraemia, only includes the suspected source in the database once the bacteraemia has been detected. In other words, the physician decides that writing down the source of infection is of no interest for non-bacteraemia cases. This feature is removed from both datasets.
This section presents the number and distribution of missing data per feature. Figure 2 illustrates the percentage of missing values for the features in \(\mathbb {F}\). The percentage is above 70% for the worst feature (number of days in ICU previous to culture) and between 40 and 37% for the following three features: the suspected origin of the bacteraemia previous to culture, the results of PCR testing and the source of bacteraemia in the last hospital department. Following them, there are 50 features with missing-data percentages from 30 to 20%.
We evaluate three different approaches to handle the high number of missing data [49]. The complete case data approach removes the instances with missing data to obtain a new dataset without misses. If we apply this approach on our original dataset, then the new dataset only contains 476 complete instances out of 4357. Hence, this approach is inappropriate due to the large volume of data lost. Nevertheless, we evaluated its achievements to classify the bacteraemias accurately.
The second approach removes the features with a higher number of missing data. Figure 3 illustrates the evolution of the total volume of data in all complete instances versus the number of complete instances. In our case, the optimal number is 51 features with 2760 instances, totalling 140,760 non-missing values in the dataset. As in the previous approach, we think this is also inappropriate because (i) it removes critical features from datasets such as, for example, the suspected medical source of the patient's infection, and (ii) it removes 33.8% of the features and 44.6% of the number of instances. Nevertheless, we also evaluated its achievements to classify the bacteraemias accurately.
Number of features versus number of non-missing values in dataset
Thirdly, the separate class method [48] was evaluated to handle missing data. The separate class method defines a new category to represent the missing data of a feature so that each feature has its own category to represent its misses. In the case of numeric type features, the missing data receive a value that is outside the range of the feature's values. In this way, the required separation between the missing data and the correct values is created.
The performance of the three missing-data methods was compared using RF as the testbench. In these comparisons, the renormalised separate class method obtains the best performance, and for that reason, it is the method of choice in this work.
Prediction results
The three ML techniques have been evaluated using the same procedure: (i) the dataset is split into 80/20 training/testing sets, (ii) grid-search 10-fold cross-validation is run on training data for the ML techniques to find their best hyperparameters, and (iii) the best hyperparameters are applied on the testing split of the dataset.
The hyperparameters of the SVM model are swept in the ranges C = {0.1,0.2,…,1,2,…,10,20,…,100} and \(\gamma = \left \{ \frac {1}{L^{\prime }}, \frac {1}{ L^{\prime } \cdot \sigma }, 0.1, 0.2, \ldots ,1 \right \}\) with σ being the data variance, by using the Gaussian Radial Basis Function.
The hyperparameters for the best pre_culture SVM model are \(\gamma = \frac {1}{L^{\prime }}\) and C = 9, which implies that the instances are separable. Table 1 summarises key metrics to evaluate the predictive capacity of the model: accuracy, sensitivity, specificity, positive predictive value (PPV) and negative predictive value (NPV). The average accuracies of the best pre-culture SVM model are 76.9 ± 1.7% in the training phase and 75.9% in the testing phase. Accuracy in the testing phase is only 1.0% lower, proving the good generalisation capabilities of the model. This model has a sensitivity of 80.7% with a specificity of 71.4%, PPV of 72.8% and NPV of 79.6%.
Table 1 Accuracy, specificity, sensitivity, positive predictive value (PPV), negative predictive value (NPV) and area under the curve (AUC) of the models
The features' importance has been evaluated using importance sampling, and the left two columns in Table 2 present the top 10 most important features of this SVM model. Among them, the top 3 to predict bacteraemia are a chronic respiratory disease, the number of days in ICU before blood extraction and the presence of catheters.
Table 2 Feature importance for SVM
The mid_culture SVM model was designed using the same procedure. In this case, the hyperparameters of the best model are \(\gamma = \frac {1}{L^{\prime }}\) and C = 8, which implies that the instances are slightly more separable than in the pre-culture dataset. The average accuracy of the training phase is 83.0 ± 1.4% and the testing phase achieves an overall accuracy of 80.5%, sensitivity of 81.3%, specificity of 79.7%, PPV of 80.5% and NPV of 80.5%. The usage of intermediate results of the blood culture increases all the metrics from 5 to 8%. Table 2 illustrates the most relevant features to predict bacteraemia using the importance sampling method. According to this table, three out of the four new features rank in the top 5 most relevant features: growth in anaerobic and aerobic vials, and the number of days until CO2 detection.
Figure 4 presents the ROC of the three ML techniques evaluated for the two datasets. The mid_culture SVM ROC has an area under the curve (AUC) of 0.88, performing better than the pre-culture SVM model, which has an AUC of 0.85.
ROC for the best SVM, RF and KNN for models
We have not constrained either the maximum depth, the minimum number of samples in a node or any other of the hyperparameters stated in "Random forest", and we use the Gini impurity metric. The only hyperparameter of the model evaluated in the grid-search exploration is the number of trees, which is found in {1,2,…,90}.
The best pre_culture RF model averages an accuracy of 79.5 ± 1.4% in the grid-search 10-fold cross-validation with 86 trees, and an accuracy of 78.2% during the testing phase. As for SVM models, the variation in accuracy refutes the overfitting of the model. Table 1 summarises the key metrics that clinical practitioners use to evaluate the models' predictive capacity. The features' importance has been evaluated using the permutation importance algorithm, and Table 3 presents the most critical features of the model.
Table 3 Feature importance for RF
The mid_culture RF model uses 68 trees and obtains an average accuracy of 85.6 ± 1.4% in the training phase and reduces the size of the RF model by 34.9%. This model performs better than the pre-culture one, improving all the predictive metrics: it increases accuracy 6.1% in the training phase -a value similar to that observed in SVM models- and 7.7% in the testing phase -an improvement higher than that observed in the SVM models-, sensitivity by 1.3%, specificity by 13.7%, PPV by 12.6% and NPV by 2.3%.
Table 3 illustrates the most critical features to predict bacteraemia for this model. As for the SVM models, the new features are ranked among the top ones. Hence, the top-ranked feature is the number of days at CO2 detection followed by the positive in anaerobic vials, the first blood culture vial with growth and the positive in aerobic vials. Regarding the distribution of values in the rankings, the two RF rankings are more unbalanced than the SVM ones, with an outstanding feature in both cases, which doubles the importance of the second feature in the pre-culture model and which is 8 × for the mid-culture model.
The only hyperparameter for this classifier is k which, in this study, is found in {1,2,…,20}.
The best pre_culture KNN model uses k = 15 neighbours, and the best mid_culture model uses k = 9. Table 1 summarises the key metrics to evaluate the predictive capacity of the KNN models. The best pre-culture KNN model averages an accuracy of 76.5% during the testing phase. As in previous models, the inclusion of mid-culture features improves the KNN model's performance, although less significantly -only a 1.9% increment in testing accuracy- and it even has a slight decrease of 2.2% in sensitivity and of 2.7% in NPV. Moreover, similar to RF models, the inclusion of new features reduces the size of the model, in this case the number of relevant neighbours.
Table 4 presents the top 10 most important features in the KNN model according to importance sampling criteria.
Table 4 Feature importance for KNN
Finally, Fig. 4 graphs the ROC of the two KNN models with AUCs of 0.85 and 0.88. Hence, this technique has a predictive power lower than the previous ones.
Discussion of the results
Typically, medical records contain missing data that can bias the conclusions of the ML techniques. The separate class method provides a mechanism to handle the missing data, preserving the number of patients in the study and providing good metrics in the classifiers. We did not evaluate imputation methods based on ML algorithms, such as KNN, to predict the missing values in the training data because they can infer relationships among the features that could distort the data structure [58] or such as the more efficient imputation method missForest [59] because this iterative imputation method must be run with every single new patient, which would increase the computational cost of every new prediction when the system is in production.
The importance rankings of the three ML techniques provide a significant ratio of common top features for both datasets. Hence, for the pre-culture models, the number of days in ICU before blood culture extraction, the presence of catheters, fever and the presence of symptoms related to the source of fever and the presence of urine sediments are critical features of major importance. The month of the blood culture appears for the pre-culture KNN and RF models. Hence, both techniques detect seasonality in the bacteraemia, although it has a low importance in both techniques.
Regarding the models for the mid_culture dataset, the new features in this dataset are the most important for an accurate prediction of the bacteraemia, displacing the top features of the pre_culture model. Indeed, their importance in the model exceeds the importance of all the features in the pre_culture model. In particular, the mid_culture RF model ranks the four new features among the top of the ranking, whereas the other two techniques only include three out of the four new features.
This consistency highlights that prediction capability is a characteristic intrinsically related to the data already available in most of the hospital health records.
The feature importance for the pre_culture SVM and KNN models is balanced. The top 3 feature importances are within a range of 10.0% of the most important one, and then the importance is reduced softly for the remaining seven features. The high number of features taken into account for the models to generate a prediction justifies physicians' difficulty in generating accurate predictions: they cannot handle such a large number of variables. In particular, the two KNN rankings are the most balanced of the three ML techniques. The first five features in the pre-culture model and the first three features in the mid-culture model have very similar values, although the dispersion of accuracy in the training stage doubles the dispersion values of the other ML techniques, which justifies why the KNN technique produces less predictable accuracy for the model.
On the other hand, the feature importance of the pre_culture RF model is less balanced, with a critical feature then two less relevant features, and the remainder are mostly irrelevant. This behaviour is exacerbated in the mid-culture model in which new features dominate the classification. For this reason, in the presence of these features, the physician could make a prediction based on a lower number of features. Nevertheless, the features, as stated above, coincide in almost half of the cases.
The test accuracy of the ML techniques on the pre_culture dataset ranges between 75.9% for SVM and 78.2% for RF. These values are increased by around 9.8% when using the new features in the mid-culture dataset, with mid-culture RF model obtaining an accuracy of 85.9% . Hence, the accuracy of ML techniques is 8 × human accuracy (from 3.6 to 10% according to [22]).
Regarding the key metrics to evaluate the predictive capacity of the model, their values range from 80.7 to 89.6% for sensitivity, 65.2 to 84.4% in specificity, 69.0 to 85.2% for PPV and 79.6 to 86.6% for NPV, with the mid-culture RF model outperforming the other models and achieving an average accuracy of 85.9 ± 1.4%, sensitivity 87.4%, specificity 84.4%, PPV 85.2%, NPV 86.6% and an outstanding AUC of 0.93 with improvements of 6.7% with regard to the accuracy of the second best technique, SVM, 6.1% in sensitivity, 4.7% in specificity, 4.7% in PPV and 6.1% in NPV.
AUC is above 0.85 for all models, and the presence of the new features increases the AUC from 3.5 to 8.1% with respect to the pre_culture AUCs. A predictive model in the medical practice must have an AUC greater than 0.7, and a good predictive model has AUC≥ 0.8. The previous results in the literature using classical modelling techniques in specific types of bacteraemia are as follows: pneumonia [32] with AUC 0.79, skin-related [33] with AUC 0.71 or any type [34] with AUC 0.77. Therefore, the ML values of AUC, sensitivity, specificity, predictive positive and negative values exceed the results described in the literature.
Previous results indicate that bacteraemia prediction can be achieved using already available hospital records with better figures of merit than the physicians' predictions. These predictions can help physicians make an appropriate diagnosis and prevent complications, where, in this context, 'appropriate' means both in time, i.e. as soon as possible, and in type, with the more specific and personalised antibiotics and treatment for each patient.
Interplay between COVID-19 and bacteraemia
Nowadays, we are experiencing the COVID-19 pandemic, so it is necessary to refer to the possible association between COVID-19 and bacteraemia and the utility of ML techniques in this kind of patient. In this context, bacteraemia is rare for COVID-19 patients, which supports the judicious use of blood cultures in the absence of compelling evidence for bacterial co-infection [60]. In some reports, bacteraemia with S. aureus is associated with high mortality rates in patients hospitalised with COVID-19. S. aureus infections are a known complication of other viral pandemics, such as the Spanish flu in 1918–1919 and the H1N1 influenza pandemic in 2009–2010, suggesting that the interaction of S. aureus with SARS CoV-2 is similar to that in influenza [61]. The proposed mechanisms of viral-induced bacterial co-infections include the viral modification of airway structures, as well as the initiation of immune-suppressive responses [62]. A similar mechanism has been described in another report of oral infections where the authors suggest that poor oral hygiene and periodontal disease could produce the aggravation of COVID-19 [63].
Secondary bacteraemia has been developed in 37% (27/73) of patients with acute respiratory distress syndrome [64]. However, it has not been defined whether bacteraemias were secondary to pneumonia or typical hospital-acquired infection.
In this sense, ML techniques could help physicians predict bacteraemia as a secondary infection in COVID-19 patients, mostly in critical COVID-19 patients, who suffer these secondary infections more frequently [65].
Conclusions and recommendations in the framework of 3P medicine
The three ML supervised classifiers create accurate predictive models of the blood culture outcome using hospital electronic health records, i.e. data previous to blood extraction and data measured in the first hours/days of the blood culture. The concordance in the results of the three classifiers increases the power of the conclusions and confirms the viability of ML techniques as a key technology for applying the PPPM/3PM principles to improving patients' survival rates significantly and providing more cost-effective management of the disease.
Expert recommendations
Bacteraemia is an entity with high morbidity and mortality. Its early diagnosis and an appropriate early antibiotic treatment are critical. For these reasons, in this kind of pathology, it is essential to combine predictive techniques and personalised treatments in which ML techniques can help physicians diagnose, reduce time to treatment and manage bacteraemia. ML techniques could help determine preventive actions to avoid this entity, and secondly, to optimise the cost of the disease. If physicians could predict bacteraemia, then they could avoid the intervention to obtain blood samples, the use of four to six bottles for blood culture per patient, the time lapse devoted to the culture and the procedures to identify possible contaminant microorganisms with their associated cost in time and money.
Regarding the selection of antibiotic treatment and its duration, both could change depending on whether the patient is suffering from bacteraemia or not. Usually, diseases associated with bacteraemia need a longer duration of antibiotic treatments. This duration could be optimised if physicians could predict whether a patient has or does not have bacteraemia. If we could shorten the duration of antibiotic treatment, we would spend less money on each patient and avoid secondary effects associated with longer antibiotic treatment, such as antibiotic resistance [66].
Therefore, continuous data extraction from electronic medical records could help physicians identify bacteraemia and the progression to a severe disease earlier and provide timely interventions, such as appropriate antibiotic treatment, to reduce mortality and morbidity [67, 68].
The adoption of ML technologies in the framework of 3P medicine depends entirely on the accuracy of their models, which is related to the availability of datasets with low missing value rates and no bias in the missing values because of the physician's a priori interpretation of the data. Patient databases play a central role in 3P medicine [1], and it is critical to ensure their completeness and avoid depending on the physician's discretion at the time of completing the database records. This requirement should be included in database design specifications and the design of database user interfaces.
The application of ML techniques also depends on the availability of structured datasets. Most hospital records store health information according to the European Commission's Recommendation on Electronic Health Records [69], but data would have to be stored in a format suitable for the automatic manipulation of the features, avoiding as much as possible those features expressed in natural language that hinder the extraction of structured information.
Predictive models play a key role in bolstering decision systems, and ML techniques have outstanding potential to create models with an excellent level of accuracy [70]. They have been used to identify useful correlations between biometric, genetic and environmental data with the potential risks and benefits of certain therapeutic choices [71]. They also have great potential to exceed the performance of physicians' heuristics, reducing lags in diagnosis and treatment costs when their application is extended from the genomic and biometric data to the clinical and demographic data in the patient's records.
Our future work will focus on studying non-structured features (medical texts described in natural language), also included in the database, that could improve the model's accuracy. Additionally, we will validate these findings using independently collected databases and, subsequently, under regulatory approval, we will develop an app for mobile devices that enables the translation of these results to the hospital practice by providing a prediction to the physician at the bedside based on the latest available patient records.
These ideas are directed to improve predictive and personalised treatment in a disease as bacteraemia that currently continues producing a high level of mortality.
The code used in this study is available from the corresponding author on reasonable request.
The dataset is available from the corresponding author on reasonable request.
Golubnitschaja O, Kinkorova J, Costigliola V. Predictive, preventive and personalised medicine as the hardcore of 'horizon 2020': Epma position paper. EPMA J 2014;5(1):6–6. ISSN 1878-50771878-5085.
Yu JC, Khodadadi H, Baban B. Innate immunity and oral microbiome: a personalized, predictive, and preventive approach to the management of oral diseases. EPMA J 2019;10(1):43–50. ISSN 1878-5085.
Pien BC, Sundaram P, Raoof N, Costa SF, Mirrett S, Woods CW, Reller LB, Weinstein MP. The clinical and prognostic importance of positive blood cultures in adults. Amer J Med 2010;123 (9):819–828. ISSN 0002-9343.
Fleischmann C, Scherag A, Adhikari NKJ, Hartog CS, Tsaganos T, Schlattmann P, Angus DC, Reinhart K. Assessment of global incidence and mortality of hospital-treated sepsis. current estimates and limitations. Am J Respir Crit Care Med 2016;193(3):259–272.
CAS PubMed Article Google Scholar
Gudiol F, Aguado JM, Almirante B, Bouza E, Cercenado E, Ángeles Domínguez M, Gasch O, Lora-Tamayo J, Miró J M, Palomar M, Pascual A, Pericas JM, Pujol M, Rodríguez-Baño J, Shaw E, Soriano A, Vallés J. Diagnosis and treatment of bacteremia and endocarditis due to staphylococcus aureus. a clinical guideline from the spanish society of clinical microbiology and infectious diseases (seimc). Enfermedades Infecciosas Microbiol Clín 2015;33(9):625.e1–625.e23. ISSN 0213-005X.
Sakarikou C, Altieri A, Bossa MC, Minelli S, Dolfa C, Piperno M, Favalli C. Rapid and cost-effective identification and antimicrobial susceptibility testing in patients with gram-negative bacteremia directly from blood-culture fluid. J Microbiol Meth 2018;146:7–12. ISSN 0167-7012.
Wilson ML. Critical factors in the recovery of pathogenic microorganisms in blood. Clin Microbiol Infect 2020;26(2):174–179.
Pai S, Enoch DA, Aliyu SH. Bacteremia in children: epidemiology, clinical diagnosis and antibiotic treatment. Expert Rev Anti-infect Therapy 2015;13(9):1073–1088. ISSN 1534-6277.
Song Y, Himmel B, hrmalm L, Gyarmati P. The microbiota in hematologic malignancies. Curr Treat Opt Oncol 2020;21(1):2. ISSN 1534-6277.
Phua A I-H, Hon KY, Holt A, O'Callaghan M, Bihari S. Candida catheter-related bloodstream infection in patients on home parenteral nutrition - rates, risk factors, outcomes, and management. Clin Nutrition ESPEN 2019;31:1–9. ISSN 2405-4577.
Smith DA, Nehring SM. 2019. Bacteremia. StatPearls Publishing, Treasure Island (FL). http://europepmc.org/books/NBK441979.
Schaefer G, Campbell W, Jenks J, Beesley C, Katsivas T, Hoffmaster A, Mehta SR, Reed S. Persistent bacillus cereus bacteremia in 3 persons who inject drugs, san diego, california, usa. Emerg Infect Diseas 2016;22(9):1621–1623. ISSN 1080-6059 1080-6040.
Cisneros-Herreros JM, Cobo-Reinoso J, Pujol-Rojo M, Rodríguez-Baño J, Salavert-Llet M. Guía para el diagnóstico y tratamiento del paciente con bacteriemia. guís de la sociedad española de enfermedades infecciosas y microbiologa clnica (seimc). Enfermedades Infecciosas Microbiol Clín 2007;25(2):111–130. ISSN 0213005X.
Laupland KB, Church DL. Population-based epidemiology and microbiology of community-onset bloodstream infections. Clin Microbiol Rev 2014;27(4):647–664. ISSN 0893-8512.
Lee C-C, Lee C-H, Hong M-Y, Tang H-J, Ko W-C. Timing of appropriate empirical antimicrobial administration and outcome of adults with community-onset bacteremia. Crit Care (London, England) 2017;21(1):119–119. ISSN 1466-609X1364-8535.
Golubnitschaja O, Topolcan O, Kucera R, Costigliola V, Akopyan M, et al. 10th anniversary of the european association for predictive, preventive and personalised (3p) medicine - epma world congress supplement 2020. EPMA J 2020;11(1):1–133. ISSN 1878-5085.
PubMed Central Article Google Scholar
Stanski NL, Wong HR. Prognostic and predictive enrichment in sepsis. Nat Rev Nephrol 2020; 16(1):20–31. ISSN 1759-507X.
PubMed Article PubMed Central Google Scholar
Mylotte JM, Tayara A. Blood cultures: clinical aspects and controversies. Eur J Clin Microbiol Infectious Dis Official Publ Eur Soc Clin Microbiol 2000;19(3):157–163.
Ortiz E, Sande MA. Routine use of anaerobic blood cultures: are they still indicated?. Amer J Med 2000;108(6):445–447. ISSN 0002-9343.
CAS PubMed Article PubMed Central Google Scholar
Makadon HJ, Bor D, Friedland G, Dasse P, Komaroff AL, Aronson MD. Febrile inpatients. J Gen Intern Med 1987;2(5):293–297. ISSN 1525-1497.
Bates DW, Cook EF, Goldman L, Lee TH. Predicting bacteremia in hospitalized patients: a prospectively validated model. Ann Intern Med 1990;113(7):495–500.
Linsenmeyer K, Gupta K, Strymish JM, Dhanani M, Brecher SM, Breu AC. Culture if spikes? indications and yield of blood cultures in hospitalized medical patients. J Hospital Med 2016;11 (5):336–340.
Perl B, Gottehrer NP, Raveh D, Schlesinger Y, Rudensky B, Yinnon AM. Cost-Effectiveness of Blood Cultures for Adult Patients with Cellulitis. Clin Infect Dis 1999;29(6):1483–1488. ISSN 1058-4838.
Ratzinger F, Dedeyan M, Rammerstorfer M, Perkmann T, Burgmann H, Makristathis A, Dorffner G, Lötsch F, Blacky A, Ramharter M. A risk prediction model for screening bacteremic patients: A cross sectional study. PLOS ONE 2014;9(9):1–10.
van der Heijden YF, Miller G, Wright PW, Shepherd BE, Daniels TL, Talbot TR. Clinical impact of blood cultures contaminated with coagulase-negative staphylococci at an academic medical center. Infect Control Hospital Epidemiol 2011;32(6):623–625.
Qamruddin A, Khanna N, Orr D. Peripheral blood culture contamination in adults and venepuncture technique: prospective cohort study. J Clin Pathol 2008;61(4):509–513. ISSN 0021-9746.
Alahmadi YM, Aldeyab MA, McElnay JC, Scott MG, Darwish Elhajji FW, Magee FA, Dowds M, Edwards C, Fullerton L, Tate A, Kearney MP. Clinical and economic impact of contaminated blood cultures within the hospital setting. J Hosp Infect 2011;77(3):233–6. ISSN 0195-6701.
Wildi K, Tschudin-Sutter S, Dell-Kuster S, Frei R, Bucher HC, Nüesch R. Factors associated with positive blood cultures in outpatients with suspected bacteremia. Eur J Clin Microbiol Infect Diseas 2011;30(12):1615–1619. ISSN 1435-4373.
Shipe ME, Deppen SA, Farjah F, Grogan EL. 2019. Developing prediction models for clinical use using logistic regression: an overview. J Thoracic Diseas. 2019;11(4). ISSN 2077-6624.
Hendriksen JMT, Geersing GJ, Moons KGM, de Groot JAH. Diagnostic and prognostic prediction models. J Thromb Haemost 2013;11(s1):129–141.
Kim B, Choi J, Kim K, Jang S, Shin TG, Kim WY, Kim J-Y, Park YS, Kim SH, Lee HJ, Shin J, You JS, Kim KS, Chung SP. Bacteremia prediction model for community-acquired pneumonia: External validation in a multicenter retrospective cohort. Acad Emerg Med 2017;24(10):1226–1234.
Lee J, Hwang SS, Kim K, Jo YH, Lee JH, Kim J, Rhee JE, Park C, Chung H, Jung JY. Bacteremia prediction model using a common clinical test in patients with community-acquired pneumonia. Amer J Emerg Med 2014;32(7):700–704. ISSN 0735-6757.
Lipsky BA, Kollef MH, Miller LG, Sun X, Johannes RS, Tabak YP. Predicting bacteremia among patients hospitalized for skin and skin-structure infections: derivation and validation of a risk score. Infect Control Hospital Epidemiol 2010;31(8):828–837.
Lizarralde Palacios E, Gutiérrez Macías A, Martínez Odriozola P, Franco Vicario R, García Jiménez N, Miguel de la Villa F. Bacteriemia adquirida en la comunidad: elaboración de un modelo de predicción clínica en pacientes ingresados en un servicio de medicina interna. Med Clín 2004;123(7): 241–246. ISSN 0025-7753.
Ramesh AN, Kambhampati C, Monson JRT, Drew PJ. Artificial intelligence in medicine. Ann R Coll Surg Engl 2004;86(5):334–338.
CAS PubMed PubMed Central Article Google Scholar
Sidey-Gibbons JAM, Sidey-Gibbons CJ. Machine learning in medicine: a practical introduction. BMC Med Res Methodol 2019;19(1):64. ISSN 1471-2288.
Catto JW, Linkens DA, Abbod MF, Chen M, Burton JL, Feeley KM, Hamdy FC. Artificial intelligence in predicting bladder cancer outcome: a comparison of neuro-fuzzy modeling and artificial neural networks. Clin Cancer Res 2003;9(11):4172–7. ISSN 1078-0432 (Print) 1078-0432.
Martínez-Romero M, Vázquez-Naya J M, Rabuñal J R, Pita-Fernández S, Macenlle R, Castro-Alvariño J, López-Roses L, Ulla JL, Martínez-Calvo A V, Vázquez S, Pereira J, Porto-Pazos AB, Dorado J, Pazos A, Munteanu CR. Artificial intelligence techniques for colorectal cancer drug metabolism: ontology and complex network. Curr Drug Metab 2010;11(4):347–68. ISSN 1389-2002.
Wei L, Wan S, Guo J, Wong KKL. A novel hierarchical selective ensemble classifier with bioinformatics application. Artif Intell Med 2017;83:82–90. ISSN 0933-3657.
Gandomkar Z, Brennan PC, Mello-Thoms C. Mudern: Multi-category classification of breast histopathological image using deep residual networks. Artif Intell Med 2018;88:14–24. ISSN 0933-3657.
Jovanovic M, Radovanovic S, Vukicevic M, Van Poucke S, Delibasic B. Building interpretable predictive models for pediatric hospital readmission using tree-lasso logistic regression. Artif Intell Med 2016;72: 12–21. ISSN 0933-3657.
Wu Y, McLeod C, Blyth C, Bowen A, Martin A, Nicholson A, Mascaro S, Snelling T. Predicting the causative pathogen among children with osteomyelitis using bayesian networks improving antibiotic selection in clinical practice. Artif Intell Med 2020;107:101895. ISSN 0933-3657.
Schetinin V, Jakaite L, Krzanowski W. Bayesian averaging over decision tree models for trauma severity scoring. Artif Intell Med 2018;84:139–145. ISSN 0933-3657.
Mahfouz MA, Shoukry A, Ismail MA. Eknn: Ensemble classifier incorporating connectivity and density into knn with application to cancer diagnosis. Artif Intell Med. 2020;101985. ISSN 0933-3657.
Lin J, Chen H, Li S, Liu Y, Li X, Yu B. Accurate prediction of potential druggable proteins based on genetic algorithm and bagging-svm ensemble classifier. Artif Intell Med 2019;98:35–47. ISSN 0933-3657.
Khalilia M, Chakraborty S, Popescu M. Predicting disease risks from highly imbalanced data using random forest. BMC Med Inf Decis Making 2011;11(1):51. ISSN 1472-6947.
Little RJA, Rubin DB. Statistical analysis with missing data. USA: Wiley; 2002. ISBN 9780471183860.
Ding Y, Simonoff JS. An investigation of missing data methods for classification trees applied to binary response data. J Mach Learn Res 2010;11(6):131–170.
Guyon IM, Elisseeff A. An introduction to variable and feature selection. J Mach Learn Res 2003;3:1157–1182.
Batista GEAPA, Monard MC. An analysis of four missing data treatment methods for supervised learning. Appl Artif Intell 2003;17(5-6):519–533.
Boser BE, Guyon IM, Vapnik VN. A training algorithm for optimal margin classifiers. COLT '92: Proceedings of the fifth annual workshop on Computational learning theory. New York: Association for Computing Machinery; 1992. p. 144–152. ISBN 089791497X.
Cristianini N, Shawe-Taylor J. Support vector machines. An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press; 2000. p. 93–124.
Scholkopf B, Kah-Kay Sung, Burges CJC, Girosi F, Niyogi P, Poggio T, Vapnik V. Comparing support vector machines with gaussian kernels to radial basis function classifiers. IEEE Trans Signal Process 1997;45(11):2758–2765.
Breiman L. Random forests. Mach Learn 2001;45(1):5–32.
Menze BH, Kelm BM, Masuch R, Himmelreich U, Bachert P, Petrich W, Hamprecht FA. A comparison of random forest and its gini importance with standard chemometric methods for the feature selection and classification of spectral data. BMC Bioinforma 2009;10(1):213. ISSN 1471-2105.
Bernard S, Heutte L, Adam S. Influence of hyperparameters on random forest accuracy. Proceedings of the 8th International Workshop on Multiple Classifier Systems, MCS '09. Berlin: Springer; 2009. p. 171–180. ISBN 9783642023255.
Peterson LE. K-nearest neighbor. Scholarpedia 2009;4(2):1883.
Beretta L, Santaniello A. Nearest neighbor imputation algorithms: a critical evaluation. BMC Med Inf Decis Making 2016;16(3):74. ISSN 1472-6947.
Stekhoven DJ, Bühlmann P. MissForest—non-parametric missing value imputation for mixed-type data. Bioinformatics 2011;28(1):112–118. ISSN 1367-4803.
PubMed Article CAS PubMed Central Google Scholar
Sepulveda J, Westblade LF, Whittier S, Satlin MJ, Greendyke WG, Aaron JG, Zucker J, Dietz D, Sobieszczyk M, Choi JJ, Liu D, Russell S, Connelly C, Green DA, Carroll KC. Bacteremia and blood culture utilization during covid-19 surge in new york city. J Clin Microbiol 2020; 58(8):e00875–20.
Morens DM, Taubenberger JK, Fauci AS. Predominant Role of Bacterial Pneumonia as a Cause of Death in Pandemic Influenza: Implications for Pandemic Influenza Preparedness. The J Infect Diseas 2008;198 (7):962–970. ISSN 0022-1899.
Goncheva MI, Conceicao C, Tuffs SW, Lee H-M, Quigg-Nicol M, Bennet I, Sargison F, Pickering AC, Hussain S, Gill AC, Dutia BM, Digard P, Fitzgerald JR, Palese P. Staphylococcus aureus lipase 1 enhances influenza a virus replication. mBio 2020;11(4):e00975–20.
Tachalov VV, Orekhova LY, Kudryavtseva TV, Loboda ES, Pachkoriia MG, Berezkina IV, Golubnitschaja O. Making a complex dental care tailored to the person: population health in focus of predictive, preventive and personalised (3p) medical approach. EPMA J 2021;12(2):129–140. ISSN 1878-5085.
Zangrillo A, Beretta L, Scandroglio AM, Monti G, Fominskiy E, Colombo S, Morselli F, Belletti A, Silvani P, Crivellari M, Monaco F, Azzolini ML, Reineke R, Nardelli P, Sartorelli M, Votta CD, Ruggeri A, Ciceri F, De Cobelli F, Tresoldi M, Dagna L, Rovere-Querini P, Serpa Neto A, Bellomo R, Landoni G, COVID-BioB Study Group. Characteristics, treatment, outcomes and cause of death of invasively ventilated patients with covid-19 ards in Milan, Italy. Crit Care Resuscit J Austral Acad Crit Care Med 2020;22(3):200–211. ISSN 1441-2772.
Lai C-C, Wang C-Y, Hsueh P-R. Co-infections among patients with covid-19: The need for combination therapy with non-anti-sars-cov-2 agents?. J Microbiol Immunol Infect 2020;53(4):505–512. ISSN 1684-1182.
Davey P, Marwick CA, Scott CL, Charani E, McNeil K, Brown E, Gould IM, Ramsay CR, Michie S. Interventions to improve antibiotic prescribing practices for hospital inpatients. Cochrane Database Syst Rev. 2017;2. ISSN 1465–1858.
Murdoch TB, Detsky AS. The Inevitable Application of Big Data to Health Care. JAMA 2013; 309(13):1351–1352. ISSN 0098-7484.
Lee KH, Dong JJ, Jeong SJ, Chae M-H, Lee BS, Kim HJ, Ko SH, Song YG. Early detection of bacteraemia using ten clinical variables with an artificial neural network approach. J Clin Med. 2019;8(10). ISSN 2077-0383.
European Commission. Commission recommendation (eu) 2019/243 of 6 february 2019 on a european electronic health record exchange format. Technical Report MSU-CSE-06-2, European Commission. 2019. https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=uriserv:OJ.L_.2019.%039.01.0018.01.ENG.
Lella L, Licata I, Minati G, Pristipino C, Belvis AGD, Pastorino R. Predictive AI models for the personalized medicine. Proceedings of the 12th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2019) - Volume 5: HEALTHINF. In: Moucek R, Fred ALN, and Gamboa H, editors. Prague: SciTePress; 2019. p. 396–401.
Nardini C, Osmani V, Cormio PG, Frosini A, Turrini M, Lionis C, Neumuth T, Ballensiefen W, Borgonovi E, D'Errico G. The evolution of personalized healthcare and the pivotal role of european regions in its implementation. Person Med 2021;18(3):283–294.
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was supported by Fundación Eugenio Rodríguez Pascual 2019 grant –Development of adaptive and bioinspired systems for glycaemic control with continuous subcutaneous insulin infusions and continuous glucose monitors; the Spanish Ministerio de Innovación, Ciencia y Universidad –grant RTI2018-095180-B-I00; Madrid Regional Government –FEDER grants B2017/BMD3773 (GenObIA-CM) and Y2018/NMT- 4668 (Micro-Stress- MAP-CM).
Departamento de Arquitectura de Computadores, Universidad Complutense de Madrid, Madrid, Spain
Oscar Garnica & J. Ignacio Hidalgo
Universidad Complutense de Madrid, Madrid, Spain
Diego Gómez & Víctor Ramos
Departamento de Medicina Interna, Hospital Universitario de Fuenlabrada, Madrid, Spain
José M. Ruiz-Giardín
Oscar Garnica
Diego Gómez
Víctor Ramos
J. Ignacio Hidalgo
– O. Garnica: conceptualisation, methodology, investigation, writing (original draft preparation, reviewing and editing)
– D. Gómez: data curation, software
– V. Ramos: data curation, software
– J.I. Hidalgo: funding acquisition
– J.M. Ruiz-Giardín: conceptualisation, resources, writing (original draft preparation, reviewing, validation)
Correspondence to Oscar Garnica.
This is an observational and retrospective study. This study is in accordance with the ethical standards of the 1964 Helsinki declaration and its later amendments or comparable ethical standards. For this type of study, formal consent is not required. This is an observational retrospective study without interventions and medicaments.
Appendix A: Features in the study
Table 5 presents the description of the features used in this work.
Table 5 Features in the study sorted according to the number of missing values
Garnica, O., Gómez, D., Ramos, V. et al. Diagnosing hospital bacteraemia in the framework of predictive, preventive and personalised medicine using electronic health records and machine learning classifiers. EPMA Journal 12, 365–381 (2021). https://doi.org/10.1007/s13167-021-00252-3
Issue Date: September 2021
Preventive and personalised medicine (PPPM/3PM)
Bacteraemia diagnosis
Bacteraemia prediction
Blood culture's outcome prediction
Individualised electronic patient record analysis
Personalised antibiotic treatment
Healthcare economy | CommonCrawl |
\begin{document}
\title{The Mean First Rotation Time of a planar polymer} \begin{abstract} We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of $n$ rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on $\sqrt{n}$ and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations. \end{abstract}
\section{Introduction}
\renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} This paper focusses mainly on some properties of a planar polymer motion and in particular, on the mean time that a rotation is completed around a fixed point. This mean rotation time (MRT) provides a quantification for the transition between a free two dimensional Brownian motion and a restricted motion. To study the distribution of this time, we shall use a simplified model of a polymer made of a collection of $n$ two dimensional connected rods with Brownian random angles (Figure \ref{figure1}). To estimate the mean rotation time, we shall fix one end of the polymer. We shall examine how the MRT depends on various parameters such as the diffusion constant, the number of rods or their common length. Using some approximations and numerical simulations, the mean time for the two polymer ends to meet was estimated in dimension three \cite{WiF74,PZS96}.
Although the windings of a (planar) Brownian motion, that is the number of rotations around one point in dimension 2 or a line in dimension three and its asymptotic behavior have been studied quite extensively \cite{Spi58,PiY86,LeGY87,LeG90,ReY99}, little seems to be known about the mean time for a rotation to be completed for the first time. For an Ornstein-Uhlenbeck process, the first time that it hits the boundary of a given cone was recently estimated \cite{Vak10}.
The paper is organized as follows: in section 2, we present the polymer model. In section 3, we study a sum of i.i.d. exponentials of one dimensional Brownian motions. Interestingly, using a Central Limit Theorem, we obtain a new stochastic equation for the limit process. This equation describes the motion of the free polymer end. In section 4, we obtain our main result which is an asymptotic formula for $E\left[\tau_{n}\right]$ the MRT when the polymer is made of $n$ rods of equal lengths $l_{0}$ with the first end fixed at a distance $L$ from the origin and the Brownian motion is characterized by its rotational diffusion constant $D$. We find that the MRT depends logarithmically on the mean initial configuration and for $nl_{0} >> L$ and $n\geq 3$, the leading order term is given by: \begin{enumerate}
\item { for a general initial configuration: \begin{eqnarray*} E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \left[2\ln \left( \frac{E \left(\sum^{n}_{k=1} e^{\frac{i}{\sqrt{2D}} \theta_{k}(0)}\right)}{\sqrt{n}}\right)+Q\right], \end{eqnarray*} where $\left(\theta_{k}(0),1\leq k\leq n\right)$ are the initial angles of the polymer and $Q\approx9.56$, } \item { for an initially stretched polymer: \begin{eqnarray*} E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \; \left(\ln (n)+Q\right), \end{eqnarray*} } \\
\item {for an average over uniformly distributed initial angles: \begin{eqnarray*} E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \; \tilde{Q}, \end{eqnarray*} where $\tilde{Q}\approx9.62$. } \end{enumerate} We confirm our analytical results with generic Brownian simulations with normalized parameters . Finally, in section 5 we discuss some related open questions.
\section{Stochastic modeling of a planar polymer}
Various models are available to study polymers: the Rouse model consists of a collection of beads connected by springs, while more sophisticated models account for bending, torsion and specific mechanical properties \cite{Rou53,SSS80,DoE94}. Here, we consider a very crude approximation where a planar polymer is modeled as a collection of $n$ rigid rods, with equal fixed lengths $l_{0}$ and we denote their extremities by $\left(X_{0},X_{1},X_{2},\ldots,X_{n} \right)$ (Figure \ref{figure1}) in a framework with origin ${\bf 0}$. We shall fix one of the polymer ends $X_{0}=(L,0)$ (where $L>0$) on the $x$-axis.
\begin{figure}
\caption{ {\bf Schematic representation of a planar polymer winding
around the origin}.(a) A random configuration,
(b) MRT when the $n$-th bead reaches $2\pi$.}
\label{figure1}
\end{figure}
The dynamics of the $i$-th rod is characterized by its angle $\theta_{i}(t)$ with respect to the $x$-axis. The overall polymer dynamics is thus characterized by the angles $\left(\theta_{1}(t),\theta_{2}(t),\ldots,\theta_{n}(t), t\geq 0 \right)$. Due to the thermal collisions in the medium, each angle follows a Brownian motion. Thus, with $\stackrel{(law)}{=}$ denoting equality in law, \begin{eqnarray*} \left( \theta_{i}(t),i\leq n \right) \stackrel{(law)}{=} \sqrt{2D} \left( B_{i}(t),i\leq n \right) \ \mathrm{that} \ \mathrm{is:} \ \ d\theta_{i}(t) = \sqrt{2D} \: dB_{i}(t), \ \ i\leq n, \end{eqnarray*} where $D$ is the rotational diffusion constant and $\left( B_{1}(t),\ldots,B_{n}(t), t\geq 0 \right)$ is an $n$-dimensional Brownian motion (BM). The position of each rod can now be obtained as: \begin{eqnarray}\label{2Xi}
X_{1}(t) = L + l_{0} e^{i\theta_{1}(t)}, \ X_{2}(t) = X_{1}(t) + l_{0} e^{i\theta_{2}(t)}, \ \ldots, \
X_{n}(t) = X_{n-1}(t) + l_{0} e^{i\theta_{n}(t)}. \nonumber \\ \end{eqnarray} In particular, the moving end is given by: \begin{eqnarray}\label{2Xn}
X_{n}(t) = L + l_{0} \sum^{n}_{k=1} e^{i\theta_{k}(t)} = L + l_{0} \sum^{n}_{k=1} e^{i\sqrt{2D} B_{k}(t)}, \nonumber \\ \end{eqnarray} which can be written as: \begin{eqnarray}\label{2XnRphi}
X_{n}(t) = R_{n}(t) e^{i \varphi_{n}(t)}. \end{eqnarray} Thus $\varphi_{n}(t)$ accounts for the rotation of the polymer with respect to the origin ${\bf 0}$ and $R_{n}$ is the distance to the origin.
In order to compute the MRT, we shall study a sum of exponentials of Brownian motions, a topic which often leads to surprising computations \cite{Yor01}. First, we scale the space and time variables as follows: \begin{eqnarray}\label{rescaling}
\tilde{l}=\frac{L}{l_{0}} \ \ \mathrm{and} \ \ \tilde{t}=\frac{t}{2D}. \end{eqnarray} Equation (\ref{2Xn}) becomes: \begin{eqnarray}\label{2Xnrescaled}
X_{n}(t) = \tilde{l} + \sum^{n}_{k=1} e^{i\tilde{B}_{k}(t)}, \end{eqnarray} where $\left(\tilde{B}_{1}(t),\ldots,\tilde{B}_{n}(t), t\geq 0 \right)$ is an $n$-dimensional Brownian motion (BM), and for $k=1,\ldots,n$, using the scaling property of Brownian motion, we have: \begin{eqnarray*}
\tilde{B}_{k}(t)\equiv \frac{1}{\sqrt{2D}} B_{k}(t) \stackrel{(law)}{=} B_{k}\left(\frac{t}{2D}\right)=B_{k}(\tilde{t}). \end{eqnarray*}
Before describing our approach, we first discuss the mean initial configuration of the polymer. It is given by: \begin{eqnarray}\label{initialbis2} c_{n}= E \left( \sum^{n}_{k=1} e^{i\theta_{k}(0)} \right), \end{eqnarray} where the initial angles $\theta_{k}(0)$ are such that the polymer has not already made a loop. After scaling, the mean initial configuration becomes: \begin{eqnarray}\label{initial} c_{n}= E \left( \sum^{n}_{k=1} e^{i \: \tilde{\theta}_{k}(0)} \right), \end{eqnarray} with \begin{eqnarray}\label{rescalinginitialangle}
\tilde{\theta}_{k}(0)=\tilde{B}_{k}(0). \end{eqnarray} From now on, we use $\theta_{k}$ instead of $\tilde{\theta}_{k}$, $B$ instead of $\tilde{B}$ and $t$ instead of $\tilde{t}$. Any segment in the interior of the polymer can hit the angle $2\pi$ around the origin, but we will not consider this as a winding event, although we could and in that case, the MRT would be different. Rather, we shall only consider that, given an initial configuration $c_{n}$, the MRT is defined as:
\begin{eqnarray} MRT \equiv E\{\tau_{n}|c_{n} \} \equiv E\left[\tau_{n}\right], \end{eqnarray} where \begin{eqnarray}\label{deftaun}
\tau_{n} \equiv \inf\{ t>0, |\varphi_{n}(t)|=2\pi \}. \end{eqnarray} Thus, an initial configuration is not winding when \begin{eqnarray}
\left| \varphi_{n}(0) \right| <2\pi. \end{eqnarray} Then, we can define the winding event using a one dimensional variable only. In general, winding is a rare event and we expect that the MRT will depend crucially on the length of the polymer which will be quite long. Interestingly, the rotation is accomplished when the angle $\varphi_{n}(t)$ reaches $2 \pi$ or $-2 \pi$, but the distance of the free end point to the origin is not fixed, leading to a one dimensional free parameter space. This undefined position is in favor of a winding time that is not too large when compared to any narrow escape problem where a Brownian particle has to find a small target in a confined domain \cite{WaK93,HoS04,SSHE06,SSH07,BKH07,PWPSK09}.
In this study, we consider not only that the initial condition satisfies $\left| \varphi_{n}(0) \right| <2\pi$, but we impose that $\varphi_{n}(0)$ is located far enough from $2\pi$, to avoid studying any boundary layer effect, which would lead to a different MRT law. Indeed, starting inside the boundary layer for a narrow escape type problem leads to specific escape laws \cite{SSHE06}. Given a small $\varepsilon>0$ we shall consider the space of configurations $\Omega_{\varepsilon}$
such that $\left|\varphi_{n}(0) \right| <2\pi-\varepsilon$. We shall mainly focus on the stretched polymer, \begin{eqnarray} \left(\theta_{1}(0),\theta_{2}(0),\ldots,\theta_{n}(0) \right)=\left(0,0,\ldots,0\right), \end{eqnarray} and thus $c_{n}=n >0$ (in this case, $\varphi_{n}(0)=0$). Finally, it is quite obvious that winding occurs only when the condition: \begin{eqnarray} n l_{0}>L \end{eqnarray} is satisfied, which we assume all along.
The outline of our method is: first we show that the sum $X_{n}(t)$ converges (eq. (\ref{2Xn})) and we obtain a Central Limit Theorem. Using It\^{o} calculus, we study the sequence: \begin{eqnarray} \frac{1}{\sqrt{n}} \overline{X}_{n}(t) = \frac{1}{\sqrt{n}} \left[ X_{n}(t) - E\left( X_{n}(t) \right) \right]. \end{eqnarray} and prove that $\frac{1}{\sqrt{n}} \overline{X}_{n}(t)$, for $n$ large, converges to a stochastic process which is a generalization of an Ornstein-Uhlenbeck process (GOUP), containing a time dependent deterministic drift $c_{n}e^{-t}$. This GOUP is driven by a martingale $\left(M_{t}^{(n)}, t\geq 0 \right)$ that we further characterize. Interestingly, the two cartesian coordinates of
$\left(M_{t}^{(n)}, t\geq 0 \right)$ converge to two independent Brownian motions with two different time scale functions. To obtain an asymptotic formula for the MRT, we show that in the long time asymptotics, where winding occurs, the GOUP can be approximated by a standard Ornstein-Uhlenbeck process (OUP). Using some properties of the GOUP \cite{Vak10}, we finally derive the MRT for the polymer which is the mean time that $|\varphi_{n}(t)|=2\pi$.
\section{Properties of the free polymer end $X_{n}(t)$ using a Central Limit Theorem}
In this section we study some properties of the free polymer end $X_{n}(t)$. In particular, using a Central Limit Theorem, we show that the limit process satisfies a stochastic equation of a new type. To study the random part of $X_{n}(t)$, we shall remove from it its first moment and we shall now consider the asymptotic behavior of the drift-less sequence: \begin{eqnarray} \label{2Xnbar1}
\frac{1}{\sqrt{n}} \overline{X}_{n}(t) =
\frac{1}{\sqrt{n}} \left[ X_{n}(t) - E\left( X_{n}(t) \right) \right]. \end{eqnarray} We start by computing the first moment $E\left( X_{n}(t) \right)$. Because (see (\ref{2Xn})), $\left(\theta_{i}(t),t\geq0\right)$ are assumed to be $n$ independent identically distributed (iid) Brownian motions with variance $2D$, after rescaling, we obtain: \begin{eqnarray} \label{meanX_{n}(t)}
E\left( X_{n}(t) \right) &=& E \left[\tilde{l} + \sum^{n}_{k=1} e^{i B_{k}(t)}\right] \nonumber \\
&=& \tilde{l} + \left(\sum^{n}_{k=1} E \left[e^{i\left(B_{k}(t)-B_{k}(0)\right)}\right] \: E \left[e^{i\left(B_{k}(0)\right)}\right]\right) \nonumber \\
&=& \tilde{l} + c_{n} e^{-\frac{t}{2}}. \end{eqnarray} where $c_{n}$ is defined by (\ref{initial}) and we have used that: \begin{eqnarray}
E \left[e^{i\left(B_{k}(t)-B_{k}(0)\right)}\right]= e^{-\frac{t}{2}} \ . \end{eqnarray} We study the sequence (\ref{2Xnbar1}) as follows: \begin{eqnarray} \label{2Xnbar}
\frac{1}{\sqrt{n}} \overline{X}_{n}(t) &=& \frac{1}{\sqrt{n}} \left[ \sum^{n}_{k=1} e^{iB_{k}(t)} - E\left( \sum^{n}_{k=1} e^{i B_{k}(t)} \right) \right] \nonumber \\
&=& \frac{1}{\sqrt{n}} \sum^{n}_{k=1} F_{k} (t), \end{eqnarray} where $F_{k} (t) = e^{iB_{k}(t)} - E\left(e^{iB_{k}(t)} \right)$. Applying It\^{o}'s formula to \begin{eqnarray}\label{ZsumF} Z_{t}^{(n)} = \frac{1}{\sqrt{n}} \sum^{n}_{k=1} F_{k} (t) \ , \end{eqnarray} with $Z_{0}^{(n)}=0$, we obtain: \begin{eqnarray} Z_{t}^{(n)} &=& \frac{i}{\sqrt{n}} \int^{t}_{0} \sum^{n}_{k=1} e^{i B_{k}(s)} dB_{k}(s) - \frac{1}{2\sqrt{n}} \int^{t}_{0} \sum^{n}_{k=1} \left( e^{i B_{k}(s)} - E\left( e^{i B_{k}(s)} \right) \right) ds \nonumber \\ \label{2Zn}
&=& M_{t}^{(n)} - \frac{1}{2}\int^{t}_{0} Z_{s}^{(n)} \: ds, \end{eqnarray} where: \begin{eqnarray}\label{2MSC} M_{t}^{(n)} &=& \frac{i}{\sqrt{n}} \int^{t}_{0} \sum^{n}_{k=1} e^{i B_{k}(s)} \: dB_{k}(s) \nonumber \\
&=& \frac{1}{\sqrt{n}} \int^{t}_{0} \sum^{n}_{k=1} \left( i\cos(B_{k}(s))- \sin( B_{k}(s)) \right) \: dB_{k}(s) \nonumber \\
&=& -S_{t}^{(n)} + i C_{t}^{(n)}. \end{eqnarray} We shall now study the asymptotic limit of the martingales $M_{t}^{(n)}$ as $n\rightarrow\infty$ and summarize our result in the following theorem; the convergence in law being considered there is associated with the topology of the uniform convergence on compact sets of the functions in $C(\mathbb{R}_{+},\mathbb{R}^{2})$.
\begin{theo}\label{CLT} The sequence $(M_{t}^{(n)},t\geq 0)$ converges in law to a Brownian motion in dimension 2, with 2 deterministic time changes. More precisely, \begin{eqnarray} \label{2CLTcvg}
(S^{(n)}_{t},C^{(n)}_{t}, t\geq0) \overset{{(law)}}{\underset{n\rightarrow\infty}\longrightarrow} ( \sigma_{\left(\frac{1}{2}\int^{t}_{0} ds \: (1-e^{-2s})\right)} , \gamma_{\left(\frac{1}{2}\int^{t}_{0} ds \: (1+e^{-2s})\right)} , t\geq0), \end{eqnarray} where $(\sigma_{u},\gamma_{u}, u\geq0)$ are two independent Brownian motions. \end{theo}
\begin{rem}\label{remTCL} $\left.a\right)$ The convergence in law (\ref{2CLTcvg}) is a new result and it shows that the sum of the complex exponentials of i.i.d. Brownian motions can be approximated by a two dimensional Brownian motion, with a different time scale for each coordinate. \\ $\left.b\right)$ An extension of Theorem \ref{CLT} when $\left(\exp\left(iB(s)\right),s\geq0\right)$ a Brownian motion on the unit circle is replaced by a BM on the unit sphere in $\mathbb{R}^{n}$ is obtained in \cite{HVY11}. \end{rem}
{\noindent \textbf{Proof of Theorem \ref{CLT}}} See Appendix \ref{apTCL}.
\quad\hbox{\hskip 4pt\vrule width 5pt height 6pt depth 1.5pt}
\noindent We conclude that the sequence $(M_{t}^{(n)},t\geq 0) $ converges in law: \begin{eqnarray}\label{2Mcvg}
(M_{t}^{(n)},t\geq 0) \overset{{(law)}}{\underset{n\rightarrow\infty}\longrightarrow} (\sigma_{\left(\frac{t}{2}-\frac{1-e^{-2t}}{4}\right)} + i \gamma_{\left(\frac{t}{2}+\frac{1-e^{-2t}}{4}\right)},t\geq 0). \end{eqnarray} The process $(Z_{t}^{(n)},t\geq 0)$, which we defined in (\ref{2Zn}), is a generalization of the classical Ornstein-Uhlenbeck process; it is driven by $(M_{t}^{(n)},t\geq 0)$ and, from (\ref{2Zn}), we obtain: \begin{eqnarray}\label{2OU}
Z_{t}^{(n)} &=& e^{-\frac{t}{2}} \int^{t}_{0} e^{\frac{s}{2}} dM_{s}^{(n)}. \end{eqnarray}
\begin{corr}\label{2Corr} $\left.\mathrm{a}\right)$ The sequence $(Z_{t}^{(n)},t\geq 0)$ converges in law: \begin{eqnarray} Z_{t}^{(n)}\overset{{(law)}}{\underset{n\rightarrow\infty}\longrightarrow}Z_{t}^{(\infty)}, \end{eqnarray} with: \begin{eqnarray}\label{cvgZn}
Z_{t}^{(\infty)} = e^{-\frac{t}{2}} \int^{t}_{0} \sqrt{\sinh(s)} \; d\delta_{s} + i \; e^{-\frac{t}{2}} \int^{t}_{0} \sqrt{\cosh(s)} \; d\tilde{\delta}_{s}, \end{eqnarray} where $\left(\delta_{t}, \tilde{\delta}_{t},t\geq0\right)$ are two independent 1-dimensional Brownian motions. $\left.\mathrm{b}\right)$ $Z_{t}^{(\infty)}\overset{{(law)}}{\underset{t\rightarrow\infty}\longrightarrow}\frac{1}{\sqrt{2}}\left(N+i\tilde{N}\right)$, where $N$ and $\tilde{N}$ are two centered and reduced Gaussian variables. \end{corr}
{\noindent \textbf{Proof of Corollary \ref{2Corr}}} \\ $\left.\mathrm{a}\right)$ From (\ref{2MSC}), (\ref{2CLTcvg}) and (\ref{2OU}) we deduce: \begin{eqnarray} Z_{t}^{(n)} &=& e^{-\frac{t}{2}} \int^{t}_{0} e^{\frac{s}{2}} dM_{s}^{(n)}\label{2Zncvg} \\
&\overset{{(law)}}{\underset{n\rightarrow\infty}\longrightarrow}& e^{-\frac{t}{2}} \int^{t}_{0} \underbrace{e^{\frac{s}{2}} \sqrt{\frac{1-e^{-2s}}{2}}}_{\sqrt{\sinh(s)}} \; d\delta_{s}
+ i e^{-\frac{t}{2}} \int^{t}_{0} \underbrace{e^{\frac{s}{2}} \sqrt{\frac{1+e^{-2s}}{2}}}_{\sqrt{\cosh(s)}} \; d\tilde{\delta}_{s} \equiv Z_{t}^{(\infty)}. \nonumber \\
\label{2Zncvgfinal} \end{eqnarray} $\left.\mathrm{b}\right)$ From (\ref{2Zncvgfinal}), we change variables: $u=t-s$ and we obtain: \begin{eqnarray}
Z_{t}^{(\infty)} &=& \int^{t}_{0} e^{-\frac{t-s}{2}} \sqrt{\frac{1-e^{-2s}}{2}} \; d\delta_{s}
+ i \int^{t}_{0} e^{-\frac{t-s}{2}} \sqrt{\frac{1+e^{-2s}}{2}} \; d\tilde{\delta}_{s} \nonumber \\
&\overset{{u=t-s}}{\underset{(law)}=}& \int^{t}_{0} e^{-\frac{u}{2}} \sqrt{\frac{1}{2}-\frac{e^{-2(t-u)}}{2}} \; d\delta_{u}
+ i \int^{t}_{0} e^{-\frac{u}{2}} \sqrt{\frac{1}{2}+\frac{e^{-2(t-u)}}{2}} \; d\tilde{\delta}_{u} \nonumber \\
&& \label{2Zncvgfinalbis} \\
&\overset{{L^{2}}}{\underset{t\rightarrow\infty}\longrightarrow}& \frac{1}{\sqrt{2}} \int^{\infty}_{0} e^{-\frac{u}{2}}d\delta_{u} + i \; \frac{1}{\sqrt{2}} \int^{\infty}_{0} e^{-\frac{u}{2}} d\tilde{\delta}_{u} \ , \label{2Zncvgfinalbisbis} \end{eqnarray} where the two variables on the RHS of (\ref{2Zncvgfinalbisbis}) are centered Gaussian with variance $1/2$ and the convergence in $L^{2}$ for $t\rightarrow\infty$ may be proved by using the dominated convergence theorem.
\quad\hbox{\hskip 4pt\vrule width 5pt height 6pt depth 1.5pt}
\begin{rem}\label{2ItoOchi} In \cite{It\^{o}83} and \cite{Ochi85} there is a more complete asymptotic study for $Z_{t}^{(n)}(\varphi)$, as $n\rightarrow\infty$, for a large class of functions $\varphi$, but we do not pursue this line of research here. \end{rem}
On one hand, from the identities (\ref{2Xnbar1}), (\ref{meanX_{n}(t)}), (\ref{2Xnbar}) and (\ref{ZsumF}), we obtain the following expansion: \begin{eqnarray}\label{2Xnfinal}
X_{n}(t) &=& \overline{X}_{n}(t) + E\left[ X_{n}(t) \right] \nonumber \\
&=& \sqrt{n} \: Z_{t}^{(n)} + c_{n} e^{-\frac{t}{2}} + \tilde{l} \nonumber \\
\Rightarrow \tilde{X}_{n}(t)&\equiv& X_{n}(t)-\tilde{l}=\sqrt{n} \: Z_{t}^{(n)} + c_{n} e^{-\frac{t}{2}} , \end{eqnarray} with $n$ the number of rods/beads, $Z_{t}^{(n)}$ a GOUP driven by $M_{t}^{(n)}$ which is given by (\ref{2Mcvg}), $c_{n}=E \left( \sum^{n}_{k=1} e^{i\theta_{k}(0)} \right)$ a constant depending on the mean initial configuration and $\tilde{l}=\frac{L}{l_{0}}$ the rescaled distance of the fixed end from the origin ${\bf 0}$ ($l_{0}$ is the fixed length of the rods). \\ On the other hand, with Corollary \ref{2Corr}(a), we deduce that: \begin{eqnarray} \label{2Xnbar2}
Z^{(n)}_{t} &\equiv& \frac{1}{\sqrt{n}} \overline{X}_{n}(t) \equiv \frac{1}{\sqrt{n}} \left[ X_{n}(t) - E\left( X_{n}(t) \right) \right] \nonumber \\
&=& \frac{1}{\sqrt{n}} \left[ X_{n}(t) - c_{n} e^{-\frac{t}{2}} - \tilde{l}\right] \nonumber \\
&\overset{{(law)}}{\underset{n\rightarrow\infty}\longrightarrow}& e^{-\frac{t}{2}} \int^{t}_{0} \sqrt{\sinh(s)} \; d\delta_{s} + i \; e^{-\frac{t}{2}} \int^{t}_{0} \sqrt{\cosh(s)} \; d\tilde{\delta}_{s}\equiv Z^{(\infty)}_{t}. \end{eqnarray}
\section{Asymptotic expression for the MRT}\label{2secMRT}
We now study more precisely the different time scales of the two Brownian motions in (\ref{2Mcvg}) and (\ref{2Xnbar2}). We estimate $Z_{t}^{(\infty)}$ for $t$ large, the regime for which the rotation will be accomplished. \\ We introduce here the following notation: $\stackrel{L^{2}}{\approx}$ denotes closeness in the $L^{2}-$norm: for two stochastic processes $(W^{(1)}_{t},t\geq0)$ and $(W^{(2)}_{t},t\geq0)$, the notation $W^{(1)}_{t}\stackrel{L^{2}}{\approx}W^{(2)}_{t}$ means that
${\underset{t\rightarrow \infty}\lim} E\left[\left|W^{(1)}_{t}-W^{(2)}_{t}\right|^{2}\right]=0$. \\ We shall show that, with $\left(\mathbb{B}_{t} = \delta_{t}+i \tilde{\delta}_{t},t\geq0\right)$ a 2-dimensional Brownian motion starting from $1$: \begin{eqnarray}\label{2I}
Z_{t}^{(\infty)} &=& e^{-\frac{t}{2}} \int^{t}_{0} \sqrt{\sinh(s)} \; d\delta_{s} + i \; e^{-\frac{t}{2}} \int^{t}_{0} \sqrt{\cosh(s)} \; d\tilde{\delta}_{s} \nonumber \\
&\stackrel{L^{2}}{\approx}& e^{-\frac{t}{2}} \int^{t}_{0} \frac{e^{\frac{s}{2}}}{\sqrt{2}} d\mathbb{B}_{s}. \end{eqnarray} For this, it suffices to use the expression (\ref{2Zncvgfinalbisbis}) and the following Proposition, which reinforces the $\stackrel{L^{2}}{\approx}$ result in (\ref{2I}).
\begin{prop}\label{propositionZ} As $t\rightarrow\infty$, the Gaussian martingales \begin{eqnarray*}
\left(\int^{t}_{0} \sqrt{\sinh(s)} \; d\delta_{s}-\int^{t}_{0} \frac{e^{s/2}}{\sqrt{2}} \; d\delta_{s}, t\geq0\right), \end{eqnarray*} and \begin{eqnarray*}
\left(\int^{t}_{0} \sqrt{\cosh(s)} \; d\tilde{\delta}_{s}-\int^{t}_{0} \frac{e^{s/2}}{\sqrt{2}} \; d\tilde{\delta}_{s}, t\geq0\right) \end{eqnarray*} converge a.s. and in $L^{2}$. The limit variables are Gaussian with variances $\frac{\pi-3}{2}$ and $-1+2\sqrt{2}-2a_{s}(1)\approx0,033$, where $a_{s}(x)\equiv\arg\sinh (x) \equiv \log (x+\sqrt{1+x^{2}}), \ x \in \mathbb{R}$, respectively. \end{prop}
Thus, by multiplying both processes by $e^{-\frac{t}{2}}$, we obtain (\ref{2I}).\\
{\noindent \textbf{Proof of Proposition \ref{propositionZ}}} The Gaussian martingale $\int^{t}_{0} \left(\sqrt{\sinh(s)} - \frac{e^{s/2}}{\sqrt{2}}\right) \; d\delta_{s}$ has increasing process \begin{eqnarray*}
\int^{t}_{0} \left(\sqrt{\sinh(s)}-\frac{e^{s/2}}{\sqrt{2}}\right)^{2} \; ds = \int^{t}_{0} \frac{e^{s}}{2}\left(\sqrt{1-e^{-2s}}-1 \right)^{2} \; ds \ , \end{eqnarray*} which converges as $t\rightarrow\infty$. Hence, the limit variable $\int^{\infty}_{0} \left(\sqrt{\sinh(s)} - \frac{e^{s/2}}{\sqrt{2}}\right) \; d\delta_{s}$ is Gaussian, and its variance is given by (we change variables: $u=e^{-2s}$ and $B(a,b)$ denotes the Beta function with arguments $a$ and $b$\footnote[7]{We recall that if $\left(\Gamma(x),x\geq0\right)$ denotes the Gamma function, then $B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$.}): \begin{eqnarray*}
&& \int^{\infty}_{0} \frac{ds \; e^{s}}{2} \left(\sqrt{1-e^{-2s}} - 1\right)^{2} = \frac{1}{4} \int^{1}_{0} du \; u^{-3/2} \left(\sqrt{1-u} - 1\right)^{2} \\
&=& \frac{1}{4} \left[ \int^{1}_{0} du \; u^{-3/2} \left((1-u)-2\sqrt{1-u}+1\right)\right] \\
&=& \frac{1}{4} \left\{B\left(-\frac{1}{2},2\right)-2B\left(-\frac{1}{2},\frac{3}{2}\right)-2\right\} = \frac{\pi-3}{2} \ . \end{eqnarray*} To be rigorous, the integral $\int^{1}_{0} du \; u^{-\alpha} \left(\sqrt{1-u} - 1\right)^{2}$, which is well defined for $0<\alpha<1$, can be extended analytically for any complex $\alpha$ with $\mathrm{Re}(\alpha)<3$. \\ For the convergence of the second process, it suffices to replace $\sinh(s)$ by $\cosh(s)\equiv\frac{e^{s}}{2}(1+e^{-2s})$. The limit variable $\int^{\infty}_{0} \left(\sqrt{\cosh(s)} - \frac{e^{s/2}}{\sqrt{2}}\right) \; d\tilde{\delta}_{s}$ is also Gaussian, and, repeating the previous calculation, we easily compute its variance.
\quad\hbox{\hskip 4pt\vrule width 5pt height 6pt depth 1.5pt}
\subsubsection*{Asymptotic expression for the MRT}\label{2asymptMRT}
For $t$ large, we derive an asymptotic value for the MRT. First, from (\ref{2OU}) and (\ref{2Xnfinal}): \begin{eqnarray}\label{2Xnfinal3}
\tilde{X}_{n}(t) &=& \sqrt{n}Z^{(n)}_{t}+c_{n}e^{-\frac{t}{2}} \nonumber \\
&=& \sqrt{n}e^{-\frac{t}{2}} \left( \tilde{c}_{n} + \int^{t}_{0} e^{\frac{s}{2}} dM^{(n)}_{s} \right), \end{eqnarray} where, the sequence $\tilde{c}_{n}$ is: \begin{eqnarray}\label{2cn'} \tilde{c}_{n}\equiv \frac{c_{n}}{\sqrt{n}}, \end{eqnarray} $n$ is the number of rods/beads, and $c_{n} \equiv E \left( \sum^{n}_{k=1} e^{i \theta_{k}(0)} \right)$ is a constant depending on the mean initial configuration. Thus, from (\ref{2Xnfinal}), (\ref{2I}), (\ref{2Xnfinal3}) and also using the scaling property of Brownian motion: \begin{eqnarray}\label{2Xnfinal3bis}
\tilde{X}_{n}(t) \overset{{(law)}}{\underset{n:large}\approx} \sqrt{n}Y_{t}^{(n)}, \end{eqnarray} where \begin{eqnarray}\label{2Xnfinal4}
Y_{t}^{(n)}\equiv e^{-\frac{t}{2}} \left( \tilde{c}_{n} + \int^{t}_{0} e^{\frac{s}{2}} d\mathbb{B}_{s/2} \right). \end{eqnarray}
\subsubsection*{Changing time and expression of the MRT}\label{2asymptMRTpp}
To express the MRT, we now apply deterministic time changes. To make our writing simple, we denote $Y_{t}$ for $Y_{t}^{(n)}$, and changing variables $u=\frac{s}{2}$ in (\ref{2Xnfinal4}), we obtain: \begin{eqnarray}\label{2OUz0DeB2}
Y_{2t} = e^{-t} \left( \tilde{c}_{n} + \int^{t}_{0} e^{u} d\mathbb{B}_{u} \right). \end{eqnarray} Now, there is another BM $(\tilde{\mathbb{B}}_{t},t\geq0)$, starting from $\tilde{c}_{n}$, such that: \begin{eqnarray}\label{2OUz0DeBDS}
Y_{2t} = e^{-t} \left( \tilde{\mathbb{B}}_{\alpha_{t}} \right), \end{eqnarray} where: \begin{eqnarray*}
\alpha_{t}= \int^{t}_{0} e^{2s} ds = \frac{e^{2t}-1}{2}, \end{eqnarray*} hence: \begin{eqnarray}
\alpha^{-1}(t)= \frac{1}{2} \ln \left( 1+2t\right).\label{eeeeq} \end{eqnarray} Applying It\^{o}'s formula to (\ref{2OUz0DeBDS}), we obtain: \begin{eqnarray*} dY_{2s} = -e^{-s} \tilde{\mathbb{B}}_{\alpha_{s}} ds + e^{-s} d\left(\tilde{\mathbb{B}}_{\alpha_{s}}\right). \end{eqnarray*} We divide by $Y_{2s}$ and we obtain: \begin{eqnarray*} \frac{dY_{2s}}{Y_{2s}} = - ds +\frac{d\tilde{\mathbb{B}}_{\alpha_{s}}}{\tilde{\mathbb{B}}_{\alpha_{s}}}. \end{eqnarray*} Thus: \begin{eqnarray*} \mathrm{Im} \left(\frac{dY_{2s}}{Y_{2s}}\right) = \mathrm{Im} \left(\frac{d\tilde{\mathbb{B}}_{\alpha_{s}}}{\tilde{\mathbb{B}}_{\alpha_{s}}}\right), \end{eqnarray*} which means that, if we denote: \begin{eqnarray} \theta^{Z}_{t}\equiv \mathrm{Im}(\int^{t}_{0}\frac{dZ_{s}}{Z_{s}}), t\geq0, \end{eqnarray} the continuous winding process associated to a generic stochastic process $Z$, then: \begin{eqnarray*} \theta^{Y}_{2t} = \theta^{\tilde{\mathbb{B}}}_{\alpha_{t}}. \end{eqnarray*} Thus, the first hitting times of the symmetric conic boundary of angle $c$ \begin{eqnarray}\label{2Tthetac}
T^{|\theta^{Y}|}_{c}\equiv
\inf \left\{t\geq 0 : \left|\theta^{Y}_{t}\right|=c \right\}, \end{eqnarray} and \begin{eqnarray}\label{2Tthetacc}
T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}\equiv\inf \left\{t\geq 0 : \left|\theta^{\tilde{\mathbb{B}}}_{t}\right|=c \right\}, \end{eqnarray} for an Ornstein-Uhlenbeck process $Y$ with parameter $\lambda=1$, and for a Brownian motion $\tilde{\mathbb{B}}$ respectively, with relation (\ref{eeeeq}), satisfy: \begin{eqnarray}\label{2bisTchat}
2T^{|\theta^{Y}|}_{c}=\frac{1}{2}\ln
\left(1+2T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}\right). \end{eqnarray} Finally, \begin{eqnarray}\label{ETchatz0Dasympbis}
E\left[2T^{|\theta^{Y}|}_{c}\right] &=& \frac{1}{2} E\left[\ln \left(1+2 T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}\right) \right] \\
&=& \frac{\ln 2}{2} + \frac{1}{2} E\left[\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}+\frac{1}{2}\right)\right] , \end{eqnarray} and equivalently: \begin{eqnarray}\label{ETchatz0Dasymp2}
E\left[T^{|\theta^{Y}|}_{c}\right] &=& \frac{\ln 2}{4} + \frac{1}{4} E\left[\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}+\frac{1}{2}\right)\right].
\end{eqnarray} Thus, by taking $c=2\pi$, for $n$ large, with $\tilde{\varphi}_{n}(t)$ denoting the total angle of $\tilde{X}_{n}(t)$, the mean time $E\left[\tilde{\tau}_{n}\right]$, where $\tilde{\tau}_{n}\equiv \inf\{ t>0, |\tilde{\varphi}_{n}(t)|=2\pi \}$, that $\tilde{X}_{n}(t)$ rotates around ${\bf 0}$, is: \begin{eqnarray}\label{2taunapprox}
E\left[\tilde{\tau}_{n}\right] &\approx& \frac{\sqrt{n} }{4} \left( \ln 2 + E\left[\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{2\pi}+\frac{1}{2}\right)\right]\right). \end{eqnarray} By using the series expansion of $\log(1+x)$, we obtain informally: \begin{eqnarray}
E\left[\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}+\frac{1}{2}\right)\right] - E\left[\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}\right)\right] &=& E\left[\ln\left(1+\frac{1}{2T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}}\right)\right] \label{2logseries2} \\
&=& \frac{1}{2} E\left[\frac{1}{T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}}\right]-\frac{1}{8}E\left[\left(\frac{1}{T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}}\right)^{2}\right]+\ldots, \nonumber \\ \label{2logseries}
\end{eqnarray} where this expansion should be understood as follows: we obtain an alternate series of negative moments of $T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}$, which converges \cite{VaY11}. For our purpose, we will truncate the series. We shall estimate and use only the first moment of $1/T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}$. By numerical calculations, the other moments can be neglected.
From the skew product representation \cite{KaSh88,ReY99} there is another planar Brownian motion $(\beta_{u}+i \gamma_{u},u\geq0)$, starting from $\log \tilde{c}_{n}+i0$, such that: \begin{eqnarray}\label{2skew-product}
\log\left|\tilde{\mathbb{B}}_{t}\right|+i\theta_{t}\equiv\int^{t}_{0}\frac{d\tilde{\mathbb{B}}_{s}}{\tilde{\mathbb{B}}_{s}}=\left( \beta_{u}+i\gamma_{u}\right)
\Bigm|_{u=H_{t}\equiv\int^{t}_{0}\frac{ds}{\left|\tilde{\mathbb{B}}_{s}\right|^{2}}}, \end{eqnarray} and equivalently: \begin{eqnarray}\label{2skew-product2}
\log\left|\tilde{\mathbb{B}}_{t}\right|=\beta_{H_{t}}; \ \ \theta_{t}=\gamma_{H_{t}}.
\end{eqnarray} Thus, with $T^{|\gamma|}_{c}\equiv\inf \left\{t\geq 0 :\left|\gamma_{t}\right|=c \right\}$ and
$T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}\equiv\inf \left\{t\geq 0 :\left|\theta^{\tilde{\mathbb{B}}}_{t}\right|=c \right\}$, because $\theta_{T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}}=\gamma_{H_{T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}}}$:
\begin{eqnarray*} T^{|\gamma|}_{c}=H_{T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}},
\end{eqnarray*} hence $T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}=H^{-1}_{u}\Bigm|_{u=T^{|\gamma|}_{c}}$, where: \begin{eqnarray}\label{2Hinverse} H^{-1}_{u}\equiv \inf\{ t:H_{t}>u \} = \int^{u}_{0}ds \exp(2\beta_{s})\equiv A_{u},
\end{eqnarray} and for $u=T^{|\gamma|}_{c}$, we obtain: \begin{eqnarray}\label{2negmom}
T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}=A_{T^{|\gamma|}_{c}}. \end{eqnarray} It is of interest for our purpose to look at the first negative moment of $A_{t}$ which has the following integral representation for any $t>0$ (\cite{Duf00}, \cite{D-MMY00}, p.49, Prop. 7, formula (15)): \begin{eqnarray}\label{2negmom2}
E\left[\frac{1}{A_{t}}\right]=\int^{\infty}_{0} y e^{-y^{2}t/2} \coth\left(\frac{\pi}{2}y\right) \; dy . \end{eqnarray} With $\beta_{s}=\log(\tilde{c}_{n})+\beta^{(0)}_{s}$, where $(\beta^{(0)}_{s},s\geq0)$ is a one-dimensional Brownian motion starting from 0, we obtain: \begin{eqnarray} \label{Tcthetacn}
T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}&=& H^{-1}(T^{|\gamma|}_{c}) \equiv \int^{T^{|\gamma|}_{c}}_{0} ds \; \exp\left(2 \beta_{s}\right) \nonumber \\
&=& (\tilde{c}_{n})^{2} \left(\int^{T^{|\gamma|}_{c}}_{0} ds \; \exp\left(2 \beta^{(0)}_{s}\right)\right) \nonumber \\
&\equiv& (\tilde{c}_{n})^{2}T^{|\theta^{\mathbb{B}^{(1)}}|}_{c} \ \ ,
\end{eqnarray} where $T^{|\theta^{\mathbb{B}^{(1)}}|}_{c}\equiv \inf \left\{t\geq 0 :
\left|\theta^{\mathbb{B}^{(1)}}_{t}\right|=c \right\}$ is the first hitting time of the symmetric conic boundary of angle $c$ of a Brownian motion $\mathbb{B}^{(1)}$ starting from $1+i0$. Hence, from (\ref{2negmom}-\ref{2negmom2}) and by using the Laplace transform for the hitting time
$T^{|\gamma|}_{c}$ \cite{ReY99} (Chapter II, Prop. 3.7) or \cite{PiY03}(p.298):
\begin{eqnarray} E\left[e^{-\frac{y^{2}}{2}T^{|\gamma|}_{c}}\right]=\frac{1}{\cosh(y c)}, \end{eqnarray} we get: \begin{eqnarray}\label{21/T}
E\left[\frac{1}{T^{|\theta^{\mathbb{B}^{(1)}}|}_{c}}\right] &=& \int^{\infty}_{0} y E\left[e^{-\frac{y^{2}}{2}T^{|\gamma|}_{c}}\right] \coth\left(\frac{\pi}{2}y\right) \; dy \nonumber \\
&=& \int^{\infty}_{0} \frac{y}{\cosh(y c)} \coth\left(\frac{\pi}{2}y\right) \; dy\equiv G(c) . \end{eqnarray} For $c=2\pi$, we obtain the numerical result: \begin{eqnarray}\label{num}
G(2\pi)\equiv E\left[\frac{1}{T^{|\theta^{\mathbb{B}^{(1)}}|}_{2\pi}}\right]\approx 0.167. \end{eqnarray} Thus, with (\ref{Tcthetacn}), \begin{eqnarray} \label{ETchatz0Dbisb}
E\left[\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}\right)\right] &=& 2\ln(\tilde{c}_{n}) + E\left[\ln\left(T^{|\theta^{\mathbb{B}^{(1)}}|}_{c}\right)\right], \end{eqnarray} and from (\ref{2logseries}), we have: \begin{eqnarray}\label{2log1/2approx}
E\left[\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{2 \pi}+\frac{1}{2}\right)\right] \approx 2\ln(\tilde{c}_{n}) + E\left[\ln\left(T^{|\theta^{\mathbb{B}^{(1)}}|}_{2\pi}\right)\right] + \frac{1}{2\tilde{c}^{2}_{n}}E\left[\frac{1}{T^{|\theta^{\mathbb{B}^{(1)}}|}_{2\pi}}\right]. \nonumber \\ \end{eqnarray} For the first moment of
$\ln\left(T^{|\theta^{\tilde{\mathbb{B}}}|}_{c}\right)$, for an angle $c$, using Bougerol's identity \cite{Bou83,ReY99}, there is the integral representation \cite{CMY98,Vak10}: \begin{eqnarray} \label{ETchatz0Dbisb1}
E\left[\ln\left(T^{|\theta^{\mathbb{B}^{(1)}}|}_{c}\right)\right]=2F(c) + \ln\left(2\right) + c_{E}, \end{eqnarray} where\footnote[8]{We note that there is a simple relation between $F$ and $G$: $\frac{\pi^{2}}{4c} \ F'\left(\frac{\pi^{2}}{4c}\right)=c \ G(c)$. For a more complete discussion, see \cite{Vak10}.}: \begin{eqnarray}\label{2F(c)} F(c)=\int^{\infty}_{0} \frac{dz}{\cosh \left(\frac{\pi z}{2}\right)} \ln\left(\sinh\left(cz\right)\right), \end{eqnarray} and $c_{E}\approx 0.577$ denotes Euler's constant.\\ For $c=2\pi$, we have $F(2 \pi)\approx3.84$. In Figure \ref{figFc} we plot $F$ with respect to the angle $c$.
\begin{figure}
\caption{{\bf $F$ as a function of the angle $c$.} }
\label{figFc}
\end{figure}
\\ In summary, from (\ref{ETchatz0Dasymp2}), (\ref{num}), (\ref{2log1/2approx}) and
(\ref{ETchatz0Dbisb1}), we approximate $E\left[T^{|\theta^{Y}|}_{2\pi}\right]$: \begin{eqnarray}\label{2EThat}
E\left[T^{|\theta^{Y}|}_{2\pi}\right] \approx \frac{1}{4} \left( 2\ln(\tilde{c}_{n}) + Q +
\frac{1}{2\tilde{c}^{2}_{n}}E\left[\frac{1}{T^{|\theta^{\mathbb{B}^{(1)}}|}_{2\pi}}\right] \right), \end{eqnarray} where: \begin{eqnarray} Q=2F(2\pi) + 2\ln 2 + c_{E} \end{eqnarray} is a constant with $F(2\pi)\approx3.84$, $c_{E}\approx 0.577$, and
$E\left[\frac{1}{T^{|\theta^{\mathbb{B}^{(1)}}|}_{2\pi}}\right]\approx 0.167$, thus: \begin{eqnarray} Q\approx 9.54. \end{eqnarray} Thus, by taking $c=2\pi$, for $n$ large, the mean time $E\left[\tilde{\tau}_{n}\right]$ that $\tilde{X}_{n}(t)$ rotates around ${\bf 0}$, is: \begin{eqnarray}\label{2taunapproxbis} E\left[\tilde{\tau}_{n}\right] &\approx& \frac{\sqrt{n}}{4} \left( 2\ln(\tilde{c}_{n}) + \frac{0.08}{\tilde{c}^{2}_{n}}+ Q \right), \end{eqnarray} and since: \begin{eqnarray}\label{2constantrandom} \tilde{c}_{n}\equiv\frac{c_{n}}{\sqrt{n}}, \end{eqnarray} the MRT of $\tilde{X}_{n}(t)$ is given by the formula: \begin{eqnarray}\label{2ETfinalrandomIC}
E\left[\tilde{\tau}_{n}\right] \approx \frac{\sqrt{n}}{4} \left[ 2 \ln \left( \frac{c_{n}}{\sqrt{n}}\right)+ 0.08\frac{n}{c^{2}_{n}}+ Q \right]. \end{eqnarray} For a long enough polymer, such that $nl_{0}>>L\Rightarrow n>> \tilde{l}$, thus $\tilde{l}=\frac{L}{l_{0}}$ is negligible with respect to $X_{n}(t)$ and from (\ref{2Xnfinal}) $X_{n}(t)\approx\tilde{X}_{n}(t)$. For a mean initial configuration $c_{n}\equiv E \left( \sum^{n}_{k=1} e^{i \theta_{k}(0)} \right)$, we obtain that the MRT of the polymer is given by the formula: \begin{eqnarray}\label{2ETfinalrandomIClong}
E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{4} \left[ 2 \ln \left( \frac{E \left( \sum^{n}_{k=1} e^{i \theta_{k}(0)}\right)}{\sqrt{n}} \right)+\frac{0.08 n}{\left(E \left( \sum^{n}_{k=1} e^{i\theta_{k}(0)}\right)\right)^{2}}+ Q \right]. \end{eqnarray} Finally, using the unscaled variables (\ref{rescaling}), we obtain: \begin{eqnarray}\label{2ETfinaluncsaledrandomIClong}
E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \left[ 2 \ln \left( \frac{E \left( \sum^{n}_{k=1} e^{\frac{i}{\sqrt{2D}}\theta_{k}(0)}\right)}{\sqrt{n}} \right)+ \frac{0.08 n}{\left(E \left( \sum^{n}_{k=1} e^{\frac{i}{\sqrt{2D}}\theta_{k}(0)}\right)\right)^{2}}+ Q \right]. \nonumber \\ \end{eqnarray} Expressions (\ref{2ETfinalrandomIClong}) and (\ref{2ETfinaluncsaledrandomIClong}) show that the leading order term of the MRT depends on the initial configuration, however this dependence is weak.
We consider now that the polymer is initially stretched $\left(\theta_{k}(0)=0, \ \forall k=(1,...,n)\right)$, hence $c_{n}=n$. Thus, from (\ref{2ETfinaluncsaledrandomIClong}), the MRT is approximately: \begin{eqnarray}\label{2ETfinaluncsaled}
E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \left[ \ln \left(n \right)+ 0.08\frac{1}{n}+ Q \right]. \end{eqnarray}
In order to check the range of validity of formula (\ref{2ETfinaluncsaled}), we ran some Brownian simulations. In Figure \ref{2E[T]}, we simulated the MRT with a time step $dt=0.01$ for $n=50$ to $300$ rods in steps of 5, and for each $n$, we took 300 samples and averaged over all of them. The parameters we chose were $D=10$ for the diffusion coefficient, $L=0.3$ for the distance from the origin ${\bf 0}$ and $l_{0}=0.25$ for the length of each rod, and for the initial condition we chose a stretched polymer located on the half line $\overrightarrow{{\bf 0}x}$, $\theta_{k}(0)=0$, $\forall k=(1,...,n)$ (hence $c_{n}=L+nl_{0}$), and then we computed the MRT $E\left[\tau_{n}\right]$.
\begin{figure}
\caption{{\bf MRT of the free polymer end as a function of the number of beads $n$ (Brownian simulations)}.}
\label{2E[T]}
\end{figure}
In Figure \ref{2comparisonimpr}, we plot both the results from Brownian simulations and the formula (\ref{2ETfinaluncsaled}). We considered values from $n=50$ to $300$ rods in steps of 10, $D=10$, $L=0.3$, $l_{0}=0.25$, thus the condition $\sqrt{n} \: l_{0}>>L$ is satisfied. {For each numerical computation, we performed 300 runs with a time step $dt=0.01$. By comparing the numerical simulations (Fig. \ref{2comparisonimpr}), with the analytical formula for the MRT, we see an overshoot.
\begin{figure}
\caption{ {\bf Comparing the Analytical Formula and the Brownian simulations of the MRT. } The MRT of the polymer end is depicted with respect to the number of beads $n$. }
\label{2comparisonimpr}
\end{figure}
The straight initial configuration is no restriction to the generality of our study: we have $c_{n}\leq n$ and the upper bound is achieved for a straight initial configuration. In Figure \ref{straightrandom}, we simulate the MRT $E\left[\tau_{n}\right]$ for both a random and an initially straight configuration (Brownian simulations with the same values for the parameters as above, after formula (\ref{2ETfinaluncsaled})).
\begin{figure}
\caption{ {\bf MRT of the free polymer end as a function of the number of beads $n$
for the straight and for a random initial configuration (Brownian simulations)}.}
\label{straightrandom}
\end{figure}
\subsubsection*{Uniformly distributed initial angles}
When the initial angles $\left(\theta_{1}(0),\theta_{2}(0),\ldots,\theta_{n}(0)\right)$ are uniformly distributed over $\left[0,2\pi\right]$, by averaging over all possible initial configurations, from (\ref{initial}) and (\ref{rescalinginitialangle}), we obtain: \begin{eqnarray}
c_{n}=E\left[\sum^{n}_{k=1} e^{i \theta_{k}(0)}\right]=E\left[\sum^{n}_{k=1} \left(\cos(\theta_{k}(0)) +i \; \sin(\theta_{k}(0))\right)\right]=0, \end{eqnarray} hence, from (\ref{meanX_{n}(t)}), \begin{eqnarray}\label{2initialaverage}
E\left[X_{n}(t)\right] \equiv \tilde{l}+c_{n} = \tilde{l}. \end{eqnarray} We define: \begin{eqnarray}\label{XhattildeZ}
\hat{X}_{n}(t)\equiv\frac{1}{\sqrt{n}} \left(X_{n}(t)-\tilde{l}\right)+1=1+Z^{(n)}_{t}. \end{eqnarray} We know that, for $n$ large, with $\hat{\varphi}_{n}(t)$ denoting the total angle of $\hat{X}_{n}(t)$
the mean time $E\left[\hat{\tau}_{n}\right]$, where $\hat{\tau}_{n}\equiv \inf\{ t>0, |\hat{\varphi}_{n}(t)|=2\pi \}$, that $\hat{X}_{n}(t)$ rotates around ${\bf 0}$, is: \begin{eqnarray*}
E\left[\hat{\tau}_{n}\right]\approx \frac{\sqrt{n}}{8D} \tilde{Q}. \end{eqnarray*} Finally, for a long enough polymer, such that $nl_{0}>>L\Rightarrow n>> \tilde{l}$ and $\sum^{n}_{k=1} e^{i \theta_{k}(t)}>>\sqrt{n}$, thus $\tilde{l}=\frac{L}{l_{0}}$ and $\sqrt{n}$ are negligible with respect to $X_{n}(t)$ and from (\ref{XhattildeZ}) $X_{n}(t)\approx\hat{X}_{n}(t)$. Using the unscaled variables (\ref{rescaling}), with $\tilde{Q}\approx 9.62$, the MRT satisfies: \begin{eqnarray}\label{2MRTrandominitial}
E\left[\tau_{n}\right] \approx E\left[\hat{\tau}_{n}\right]\approx \frac{\sqrt{n}}{8D} \tilde{Q}. \end{eqnarray}
\begin{rem}\label{2RealCase} Formula (\ref{2ETfinaluncsaledrandomIClong}) or formulas (\ref{2ETfinaluncsaled}) and (\ref{2MRTrandominitial}) provide the asymptotic expansion for the MRT when $\theta_{1}\in \mathbb{R_{+}}$. In fact, $\theta_{1}$ is a reflected Brownian motion in $[0,2\pi]$, thus a better characterization is to estimate the MRT by using the probability density function for $\theta_{1}$ in the one dimensional torus $[0,2\pi]$. In the Appendix of \cite{Vakth11}, we derive this probability density function and by repeating the previous calculations, we show that, e.g. formula (\ref{2MRTrandominitial}), remains valid. \end{rem}
\subsection{The Minimum Mean First Rotation Time}
The Minimum Mean Rotation Time (MMRT) is the first time that any of the segments of the polymer loops around the origin,
\begin{eqnarray} MMRT \equiv \min_{\mathcal{E}_n} E\{\tau_{n}|c_{n} \}\equiv E\left[\tau_{min}\right], \end{eqnarray} where $\mathcal{E}_n$ is the ensemble of rods which can travel up to the origin. The MMRT is now a decreasing function of $n$. In Figure \ref{figETvar}, we present some simulations for the MMRT as a function of $n$ (100 simulations per time step $dt=0.01$ with $n=4$ to $15$ rods, $D=10$ for the diffusion constant , $L=0.3$ for the distance from the origin
${\bf 0}$ and $l_{0}=0.25$ the length of each rod). The initial configuration is such that $\left| \varphi_{n}(0)
\right| <2\pi-\varepsilon$, e.g. the straight initial configuration: $\theta_{k}(0)=0, \: \forall k=(1,...,n)$.
\begin{figure}
\caption{ {\bf Brownian simulations of the MMRT as a function of the number of beads $n-2$ for $D=10$.} }
\label{figETvar}
\end{figure}
In Figure \ref{figDmanynL}, we present some Brownian simulations for the MMRT as a function of $D$ and of $L$ (Figure \ref{figDmanynL}(a) and Figure \ref{figDmanynL}(b) respectively). $L$ and $l_{0}$ satisfy the rotation compatibility condition $n l_{0}>L$. $E\left[\tau_{min}\right]$ decreases with $D$ and increases with the distance from the origin $L$. It remains an open problem to compute the MMRT asymptotically for $n$ large.
\begin{figure}
\caption{ {\bf Brownian simulations of the
MMRT as a function of the number of beads $n-2$. } (a)
For several values of $D$ with $L=0.3$, (b) for several values of $L$ with $D=10$. }
\label{figDmanynL}
\end{figure}
\subsection{Initial configuration in the boundary layer}\label{2inconf}
When the polymer has initially almost made a loop, we expect the MRT to have a different behavior. We look at this numerically, and we start with an initial polymer configuration in the boundary layer :
\begin{eqnarray*} 2\pi-\varepsilon \leq \left|
\varphi_{n}(0) \right| < 2\pi.
\end{eqnarray*} where $\varphi_n$ is defined in eq. (\ref{2XnRphi}). The rotation of the polymer will be completed very fast and using Brownian simulations (100 runs per point with $n=10$ rods and $D=1$ for the diffusion constant, $L=0.1$ for the distance from the origin ${\bf 0}$ and $l_{0}=0.2$ for the length of each rod), we plotted in Figure \ref{InitialTotalAngle} the results showing that when the initial total angle $\varphi_{n}(0)$ tends to $2\pi$, the MRT tends to zero, with the precise asymptotic remaining to be completed. Numerically, we postulate that there is a threshold for an initial total angle $\left| \varphi_{n}(0) \right|
=\frac{\pi}{2}$. When $\left| \varphi_{n}(0) \right|<\frac{\pi}{2}$, the MRT appears to be independent from this angle, whereas for $\left| \varphi_{n}(0)
\right|>\frac{\pi}{2}$, the MRT decreases to zero. However, this needs further investigation.
\begin{figure}\label{InitialTotalAngle}
\end{figure}
\section{Discussion and conclusion}
In the present paper, we studied the MRT for a planar random polymer consisting of $n$ rods of length $l_0$. The first end is fixed at a distance $L$ from the origin, while the other end moves as a Brownian motion. Interestingly, we have shown here that the motion of the free polymer end satisfies a new stochastic equation (\ref{2Xnfinal}), containing a nonlinear time-dependent deterministic drift. When $n$ is large, the limit process is an Ornstein-Uhlenbeck process, with different time scales for each of the two coordinates.
We found that the MRT $E\left[\tau_{n}\right]$ actually depends on the mean initial configuration of the polymer. This result is in contrast with the one of the small hole theory \cite{WaK93,WHK93,HoS04,SSHE06,SSH06a,SSH06b,SSH07,PRE2008} where the leading order term of the MFPT for a Brownian particle to reach a small hole does not depend on the initial configuration. Although the MRT is not falling exactly into the narrow escape problems, it is a rare event and the polymer completes a rotation when the free end reaches any point of the positive $x$-axis. The reason why the initial configuration survives in the large time asymptotic regime, is due to the dynamics of the free moving end, approximated as a sum of i.i.d. variables, which is not Markovian. It leads to a process with memory. In summary, for $n l_0>>L$, the leading order term of the MRT is given by: \begin{enumerate}
\item {for a general initial configuration: \begin{eqnarray*} E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \left[2\ln \left(\frac{c_{n}}{\sqrt{n}}\right)+0.08\frac{n}{c_{n}^{2}}+Q\right], \end{eqnarray*} where $c_{n}\equiv E \left( \sum^{n}_{k=1} e^{\frac{i}{\sqrt{2D}} \theta_{k}(0)}\right)$, $\left(\theta_{k}(0),1\leq k\leq n\right)$ is the sequence of the initial angles and $Q\approx9.54$,} \item { for a stretched initial configuration: \begin{eqnarray*} E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \; \left(\ln (n)+Q\right), \end{eqnarray*} } \\
\item {for an average over uniformly distributed initial angles: \begin{eqnarray*} E\left[\tau_{n}\right] \approx \frac{\sqrt{n}}{8D} \; \tilde{Q}, \end{eqnarray*} where $\tilde{Q}\approx9.62$. } \end{enumerate} As we have shown, these formulas are in very good agreement with Brownian simulations (Fig. \ref{2comparisonimpr}). \\ After completion of this work, several questions arise naturally, namely:\\ - When the polymer has already made one loop, what is the probability that it makes a second loop before unwrapping and in that case, what is the MRT? \\ - How can we extend our study in dimension 3 and higher dimensions? See \cite{HVY11} for some first results.
\appendix
\section{Appendix: Proof of Theorem \ref{CLT}} \label{apTCL}
We show the convergence of the sequence $(M_{t}^{(n)},t\geq 0)$ to a Brownian motion in dimension 2, with a different time scale function for each coordinate. This classically involves 2 steps (see e.g. \cite{Bil68,Bil78} or \cite{ReY99} (Chapter XIII)): \begin{enumerate}
\item {the convergence of the finite dimensional distributions, and}
\item {the tightness of the sequence $(M_{t}^{(n)},t\geq0)$} \end{enumerate} $\left.a\right)$ The martingales $S_{t}^{(n)}$ and $C_{t}^{(n)}$ may be written in the form of stochastic integrals, as: \begin{eqnarray*} S_{t}^{(n)} &=& \frac{1}{\sqrt{n}} \int^{t}_{0} \sum^{n}_{k=1} \sin( B_{k}(s)) \: dB_{k}(s) \\ C_{t}^{(n)} &=& \frac{1}{\sqrt{n}} \int^{t}_{0} \sum^{n}_{k=1} \cos( B_{k}(s)) \: dB_{k}(s). \end{eqnarray*} Consequently, with $<M>_{t}$ denoting the quadratic variation \cite{ReY99} of the martingale $\left(M_{t},t\geq0\right)$, we obtain: \begin{eqnarray}
\left\langle S^{(n)}\right\rangle_{t} &=& \frac{1}{n} \int^{t}_{0} \sum^{n}_{k=1} \sin^{2}( B_{k}(s)) \: ds
\overset{{a.s.}}{\underset{n\rightarrow\infty}\longrightarrow} \int^{t}_{0} E\left[ \sin^{2}(B_{1}(s)) \right] \: ds, \label{2crochetS} \\
\left\langle C^{(n)}\right\rangle_{t} &=& \frac{1}{n} \int^{t}_{0} \sum^{n}_{k=1} \cos^{2}(B_{k}(s)) \: ds \overset{{a.s.}}{\underset{n\rightarrow\infty}\longrightarrow} \int^{t}_{0} E\left[ \cos^{2}(B_{1}(s)) \right] \: ds, \label{2crochetC} \\
\left\langle S^{(n)},C^{(n)} \right\rangle_{t} &=& -\frac{1}{2n} \int^{t}_{0} \sum^{n}_{k=1} \sin(2B_{k}(s)) \: ds
\overset{{a.s.}}{\underset{n\rightarrow\infty}\longrightarrow} \int^{t}_{0} E\left[ \sin(2B_{1}(s)) \right] \: ds, \nonumber \\
\label{2crochetSC} \end{eqnarray} where the classical ergodic theorem yields the a.s. convergence \cite{Bil68}. We now give explicit expressions for the 3 right-hand sides. \\ Consider: \begin{eqnarray*} E \left[\left(\exp(i B_{1}(s))\right)^{2}\right] = E \left[\exp(2i B_{1}(s))\right]=e^{-2s}, \end{eqnarray*} from which we deduce: \begin{eqnarray*} E \left[ \cos^{2}( B_{1}(s)) - \sin^{2}( B_{1}(s)) \right]=E \left[ \cos( 2B_{1}(s))\right]=e^{-2s}, \end{eqnarray*} and \begin{eqnarray*} E \left[ \sin( 2B_{1}(s))\right]=0. \end{eqnarray*} Finally, we obtain: \begin{eqnarray}\label{2Ecossquared}
E \left[ \cos^{2}( B_{1}(s)) \right] = \frac{1+e^{-2s}}{2}, \end{eqnarray} \begin{eqnarray}\label{2Esinsquared}
E \left[ \sin^{2}( B_{1}(s)) \right] = \frac{1-e^{-2s}}{2}. \end{eqnarray} $\left.b\right)$ The previous results allow to obtain the convergence in law for $\left(M^{(n)}_{t_{1}},\ldots,M^{(n)}_{t_{k}}\right)$, say, as $n\rightarrow\infty$. Indeed, by taking $f,g:\mathbb{R}_{+}\rightarrow\mathbb{R}$, simple functions, we may write: \begin{eqnarray}
E \left[ e^{i\left(\int^{\infty}_{0} f(u) dS^{(n)}_{u} + \int^{\infty}_{0} g(u) dC^{(n)}_{u} \right)} \right] \equiv E \left[ \exp\left(i \; \Sigma^{(n)}_{\infty}\right)\right], \end{eqnarray} where: \begin{eqnarray}
\Sigma^{(n)}_{t}= \int^{t}_{0} \left(f(u) dS^{(n)}_{u} + g(u) dC^{(n)}_{u} \right), \end{eqnarray} as follows: \begin{eqnarray}\label{2characteristicfunction} E \left[ \exp\left(i \; \Sigma^{(n)}_{\infty}\right)\right]=E \left[ \exp\left(i \; \Sigma^{(n)}_{\infty}+\frac{1}{2}\; <\Sigma^{(n)}>_{\infty}\right)\exp\left(-\frac{1}{2} \; <\Sigma^{(n)}>_{\infty}\right)\right]. \end{eqnarray} The results obtained in part $\left.a\right)$, together with the fact that \cite{ReY99}: \begin{eqnarray} E \left[ \exp\left(i \; \Sigma^{(n)}_{\infty}+\frac{1}{2}\; <\Sigma^{(n)}>_{\infty}\right)\right]=1 \end{eqnarray} now yield: \begin{eqnarray} E \left[ \exp\left(i \; \Sigma^{(n)}_{\infty}\right)\right]{\underset{n\rightarrow\infty}\longrightarrow} \exp\left( -\frac{1}{2} \int^{\infty}_{0} \left(f^{2}(u) \frac{1-e^{-2u}}{2} + g^{2}(u) \frac{1+e^{-2u}}{2}\right) \: du \right). \end{eqnarray} $\left.c\right)$ It now remains to prove the tightness \cite{Bil78} of the distributions of the sequence $M^{(n)}$, which follows from a classical application of Kolmogorov's criterion; indeed, for $\beta>0$ and $c_{\beta}$ a positive constant, \begin{eqnarray}\label{2tightness3}
&& E \left[ \left| M_{t}^{(n)}-M_{s}^{(n)} \right|^{2\beta} \right] \nonumber \\
&\leq& c_{\beta} \left\{E \left[ \left(\left\langle S^{(n)}\right\rangle_{t}-\left\langle S^{(n)}\right\rangle_{s}\right)^{\beta} \right]
+ E \left[ \left(\left\langle C^{(n)}\right\rangle_{t}-\left\langle C^{(n)}\right\rangle_{s}\right)^{\beta} \right]\right\} \nonumber \\
&\leq& 2 c_{\beta} \left| t-s \right|^{\beta}. \end{eqnarray} We refer the reader who may want more details about the arguments used to \cite{ReY99}, Chapter XIII, where convergence in distribution on the canonical space $C(\mathbb{R}_{+},\mathbb{R})$ is discussed.
\quad\hbox{\hskip 4pt\vrule width 5pt height 6pt depth 1.5pt}
\end{document} | arXiv |
WHAT IS A NON EXAMPLE OF A FRACTION
You are here: Home > Stewiacke > What is a non example of a fraction
Converting Among Non Negative Fractions Decimals and. Illustrated definition of Common Fraction: A fraction where both top and bottom numbers are integers. Example: sup1supsub2sub and sup3supsub4sub..., A number written as $\frac{a}{b}$, where $a$ is an integer and $b$ is a non-zero integer, is called a fraction. Examples of fractions. Example 1: Becky,.
Unit fraction Wikipedia
Compound Fractions S.O.S. Math. These rational numbers when converted into decimal fractions can be both terminating and non-terminating which is an example of terminating decimal fraction., non examples of like fractions = fractions that are not equal in value to other fractions. For example: 1/2 and 3/4..
non examples of like fractions = fractions that are not equal in value to other fractions. For example: 1/2 and 3/4. Numerator and Denominator In fractions x/y, Mixed fraction is a combination of a whole number and a fraction. For example 7 1 Formal, Informal & Non-formal;
Then construct the nonв€'negative fractions, Because pronumerals are just numbers, we can cancel both pronumerals and numbers in an algebraic fraction. For example, Definition of Non-terminating Decimal: While expressing a fraction in the decimal form, when we perform division we get some remainder. If the division process does
Fraction fiddle: comparing non-unit fractions – a kookaburra and a magpie each gobble For example, identify the fraction of a cake remaining after it has had Written for teachers, this article describes four basic approaches children use in understanding fractions as equal parts of a whole.
Example #2: A solution is prepared by mixing 25.0 g of water, H 2 O, and 25.0 g of ethanol, C 2 H 5 OH. Determine the mole fractions of each substance. What is friction? Until now in physics For example, a person sliding equals, start fraction, start color greenD, T, start subscript, x, end subscript, end
Comparing non-unit fractions. Good Part and wholes. students solve problems to determine fractions of collections and multiples of those fractions. For example: Different types of fraction Here are some examples of non-unit fractions. A non-unit fraction is many parts of a whole that is divided into equal parts.
Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction, Let's look at some more examples of fractions. In examples 1 through 4 below, we have identified the numerator and the denominator for each shaded circle.
Find here good examples of complex fractions and some basic or complicated operations that can be performed with them. Illustrated definition of Common Fraction: A fraction where both top and bottom numbers are integers. Example: sup1supsub2sub and sup3supsub4sub...
Fractions with Terminating and Non-Terminating Decimal
Fraction Bar nzmaths. A number written as $\frac{a}{b}$, where $a$ is an integer and $b$ is a non-zero integer, is called a fraction. Examples of fractions. Example 1: Becky,, Comparing non-unit fractions. Good Part and wholes. students solve problems to determine fractions of collections and multiples of those fractions. For example:.
What is Unit Fraction YouTube
What is a Fraction? Definition and Types - Video. A fraction is a number that represents part of a whole....Complete information about the fraction, definition of an fraction, examples of an fraction, step by step In a fraction, two numbers are separated by a horizontal...complete information about numerator, definition of an numerator, examples of an numerator, step by step.
Fraction Define Fraction at Dictionary.com
What are non examples of like fractions? science.answers.com
Converting Among Non Negative Fractions Decimals and
These rational numbers when converted into decimal fractions can be both terminating and non-terminating which is an example of terminating decimal fraction. Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction,
See how the top number is smaller than the bottom number in each example? That makes it a Proper Fraction. Three Types of Fractions. Proper Fractions. So, For example, the fraction 6/9 can be simplified to 2/3 since we can divide the 6 by 3 and the 9 What is a Fraction? - Definition and Types Related Study Materials
An infinite repeating decimal is one that has a specified sequence start by putting the fraction in will give an infinite repeating decimal. For example: What are some examples of non-integer rational numbers? How are What are some examples of non-integer the fraction, then you have an example of a non-integer
Decimal to Fraction Calculator. See the following table for examples: Type of number Example What to enter the non repeating portion is converted as explained Sometimes the numerator of a fraction will divide evenly into the denominator. The fraction can be reduced by replacing the numerator with a 1 and dividing the
Find here good examples of complex fractions and some basic or complicated operations that can be performed with them. Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction,
For example, the fraction 3/4 denotes "3 of 4 equal parts." 3 is the numerator, and 4 is the denominator. Proper Fractions and Improper Fractions Watch videoВ В· Equivalent fraction word problem example 3. Equivalent fraction word problem example 4. Use 48 as the denominator and find an equivalent fraction to 1/3.
Decimal to Fraction Calculator. See the following table for examples: Type of number Example What to enter the non repeating portion is converted as explained Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction,
Definition and examples of numerator define numerator
Converting Among Non Negative Fractions Decimals and. Comparing non-unit fractions. Good Part and wholes. students solve problems to determine fractions of collections and multiples of those fractions. For example:, Examples of fractions belonging to but with a non-repeating sequence of digits that So this particular repeating decimal corresponds to the fraction 1/.
Fraction Define Fraction at Dictionary.com. These non-examples were selected to be "near-misses," very close to the image people have of triangles. Number: whole, fraction, decimal, negative (8) Pacing (1), Sometimes the numerator of a fraction will divide evenly into the denominator. The fraction can be reduced by replacing the numerator with a 1 and dividing the.
Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction, Fraction definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. Look it up now! Historical Examples. of fraction.
Fraction definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. Look it up now! Historical Examples. of fraction. Unit and non-unit fractions. The first comparison tasks usually encountered by students involve unit fractions. An example of a comparison task is.
Fraction fiddle: comparing non-unit fractions – a kookaburra and a magpie each gobble For example, identify the fraction of a cake remaining after it has had Find here good examples of complex fractions and some basic or complicated operations that can be performed with them.
Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction, What are some examples of non-integer rational numbers? How are What are some examples of non-integer the fraction, then you have an example of a non-integer
Provides worked examples demonstrating two methods for simplifying complex fractions. Numerator and Denominator In fractions x/y, Mixed fraction is a combination of a whole number and a fraction. For example 7 1 Formal, Informal & Non-formal;
Guide to help understand and demonstrate Converting Among Non Negative Fractions, Decimals, For example, reducing fractions is done in the following manner: Comparing non-unit fractions. Good Part and wholes. students solve problems to determine fractions of collections and multiples of those fractions. For example:
See how the top number is smaller than the bottom number in each example? That makes it a Proper Fraction. Three Types of Fractions. Proper Fractions. So, Watch videoВ В· Equivalent fraction word problem example 3. Equivalent fraction word problem example 4. Use 48 as the denominator and find an equivalent fraction to 1/3.
What is friction? Until now in physics For example, a person sliding equals, start fraction, start color greenD, T, start subscript, x, end subscript, end How do you take a random, non-terminating, non-repeating decimal into a fraction? For example, 3/7 repeats in 7 or less digits: 0.428571428571428571
Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction, Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction,
An infinite repeating decimal is one that has a specified sequence start by putting the fraction in will give an infinite repeating decimal. For example: Having fun with unit fractions. By . of unit fractions adding up to 1 longer by decomposing each individual unit fraction into a sum of unit fractions. For example,
Then construct the nonв€'negative fractions, Because pronumerals are just numbers, we can cancel both pronumerals and numbers in an algebraic fraction. For example, Partial Fractions. Adding rational expressions and simplifying is relatively The partial fractions form of this example, for instance, is $$\frac{A}{x-2}+\frac
We took a look at an example and non-example type of comparison to help in our understanding of Using Example and Non-Example in Math. By Jessica Boschen Then construct the nonв€'negative fractions, Because pronumerals are just numbers, we can cancel both pronumerals and numbers in an algebraic fraction. For example,
Numerator and Denominator In fractions x/y, Mixed fraction is a combination of a whole number and a fraction. For example 7 1 Formal, Informal & Non-formal; A compound fraction is sometimes called a mixed number. Recall that , , and are examples of compound fractions. To manipulate compound fractions, just convert them
Non-equivalent fractions are fractions that are not equal. Go. Here are some more examples : a fraction equivalent to 1/2 would be 7/14. to reduce a fraction, What Is Fraction? That boy who holds (non-trivial) factors, there is always another For example, $81/135 = 3/5.$ Fractions that represent the same number are
Having fun with unit fractions. By . of unit fractions adding up to 1 longer by decomposing each individual unit fraction into a sum of unit fractions. For example, Then construct the nonв€'negative fractions, Because pronumerals are just numbers, we can cancel both pronumerals and numbers in an algebraic fraction. For example,
Non-unit Fractions. Keywords: They use the numbers to form a fraction, For example, if the "3" and "4" cards are picked up, the student might make or . Non-Unit Fraction A non-unit fraction is a fraction where the numerator (the number on the top half of the fraction) is greater than 1. For example, 3/4 is a non-unit
Partial Fractions. Adding rational expressions and simplifying is relatively The partial fractions form of this example, for instance, is $$\frac{A}{x-2}+\frac 2/07/2012В В· Rational Exponents Non Unit Fraction Exponent with a Whole Number Base Q3 W2 #3a - Duration: 5:05. Mr Rague 419 views. 5:05.
Definition and examples of numerator define numerator. Examples of fractions belonging to but with a non-repeating sequence of digits that So this particular repeating decimal corresponds to the fraction 1/, Sometimes the numerator of a fraction will divide evenly into the denominator. The fraction can be reduced by replacing the numerator with a 1 and dividing the.
Different types of fraction BBC
Unit fraction Wikipedia. Watch videoВ В· Equivalent fraction word problem example 3. Equivalent fraction word problem example 4. Use 48 as the denominator and find an equivalent fraction to 1/3., Non-Unit Fraction A non-unit fraction is a fraction where the numerator (the number on the top half of the fraction) is greater than 1. For example, 3/4 is a non-unit.
Non-Terminating Decimal Mixed Recurring Decimals Pure. What are some examples of non-integer rational numbers? How are What are some examples of non-integer the fraction, then you have an example of a non-integer P-chart (fraction non-conforming) C-chart The fraction non-conforming sample size and control limits..
12/05/2009В В· A non-example is the opposite, or something that is the subject has nothing to do with. So whats the opposite of a fraction? Would..say, Zac Efron be a non Unit and non-unit fractions. The first comparison tasks usually encountered by students involve unit fractions. An example of a comparison task is.
ing fractions В· Multiplying Examples : Workout: Adding and subtracting fractions Find here good examples of complex fractions and some basic or complicated operations that can be performed with them.
Benchmark fractions are common fractions that are used for comparison to other numbers. For example, the benchmark fraction 1/10 is... A list of benchmark fractions Unit and non-unit fractions. The first comparison tasks usually encountered by students involve unit fractions. An example of a comparison task is.
Benchmark fractions are common fractions that are used for comparison to other numbers. For example, the benchmark fraction 1/10 is... A list of benchmark fractions ... them work out a non-unit fraction of that number. For example 3/4 Fraction Bar to work with fractions into fractions. For example if they fold
Comparing and contrasting examples and non-examples of unit fractions allows students to develop a conceptual understanding of fractions. Plan your 60-minute lesson Non-unit Fractions. Keywords: They use the numbers to form a fraction, For example, if the "3" and "4" cards are picked up, the student might make or .
Decimal to Fraction Calculator. See the following table for examples: Type of number Example What to enter the non repeating portion is converted as explained non examples of like fractions = fractions that are not equal in value to other fractions. For example: 1/2 and 3/4.
non examples of like fractions = fractions that are not equal in value to other fractions. For example: 1/2 and 3/4. See how the top number is smaller than the bottom number in each example? That makes it a Proper Fraction. Three Types of Fractions. Proper Fractions. So,
These rational numbers when converted into decimal fractions can be both terminating and non-terminating which is an example of terminating decimal fraction. Then construct the nonв€'negative fractions, Because pronumerals are just numbers, we can cancel both pronumerals and numbers in an algebraic fraction. For example,
24/11/2015В В· Some complex fractions can be modified easily to be non-complex. For example, 1/1.5 equals 2/3. What is an "Imperial Fraction"? 24/11/2015В В· Some complex fractions can be modified easily to be non-complex. For example, 1/1.5 equals 2/3. What is an "Imperial Fraction"?
Let's look at some more examples of fractions. In examples 1 through 4 below, we have identified the numerator and the denominator for each shaded circle. What are examples of non-terminating repeating decimals? Examples of fractions that result in non-terminating repeating decimals are 1/3, 1/9,
Example #2: A solution is prepared by mixing 25.0 g of water, H 2 O, and 25.0 g of ethanol, C 2 H 5 OH. Determine the mole fractions of each substance. Having fun with unit fractions. By . of unit fractions adding up to 1 longer by decomposing each individual unit fraction into a sum of unit fractions. For example,
Using jigsaw puzzles to introduce the Continued Fraction For the continued fraction examples and so use this Calculator for fractions and non-periodic CF Step One: List all of the factors and circle common ones for the numerator and denominator. 6/8 is the fraction. Factors of 6: 1, 2, 4, 6
What is friction? Until now in physics For example, a person sliding equals, start fraction, start color greenD, T, start subscript, x, end subscript, end Using jigsaw puzzles to introduce the Continued Fraction For the continued fraction examples and so use this Calculator for fractions and non-periodic CF
← Example Of Finding Sum Of Geometric Series
Capital Cash Flow Method Example →
Fully Qualified Class Name Java Example
Win32 Custom Class Example C
Single Responsibility Principle Simple Example C#
Example Of Water Being In An Equilibrium Expression
Creating Postgresql Guc Variables Example
What Is Rent Control An Example Of
Cursor In Sql Server Stored Procedure Example
What Is A Prime Polynomial Example
Bootstrap Price Range Slider Example
Type Of Activity For Example Cashier Customer Service Craigslist
Which Is An Example Of Debt Financing
Quoting Poetry In An Essay Example
Practical Example Of Right Understanding
How Much Does She Earn Per Hour Example
Example Of Oppression In The Philippines
What Is Transfer Pricing Example | CommonCrawl |
\begin{document}
\swapnumbers \theoremstyle{definition} \newtheorem{defi}{Definition}[section] \newtheorem{rem}[defi]{Remark} \newtheorem{ques}[defi]{Question} \newtheorem{expl}[defi]{Example} \newtheorem{conj}[defi]{Conjecture} \newtheorem{claim}[defi]{Claim} \newtheorem{nota}[defi]{Notation} \newtheorem{noth}[defi]{}
\theoremstyle{plain} \newtheorem{prop}[defi]{Proposition} \newtheorem{lemma}[defi]{Lemma} \newtheorem{cor}[defi]{Corollary} \newtheorem{thm}[defi]{Theorem}
\renewcommand{\textsl{\textbf{Proof}}}{\textsl{\textbf{Proof}}}
\baselineskip=14pt
\begin{center} {\bf\Large Vertices of Simple Modules of Symmetric Groups \\ Labelled by Hook Partitions}
Susanne Danz and Eugenio Giannelli
\today
\begin{abstract} \noindent In this article we study the vertices of simple modules for the symmetric groups in prime characteristic $p$. In particular, we complete the classification of the vertices of simple $F\mathfrak{S}_n$-modules labelled by hook partitions.
\noindent {\bf Mathematics Subject Classification (2010):} 20C20, 20C30.
\noindent {\bf Keywords:} symmetric group, simple module, hook partition, vertex.
\end{abstract} \end{center}
\section{Introduction}\label{sec intro}
Introduced by J.~A.~Green in 1959 \cite{Gr}, {\sl vertices} of indecomposable modules over modular group algebras have proved to be important invariants linking the global and local representation theory of finite groups over fields of positive characteristic. Given a finite group $G$ and a field $F$ of characteristic $p>0$, by Green's result, the vertices of every indecomposable $FG$-module form a $G$-conjugacy class of $p$-subgroups of $G$. Moreover, vertices of simple $FG$-modules are known to satisfy a number of very restrictive properties, most notably in consequence of Kn\"orr's Theorem \cite{Kn}. The latter, in particular, implies that vertices of simple $FG$-modules in blocks with abelian defect groups have precisely these defect groups as their vertices. Despite this result, the precise structure of vertices of simple $FG$-modules is still poorly understood, even for very concrete groups and modules.
The aim of this paper is to complete the description of the vertices of a distinguished class of simple modules of finite symmetric groups. Throughout, let $n\in\mathbb{N}$, and let $\mathfrak{S}_n$ be the symmetric group of degree $n$.
Then, as is well known, the isomorphism classes of simple $F\mathfrak{S}_n$-modules are labelled by the $p$-regular partitions of $n$. We denote the simple $F\mathfrak{S}_n$-module corresponding to a $p$-regular partition $\lambda$ by $D^\lambda$. If $\lambda=(n-r,1^r)$, for some $r\in\{0,\ldots,p-1\}$, then $\lambda$ is called a $p$-regular {\sl hook partition} of $n$. Whilst, in general, even the dimensions of the simple $F\mathfrak{S}_n$-modules are unknown, one has a neat description of an $F$-basis of $D^{(n-r,1^r)}$; we shall comment on this in \ref{noth exterior} below.
The problem of determining the vertices of the simple $F\mathfrak{S}_n$-module $D^{(n-r,1^r)}$ has been studied before by Wildon in \cite{W}, by M\"uller and Zimmermann in \cite{MZ}, and by the first author in \cite{D}. In consequence of these results, the vertices of $D^{(n-r,1^r)}$ have been known, except in the case where $p>2$, $r=p-1$ and $n\equiv p\pmod{p^2}$. In Section~\ref{sec proof} of the current paper we shall now prove the following theorem, which together with \cite[Corollary~5.5]{D} proves \cite[Conjecture~1.6(a)]{MZ}.
\begin{thm}\label{thm main} Let $p>2$, let $F$ be a field of characteristic $p$, and let $n\in\mathbb{N}$ be such that $n\equiv p\pmod{p^2}$. Then the vertices of the simple $F\mathfrak{S}_n$-module $D^{(n-p+1,1^{p-1})}$ are precisely the Sylow $p$-subgroups of $\mathfrak{S}_n$. \end{thm}
Our key ingredients for proving Theorem~\ref{thm main} will be the Brauer construction in the sense of Brou\'e \cite{Br} and Wildon's result in \cite{W}. Both of these will enable us to obtain lower bounds on the vertices of $D^{(n-p+1,1^{p-1})}$, which together will then provide sufficient information to deduce Theorem~\ref{thm main}.
To summarize, the abovementioned results in \cite{D,MZ,W} and Theorem~\ref{thm main} lead to the following exhaustive description of the vertices of the modules $D^{(n-r,1^r)}$:
\begin{thm}\label{thm vertices} Let $F$ be a field of characteristic $p>0$, and let $n\in\mathbb{N}$. Let further $r\in\{0,1\ldots,p-1\}$, and let $Q$ be a vertex of the simple $F\mathfrak{S}_n$-module $D^{(n-r,1^r)}$.
{\rm (a)}\, If $p\nmid n$ then $Q$ is $\mathfrak{S}_n$-conjugate to a Sylow $p$-subgroup of $\mathfrak{S}_{n-r-1}\times \mathfrak{S}_r$.
{\rm (b)}\, If $p=2$, $p\mid n$ and $(n,r)\neq (4,1)$ then $Q$ is a Sylow $2$-subgroup of $\mathfrak{S}_n$.
{\rm (c)}\, If $p=2$, $n=4$ and $r=1$ then $Q$ is the unique Sylow $2$-subgroup of $\mathfrak{A}_4$.
{\rm (d)}\, If $p>2$ and $p\mid n$ then $Q$ is a Sylow $p$-subgroup of $\mathfrak{S}_n$. \end{thm}
In the case where $p\nmid n$, the simple module $D^{(n-r,1^r)}$ is isomorphic to the Specht $F\mathfrak{S}_n$-module $S^{(n-r,1^r)}$, by work of Peel \cite{P}. Thus assertion~(a) follows immediately from \cite[Theorem 2]{W}. Assertions (b) and (c) have been established by M\"uller and Zimmermann \cite[Theorem~1.4]{MZ}. Moreover, if $p>2$, $p\mid n$ and $r<p-1$ then assertion (d) can also be found in \cite[Theorem~1.2]{MZ}. The case where $p>2$, $p\mid n$, $r=p-1$ was treated in \cite[Corollary~5.5]{D}, except when $n\equiv p\pmod{p^2}$, which is covered by Theorem~\ref{thm main} above.
We should also like to comment on the sources of the simple $F\mathfrak{S}_n$-modules $D^{(n-r,1^r)}$. For $r=0$, we get the trivial $F\mathfrak{S}_n$-module $D^{(n)}$, which has of course trivial source.
If $p\mid n$, then the module $D^{(n-1,1)}$ restricts indecomposably to its vertices, by \cite[Theorems~1.3, 1.5]{MZ}, except when $p=2$ and $n=4$. For $p=2$, the simple $F\mathfrak{S}_4$-module $D^{(3,1)}$ has trivial source, by \cite[Theorem~1.5]{MZ}. If $p\nmid n$ then $D^{(n-r,1^r)}\cong S^{(n-r,1^r)}$ has always trivial sources; see, for instance
\cite[Theorem~1.3]{MZ}. However, in the case where $p>2$, $p\mid n$ and $r>1$, we do not know
the sources of $D^{(n-r,1^r)}$. In these latter cases, the restrictions of $D^{(n-r,1^r)}$
to its vertices should, conjecturally, be indecomposable, hence should be sources of $D^{(n-r,1^r)}$; see \cite[Conjecture~1.6(b)]{MZ}. This conjecture has been verified computationally in several cases, see \cite{D,MZ}, but remains still open in general.
\noindent {\bf Acknowledgements:} The first author has been supported through DFG Priority Programme `Representation Theory' (Grant \# DA1115/3-1), and a Marie Curie Career Integration Grant (PCIG10-GA-2011-303774). The results of this article were achieved during the visit of the second author to the University of Kaiserslautern. He gratefully acknowledges his PhD supervisor Dr.~Mark Wildon for supporting the visit. He also thanks the research group \textsl{Algebra, Geometry and Computer Algebra} at Kaiserslautern for their kind hospitality.
\section{Prerequisites}\label{sec pre}
Throughout this section, let $F$ be a field of characteristic $p>0$. We begin by introducing some basic notation that we shall use repeatedly throughout subsequent sections. Whenever $G$ is a finite group, $FG$-modules are always understood to be finite-dimensional left modules. Whenever $H$ and $K$ are subgroups of $G$ such that $H$ is $G$-conjugate to a subgroup of $K$, we write $H\leqslant_G K$. If $H$ and $K$ are $G$-conjugate then we write $H=_G K$. For $g\in G$, we set ${}^gH:=gHg^{-1}$.
We assume the reader to be familiar with the basic concepts of the representation theory of the symmetric groups. For background information we refer to \cite{J,JK}. As usual, for $n\in\mathbb{N}$, we shall denote the Specht $F\mathfrak{S}_n$-module labelled by a partition $\lambda$ of $n$ by $S^\lambda$. If $\lambda$ is a $p$-regular partition of $n$ then we shall denote the simple $F\mathfrak{S}_n$-module $S^\lambda/\Rad(S^\lambda)$ by $D^\lambda$.
\begin{noth}{\bf Brauer constructions and vertices.}\label{noth Brauer} (a)\, Let $G$ be a finite group, let $M$ be an $FG$-module, and let $P$ be a $p$-subgroup of $G$. The {\sl Brauer construction} of $M$ with respect to $P$ is defined as \begin{equation}\label{eqn M(P)} M(P):=M^P/\sum_{Q<P}\Tr_Q^P(M^Q)\,, \end{equation} where $M^P$ denotes the set of $P$-fixed points of $M$, and $\Tr_Q^P:M^Q\to M^P,\; m\mapsto \sum_{xQ\in P/Q} xm$ denotes the relative trace map. The latter is independent of the choice of representatives of the left cosets $P/Q$. The $FG$-module structure of $M$ induces an $FN_G(P)$-module structure on the $F$-vector space $M(P)$, and $P$ acts trivially on $M(P)$. Set $\Tr^P(M):=\sum_{Q<P}\Tr_Q^P(M^Q)$.
Moreover, if $R<Q<P$ then $\Tr_R^P=\Tr_Q^P\circ\Tr_R^Q$. Thus $M(P)=M^P/\sum_{Q<_{\mathrm{max}}P}\Tr_Q^P(M^Q)$, where $Q<_{\mathrm{max}}P$ denotes a maximal subgroup of $P$. If $Q<_{\mathrm{max}}P$ then every element $g\in P\smallsetminus Q$ has the property that $\{1,g,g^2,\ldots,g^{p-1}\}$ is a set of representatives of the left cosets of $Q$ in $P$; in particular, we get $\Tr_Q^P(m)=m+gm+\cdots +g^{p-1}m$, for $m\in M^Q$.
(b)\, Suppose that $M$ is an indecomposable $FG$-module. Then a {\sl vertex} of $M$ is a subgroup $Q$ of $G$ that is minimal with respect to the property that $M$ is isomorphic to a direct summand of $\ind_Q^G(\res_Q^G(M))$. By \cite{Gr}, the vertices of $M$ form a $G$-conjugacy class of $p$-subgroups of $G$. Moreover, if $R\leqslant G$ is a $p$-subgroup such that $M(R)\neq \{0\}$ then $R\leqslant_G Q$, by \cite[(1.3)]{Br}. The converse is, however, not true in general. \end{noth}
For proofs of the abovementioned properties of Brauer constructions, see \cite{Br}. Details on the theory of vertices of indecomposable $FG$-modules can be found in \cite[Section~9]{Alperin} or \cite[Section~4.3]{NT}. The following will be very useful for proving Theorem~\ref{thm main} in Section~\ref{sec proof} below. The proof is straightforward, and is thus left to the reader.
\begin{prop}\label{prop Brauer} Let $G$ be a finite group, let $M$ be an $FG$-module with $F$-basis $B$, and let $P\leqslant G$ be a $p$-group. Suppose that there is some $b_0\in B$ satisfying the following properties:
{\rm (i)}\, $b_0\in M^P$;
{\rm (ii)}\, whenever $Q<_{\max} P$, $u\in M^Q$ and $\Tr_Q^P(u)=\sum_{b\in B}a_b(u) b$, for $a_b(u)\in F$, one has $a_{b_0}(u)=0$.
\noindent Then $b_0+\Tr^P(M)\in M(P)\smallsetminus\{0\}$. \end{prop}
Next we shall recall some well-known properties of the simple $F\mathfrak{S}_n$-modules labelled by hook partitions $(n-r,1^r)$, for $r\in\{0,\ldots,p-1\}$, that we shall need repeatedly in the proof of Theorem~\ref{thm main}. In particular, we shall fix a convenient $F$-basis of $D^{(n-r,1^r)}$. In light of Theorem~\ref{thm main} we shall only be interested in the case where $p\mid n$ and $p>2$.
\begin{noth}{\bf Exterior powers of the natural $F\mathfrak{S}_n$-module.}\label{noth exterior} (a)\, Let $p>2$, let $n\in\mathbb{N}$ be such that $p\mid n$, and let $M:=M^{(n-1,1)}$ be the natural permutation $F\mathfrak{S}_n$-module, with natural permutation basis $\Omega=\{\omega_1,\ldots,\omega_n\}$. Since $p\mid n$, the module $M$ is uniserial with composition series $\{0\}\subset M_2\subset M_1\subset M$, where $M_1=\{\sum_{i=1}^na_i\omega_i: a_1,\ldots,a_n\in F,\, \sum_{i=1}^na_i=0\}$ and $M_2=\{a\sum_{i=1}^n\omega_i: a\in F\}$; see, for instance, \cite[Example~5.1]{J}.
Furthermore, $M_1=S^{(n-1,1)}$, and $M_1/M_2=:\Hd(S^{(n-1,1)})\cong D^{(n-1,1)}$; in particular, $\dim_F(D^{(n-1,1)})=n-2$. One sometimes calls $D^{(n-1,1)}$ the {\sl natural (simple) $F\mathfrak{S}_n$-module}. An $F$-basis of $M_1$ is given by the elements $\omega_i-\omega_1$, where $i\in\{2,\ldots,n\}$. In the following, we shall identify the module $D^{(n-1,1)}$ with $M_1/M_2$.
Consider the natural epimorphism ${}^-:M_1\to M_1/M_2$, and set $e_i:=\overline{\omega_i-\omega_1}$, for $i\in\{1,\ldots,n\}$. Then $e_n=-e_2-e_3-\cdots -e_{n-1}$, and the elements $e_2,\ldots, e_{n-1}$ form an $F$-basis of $D^{(n-1,1)}$.
(b)\, Let $r\in\{0,\ldots,n-1\}$. By \cite[Proposition~2.2]{MZ}, there is an $F\mathfrak{S}_n$-isomorphism $S^{(n-r,1^r)}\cong \bigwedge^rS^{(n-1,1)}$. Moreover, if $r\leqslant n-2$ then, in consequence of \cite{P}, $\Hd(\bigwedge^rS^{(n-1,1)})\cong \bigwedge^r \Hd(S^{(n-1,1)})\cong \bigwedge^r D^{(n-1,1)}=:D_r$ is simple. Thus $D_r$ has $F$-basis \begin{equation}\label{eqn B_r} \mathcal{B}_r:=\{e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_r}: 2\leqslant i_1<i_2<\cdots < i_r\leqslant n-1\}\,. \end{equation} If $r\leqslant p-1$ then $\bigwedge^r D^{(n-1,1)}\cong D^{(n-r,1^r)}$. \end{noth}
\section{Symmetric Groups and $p$-Subgroups}\label{sec sym}
Throughout this section, let $n\in\mathbb{N}$, and let $p$ be a prime number. Permutations in the symmetric group $\mathfrak{S}_n$ will be composed from right to left, so that, for instance, we have $(1,2)(2,3)=(1,2,3)\in\mathfrak{S}_3$.
\begin{defi}\label{defi supp} Given an element $\sigma\in\mathfrak{S}_n$, we call $\supp(\sigma):=\{i\in\{1,\ldots,n\}: \sigma(i)\neq i\}$ the {\sl support of $\sigma$}. If $H\leqslant\mathfrak{S}_n$ then we call $\supp(H):=\bigcup_{\sigma\in H}\supp(\sigma)$ the {\sl support of $H$}. \end{defi}
\begin{noth}{\bf Sylow subgroups of symmetric groups.}\label{noth sylow} (a)\, Let $P_p$ be the cyclic group $\langle (1,2,\ldots,p)\rangle\leqslant \mathfrak{S}_p$ of order $p$. Let further $P_1:=\{1\}$ and, for $d\geqslant 1$, we set $$P_{p^{d+1}}:=P_{p^d}\wr P_p:=\{(\sigma_1,\ldots,\sigma_p;\pi): \sigma_1,\ldots,\sigma_p\in P_{p^d},\, \pi\in P_p\}\,.$$ Recall that, for $d\geqslant 2$, the multiplication in $P_{p^{d}}$ is given by $(\sigma_1,\ldots,\sigma_p;\pi)(\sigma_1',\ldots,\sigma_p';\pi')=(\sigma_1\sigma_{\pi^{-1}(1)}',\ldots,\sigma_p\sigma_{\pi^{-1}(p)}';\pi\pi')\,,$ for $(\sigma_1,\ldots,\sigma_p;\pi), \, (\sigma_1',\ldots,\sigma_p';\pi')\in P_{p^{d}}$.
We shall always identify $P_{p^d}$ with a subgroup of $\mathfrak{S}_{p^d}$ in the usual way. That is, $(\sigma_1,\ldots,\sigma_p;\pi)\in P_{p^d}$ is identified with the element $\overline{(\sigma_1,\ldots,\sigma_p;\pi)}\in\mathfrak{S}_{p^d}$ that is defined as follows: if $j\in\{1,\ldots,p^d\}$ is such that $j=p^{d-1}(a-1)+b$, for some $a\in\{1,\ldots,p\}$ and some $b\in\{1,\ldots,p^{d-1}\}$ then $\overline{(\sigma_1,\ldots,\sigma_p;\pi)}(j):=p^{d-1}(\pi(a)-1)+\sigma_{\pi(a)}(b)$. Via this identification, $P_{p^d}$ can be generated by the elements $g_1,\ldots, g_d\in\mathfrak{S}_{p^d}$, where \begin{equation}\label{eqn g_j} g_j:=\prod_{k=1}^{p^{j-1}}(k,k+p^{j-1},k+2p^{j-1},\ldots, k+(p-1)p^{j-1})\quad (1\leqslant j\leqslant d)\,. \end{equation} In particular, with this notation we have $P_p\leqslant P_{p^2}\leqslant \cdots \leqslant P_{p^{d-1}}\leqslant P_{p^d}$, and
the base group of the wreath product $P_{p^{d-1}}\wr P_p$ has the form $\prod_{i=0}^{p-1} g_d^i\cdot P_{p^{d-1}} \cdot g_d^{-i}$.
(b)\, Now let $n\in \mathbb{N}$ be arbitrary, and consider the $p$-adic expansion $n=\sum_{i=0}^r n_ip^i$ of $n$, where $0\leqslant n_i\leqslant p-1$ for $i\in\{0,\ldots, r\}$, and where we may suppose that $n_r\neq 0$.
By \cite[4.1.22, 4.1.24]{JK}, the Sylow $p$-subgroups of $\mathfrak{S}_n$ are isomorphic to the direct product $\prod_{i=0}^r(P_{p^i})^{n_i}$. For subsequent computations it will be useful to fix a particular Sylow $p$-subgroup $P_n$ of $\mathfrak{S}_n$ as follows: for $i\in\{t\in\mathbb{N}\ |\ n_t\neq 0\}$ and $1\leqslant j_i\leqslant n_i$, let $k(j_i):=\sum_{l=0}^{i-1}n_lp^l+(j_i-1)p^i$ and $$P_{p^i,j_i}:= (1,1+k(j_i))\cdots (p^i,p^i+k(j_i))\cdot P_{p^i}\cdot (1,1+k(j_i))\cdots (p^i,p^i+k(j_i))\,.$$ Now set $$P_n:=P_{p,1}\times\cdots\times P_{p,n_1}\times\cdots\times P_{p^r,1}\times\cdots\times P_{p^r,n_r}\,.$$ Given this convention, we shall then also write $P_n=\prod_{i=0}^r(P_{p^i})^{n_i}$, for simplicity. \end{noth}
\begin{expl}\label{expl sylow} Suppose that $p=3$. Then $P_3=\langle g_1\rangle$, $P_9=\langle g_1,g_2\rangle$ and $P_{27}=\langle g_1,g_2,g_3\rangle$, where \begin{align*} g_1&=(1,2,3)\,,\\ g_2&=(1,4,7)(2,5,8)(3,6,9)\,,\\ g_3&=(1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)\,. \end{align*} Moreover, $P_{51}=P_3\times P_3\times P_9\times P_9\times P_{27}$. \end{expl}
\begin{noth}{\bf Elementary abelian groups.}\label{noth E_n} (a)\, Suppose again that $n=p^d$, for some $d\in\mathbb{N}$. We shall denote by $E_n$ the following elementary abelian subgroup of $P_n$ that acts regularly on $\{1,\ldots,n\}$: let $g_1,\ldots,g_d$ be the generators of $P_n$ fixed in (\ref{eqn g_j}). For $j\in\{1,\ldots,d-1\}$, let $g_{j,j+1}:=\prod_{i=0}^{p-1} g_{j+1}^i g_j g_{j+1}^{-i}$, and for $l\in\{1,\ldots,d-j-1\}$, we inductively set $$g_{j,j+1,\ldots,j+l+1}:=\prod_{i=0}^{p-1} g_{j+l+1}^i \cdot g_{j,j+1,\ldots,j+l} \cdot g_{j+l+1}^{-i}\,.$$
Then $E_n:=\langle g_{1,\ldots,d}, g_{2,\ldots, d},\ldots, g_{d-1,d}, g_d\rangle$, and $|E_n|=n=p^d$.
(b)\, Let $n\in\mathbb{N}$ be arbitrary with $p\mid n$, and let $t,m_1,\ldots,m_t\in\mathbb{N}_0$ be such that $n=\sum_{i=1}^t m_ip^i$. For $i\in\{s\in\mathbb{N}\ |\ m_s\neq 0\}$ and $1\leqslant j_i\leqslant m_i$, we set $k(j_i):=\sum_{l=0}^{i-1}m_lp^l+(j_i-1)p^i$ and $$E_{p^i,j_i}:=(1,1+k(j_i))\cdots (p^i,p^i+k(j_i))\cdot E_{p^i}\cdot (1,1+k(j_i))\cdots (p^i,p^i+k(j_i))\,.$$ Then $E(m_1,\ldots,m_t)\leqslant \mathfrak{S}_n$ denotes the elementary abelian group $$E_{p,1}\times\cdots \times E_{p,m_1}\times \cdots \times E_{p^t,1}\times\cdots \times E_{p^t,m_t}\,.$$
We emphasize that, unlike in \ref{noth sylow}, the integers $m_1,\ldots,m_t$ need not be less than $p$. \end{noth}
\begin{expl}\label{expl E_n} Suppose that $p=3$ and $n=27$. Then $E_n=E_{27}$ is generated by the elements \begin{align*} g_{1,2,3}&=(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)\,,\\ g_{2,3}&=(1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)\,,\\ g_3&=(1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)\,. \end{align*} \end{expl}
We recall the following lemma from \cite{D}, which will be useful for our subsequent considerations.
\begin{lemma}[\protect{\cite[Lemma~2.1, Remark~2.2]{D}}]\label{lemma D 2.1} Let $n\in\mathbb{N}$ with $p$-adic expansion $n=\sum_{i=0}^r n_i p^i$, as in \ref{noth sylow}. Let $P\leqslant P_n$ be such that $P=_{\mathfrak{S}_n} P_{p^i}$, for some $i\in\{1,\ldots,r\}$. Then $P\leqslant P_{p^l,j_l}$, for some $l\in\{i,\ldots,r\}$ and some $1\leqslant j_l\leqslant n_l$. Moreover, $P_{p^l,j_l}$ has precisely $p^{l-i}$ subgroups that are $\mathfrak{S}_n$-conjugate to $P_{p^i}$, and these are pairwise $P_{p^l,j_l}$-conjugate to each other. \end{lemma}
\begin{rem}\label{rem E_n P_n} Let again $n\in\mathbb{N}$ with $p$-adic expansion $n=\sum_{i=0}^r n_i p^i$.
(a)\, Let $P\leqslant P_n$ be such that $P=_{\mathfrak{S}_n} P_{p^i}$, for some $i\in\{1,\ldots,r\}$, so that $P\leqslant P_{p^l,j_l}$, for some $l\in\{i,\ldots,r\}$ and some $1\leqslant j_l\leqslant n_l$, by Lemma~\ref{lemma D 2.1}. Note that the subgroups of $P_{p^l,j_l}$ that are $\mathfrak{S}_n$-conjugate to $P_{p^i}$ are uniquely determined by their supports. In particular, if $i=1$ then $P$ is generated by one of the $p$-cycles $(1,\ldots,p),\ldots, (n-n_0-p+1,\ldots,n-n_0)\in P_n$.
(b)\, Suppose that $E\leqslant P_n$ is such that $E=_{\mathfrak{S}_n} E_{p^i}$, for some $i\in\{1,\ldots,r\}$. Since $E$ has precisely one non-trivial orbit, we then also get $E\leqslant P_{p^l,j_l}$, for some $l\in\{i,\ldots,r\}$ and some $1\leqslant j_l\leqslant n_l$. Moreover, arguing by induction on $l-i$ as in the proof of \cite[Lemma 2.1]{D}, we deduce that $E$ then has to be contained in one of the $p^{l-i}$ subgroups of $P_{p^l,j_l}$ that are $\mathfrak{S}_n$-conjugate to $P_{p^i}$. \end{rem}
\begin{lemma}\label{lemma E_n} Let $n,d\in\mathbb{N}$, and let $P\leqslant P_{p^d}\leqslant \mathfrak{S}_n$. Suppose that $P$ contains an $\mathfrak{S}_n$-conjugate of $P_{p^{d-1}}$. Suppose further that $P$ contains an elementary abelian group $E$ of order $p^d$ acting regularly on $\{1,\ldots,p^d\}$. Then $P=P_{p^d}$. \end{lemma}
\begin{proof} If $d=1$ then $P_{p^d}=P_p=E$. From now on we may suppose that $d\geqslant 2$. Recall that $P_{p^d}$ is generated by the elements $g_1,\ldots,g_d$ introduced in (\ref{eqn g_j}). Moreover, $P_{p^d}$ acts imprimitively on the set $\{1,\ldots,p^d\}$, a system of imprimitivity being given by $\Delta:=\{\Delta_1,\ldots,\Delta_p\}$, where $\Delta_s:=\{(s-1)p^{d-1}+1,\ldots,sp^{d-1}\}$, for $s\in\{1,\ldots,p\}$. Since $E$ acts transitively on $\{1,\ldots,p^d\}$, there is some $g\in E$ such that $g(1)=p^{d-1}+1$; in particular, $g\cdot \Delta_1=\Delta_2$. Since $p^{d-1}+1\neq 1$, we have $g\neq 1$, hence $g$ is an element of order $p$. Moreover, the group $\langle g\rangle$ acts on $\Delta$, so that we obtain a group homomorphism $\varphi:\langle g\rangle\to \mathfrak{S}(\Delta)\cong \mathfrak{S}_p$. Since $g\cdot \Delta_1=\Delta_2\neq \Delta_1$, $\varphi$ must be injective. Thus $\varphi(g)$ has order $p$, implying $g\cdot \Delta_1=\Delta_2$, $g\cdot \Delta_2=\Delta_{i_3},\ldots, g\cdot \Delta_{i_p}=\Delta_1$, for $\{1,2,i_3,\ldots,i_p\}=\{1,\ldots,p\}$.
Let $R:={}^{\sigma}P_{p^{d-1}}\leqslant P$, for some $\sigma\in \mathfrak{S}_n$. By Lemma~\ref{lemma D 2.1}, we know that $R=g_d^i P_{p^{d-1}} g_d^{-i}$, for some $i\in\{0,\ldots,p-1\}$. Thus $\supp(R)=\Delta_{i+1}$. So, for $s\in\{0,\ldots, p-1\}$, the group ${}^{g^s}R$ has support $g^s\cdot \Delta_{i+1}$. As we have just seen, the sets $\Delta_{i+1}, g\cdot \Delta_{i+1},\ldots, g^{p-1}\cdot\Delta_{i+1}$ are pairwise disjoint. Consequently, the groups $R$, ${}^gR,\ldots,{}^{g^{p-1}}R$ are precisely the different subgroups of $P_{p^d}$ that are $P_{p^d}$-conjugate to $P_{p^{d-1}}$, $B:=\prod_{s=0}^{p-1}{}^{g^s}R$ is the base group of $P_{p^d}$, and is contained in $P$. Clearly $g\notin B$, since $g(1)\notin \Delta_1$. Since $[P_{p^d}:B]=p$, this implies $P_{p^d}=\langle B, g\rangle\leqslant P\leqslant P_{p^d}$, and the proof is complete. \end{proof}
\begin{lemma}\label{lemma max in E} Let $n,t\in\mathbb{N}$ and let $m_1,\ldots,m_t\in\mathbb{N}_0$ be such that $m_t\neq 0$ and $n=\sum_{i=1}^tm_i p^i$. Suppose that $m_1=1$ and $t\geqslant 2$. Let $P$ be a maximal subgroup of $E(m_1,\ldots,m_t)$ such that $E_{p,1}\not\leqslant P$. Then $P$ contains a subgroup $Q\leqslant \prod_{i=2}^t \prod_{j=1}^{m_i} E_{p^i,j}$ that acts fixed point freely on $\{p+1,\ldots,n\}$. \end{lemma}
\begin{proof} For convenience, set $E':=\prod_{i=2}^t \prod_{j=1}^{m_i} E_{p^i,j}$, so that $E(m_1,\ldots,m_t)=E_p\times E'\geqslant P$. By Goursat's Lemma, we may identify $P$ with the quintuple $(P_1,K_1,\eta,P_2,K_2)$, where $P_1$ and $P_2$ are the projections of $P$ onto $E_p$ and onto $E'$, respectively, $K_1:=\{g\in E_p: (g,1)\in P\}\trianglelefteqslant P_1$, $K_2:=\{h\in E': (1,h)\in P\}$, and $\eta: P_2/K_2\to P_1/K_1$ is a
group isomorphism. Since $|E_p|=p$, there are precisely three possibilities for the section $(P_1,K_1)$ of $E_p$:
(i)\, $P_1=K_1=E_p$,
(ii)\, $P_1=K_1=\{1\}$,
(iii)\, $P_1=E_p$ and $K_1=\{1\}$.
\noindent Case (i) cannot occur, since we are assuming $E_p\not\leqslant P$. In case (ii) we get $P=E'$, so that the assertion then holds with $Q:=P$. So suppose that $P_1=E_p$ and $K_1=\{1\}$, so that also $[P_2:K_2]=p$. Next recall that $P/(K_1\times K_2)\cong P_1/K_1\cong P_2/K_2$; see, for instance, \cite[2.3.21]{Bouc}. This forces
$|E'|=|P|=|K_2|\cdot |P_1|=|K_1|\cdot |P_2|=|P_2|$. Thus $P_2=E'$, and $K_2$ is a maximal subgroup of $E'$. Assume that $K_2$ has a fixed point $x$ on $\{p+1,\ldots,n\}$. Then $x\in\supp(E_{p^i,j})$,
for some $i\geqslant 2$ with $m_i\neq 0$ and some $j\in\{1,\ldots,m_i\}$. But then $K_2$ has to fix the entire support of $E_{p^i,j}$, since $E_{p^i,j}$ acts regularly on its support. This implies $[P_2:K_2]\geqslant p^i\geqslant p^2$, a contradiction. Consequently, $K_2$ must act fixed point freely on $\{p+1,\ldots,n\}$, and the assertion of the lemma follows with $Q:=\{1\}\times K_2\leqslant P$. \end{proof}
The next result will be one of the key ingredients of our proof of Theorem~\ref{thm main} in Section~\ref{sec proof} below.
\begin{prop}\label{prop P and E in Q} Let $n\in\mathbb{N}$ with $p$-adic expansion $n=p+\sum_{i=2}^rn_ip^i$, where $r\geqslant 2$ and $n_r\neq 0$. Let $Q\leqslant P_n$ be such that $P_{n-2p}\leqslant_{\mathfrak{S}_n} Q$ and $E(1,n_2,\ldots,n_r)\leqslant_{\mathfrak{S}_n} Q$. Then $Q=P_n$. \end{prop}
\begin{proof} Let $2\leqslant s\leqslant r$ be minimal such that $n_s\neq 0$. Then $n-2p$ has $p$-adic expansion $n-2p=\sum_{j=1}^{s-1} (p-1)p^j+(n_s-1)p^s+\sum_{i=s+1}^rn_ip^i$. Moreover, we have $$P_n=P_{p,1}\times\prod_{i=s}^r\prod_{j=1}^{n_i} P_{p^i,j}\quad \text{ and } \quad E_n=E(1,n_2,\ldots,n_r)=E_{p,1}\times\prod_{i=s}^r\prod_{j=1}^{n_i} E_{p^i,j}\,.$$ By our hypothesis, there is some $g\in\mathfrak{S}_n$ such that ${}^gE_{p,1}\times\prod_{i=s}^r\prod_{j=1}^{n_i} {}^gE_{p^i,j}\leqslant Q\leqslant P_n$. In consequence of Lemma~\ref{lemma D 2.1} and Remark~\ref{rem E_n P_n}, we may suppose that ${}^gE_{p^i,j}\leqslant P_{p^i,j}$, for $i\geqslant 2$ and $1\leqslant j\leqslant n_i$, as well as ${}^gE_{p,1}=E_{p,1}=P_{p,1}$. Since also $P_{n-2p}\leqslant_{\mathfrak{S}_n} Q$, there exists some $R\leqslant Q\leqslant P_n$ of the form $$R=\prod_{i=1}^{s-1}\prod_{j=1}^{p-1}R_{p^i,j}\times \prod_{j=1}^{n_s-1}R_{p^s,j}\times\prod_{i=s+1}^r \prod_{j=1}^{n_i}R_{p^i,j}\,,$$ where $R_{p^k,l}=_{\mathfrak{S}_n} P_{p^k,l}$, for all possible $k$ and $l$. By Lemma~\ref{lemma D 2.1} and Remark~\ref{rem E_n P_n} again, we must have $\prod_{i=s+1}^r \prod_{j=1}^{n_i}R_{p^i,j}=\prod_{i=s+1}^r \prod_{j=1}^{n_i}P_{p^i,j}\leqslant P_n$. As well, there is some $k\in\{1,\ldots,n_s\}$ and some $m\in\{1,\ldots,p-1\}$ such that $ \prod_{j=1}^{n_s-1}R_{p^s,j}=\prod_{j=1}^{k-1}P_{p^s,j}\times \prod_{l=k+1}^{n_s} P_{p^s,l}\leqslant P_n$ and $R_{p^{s-1},m}\leqslant P_{p^s,k}$. By Lemma~\ref{lemma D 2.1}, $R_{p^{s-1},m}$ is thus $P_{p^s,k}$-conjugate to one of the $p^{s-1}$ subgroups of $P_{p^s,k}$ that are $\mathfrak{S}_n$-conjugate to $P_{p^{s-1}}$. Since $Q$ also contains the regular elementary abelian group ${}^gE_{p^s,k}\leqslant P_{p^s,k}$, Lemma~\ref{lemma E_n} now implies that $P_{p^s,k}\leqslant Q$. Altogether this shows that indeed $P_n\leqslant Q$, and the assertion of the proposition follows. \end{proof}
\section{The Proof of Theorem~\ref{thm main}}\label{sec proof}
The aim of this section is to establish a proof of Theorem~\ref{thm main}. To this end, let $F$ be a field of characteristic $p>2$, and let $n\in \mathbb{N}$ be such that $n\equiv p\pmod{p^2}$. The simple $F\mathfrak{S}_n$-module $D^{(n-p+1,1^{p-1})}$ will henceforth be denoted by $D$. If $p=n$ then the Sylow $p$-subgroups of $\mathfrak{S}_n$ are abelian, and are thus the vertices of $D$, by Kn\"orr's Theorem \cite{Kn}. From now on we shall suppose that $n\geqslant p^2+p$. Let $P_n$ be the Sylow $p$-subgroup of $\mathfrak{S}_n$ introduced in \ref{noth sylow}. In order to show that $P_n$ is a vertex of $D$, we shall proceed as follows: suppose that $Q\leqslant P_n$ is a vertex of $D$. Then:
(i)\, Building on Wildon's result in \cite[Theorem~2]{W}, it was shown in \cite[Proposition~5.2]{D} that $P_{n-2p}=P_{n-(p-1)-2}\times P_{p-1}<_{\mathfrak{S}_n} Q$.
(ii)\, Let $n=\sum_{i=2}^rn_i p^i+p$ be the $p$-adic expansion of $n$, where $r\geqslant 2$ and $n_r\neq 0$. We shall show in Proposition~\ref{prop Brauer D(E)} below that $D(E(1,n_2,\ldots,n_r))\neq \{0\}$. Here $E(1,n_2,\ldots,n_r)$ denotes the elementary abelian subgroup of $P_n$ defined in \ref{noth E_n}, and $D(E(1,n_2,\ldots,n_r))$ denotes the Brauer construction of $D$ with respect to $E(1,n_2,\ldots,n_r)$ as defined in \ref{noth Brauer}. Thus, $E(1,n_2,\ldots,n_r)\leqslant_{\mathfrak{S}_n} Q$, by \cite[(1.3)]{Br}.
(iii)\, Once we have verified (ii), we can apply Proposition~\ref{prop P and E in Q}, which will then show that $Q=P_n$.
\begin{nota}\label{nota e} (a)\, Let $\mathcal{B}:=\mathcal{B}_{p-1}$ be the $F$-basis of $D$ defined in (\ref{eqn B_r}), and let $u\in D$ be such that $u=\sum_{b\in\mathcal{B}}\lambda_b b$, for $\lambda_b\in F$. The basis element $e_2\wedge e_3\wedge \cdots \wedge e_p\in\mathcal{B}$ will from now on be denoted by $e$. Moreover, suppose that $k,x\in\{2,\ldots,n-1\}$ and that $k\leqslant p$. Then we denote the element $e_2\wedge \cdots \wedge e_{k-1}\wedge e_{k+1}\wedge\cdots \wedge e_p\wedge e_x$ of $D$ by $\hat{e}_k\wedge e_x$. In the case where $\hat{e}_k\wedge e_x\in\mathcal{B}$, the coefficient $\lambda_{e_2\wedge \cdots \wedge e_{k-1}\wedge e_{k+1}\wedge\cdots \wedge e_p\wedge e_x}$ will be abbreviated by $\lambda_{\hat{k},x}$.
Similarly, if $2\leqslant k<l\leqslant p$ and if $x,y\in\{2,\ldots,n-1\}$, then we set $\hat{e}_{k,l}\wedge e_x\wedge e_y:=e_2\wedge\cdots\wedge e_{k-1}\wedge e_{k+1}\wedge\cdots \wedge e_{l-1}\wedge e_{l+1}\wedge\cdots \wedge e_p\wedge e_x\wedge e_y\in D$. In the case where $\hat{e}_{k,l}\wedge e_x\wedge e_y\in\mathcal{B}$, we denote by $\lambda_{\widehat{k,l},x,y}$ the coefficient at $\hat{e}_{k,l}\wedge e_x\wedge e_y$ in $u$.
(b)\, Let $u\in D$ be such that $u=\sum_{b\in \mathcal{B}} \lambda_b b$, with $\lambda_b\in F$. We say that the basis element $b\in\mathcal{B}$ {\sl occurs in $u$} with coefficient $\lambda_b$.
(c)\, For $k_1,k_2\in\{2,\ldots,n-1\}$, we set \begin{equation}\label{eqn s(k)} s(k_1,k_2):= \begin{cases} k_2-(k_1-1) &\text{ if } k_1\leqslant k_2\,,\\ 0&\text{ if } k_2< k_1\,. \end{cases} \end{equation} Thus, if $k_1\leqslant k_2$ then $$s(k_1,k_2)\equiv \begin{cases} 0\pmod{ 2}&\text{ if } k_1\not\equiv k_2\pmod{2}\,,\\ 1\pmod{2}&\text{ if } k_1\equiv k_2\pmod{2}\,. \end{cases}$$
(d)\, From now on, let $t,m_2,\ldots,m_t\in \mathbb{N}$ be such that $t\geqslant 2$, $m_t\neq 0$, and $n=p+\sum_{i=2}^t m_i p^{i}$. The elementary abelian group $E(1,m_2,\ldots,m_t)\leqslant \mathfrak{S}_n$ will be denoted by $E$. Note that, by our convention in \ref{noth E_n}, we have $(1,2,\ldots,p)\in E$. In the case where $t=r$ and $m_i=n_i$, for $i=2,\ldots,r$, we, in particular, get $E=E(1,n_2,\ldots,n_r)$. \end{nota}
In the course of this section we shall have to compute explicitly the actions of elements in $E$ on our chosen basis $\mathcal{B}$ of $D$. The following lemmas will be used repeatedly in this section.
\begin{lemma}\label{lemma acts alpha} Let $\alpha:=(1,2,\ldots,p)\in \mathfrak{S}_n$. Let further $\beta:=(x_1,\ldots,x_p)\in\mathfrak{S}_n$ be such that $\{x_1,\ldots,x_p\}\cap \{1,\ldots,p\}=\emptyset$.
{\rm (a)}\, For $i\in\{2,\ldots,n-1\}$, one has $$\alpha\cdot e_i=\begin{cases} e_{i+1}-e_2&\text{ if } 2\leqslant i\leqslant p-1\,,\\ -e_2&\text{ if } i=p\,,\\ e_i-e_2&\text{ if } i\geqslant p+1\,. \end{cases}$$
{\rm (b)}\, If $n\notin\supp(\beta)$ then, for $i\in\{2,\ldots,n-1\}$, one has $$\beta\cdot e_i=\begin{cases} e_i&\text{ if } i\notin\supp(\beta)\,,\\ e_{\beta(i)}&\text{ if } i\in\supp(\beta)\,. \end{cases}$$
{\rm (c)}\, If $x_p=n$ then, for $i\in\{2,\ldots,n-1\}$, one has $$\beta\cdot e_i=\begin{cases} e_i&\text{ if } i\notin\supp(\beta)\,,\\ e_{\beta(i)}&\text{ if } i\in\{x_1,\ldots,x_{p-2}\}\,,\\ -\sum_{j=2}^{n-1}e_j&\text{ if } i=x_{p-1}\,. \end{cases}$$ \end{lemma}
\begin{proof} {\rm (a)}\, If $2\leqslant i\leqslant p-1$, then \begin{align*} \alpha \cdot e_i=&\alpha\cdot (\overline{\omega_i-\omega_1})=\overline{\alpha\cdot (\omega_i-\omega_1)} =\overline{\omega_{\alpha(i)}-\omega_{\alpha(1)}}=\overline{\omega_{i+1}-\omega_2} =\overline{(\omega_{i+1}-\omega_1)-(\omega_2-\omega_1)}\\ =& e_{i+1}-e_2. \end{align*} If $i=p$, then $\alpha \cdot e_i=\alpha\cdot (\overline{\omega_p-\omega_1})=\overline{\omega_1-\omega_2}=-e_2$. Finally, if $i\geqslant p+1$, then we have $$\alpha \cdot e_i=\overline{\omega_i-\omega_2}=\overline{(\omega_i-\omega_1)-(\omega_2-\omega_1)}=e_i-e_2\,.$$
The proofs of {\rm (b)} and {\rm (c)} are similar, and are left to the reader. \end{proof}
\begin{lemma}\label{lemma s(k)} Let $k,l\in\{2,\ldots,p\}$, and let $x\in\{p+1,\ldots,n-1\}$. Then one has
{\rm (a)}\, $e_{k+1}\wedge\cdots \wedge e_p\wedge e_2\wedge\cdots \wedge e_{k-1}\wedge e_x=(-1)^{s(k+1,p)(k-2)} \hat{e}_k\wedge e_x$;
{\rm (b)}\, $\hat{e}_k\wedge e_k=(-1)^{s(k+1,p)} e$;
{\rm (c)}\, if $k<l$ then $\hat{e}_{k,l}\wedge e_x\wedge e_l= (-1)^{s(l+1,p)+1} \hat{e}_k\wedge e_x$;
{\rm (d)}\, if $k<l$ then $\hat{e}_{k,l}\wedge e_x\wedge e_k=(-1)^{s(k+1,p)} \hat{e}_l\wedge e_x$. \end{lemma}
\begin{proof} {\rm (a)}\, For $k\in\{2,\ldots,p\}$ and $x\in\{p+1,\ldots,n-1\}$, we have
\begin{align*} &\overbrace{e_{k+1}\wedge\cdots \wedge e_p}^{s(k+1,p)}\wedge \underbrace{e_2\wedge\cdots \wedge e_{k-1}}_{k-2}\wedge e_x\\ &=(-1)^{s(k+1,p)} e_2\wedge e_{k+1}\wedge\cdots \wedge e_p\wedge e_3\wedge\cdots \wedge e_{k-1}\wedge e_x=(-1)^{s(k+1,p)(k-2)} \hat{e}_k\wedge e_x. \end{align*}
The proofs of {\rm (b)}, {\rm (c)} and {\rm (d)} are similar, and are left to the reader. \end{proof}
\begin{cor}\label{cor e} For $e:=e_2\wedge e_3\wedge \cdots \wedge e_p$, we have $e\in D^{P_n}$; in particular, $e\in D^P$, for every $P\leqslant P_n$. \end{cor}
\begin{proof} With the notation in \ref{noth sylow} we have $P_n=P_p\times\prod_{i=2}^r(P_{p^i})^{n_i}$, and $P_p=\langle \alpha\rangle$, where $\alpha:=(1,2,\ldots,p)$. If $\beta\in \prod_{i=2}^r(P_{p^i})^{n_i}$ then we clearly have $\beta\cdot e=e$. By Lemma~\ref{lemma acts alpha} and Lemma~\ref{lemma s(k)}(b), we also have $$\alpha\cdot e=(e_3-e_2)\wedge (e_4-e_2)\wedge \cdots \wedge (e_p-e_2)\wedge (-e_2)=(-1)^{s(3,p)+1} e=(-1)^2 e=e\,.$$ \end{proof}
\begin{lemma}\label{lemma sigma} Let $1\neq \sigma\in E$, and let $q\in\mathbb{N}$ be such that $$\sigma=(x_1^1,\ldots,x_p^1)\cdots (x_1^q,\ldots,x_p^q)\,,$$ where $\{x_i^s: 1\leqslant i\leqslant p\,, 1\leqslant s\leqslant q\}=\supp(\sigma)\subseteq \{p+1,\ldots,n\}$ and $x_p^q=n$. Let further $u\in D$ be such that $u=\sum_{b\in\mathcal{B}} \lambda_b b$, for $\lambda_b\in F$. Suppose that $\sigma\cdot u=u$. Then one has the following:
{\rm (a)}\, $\sum_{k=2}^p(-1)^k \lambda_{\hat{k},x_i^q}=0$, for every $i\in\{1,\ldots,p-1\}$;
{\rm (b)}\, $\sum_{k=2}^p(-1)^{k+1} \lambda_{\hat{k},x_i^s}=\sum_{k=2}^p(-1)^{k+1}\lambda_{\hat{k},x_1^s}$, for $i\in\{1,\ldots,p\}$ and $1\leqslant s\leqslant q-1$. \end{lemma}
\begin{proof} Let $x\in\{x_i^s: 1\leqslant i\leqslant p,\, 1\leqslant s\leqslant q-1\}$, and let $k\in\{2,\ldots,p\}$. Suppose that $b\in\mathcal{B}$ is such that $\hat{e}_k\wedge e_x$ occurs with non-zero coefficient in $\sigma\cdot b$. Then
(i)\, $b=\hat{e}_k\wedge e_{\sigma^{-1}(x)}$, or
(ii)\, $b=\hat{e}_k\wedge e_{x_{p-1}^q}$, or
(iii)\, $b=\hat{e}_{k,k_2}\wedge e_{\sigma^{-1}(x)}\wedge e_{x_{p-1}^q}$ and $\sigma^{-1}(x)<x_{p-1}^q$, for some $k<k_2\leqslant p$, or
(iv)\, $b=\hat{e}_{k_1,k}\wedge e_{\sigma^{-1}(x)}\wedge e_{x_{p-1}^q}$ and $\sigma^{-1}(x)<x_{p-1}^q$, for some $2\leqslant k_1<k$, or
(v)\, $b=\hat{e}_{k,k_2}\wedge e_{x_{p-1}^q}\wedge e_{\sigma^{-1}(x)}$ and $\sigma^{-1}(x)>x_{p-1}^q$, for some $k<k_2\leqslant p$, or
(vi)\, $b=\hat{e}_{k_1,k}\wedge e_{x_{p-1}^q}\wedge e_{\sigma^{-1}(x)}$ and $\sigma^{-1}(x)>x_{p-1}^q$, for some $2\leqslant k_1<k$.
\noindent If $b$ is one of the basis elements in (i)--(vi) then the following table records $\sigma \cdot b$ as well as the coefficient at $\hat{e}_k\wedge e_x$ in $\sigma \cdot b$, which is obtained using Lemma~\ref{lemma s(k)}.
\begin{tabular}{|l|l|c|}\hline $b$& $\sigma\cdot b$ & coefficient\\\hline\hline $\hat{e}_k\wedge e_{\sigma^{-1}(x)}$&$\hat{e}_k\wedge e_x$& $1$\\\hline $\hat{e}_k\wedge e_{x_{p-1}^q}$&$\hat{e}_k\wedge \sum_{y=2}^{n-1}(-e_y)$& $-1$\\\hline $\hat{e}_{k,k_2}\wedge e_{\sigma^{-1}(x)}\wedge e_{x_{p-1}^q}$&$\hat{e}_{k,k_2}\wedge e_x\wedge \sum_{y=2}^{n-1}(-e_y)$&$(-1)^{1+s(k_2+1,p)+1}$\\\hline $\hat{e}_{k_1,k}\wedge e_{\sigma^{-1}(x)}\wedge e_{x_{p-1}^q}$&$\hat{e}_{k_1,k}\wedge e_x\wedge \sum_{y=2}^{n-1}(-e_y)$&$(-1)^{1+s(k_1+1,p)}$\\\hline $\hat{e}_{k,k_2}\wedge e_{x_{p-1}^q}\wedge e_{\sigma^{-1}(x)}$&$\hat{e}_{k,k_2}\wedge \sum_{y=2}^{n-1}(-e_y)\wedge e_x$&$(-1)^{1+s(k_2+1,p)}$\\\hline $\hat{e}_{k_1,k}\wedge e_{x_{p-1}^q}\wedge e_{\sigma^{-1}(x)}$&$\hat{e}_{k_1,k}\wedge \sum_{y=2}^{n-1}(-e_y)\wedge e_x$&$(-1)^{s(k_1+1,p)}$\\\hline
\end{tabular}
Now note that $(-1)^{1+s(k_2+1,p)+1}=(-1)^{1+p-k_2+1}=(-1)^{k_2+1}$ and $(-1)^{1+s(k_1+1,p)}=(-1)^{1+p-k_1}=(-1)^{k_1}.$ Since $\sigma\cdot u=u$, this shows that \begin{equation}\label{eqn sum 5} \lambda_{\hat{k},x}=\lambda_{\hat{k},\sigma^{-1}(x)}-\lambda_{\hat{k},x_{p-1}^q}+\sum_{k_2=k+1}^p(-1)^{k_2+1}\lambda_{\widehat{k,k_2},\sigma^{-1}(x),x_{p-1}^q}+\sum_{k_1=2}^{k-1}(-1)^{k_1}\lambda_{\widehat{k_1,k},\sigma^{-1}(x),x_{p-1}^q}\, \end{equation} if $\sigma^{-1}(x)<x_{p-1}^q$ and \begin{equation}\label{eqn sum 5'} \lambda_{\hat{k},x}=\lambda_{\hat{k},\sigma^{-1}(x)}-\lambda_{\hat{k},x_{p-1}^q}-\sum_{k_2=k+1}^p(-1)^{k_2+1}\lambda_{\widehat{k,k_2},x_{p-1}^q,\sigma^{-1}(x)}-\sum_{k_1=2}^{k-1}(-1)^{k_1}\lambda_{\widehat{k_1,k},x_{p-1}^q,\sigma^{-1}(x)}\, \end{equation} if $\sigma^{-1}(x)>x_{p-1}^q$. Moreover, \begin{align*} &\sum_{k=2}^p(-1)^{k+1}\left(\sum_{k_2=k+1}^p(-1)^{k_2+1}\lambda_{\widehat{k,k_2},\sigma^{-1}(x),x_{p-1}^q}+\sum_{k_1=2}^{k-1}(-1)^{k_1}\lambda_{\widehat{k_1,k},\sigma^{-1}(x),x_{p-1}^q}\right)\\ &=\sum_{k=2}^p\sum_{l=k+1}^p ((-1)^{k+1}(-1)^{l+1}+(-1)^k(-1)^{l+1}) \lambda_{\widehat{k,l},\sigma^{-1}(x),x_{p-1}^q}=0\, \end{align*} if $\sigma^{-1}(x)<x_{p-1}^q$, and \begin{align*} &\sum_{k=2}^p(-1)^{k+1}\left(-\sum_{k_2=k+1}^p(-1)^{k_2+1}\lambda_{\widehat{k,k_2},x_{p-1}^q,\sigma^{-1}(x)}-\sum_{k_1=2}^{k-1}(-1)^{k_1}\lambda_{\widehat{k_1,k},x_{p-1}^q,\sigma^{-1}(x)}\right)\\ &=-\sum_{k=2}^p\sum_{l=k+1}^p ((-1)^{k+1}(-1)^{l+1}+(-1)^k(-1)^{l+1}) \lambda_{\widehat{k,l},x_{p-1}^q,\sigma^{-1}(x)}=0\, \end{align*} if $\sigma^{-1}(x)>x_{p-1}^q$. Hence, from (\ref{eqn sum 5}) and (\ref{eqn sum 5'}) we get \begin{equation}\label{eqn sum 2} \sum_{k=2}^p(-1)^{k+1}\lambda_{\hat{k},x_i^s}=\sum_{k=2}^p(-1)^{k+1}\lambda_{\hat{k},\sigma^{-1}(x_i^s)}+\sum_{k=2}^p(-1)^k\lambda_{\hat{k},x_{p-1}^q}\,, \end{equation} for every $i\in\{1,\ldots,p\}$ and $1\leqslant s\leqslant q-1$.
We also have $\sigma^i \cdot u=u$, for $i=1,\ldots,p-1$. To compare the coefficient at $e$ in $u$ and in $\sigma^i\cdot u$, let $i\in\{1,\ldots,p-1\}$ and suppose that $b\in\mathcal{B}$ is such that $e$ occurs in $\sigma^i\cdot b$ with non-zero coefficient. Then either $b=e$ and $e=\sigma^i\cdot e$, or $b=\hat{e}_k\wedge e_{\sigma^{-i}(x_{p}^q)}$, for some $k\in\{2,\ldots,p\}$. Moreover, in the latter case we have $\sigma^i\cdot b=\hat{e}_k\wedge (-e_2-e_3-\cdots -e_{n-1})$, where $e$ occurs with coefficient $$(-1)^{s(k+1,p)+1}=\begin{cases} 1&\text{ if } 2\mid k\,,\\ -1&\text{ if } 2\nmid k\,, \end{cases}$$ by Lemma~\ref{lemma s(k)}. So we obtain $\lambda_e=\lambda_e+\sum_{k=2}^p(-1)^k \lambda_{\hat{k},\sigma^{-i}(x_p^q)}$, for $i\in\{1,\ldots,p-1\}$, that is, \begin{equation}\label{eqn sum 3} 0=\sum_{k=2}^p(-1)^k \lambda_{\hat{k},x_j^q}\,, \end{equation} for $j\in\{1,\ldots,p-1\}$, which proves assertion~(a). Now assertion~(b) follows from (\ref{eqn sum 2}) and (\ref{eqn sum 3}) with $j=p-1$. \end{proof}
Next we shall show that $D(E)\neq \{0\}$, where $E$ is the elementary abelian group in \ref{nota e}. In order to do so, we want to apply Proposition~\ref{prop Brauer} with $b_0=e$.
\begin{lemma}\label{lemma TrP} Let $P$ be a maximal subgroup of $E$. If $u\in D^P$ then $e$ occurs in $\Tr_P^E(u)$ with coefficient $0$. \end{lemma}
\begin{proof} Set $\alpha:=(1,2,\ldots,p)$. Let $u\in D^P$, and write $u=\sum_{b\in \mathcal{B}} \lambda_b b$, where $\lambda_b\in F$. We shall treat the case where $\alpha\in P$ and the case where $\alpha\notin P$ separately.
\underline{Case 1:} $\alpha\in P$. Then there is some $1\neq g\in \prod_{i=2}^t\prod_{j=1}^{m_i} E_{p^i,j}$ with $g\notin P$. Thus $\{1,g,g^2,\ldots,g^{p-1}\}$ is a set of representatives of the left cosets of $P$ in $E$, so that we get $\Tr_P^E(u)=u+g\cdot u+\cdots +g^{p-1}\cdot u=\sum_{b\in\mathcal{B}}\sum_{i=0}^{p-1} \lambda_b (g^i \cdot b)$.
Since $g\neq 1$ and $t\geqslant 2$, we have $$g=(x_1^1,\ldots,x_p^1)\cdots (x_1^q,\ldots,x_p^q)\,,$$ for some $q\geqslant p$ and $\{x_i^s: 2\leqslant i\leqslant p,\, 1\leqslant s\leqslant q\}=\supp(g)$.
Suppose first that $n\notin\supp(g)$, and let $b\in\mathcal{B}$. Let further $i\in\{0,\ldots,p-1\}$, and suppose that $e$ occurs in $g^i \cdot b$ with non-zero coefficient. Then we must have $b=e$, in which case $\sum_{i=0}^{p-1} g^i \cdot b=pe=0$, by Corollary~\ref{cor e}; in particular, $e$ occurs in $\Tr_P^E(u)$ with coefficient 0.
So we may now suppose that $n\in\supp(g)$. Moreover, we may suppose that $x_p^q=n$. Let $i\in\{0,\ldots,p-1\}$, and let $b\in\mathcal{B}$ be such that $e$ occurs in $g^i \cdot b$ with non-zero coefficient. If $i=0$ then we must of course have $b=e=g^0\cdot e$. If $i\geqslant 1$ then $b=e$, or $b=\hat{e}_k\wedge e_{g^{-i}(x_p^q)}$, for some $k\in\{2,\ldots,p\}$. In the latter case, we have $g^i\cdot (\hat{e}_k\wedge e_{g^{-i}(x_p^q)})=\hat{e}_k\wedge (-e_2-e_3-\cdots -e_{n-1})$, in which $e$ occurs with coefficient $$(-1)^{s(k+1,p)+1} =\begin{cases} 1&\text{ if } 2\mid k\,,\\ -1&\text{ if } 2\nmid k\,, \end{cases}$$ by Lemma~\ref{lemma s(k)}. Consequently, the coefficient at $e$ in $\Tr_P^E(u)$ equals \begin{equation}\label{eqn e in Tru} p\lambda_e+\sum_{i=1}^{p-1}\left(\mathop{\sum_{k=2}^p}_{2\mid k}\lambda_{\hat{k},x_i^q}-\mathop{\sum_{l=2}^p}_{2\nmid l}\lambda_{\hat{l},x_i^q}\right)=\sum_{i=1}^{p-1}\sum_{k=2}^p (-1)^k\lambda_{\hat{k},x_i^q}\,. \end{equation} Next we use the fact that $u\in D^P$ to show that this coefficient is indeed 0. Since $\alpha\in P$, we, in particular, have $u=\alpha^i \cdot u$, for every $i\in\{1,\ldots,p-1\}$. So let $i\in\{1,\ldots,p-1\}$, and let $x\in\{x_1^q,\ldots,x_p^q\}$. Suppose that $b\in\mathcal{B}$ is such that $\hat{e}_{i+1}\wedge e_x$ occurs in $\alpha^i \cdot b$ with non-zero coefficient. Then from Lemma~\ref{lemma acts alpha} we deduce that $b=\hat{e}_{\alpha^{-i}(1)}\wedge e_x$. Moreover, we have $$\alpha^i\cdot (\hat{e}_{\alpha^{-i}(1)}\wedge e_x)=(e_{i+2}-e_{i+1})\wedge\cdots\wedge (e_p-e_{i+1})\wedge (e_2-e_{i+1})\wedge\cdots\wedge (e_i-e_{i+1})\wedge (e_x-e_{i+1})\,.$$ Thus, by Lemma~\ref{lemma s(k)}, the coefficient at $\hat{e}_{i+1}\wedge e_x$ in $\alpha^i\cdot (\hat{e}_{\alpha^{-i}(1)}\wedge e_x)$ equals $(-1)^{s(i+2,p)(i-1)}=1$. Letting $i$ vary over $\{1,\ldots,p-1\}$ and comparing the coefficient at $\hat{e}_{i+1}\wedge e_x$ in $u$ and in $\alpha^i \cdot u$, we deduce that $\lambda_{\hat{k},x}=\lambda_{\widehat{p-k+2},x}$, for $k\in\{2,\ldots, (p+1)/2\}$ and every $x\in\{x_1^q,\ldots,x_p^q\}$. Since $k$ is even if and only if $p-k+2$ is odd, we conclude that the right-hand side of (\ref{eqn e in Tru}) is 0, as claimed. This completes the proof in case 1.
\underline{Case 2:} $\alpha\notin P$, so that $\{1,\alpha,\alpha^2,\ldots,\alpha^{p-1}\}$ is a set of representatives for the cosets of $P$ in $E$, and we get $\Tr_P^E(u)=u+\alpha \cdot u+\cdots +\alpha^{p-1}\cdot u$. We determine the coefficient at $e$ in $\Tr_P^E(u)=u+\alpha \cdot u+\cdots +\alpha^{p-1} \cdot u$. Let $i\in\{0,\ldots,p-1\}$, and let $b\in\mathcal{B}$ be such that $e$ occurs in $\alpha^i \cdot b$ with non-zero coefficient. If $i=0$ then $b=e=\alpha^0 \cdot e$. So let $i\geqslant 1$. Then, by Lemma~\ref{lemma acts alpha}, we either have $b=e$, or $b=\hat{e}_{\alpha^{-i}(1)}\wedge e_x$, for some $x\in\{p+1,\ldots,n-1\}$. Moreover, in the latter case, $$\alpha^i\cdot b=(e_{i+2}-e_{i+1})\wedge (e_{i+3}-e_{i+1})\wedge\cdots \wedge (e_p-e_{i+1})\wedge (e_2-e_{i+1})\wedge\cdots\wedge (e_i-e_{i+1})\wedge (e_x-e_{i+1})\,.$$ So the coefficient at $e$ in $\alpha^i\cdot (\hat{e}_{\alpha^{-i}(1)}\wedge e_x)$ equals $$(-1)^{s(i+2,p)(i-1)+s(i+2,p)+1}=\begin{cases} 1&\text{ if } 2\nmid i\,,\\ -1&\text{ if } 2\mid i\,. \end{cases}$$ Since $i$ is even if and only if $\alpha^{-i}(1)$ is even, we deduce from this that the coefficient at $e$ in $u+\alpha \cdot u+\cdots +\alpha^{p-1}\cdot u$ equals \begin{equation}\label{eqn e in Tru a} p\lambda_e+\sum_{x=p+1}^{n-1}\sum_{k=2}^p(-1)^{k+1}\lambda_{\hat{k},x}=\sum_{x=p+1}^{n-1}\sum_{k=2}^p(-1)^{k+1}\lambda_{\hat{k},x}\,. \end{equation} To show that this coefficient is 0, we again exploit the fact that $u\in D^P$. In fact, we shall show that \begin{equation}\label{eqn sum 0} \mathop{\sum_{x\in\supp(E_{p^l,j_l})}}_{x<n} \sum_{k=2}^p (-1)^{k+1} \lambda_{\hat{k},x}=0\,, \end{equation} for every $l\in\{2,\ldots, t\}$ and $1\leqslant j_l\leqslant m_l$. For each such $l$ and $j_l$, there is, by Lemma~\ref{lemma max in E}, some element $\sigma(l,j_l)\in P$ such that $\supp(E_{p^l,j_l})\subseteq \supp(\sigma(l,j_l))\subseteq \{p+1,\ldots,n\}$. Fixing $l$ and $j_l$, we write $$\sigma:=\sigma(l,j_l)=(x_1^1,\ldots,x_p^1)\cdots (x_1^q,\ldots, x_p^q)\,,$$
for some $q\geqslant |E_{p^l,j_l}|/p$ and $\supp(\sigma)=\{x_i^j: 1\leqslant i\leqslant p,\, 1\leqslant j\leqslant q\}$.
\underline{Case 2.1:} $n\notin\supp(\sigma)$, or equivalently, $\supp(\sigma)\cap \supp(E_{p^t,m_t})=\emptyset$. Let $x\in\supp(\sigma)$, let $k\in\{2,\ldots,p\}$, and let $b\in \mathcal{B}$ be such that $\hat{e}_k\wedge e_{x}$ occurs in $\sigma \cdot b$ with non-zero coefficient. This forces $b=\hat{e}_k\wedge e_{\sigma^{-1}(x)}$, and $\sigma\cdot (\hat{e}_k\wedge e_{\sigma^{-1}(x)})=\hat{e}_k\wedge e_x$. Thus, $\lambda_{\hat{k},x}=\lambda_{\hat{k},\sigma^{-1}(x)}$. This shows that $\lambda_{\hat{k},x_1^s}=\lambda_{\hat{k},x_i^s}$, for all $i\in\{1,\ldots,p\}$ and $s\in\{1,\ldots,q\}$. By rearranging commuting $p$-cycles in $\sigma$, we may assume that there is some $1\leqslant q_0\leqslant q$ such that $\supp(E_{p^l,j_l})=\{x_i^s: 1\leqslant i\leqslant p,\, 1\leqslant s\leqslant q_0\}$. Then \begin{equation}\label{eqn sum 1} \sum_{x\in\supp(E_{p^l,j_l})}\sum_{k=2}^p(-1)^{k+1} \lambda_{\hat{k},x}=\sum_{i=1}^p\sum_{s=1}^{q_0}\sum_{k=2}^p(-1)^{k+1} \lambda_{\hat{k},x_i^s}=p\sum_{s=1}^{q_0}\sum_{k=2}^p (-1)^{k+1}\lambda_{\hat{k},x_1^s}=0\,, \end{equation} as desired.
\underline{Case 2.2:} $n\in\supp(\sigma)$. Then we may suppose that $x_p^q=n$. If $(l,j_l)\neq (t,m_t)$, then we may further suppose that there is some $1\leqslant q_1<q$ such that $\supp(E_{p^l,j_l})=\{x_i^s: 1\leqslant i\leqslant p,\, 1\leqslant s\leqslant q_1\}$. By Lemma~\ref{lemma sigma}(b), we then get \begin{equation}\label{eqn sum 4} \sum_{x\in\supp(E_{p^l,j_l})}\sum_{k=2}^p (-1)^{k+1}\lambda_{\hat{k},x}=\sum_{i=1}^p\sum_{s=1}^{q_1}\sum_{k=2}^p(-1)^{k+1} \lambda_{\hat{k},x_i^s}=p\sum_{s=1}^{q_1}\sum_{k=2}^p(-1)^{k+1} \lambda_{\hat{k},x_1^s}=0\,. \end{equation}
If $(l,j_l)=(t,m_t)$ then we may suppose that there is $1\leqslant q_2\leqslant q$ such that $\supp(E_{p^t,m_t})=\{x_i^s: 1\leqslant i\leqslant p,\, q_2\leqslant s\leqslant q\}$. In this case, Lemma~\ref{lemma sigma} gives \begin{align*} \mathop{\sum_{x\in\supp(E_{p^t,m_t})}}_{x<n}\sum_{k=2}^p(-1)^{k+1} \lambda_{\hat{k},x}&=\sum_{i=1}^p\sum_{s=q_2}^{q-1}\sum_{k=2}^p(-1)^{k+1}\lambda_{\hat{k},x_i^s}+\sum_{i=1}^{p-1}\sum_{k=2}^p(-1)^{k+1}\lambda_{\hat{k},x_i^q}\\ &=p\sum_{s=q_2}^{q-1}\sum_{k=2}^p(-1)^{k+1} \lambda_{\hat{k},x_1^s}-\sum_{i=1}^{p-1}\sum_{k=2}^p(-1)^{k}\lambda_{\hat{k},x_i^q}=0\,. \end{align*}
To summarize, we have now verified equation (\ref{eqn sum 0}), which together with (\ref{eqn e in Tru}) shows that the coefficient at $e$ in $\Tr_P^E(u)$ is 0. This now completes the proof in case 2 and, thus, of the lemma. \end{proof}
As a direct consequence of Lemma~\ref{lemma TrP}, Corollary~\ref{cor e}, Proposition~\ref{prop Brauer}, and \cite[(1.3)]{Br} we thus have proved the following
\begin{prop}\label{prop Brauer D(E)} Let $n\in\mathbb{N}$ be such that $n=p+\sum_{i=2}^tm_ip^i$, for some $t\geqslant 2$, $m_2,\ldots,m_t\in\mathbb{N}_0$ with $m_t\neq 0$. Let further $D:=D^{(n-p+1,1^{p-1})}$, and let $Q\leqslant \mathfrak{S}_n$ be a vertex of $D$. Then $D(E(1,m_2,\ldots,m_t))\neq \{0\}$; in particular, $E(1,m_2,\ldots,m_t)\leqslant_{\mathfrak{S}_n} Q$. \end{prop}
\begin{rem}\label{rem Brauer D(E)} Again consider the $p$-adic expansion $n=p+\sum_{i=2}^rn_ip^i$, where $r\geqslant 2$ and $n_r\neq 0$. Note that Proposition~\ref{prop Brauer D(E)}, in particular, holds for $t=r$ and $m_1=1,\ldots, m_r=n_r$. Thus the elementary abelian subgroup $E(1,n_2,\ldots,n_r)\leqslant P_n$ is $\mathfrak{S}_n$-conjugate to a subgroup of every vertex of $D$. This settles item (ii) at the beginning of this section and completes the proof of Theorem~\ref{thm main}. \end{rem}
\noindent {\sc S.D.: Department of Mathematics, University of Kaiserslautern,\\ P.O. Box 3049, 67653 Kaiserslautern, Germany}\\ {\sf [email protected]}
\noindent {\sc E.G.: Department of Mathematics, Royal Holloway University of London,\\ Egham TW20 0EX, United Kingdom}\\ {\sf [email protected]}
\end{document} | arXiv |
Pregnant women adherence level to antenatal care visit and its effect on perinatal outcome among mothers in Tigray Public Health institutions, 2017: cohort study
Abera Haftu1,
Hadgay Hagos1,
Mhiret-AB Mehari1 &
Brhane G/her1
To assess pregnant women adherence level to antenatal care visit and its effect on perinatal outcome among mothers in Tigray Public Health institutions, 2017.
The overall adherence level of the women towards to antenatal care visit was 49.9% and incidence of PPH, still birth, early neonatal death, late neonatal death and low birth weight complication was 4.3%, 2.3%, 2.7%, 1.9% and 7.5% respectively. PPH, preterm labor, early neonatal death and LBW complication was reduced by 81.2%, 52%, 61% and 46% respectively among women's with complete adherence to ANC visit.
Complications that happen during pregnancy and childbirth are the most leading causes of maternal mortality and morbidity among women whose age ranges from 15 to 49 in developing countries [1]. Annually around 287,000 women die secondary to pregnancy related cause in the globe, among this figure 99% of the maternal death is from underdeveloped countries [2]. Ethiopia is among the leading countries with high maternal mortality and morbidity from the developing countries [3]. In developing countries almost all pregnant women's receive antenatal care at least once, but in sub-Saharan countries the report is around 68% where women's take antenatal care (ANC) services at least ones and majority of them visit the health institutions at third visit [4, 5].
Most research finding showed that most of maternal and neonatal deaths are preventable; one the strategic and important key step for reducing of maternal related mortality and morbidity is antenatal care directly by detecting and treating of complications in earlier period starting from the onset of pregnancy till delivery [6]. The timing of starting first ANC and total number of ANC visits that pregnant women receive and not attending the recommend ANC services may lead to adverse perinatal outcomes [7].
Ethiopian Demographic Health Survey (EDHS) 2016 showed that national ANC service coverage is around 64%, even if the total number of ANC visit is good, starting ANC follow up in the earlier second trimester is low in magnitude. Research results of late ANC service booking from Addis Ababa, Metekel, Hadiya, Ambo and Gondar was 59.8%, 55.1%, 68.2%, 86.8% and 64.9% respectively [8,9,10,11,12]. In the current situation Ethiopia deliveries three tiered health care system; this is characterized by district health system, health centers and health posts which are connected to each other by referral system [13]. The need of ANC is taken as basic rights of all pregnant women's to keep safe their infants, the high maternal and neonatal mortality in Ethiopia is the result of poor utilization of ANC [14, 15]. The primary target of ANC is to detect problem, treat on time and prevention of complications by health care provision, despite of this illiteracy and low socio economic status contribute to poor ANC adherence. There have many studies which showed positive effect of ANC on perinatal outcome including reducing risk of postpartum hemorrhage (PPH), low birth weight, preterm birth and perinatal death. World health organization (WHO) recommended for all pregnant women to have four consecutive ANC visits for low risk pregnant women's [16,17,18,19,20,21,22,23,24,25].
Methods and materials
Study area and period
The study will be conducted in Tigray Public Health institutions. Tigray is located in Northern part of Ethiopia and around 783 km away from the capital city Addis Ababa. Around 5.5 million people are found in this region (census 2007). The region is the owner of 216 health centers, 15 General Hospitals and 2 Referral Hospitals. Among the selected zones (southern, Mekele & southeastern zones) there are about 61 health centers, 5 primary Hospitals 1 Referral Hospital and 6 General Hospitals. The study was conducted from July 1, 2017 to August 2018.
Prospective cohort design was employed.
Source population
All women's who gave birth in Tigray Public Health institutions.
Study population Women's who fulfill the criteria and selected in the study period.
Exposed group Are mothers coming to the health facility for delivery services where their ANC visit was complete.
Non-exposed group Are mothers coming to the health facility for delivery services where their ANC visit was incomplete.
Inclusion criteria All women coming for delivery services in the public health facilities.
Exclusion criteria Women who has known medical illness (hypertension, cardiac disease, DM, malaria, liver disease).
Sample size determination
Sample size was calculated using double population proportion formula for cohort study considering the following assumptions:
CL = 95%.
Power-80%.
A one-to-one ratio of exposure to non-exposure.
Since there is no any documented evidence in the setting, it is assumed that the complication rate will be twice as high amongst the exposed group (complete adherence) as compared to unexposed group (incomplete adherence).
By taking prevalence of pregnancy complication (PIH/preeclampsia–eclampsia) among the mothers with complete adherence to be 5.1% from previous study in Ghana [11].
$$ n_{1} = \frac{{\left[ {Z_{\alpha /2} \sqrt {\left( {1 + \frac{1}{r}} \right)P(1 - P)} - Z_{{\beta \sqrt {P_{1} (1 - P_{1} ) + \frac{{P_{2} (1 - P_{2} )}}{r}} }} } \right]}}{{(P_{1} - P_{2} )^{2} }}^{2} $$
The final total sample size is 928.
464 participants in each group.
Sampling technique
Systematic random sampling was used for this population and the sample was distributed to each facility based on proportional allocation in correspondence of delivery services. Women who have full visits were considered as exposed group where as those with incomplete follow up will be considered as non exposed group. Exposed and non exposed mothers who fulfill the inclusion criteria were enrolled to the cohort and were followed until the end of post partum period. Among the seven zones of the region 40% of them were selected by simple random selection technique. In the selected zones there are about 73 health facilities, by using simple lottery method 20 of them will be selected. The sample size will be distributed to each selected health facility by probability proportion to size (PPS) according to their ANC flow rate.
Data collection technique and process
Women's who come for delivery services in the public health institutions who met the criteria for the cohort study were enrolled and followed till the end of the postpartum period. After reviewing the women's document based on their ANC frequency they were recruited to exposed and non exposed groups, those with complete adherence ANC visits were considered as exposed groups and those incomplete ANC visits were considered as non exposed groups. Questioner was prepared from different literatures and WHO recommendations for pregnancy, delivery and post delivery continuum of care. There were about 20 BSC Midwives data collectors one data collector per each health facility. Three day data training was given to the data collectors and frequent supervision was made to each health facilities at 1 week interval. The follow up was at respective health institutions, but for those who were unable to attend the follow up health facility required information was collected by telephone. In order to ensure adherence of the follow up the community was mobilized by health extension workers
Data was entered by Epi data version 3.1 software first then exported to SPSS version 20 software for analysis purpose. Descriptive analysis was presented using mean and proportions. Tables, figures and text were used for data presentation. Determinants of maternal and neonatal complications, as well as the effect of complete adherence on pregnancy outcomes was estimated and expressed as relative risks (RRs) with their 95% confidence intervals (CI). Binary logistic regression run to see the association between variables. Significance was declared at p value < 0.05.
Data quality assurance
Standardized English version measuring questionnaire was adapted and it was translated into Tigrigna (local language) by experts. The questionnaire was reviewed by senior researchers and comments were incorporated for internal validity. In addition it was pre-tested on 10% of the calculated sample size. Data collectors and supervisors were trained for 3 days on the tools and process of data collection. Five percent of the collected data was checked by the supervisor for completeness and finally the investigators will monitor the overall quality of data collection.
Dependent variable Perinatal outcome.
Independent variable Socio-demographic factors (age, educational level, marital status and employment status).
Maternal factors (parity, trimester at first antenatal care visit, previous pregnancy history and number of times antenatal clinic was attended during pregnancy).
Neonatal factors (mode of delivery, duration of delivery, place of birth).
Operational definitions
Complete adherence women's who attend the ANC visit four and above.
Incomplete adherence women who had attended ANC visit less than or equal to three time.
A total of 1103 women were recruited and 100% of them were followed up to 6 weeks post-partum from December 2017 to July 2018. The number of women's enrolled to the exposed and non exposed groups were above the minimum size set during the proposal writing, this is done to meet the minimum number of participants to each groups in the selected health institutions.
Participants' baseline characteristics
The age of the participants in mean and standard deviation were 26.4 (5.2) years. 45.3% of the participants' age was 19–25 years. Most of the participants' educational level were secondary school and above which makes 42.3% from the total. More than half of the participants (71.8%) were from urban residence and married individuals take the highest proportions (92.7%) from the total participants (see Table 1).
Table 1 Socio demographic characteristics of study participants
Incidence of maternal and neonatal complications
Overall the incidence of postpartum hemorrhage is 4.8% of which 1.6% and 6.9% are from women's with complete and incomplete adherence respectively. Still birth and asphyxia were 2.3% and 10.3% consecutively (see Table 2).
Table 2 Incidence of the maternal and neonatal complications among the two groups (adhered versus non adhered)
Effect of complete adherence on risk of pregnancy complications
Postpartum hemorrhage complication was reduced by 81.2% among women's with complete adherence to antenatal care visit [ARR = CI 95% = 0.188 (0.088–0.404)]. Early neonatal death was reduced by 61.3% [ARR = CI 95% = 0.387 (0.162–0.928)] and low birth weight was reduced by 46.5% [ARR = CI 95% = 0.535 (0.326–0.878)] among women's with complete adherence to antenatal visit (see Table 3).
Table 3 Multivariate analysis of complete adherence's effect on maternal and neonatal complications
Many studies have shown the positive effect of antenatal care services on perinatal outcome, so giving emphasis to determine the gap with women adherence to antenatal visit and its effect on perinatal outcome is timely and significant as woman with single visit and four visits will not have similar complications.
Overall the women's adherence to complete visit to antenatal care is 49.9% and the follow up till postpartum period was 100% complete in our study where as research findings from Addis Ababa, Metekel, Hadya Zone, Ambo and Gonder show that the prevalence of ANC service booking was 59.8%, 55.1%, 68.2%, 86.8% and 64.9%, respectively [8,9,10,11,12]. This variation could due to the geographical location and the documentation system and manly the study design in general as they use the snap shoot kind of study where as in this study the focus was on the full course of the antenatal care.
In this study women's who attend at least single antenatal care was much lower than other studies (7%) which was done in Nepal, Pakistan, Bangladesh and Indian which was 28%, 28%, 33% and 60% respectively [26]. This variation again could be due to the educational level and economic status of the participants.
In this study the incidence of low birth weight was 7.5% which is a little bit higher than a study done in Nigeria which was 4.8% among women's with greater than or equal to four visits [27] and this variation could be due the nutritional consumption and geographical location.
Most studies from what we had searched showed that prevalence and determinant factor for ANC utilization, less emphasis was given to women's level of adherence to ANC visits and its effect on perinatal outcome. The incidence of developing postpartum hemorrhage among women's with complete adherence to antenatal care visit was about 1.6% where as in women's with incomplete adherence was 6.9% and incidence of adherence to postnatal visit was 30.9% and 50.9% among women's with complete and incomplete adherence respectively.
This study shows that incidence of neonatal complication is higher among the women's neonate with incomplete adherence to antenatal visit. Incidence of still birth among women's with incomplete adherence is four fold of the women's with complete adherence to antenatal visit which is 3.6 and 0.9% respectively. The incidence of early neonatal death and late neonatal death among women's with complete adherence to antenatal visit was almost similar which is 1.5 and 1.1% respectively where as among the women's with incomplete adherence to antenatal care visit the incidence of the early neonatal death is higher than the late neonatal death which is 4 and 2.7% consecutively and this could be due to the vulnerability of the neonate to many things in the earlier period than late after they customize the environment. The incidence of low birth weight among the non exposed group was almost twice of the exposed group (9.4%:5.6%) and this might be probably due to the counseling regarding nutritional methods during antenatal visit.
Conducted at different sites in which it was difficult to supervise timely.
Geographical location was one factor during the follow up.
Delayed to meet them during follow up b/c of the method of communication.
ANC:
antenatal care
EDHS:
Ethiopian Demographic Health Survey
PPH:
postpartum haemorrhage
Simkhada B, Teijlingen ER, Porter M, Simkhada P. Factors affecting the utilization of antenatal care in developing countries: systematic review of the literature. J Adv Nurs. 2008;61(3):244–60.
World Health Organization. World Health Statistics. 2012. apps.who.int/iris/bitstream/10665/44844/1/9789241564441_eng.pdf.
Berhan Y, Berhan A. Review of maternal mortality in Ethiopia: a story of the past 30 years. Ethiop J Health Sci. 2014;24(1):3–14.
Abou-Zahr CL, Tessa M. Promises, achievements and missed opportunities: an analysis of trends, levels and differentials, 1990–2001.
Antenatal Care in Developing Countries Promises, achievements and missed opportunities. http://www.unicef.org/media/files/antenatal.pdf. Accessed 4 Apr 2017.
Banda Isaac, Michelo Charles, Hazemba Alice. Factors associated with late antenatal care attendance in selected rural and urban communities of the copper belt province of Zambia. Med J Zambia. 2012;39(3):29–36.
Raatikainen K, Heiskanen N, Heinonen S. Under-attending free antenatal care is associated with adverse pregnancy outcomes. BMC Public Health. 2007;7:268.
Tariku Alemayehu, Melkamu Yilma, Kebed Zewditu. Previous utilization of service does not improve timely booking in antenatal care: cross sectional study on timing of antenatal care booking at public health facilities in Addis Ababa. Ethiop J Health Dev. 2010;24(3):226–33.
Tura G. Antenatal care service utilization and associated factors in Metekel Zone, Northwest Ethiopia. Ethiop J Health Sci. 2009;19:2.
Hamdela B, Godebo G, Gebre T. Predictors of early antenatal care booking in government health facilities of Hossana Town, Hadiya Zone, South Ethiopia: unmatched case control study. J AIDS Clin Res. 2015;6:521.
Damme TG, Workineh D, Gmariam A. Time of antenatal care booking and associated factors among pregnant women attending Ambo Town health facilities, Central Ethiopia. J Gynecol Obstetric. 2015;3(5):103–6.
Gudayu TW. Proportion and factors associated with late antenatal care booking among pregnant mothers in Gondar Town, North West Ethiopia. Afr J Reprod Health. 2015;19(2):94–100.
Federal Democratic Republic of Ethiopia Ministry of Health., Health Sector Development Program IV 2010/11–2014/15. http://www.nationalplanningcycles.org/sites/default/files/country_docs/Ethiopia/ethiopia_hsdp_iv_final_draft_2010_-2015.pdf.
Main D. The epidemiology of preterm birth. Clin Obstet Gynecol. 1988;31:5211–32.
Harrson KA. Child bearing health and social priorities: a survey of 22,774 consecutive hospital births in Zaria, Northern Nigeria. Br J Obstet Gynecol. 1985;92(5):1–119.
Editorial. Why retained traditional birth attendants? Lancet. 1983; I: 223–4.
Editorial. Maternal health in sub Saharan Africa. Lancet. 1987; I: 255–7.
Harrison KA. The influence of maternal age and parity on child bearing with special reference to primigravida age 15 years and under. Br J Obstet Gynecol. 1985;92(5):23–31.
Orvos H, Hoffmann I. The perinatal outcome of pregnancy without prenatal care: a retrospective study in Szegd, Hungary. Eur J Obstetr Gynecol Reprod Biol. 2002;100:17.
Kapoor SK, Reddaiah VP, Lobo J. Antenatal care and perinatal mortality. Indian J Pediatr. 1985;52:159–62.
McCaw Binns A, Green Wood R, Asheley D, Golding J. Antenatal care and perinatal care in Jamaica: do they reduce perinatal death rates? Paediatr Perinat Epidemiol. 1994;8(1):86–97.
Villar J, Ba'aqeel H, Piaggio G, et al. WHO antenatal care randomized trial for the evaluation of new model of routine antenatal care. Lancet. 2001;357:1551–64.
Carroli G, Villar J, Ba'aqeel H, Piaggio G, et al. WHO systematic review of randomized control trial of routine antenatal care. Lancet. 2001;357:1565–70.
Banta D. What is antenatal care? What are its boundaries? What is the effectiveness of antenatal care? What are the financial and organizational implications of antenatal care?. Geneva: WHO; 2003.
Abou-Zahr CL, Warlaw T. Antenatal care in developing countries—promises, achievements and missed opportunities: AN analysis of trends, levels and differentials, 1990–2001. Geneva: WHO; 2003.
Campbell OMR, Graham WJ. The lancet maternal survival series steering group strategies for reducing maternal mortality: getting on with what works. Lancet. 2006;368:1284–99.
Onwuhafua P, Adaze JA, et al. The effect of frequency of antenatal visit on pregnancy outcome in Kaduna, Nigeria. Trop J Obstetr Gynecol. 2017;33:155–64.
AH was involved starting from developing of the proposal, tool, giving training, data cleaning, analysis and interpretation. HH drafted the manuscript and revised it critically in the design and analysis of the research. MM was involved mainly in statistical analysis, design and revising in draft of the manuscript and interpretation of the final result. BG was involved in revising the design and analysis of the research. All authors read and approved the final manuscript.
First and foremost we would like to thank to Department of Midwifery for giving us this opportunity to conduct the research. Similarly we would appreciate our friends who helped us in preparation of the proposal, tool and writing of the manuscript.
The data sets used during the current study are available from the corresponding author on reasonable request.
Ethical clearance was obtained from Mekele University, College of Health Sciences Ethical review board. Permission letters were also sought from Tigray Regional education Bureau and the letter was distributed to each selected sites. Written Informed consent from the participants was obtained after clear explanation of the purpose of the study. For those whose age was less than 16 years old consent for participation was taken from their parents. Confidentiality and anonymity was maintained.
Mekele University with other partners was our fund agent to conduct this study. The role of Mekele University was providing appropriate training to develop the proposal, funding money to our data collectors and following how the study is going on, finally our University provides us basic training which was helpful for our study.
Department of Midwifery, Mekele University College of Health Sciences, Tigray, Ethiopia
Abera Haftu, Hadgay Hagos, Mhiret-AB Mehari & Brhane G/her
Abera Haftu
Hadgay Hagos
Mhiret-AB Mehari
Brhane G/her
Correspondence to Abera Haftu.
Haftu, A., Hagos, H., Mehari, MA. et al. Pregnant women adherence level to antenatal care visit and its effect on perinatal outcome among mothers in Tigray Public Health institutions, 2017: cohort study. BMC Res Notes 11, 872 (2018). https://doi.org/10.1186/s13104-018-3987-0 | CommonCrawl |
Individual differences in human frequency-following response predict pitch labeling ability
On the generalization of tones: A detailed exploration of non-speech auditory perception stimuli
Michael Schutz & Jessica Gillard
Auditory local–global temporal processing: evidence for perceptual reorganization with musical expertise
Patrick Susini, Sarah Jibodh Jiaouan, … Emmanuel Ponsot
Musical training refines audiovisual integration but does not influence temporal recalibration
Matthew O'Donohue, Philippe Lacherez & Naohide Yamamoto
Meter enhances the subcortical processing of speech sounds at a strong beat
Il Joon Moon, Soojin Kang, … Kyung Myun Lee
Relationships between vocal pitch perception and production: a developmental perspective
Elizabeth S. Heller Murray & Cara E. Stepp
A tonal-language benefit for pitch in normally-hearing and cochlear-implanted children
Mickael L. D. Deroche, Hui-Ping Lu, … Monita Chatterjee
Domain-specific hearing-in-noise performance is associated with absolute pitch proficiency
I-Hui Hsieh, Hung-Chen Tseng & Jia-Wei Liu
Speech perception is similar for musicians and non-musicians across a wide range of conditions
Sara M. K. Madsen, Marton Marschall, … Andrew J. Oxenham
Combination of absolute pitch and tone language experience enhances lexical tone perception
Akshay R. Maggu, Joseph C. Y. Lau, … Patrick C. M. Wong
Katherine S. Reis1,
Shannon L. M. Heald1,
John P. Veillette1,
Stephen C. Van Hedger2,3 &
Howard C. Nusbaum1
Scientific Reports volume 11, Article number: 14290 (2021) Cite this article
Human behaviour
The frequency-following response (FFR) provides a measure of phase-locked auditory encoding in humans and has been used to study subcortical processing in the auditory system. While effects of experience on the FFR have been reported, few studies have examined whether individual differences in early sensory encoding have measurable effects on human performance. Absolute pitch (AP), the rare ability to label musical notes without reference notes, provides an excellent model system for testing how early neural encoding supports specialized auditory skills. Results show that the FFR predicts pitch labelling performance better than traditional measures related to AP (age of music onset, tonal language experience, pitch adjustment and just-noticeable-difference scores). Moreover, the stimulus type used to elicit the FFR (tones or speech) impacts predictive performance in a manner that is consistent with prior research. Additionally, the FFR predicts labelling performance for piano tones better than unfamiliar sine tones. Taken together, the FFR reliably distinguishes individuals based on their explicit pitch labeling abilities, which highlights the complex dynamics between sensory processing and cognition.
Research on auditory object perception typically focuses on the cortical networks that organize the recognition process. Whether conceived of as a dual pathway1 or focused on pattern classification2, the theoretic framing is based on an ascending auditory recognition system in which frequency specific encoding in primary auditory cortex from the eighth nerve is increasingly refined in temporal cortex for abstract sound category classification and recognition. Much of the research on cortical auditory processing suggests that the site of auditory long-term memory and thus the factors that might influence representation and recognition reside in a cortical network3. This suggests that while subcortical mechanisms may be important in the ascending auditory pathway, given that these mechanisms operate below cortical memory formation and storage, they are involved in neurally-encoded auditory signal refinement and transmission but not specifically conditioned by experience.
However, research by Kraus and colleagues has suggested a very different view of the functional role of the subcortical ascending auditory system in perception. For example, their research has shown that musical expertise modifies the auditory coding of pitch in a way that benefits learning tone language patterns4. In this research, group differences in musical experience are related to the frequency-following response (FFR) for speech stimuli as well as music and thus have generalized beyond the specific context of experience. Moreover, they argue that the group difference in the auditory brainstem response (ABR) due to musical training predicts how the groups learn. While it is unclear if there is descending cortical control over the brainstem response that sharpens it, or whether there is experiential tuning of the FFR from the bottom-up, it is important that by some mechanism, the ascending auditory pathway is not just a passive signal transmission line, but it is changed in processing by experience. Indeed, there is now substantial research showing that experience can alter encoding in the FFR substantially5,6,7,8, even after a relatively short period of training9.
However, it is still not clear whether the observed experience-based changes in the FFR are reflected in behavior. Certainly, if auditory encoding increases the fidelity of the neural representation of frequency, frequency-based auditory performance should improve. Musacchia et al.10 observed that neural responses attributed to the brainstem, including the FFR, correlated with scores in certain musical skill tasks (e.g. timbre discrimination). Moreover, Marmel et al.11 found that aspects of the FFR predict the ability to discriminate between pitches in a forced-choice task. Coffey et al.12 found that individual differences in the FFR relate to pitch perception for tones with a missing fundamental frequency. Carcagno and Plack9 found FFR changes following training in a pitch discrimination task, but the observed changes in FFR strength were not specific to stimuli that shared relevant characteristics with the trained stimuli, and correlations between FFR strength and performance metrics were nonsignificant. While these studies support the notion that FFR features seem to relate to individual differences in perceptual acuity, the extent to which plasticity in early auditory structures supports cognitive abilities that are critical to behavior, such as categorization, remains an open question.
Absolute pitch (AP) or "perfect pitch" is the relatively rare ability to label a musical note without the aid of a reference note13 and can provide a model system for investigating individual differences in the relationship between auditory encoding and human performance. Given that the spectral structure of the FFR suggests that pitch information is successfully transferred from the cochlea to the central nervous system in all listeners14, it may be surprising that most humans are unable to easily utilize that information for the categorization of isolated notes. In contrast, relative pitch perception (categorizing notes in relation to other notes) is the norm among musicians. Absolute pitch possessors' tuning standards can even be shifted after listening to "detuned" music that maintains relative pitch cues15,16. The presumed rarity of AP should be striking, as it is comparable to only being able to classify colors by their relationship to other colors and not with consistent labels such as "blue." Absolute pitch has often been used as a model system for understanding the interplay between genetic and experiential factors in the development of stable cognitive-perceptual skills17—this is a largely unexplored parallel to the way in which the scalp-recorded FFR has been used to investigate the role of experience in shaping auditory encoding, something previously thought to be non-plastic. It could be the case that features of spectral encoding in the FFR may vary between listeners who perceive the pitch of notes absolutely rather than in reference to other notes, supporting the different priorities of categorical processes downstream. Given that AP represents a distinct cognitive skill, the ability to categorize notes, it provides an excellent window into the interplay between low-level encoding, reflected by the FFR, and high-level perceptual categorization.
While AP has traditionally been construed as a dichotomous ability, in which subjects either have or do not have AP17,18, recent evidence has suggested that AP ability exists along a spectrum, where AP ability is best described as a continuously distributed variable19. While there is sizable variance in pitch labelling ability in the general population20, variables that predict continuous variation in absolute pitch perception ability are largely unknown and generally viewed as a consequence of cognitive factors rather than auditory ability21. The aim of the present study, then, is to investigate the extent to which individual differences in the FFR, reflecting low-level neural auditory encoding of sounds, predicts variation in pitch labelling ability, a higher-level cognitive process.
Behavioral results
There was a reasonable spread of performance on pitch performance for sine tones for both self-reported AP possessors (M = 0.554, SD = 0.163) and other musicians (M = 0.212, SD = 0.0960), as well as for piano tones (self-reported AP possessors: M = 0.984, SD = 0.0165; other musicians: M = 0.294, SD = 0.199). See Fig. 1A for a visualization of how the scores relate to one another. The distribution of average pitch labeling ability was approximately M = 0.769, SD = 0.0814 for self-reported AP possessors and M = 0.253, SD = 0.134 for other musicians. Performance on the pitch adjustment task (measures auditory working memory precision by requiring participants to hold in mind a target note for some period of time prior to manually adjusting the final tone to match the target) for self-reported AP possessors was M = 2.978, SD = 2.507, and M = 3.311, SD = 0.822 for other musicians (see Fig. 1B). Finally, just-noticeable difference (JND) task (assesses one's ability to behaviorally discriminate between two tones of varying frequency) performance for self-reported AP possessors was M = 0.849, SD = 0.0715, and M = 0.782, SD = 0.0918 for other musicians (see Fig. 1C).
Spread of Behavioral Data. Individual data points are provided for individual subjects. Red circles represent individuals who self-report as an AP possessor, while turquoise triangles represent other musicians. (a) Comparison of performance on the AP sine tone conservative measure compared to performance on the AP piano tone conservative measure. (b) Performance on the pitch adjustment task. (c) Performance on the just-noticeable-difference task.
While previous research has found that there is a positive relationship between tonal language experience and AP ability22, we did not find such a relationship here for both the AP piano tone conservative measure (t(11.1) = 0.55, p = 0.59) and the AP sine tone conservative measure (t(10.5) = 0.74, p = 0.48). We also found no significant difference between subjects who identified their primary instrument as fixed-pitch and not fixed-pitch on both performance on the AP piano tone conservative measure (t(9.7) = − 0.50, p = 0.63), and AP sine tone conservative measure (t(9.3) = − 0.66, p = 0.53). In other words, effects reported in past research—such as that lessons on piano or other fixed-pitch instruments enhance AP abilities23 or that personal musical histories are reflected by individual performance on absolute pitch recognition tasks24—are not significantly present in our sample.
Electrophysiology results and predictive modeling
The FFR to the piano tone (r = 0.26, t(999) = 31.49, p = 9.18e-152) and the FFR to the unfamiliar complex tone (r = 0.27, t(999) = 31.91, p = 1.31e-154) both predict pitch-labelling performance better than chance, but not significantly differently from one another (t(1994.81) = − 1.19, p = 0.234). Both the piano tone FFR (t(1875.59) = 38.81, p = 2.42e-242) and complex tone FFR (t(1840.56) = 39.16) perform significantly better than the speech-evoked FFR (r = − 0.15), which performs significantly worse than chance (t(999) = − -22.71, p = 2.29e-92).
The Lasso regression yielded the following sparse models, reported with regression coefficients in normalized units for easy comparison across models. Note, in Eq. (3), that the Lasso regression selected harmonics near the formant frequencies of the spoken /da/ to include in the model; while this is encouraging with respect to the Lasso technique picking out relevant predictors, the speech model does not perform above chance, so we caution against attempting to interpret the presence or absence of particular parameters in the model.
$$ {\text{Complex}}\;{\text{tone}}:\quad \hat{y}_{{logit}} = 6.7 \times 10^{{ - 18}} - 0.33F_{0} + 0.017H_{5} $$
$$ {\text{Piano}}\;{\text{Tone}}:\quad \hat{y}_{{logit}} = ~ - 5.1 \times 10^{{ - 18}} - 0.063F_{0} - 0.45H_{1} + 0.28H_{4} $$
$$ {\text{Speech}}:\quad \hat{y}_{{logit}} = ~1.9 \times 10^{{ - 17}} + 0.15F_{0} - 0.021H_{6} + 0.022H_{{12}} $$
The piano tone FFR predicts AP classification performance for both piano tones (r = 0.29, t(999) = 31.11, p = 4.11e-149) and sine tones (r = 0.08, t(999) = 12.26, p = 2.69e-32). However, the model does predict significantly better on piano tone performance (t(1729.47) = 19.22, p = 8.70e-75), suggesting a more specific effect of auditory encoding on pitch classification ability.
$$ {\text{Piano}}\;{\text{Tones}}:\quad \hat{y}_{{logit}} = - 3.3 \times 10^{{ - 17}} - 0.013F_{0} - 0.46H_{1} - 0.0044H_{3} + 0.25H_{4} $$
$$ {\text{Sine}}\;{\text{Tones}}:\quad \hat{y}_{{logit}} = 4.4 \times 10^{{ - 17}} - 0.089H_{1} ~ + ~0.0012H_{4} $$
The frequency-following responses to the piano tone predicts AP performance better than the behavioral measures (age of music onset, tonal language experience, pitch adjustment and just-noticeable-difference scores) are able to (t(1980.05) = − 16.22, p = 1.16e-55), with the latter only performing slightly, albeit significantly, above chance (r = 0.09, t(999) = 11.69, p = 1.06e-29). Notably, combining the behavioral and electrophysiological predictors (r = 0.21) yields a model that is worse than that based on only electrophysiological predictors (t(1982.98) = − 4.52, p = 6.55e-06), but does do better than the behavioral data alone (t(1997.86) = − 12.23, p = 3.08e-33). This suggests that the behavioral measures contain little information about pitch labelling ability that is not already captured by the FFR. Interestingly, the behavioral-only model (see Eq. 6) removed all predictors except for the just-noticeable-difference score, a measure of perceptual discrimination ability, indicating that the other behavioral measures do not provide additional information about pitch labelling ability.
$$ {\text{Behavioral}}:\quad \hat{y}_{{logit}} = 8.7 \times 10^{{ - 18}} + 0.023JND $$
$$ {\text{Combined}}:\quad \hat{y}_{{logit}} = - 5.2 \times 10^{{ - 17}} - 0.39H_{1} + 0.18H_{4} + 0.20JND - 0.0038age\_onset~ $$
$$ {\text{FFR}}:\quad \hat{y}_{{logit}} = - 5.1 \times 10^{{ - 18}} - 0.063F_{0} - 0.45H_{1} + 0.28H_{4} $$
Though previous work has shown that individual changes in the FFR can arise as a result of past experience, such as musical training, the exact relationship between the FFR and behavior has remained ambiguous. Individual differences in the FFR have been related to performance on certain perceptual discrimination tasks12 and such differences have been shown to emerge following training in such a task9, but these individual differences were not specific to task-relevant spectral features and studies that relate auditory encoding to performance rarely compare the magnitude of FFR differences across stimuli from different domains. This omission is particularly problematic, as many known FFR effects persist across auditory domains; for example, musical training seems to impact the FFR encoding of speech sounds, leading some researchers to argue that experience-dependent changes in the FFR are generally domain-nonspecific25.
The present study provides compelling evidence for the domain specificity of individual differences in FFR spectral features. While our data replicate previous findings that FFRs to domain nonspecific stimuli can predict scores in an auditory task, as the predictive performance of our model deviates from chance for all stimuli, we find robust differences between the predictive power of FFRs to different stimuli. We find that the FFRs to tones predicts performance substantially better than to speech stimuli, seemingly corresponding to the subjects' experience attending to the pitch of notes regardless of the familiarity of their timbres. In contrast, the FFR to the piano tone, a familiar timbre, does not seem to predict pitch-labelling ability for piano tone stimuli any better than the FFR to the complex tone, so instrument-specific advantages in brainstem encoding do not seem to account for well documented own-instrument advantage effects in the AP literature23,24. Our subjects do, however, generally perform better on the piano tones than on the sine tones, consistent with past literature, so the observed timbre-familiarity advantage may originate from later auditory processing or during subsequent categorization.
Importantly, we find that the FFR to the piano tone predicts subjects' ability to label the pitch of piano tones significantly better than it does the pitch of sine tones. This finding points toward a view of FFR plasticity as a mechanism that can support domain-specific auditory skills above and beyond the domain-general effects previous researchers have observed.
Notably, individual differences in early sensory encoding, as reflected by the FFR, are able to predict continuous variance in AP ability. Since the variation in pitch labelling ability has largely gone unexplained since researchers have argued that AP should be considered as a graded (rather than dichotomous) ability20 this finding is novel. It has long remained an open scientific question why humans can place some types of stimulus characteristics into stable, barely changing categories (such as color) but less so others (such as pitch); understanding the relationship between individual differences in low-level sensory coding and in the higher-level cognitive ability to consistently categorize perceptual stimuli promises to shed light on broader theories of semantic memory, concepts, and categories26.
It is tempting to conclude that the mechanism for our observed effect is a difference in stimulus encoding in subcortical structures that covaries with AP ability; indeed, this is how the FFR literature has historically interpreted such results7,10,25. Of course, our ability to draw definitive conclusions from our results is limited by the nature of a between-subject design in noninvasive electrophysiology studies using correlation. A predictive relationship between the scalp-recorded FFR and AP ability need not be caused by a true change in auditory encoding in the FFR's source structures; since part of the FFR is thought to originate subcortically, any anatomical difference between those far-field sources and the recording electrode that covaries with AP ability27 could mediate the observed effect by altering volume conduction through the brain. However, such an anatomical difference would affect the scalp recorded FFR similarly for different stimuli, and we observe robust differences in predictive power between stimuli. Individual differences in brain anatomy could conceivably have a compounding influence on some true effect if, for example, changes in white matter density or microstructure, which may affect volume conduction, support higher fidelity phase locking to the acoustic stimulus. While this situation would suggest some true effect exists, it makes estimating the effect size from a scalp-recording tenuous, since the true effect could be correlated with a confounding factor. Lastly, since the FFR is now thought to originate from a distributed network of cortical and subcortical sources rather than solely from the auditory brainstem as previously thought28, a differential contribution of cortical sources, close to the recording electrode, and subcortical sources could account for any attenuation or amplification of power in the FFR. It seems difficult to tease apart this alternative from the traditional explanation with the minimalist recording montage used in most FFR experiments, but this distinction may be addressable in future research using high density electrode montages29. Nonetheless, a shift in the relative contribution of different source regions, rather than an overall change in phase-locking to the stimulus, would still speak to the overall hypothesis that differences in early auditory encoding support higher-level cognitive abilities in a domain-specific manner.
The fields of FFR research and AP research share a common interest in how long- and short-term experience interact with less malleable aspects of nervous system development, such as genetics, to alter the encoding of sound. While the mechanisms of AP have traditionally been construed as cognitive, the present study suggests that real variance in pitch labelling ability may be attributable to low-level sensory encoding differences, as reflected in the FFR30. Conversely, individual differences in the FFR appear to be much more dependent upon the development of specialized skills and the particular domain of auditory experience than previously thought. As many fields in the behavioral sciences are now discovering, it may not be possible to fully understand cognition or perception without considering their dynamic interaction.
Thirty-five individuals participated in the experiment, four subjects were removed (one for non-compliance on tasks, one for hardware issues at the time of experimentation, one for failure to meet hearing criteria, and one for a pre-existing neurological condition). Absolute pitch possessors (N = 16) and musically matched subjects (N = 15) were recruited from the Chicagoland area. By including subjects that are expected to show a range of pitch perception ability, we hope that our sample is representative of the population distribution of absolute pitch ability described by Van Hedger et al.20. Of the 31 remaining subjects, which included both males and females (16 females) with varying amounts of musical training, the average age was M = 21.6, SD = 3.01. The self-reported absolute pitch possessors reported to play an instrument for M = 15.88, SD = 3.77, years, while the other musicians reported to play an instrument for M = 14.73, SD = 4.48, years (t(27) = 0.765, p = 0.451). Three self-reported absolute pitch possessors and seven musically matched subjects were tonal language speakers. 13 self-reported absolute pitch possessors and 10 musically matched subjects identified their primary (synonymous here with first) instrument as being a fixed-pitch instrument (piano).
The study procedure was approved by the Social and Behavioral Sciences Institutional Review Board at the University of Chicago, and all research was performed in accordance with such guidelines. Informed consent was received from each subject.
FFR acquisition and preprocessing protocol
All recordings were conducted in a soundproof semi-electrically shielded booth. Brainstem electroencephalography recordings were collected while participants were presented with auditory stimuli that were presented binaurally via fitted earbuds attached to Etymotic Research ER-3a insert tube phones at 65–75 dB. Alternating polarity presentation was used to reduce the presence of the cochlear microphonic (CM) in the recorded signal. Each stimulus type was presented 3000 times, 1500 times for each polarity. During recording participants were allowed to watch a silent film, as is common for ABR studies31. Stimuli were presented using Psychtoolbox (Matlab Psychtoolbox-3; psychtoolbox.org).
Horizontal montaging32 was used using Ag–AgCl electrodes. Electrode placement included a ground electrode on the center of the forehead, an active electrode placed at Cz, and linked reference electrodes placed on both the left and right mastoid. Impedances from Cz, each mastoid individually, and the mastoids together were taken prior to experimentation, with a maximum of 5 k Ohms allowed. BrainVision PyCorder software (BrainProducts) was used to record brainstem responses with an online filter of 0.1 to 3000 Hz.
Preprocessing in BrainVision Analyzer 2.2.0 proceeded as follows. Filtering parameters were dictated by the properties of the stimuli. The EEG recordings in response to the piano and complex stimuli were bandpass filtered (Butterworth 12 dB octave roll-off) from 100 to 2000 Hz, whereas /da/ stimuli were bandpass filtered from 70 to 2000 Hz. All stimuli had an additional notch filter of 60 Hz applied.
We then applied an absolute threshold detection (± 700 mV) on the recorded audio channel via a Boolean expression that selectively finds the negative and positive peak of the start of a stimulus, and marks whichever occurs first. It is vital to use an absolute threshold rather than solely a positive or negative threshold in order to not correct for phase differences between inverted and non-inverted stimuli. By preserving such phase differences, we are able to shift our analysis to mainly examine the ABR portion of the recorded signal rather than the cochlear microphonic (CM), as the ABR is insensitive to phase differences while the CM is not. Segmentation procedures were dependent on the length of the stimulus. Piano and complex tones were 200 ms in length, and the /da/ stimulus was 80 ms in length. As a result, piano and complex segments were defined to start 50 ms prior to stimulus onset and last 250 ms post stimulus onset, /da/ segments were defined to start -10 ms prior to the stimulus onset and last 120 ms post stimulus onset.
Trials that had been contaminated by unwanted artifacts (those that exceeded a strict amplitude threshold of 35 µV) were removed from the dataset. A baseline correction transformation was performed on the 10 ms preceding the /da/ stimulus, and 50 ms preceding the piano/complex stimuli.
The piano stimulus was sampled from an acoustic piano and produced with Reason software (Propellerhead, Stockholm). The complex tone was generated in Adobe Audition, and the /da/ stimulus was generated by the implementation of a Klatt synthesizer. The fundamental of the complex tone was 207.65 Hz (G#3). The fundamental of the piano tone was 261.63 Hz (C3). The F0 of the /da/ was 100 Hz. The complex tone stimulus had a fundamental frequency of 207.65 Hz, and consisted of the 3rd, 7th, 8th, and 10th harmonics. An F0 of 100 Hz for our speech stimulus was based on prior auditory brainstem work10, and we chose fundamental frequencies for our piano and complex tone stimuli that were in a comfortable middle octave for music listening and is conveniently within the register of most commonly played instruments.
Participants were administered a sixty second hearing screening using a Welch-Allyn Otoscope equipped with an audiometer. Participants had to detect the occurrence of four tones (500, 1000, 2000, and 4000 Hz), which were presented at random intervals to prevent guessing. Participants were also checked via otoscope to make sure their ear canals were free from debris and that their eardrums were intact.
Experimental design and statistical analyses
For each subject, we began the experimental session with several questionnaires, where we assessed their musical experience (Absolute Pitch Questionnaire and Musical Experience Questionnaire) and tonal language experience (Language Experience Questionnaire). Afterwards, participants were screened for normal hearing. (Air conduction thresholds < 40 dB, see Prescreening subsection) We then recorded EEG responses to a piano tone, a complex tone with an unfamiliar timbre, and a spoken /da/. (See Stimuli and FFR Acquisition Protocol subsections, above, for more details and Fig. 2 for stimuli power spectra.).
Power Spectra of Stimuli and of Frequency-Following Responses. The nearest integer frequency to the harmonics of the stimulus is marked on each plot, except for the speech stimulus, in which every other harmonic is marked to avoid visual clutter. The EEG spectra are corrected for 1/f frequency drop-off here for visualization, but uncorrected values were used for analysis.
Then, each subject completed an explicit pitch labelling (AP) assessment. The AP assessment consisted of two different paper-pen AP tests. Both tests presented tones across a range of different octaves. The average score of these two tests is what we refer to here as the AP test score, or pitch labelling ability (see Fig. 3C–E for full distribution of AP test scores, and Fig. 4C,D for the performance distribution broken down by piano and sine AP scores). Presentation of the stimuli was controlled by E-prime software.
Performance of Lasso Regression Models Using FFR to Different Stimuli as Predictors. (a, b) For each model, a correlation between the model's predictions and true AP sine and piano performance was computed on a test set (data points not seen by the model during training) for each of 1000 cross-validation runs as an estimate of how well the model generalizes. See Eqs. (1–3) for final model specifications. (c) Predicted AP sine and piano performance values based on complex tone FFR plotted against actual, observed AP performance. Red dots represent subjects who self-reported as AP possessors. (d) Predicted AP performance values based on piano tone FFR plotted against observed AP performance. Red dots represent subjects who self-reported as AP possessors. (e) Predicted AP performance values based on speech /da/ FFR plotted against observed AP performance. Red dots represent subjects who self-reported as AP possessors.
Predictive Performance of Piano FFR on Sine Tones and Piano Tones separately. (a, b) Correlation between the predicted pitch labelling performances and the true pitch labelling performances on a test set are shown for 1000 cross-validation runs. The FFR to the piano tone predicts pitch labelling performance for piano tones better than it does for sine tones. (c) Predicted pitch labelling performance on the piano tones plotted against actual, observed pitch labelling performance. Red dots represent subjects who self-reported as AP possessors. (d) Predicted pitch labelling performance on the sine tones plotted against observed pitch labelling performance. Red dots represent subjects who self-reported as AP possessors.
Subjects subsequently completed a just-noticeable-difference (JND) assessment, which was used to examine how well participants could behaviorally discriminate between two tones. Tones were presented in four blocks of 20 trials each. A standard 1000 Hz tone was used, and in the first block, one of the notes deviated by 56 cents from the 1000 Hz tone. In the second block, the notes deviated by 28 cents, in the third block the notes deviated by 14 cents, and in the fourth block the notes deviated by seven cents. On half of the trials the two tones presented were the same 1000 Hz tone. For a given trial, participants needed to determine whether the two tones were the same 1000 Hz tone or if they were two different tones. This assessment was also graded on a 100% scale. Individual differences in JND task performance should reflect differences in fine grained pitch processing. This task was administered using E-prime software.
Subjects then performed a pitch adjustment assessment (administered using MATLAB), which was based on a task reported by Heald et al.33. In this task, participants were required to adjust the frequency of a probe sine tone to match a previously presented target sine tone. The target tone was briefly presented (200 ms) and then immediately masked by noise (1000 ms). Following the noise, a secondary tone (200 ms) was presented. The participants were then asked to try to adjust the secondary tone to match the target tone by adjusting the pitch either up or down. Ten target tones were tested from 471.58 Hz (end point − 80 cents B4) to 547.99 Hz (end point + 80 cents C5), across the B4 and C5 categories. Participants either started above or below these categories (i.e., the location of the secondary tone). Participants were able to adjust the probe tone by adjusting the pitch drawn from a stimulus series. They could adjust the probe either by 10 or 20 cent steps. Given the masking of the target tone, matching performance on this task is designed to measure auditory working memory precision, as it is necessary for participants to hold in mind the target note despite the white noise and intermediary adjustment tones. This interpretation of this task is similarly held by Kumar et al.34 and Van Hedger et al.21.
The FFR was computed from the EEG responses as follows. Preprocessing was done using BrainVision Analyzer 2.2.0. (See FFR Acquisition and Preprocessing Protocol subsection above.). This preprocessed data was then exported from BrainVision Analyzer 2.2.0 to .mat files. (All analyses after this point were scripted in MATLAB and in R; all code, from preprocessing to the generation of figures, can be found at https://github.com/apex-lab/ap-ffr.) In order to maintain an equal number of trials for inverted and noninverted stimuli, we randomly subsampled trials from whichever stimulus polarity (inverted or noninverted) had more trials so that, for each subject, we were left with an equal number of trials of each polarity. Then, all remaining trials (of both polarities) were averaged for each subject and stimulus type (piano, complex tone, speech) to obtain the FFR. This is frequently recommended in the FFR literature35 for the purpose of averaging out any stimulus artifact and attenuating the contribution of the cochlear microphonic (see FFR Acquisition and Preprocessing Protocol subsection). Next, we applied a Hanning taper to the window corresponding to the duration of each stimulus and computed the power spectrum of each FFR over that window. We then exported the power of each subject's FFR at each harmonic of its eliciting stimulus (up to 1500 Hz, see Fig. 2) for analysis in R. (These files are available for researchers who wish to reproduce our analyses.)
We then assessed whether the FFRs elicited by stimuli from a variety of auditory domains (piano, speech, and a novel complex periodic signal) were predictive of pitch labelling performance on the score (accuracy) of both AP tests. The reason for focusing on predictive performance, rather than relying on null hypothesis significance testing for inference, is that in principle all the harmonics of a stimulus (and thus the FFR) contain information about pitch. In order to avoid making any assumptions about which harmonics to include but not allow our analysis to suffer from problems inherent to high-dimensional regression (the "curse of dimensionality," Friedman, 1997)36, we employed the Lasso regression technique to fit sparse generalizable models to our data. We describe the Lasso regression technique in some detail below in the Model Fitting subsection below.
First, we fit separate models for each FFR eliciting stimulus, predicting the pitch labelling ability across both AP tests (sine and piano tones). Pitch labelling ability is operationalized by awarding 1 point for correctly labelling a note and 0.75 points if only a semitone off, then dividing total points awarded by the number of trials. This is considered a relatively conservative measure, specifically with regard to identifying intermediate AP possessors, and has been used by a number of influential studies18,37,38. However, alternative measures of AP ability, such as mean absolute deviation (MAD) in semitones and raw accuracy, are provided for interested researchers in our open dataset. (Though we found the reported results were robust to the operationalization of AP.) Since this measure is [0, 1] bounded, we logit transform it before fitting the model. For each model, we compute the correlation between model predictions and true pitch labelling ability on a test set for each of 1000 cross-validation runs. We then apply the Fisher z-transformation to these r values (since they would otherwise be [0, 1] bounded and therefore non-normal) and compare each model's performance to chance (r = 0) with a one-sample t-test. We also compare the three models to one another to test whether the auditory domain of the FFR eliciting stimulus matters when predicting pitch labelling performance. Full distributions of raw and transformed r values are reported (Fig. 3), and regression coefficients (fit on the full dataset) are reported in normalized units for easy comparison between models.
In order to assess the evidence of a specific effect of low-level auditory encoding on task performance, we then separately fit models predicting pitch labelling performance on sine tones and pitch labelling performance on piano tones from the piano elicited FFRs. We compared these models to chance and to each other using t tests on the z-transformed r values from 1000 cross-validation runs. The full distribution of r values is reported in Fig. 4.
In total, we report 12 statistical tests. In order to control for multiple comparisons, we apply a Bonferroni correction, resulting in a new significance threshold of α = 0.00417 against which the reported p-values should be compared.
While ordinary least squares regression finds regression coefficients β to minimize the loss function \(SSE\left( \beta \right) = \mathop \sum \limits_{i} \left( {\hat{y}_{i} - y_{i} } \right)^{2}\), where \(\hat{y}\) is what the model predicts, Lasso regression minimizes \(L\left( \beta \right) = SSE\left( \beta \right) + ~\lambda \mathop \sum \limits_{j} \left| {\beta _{j} } \right|\). The addition of a penalty term for the size of β means that the fit model will only include nonzero values of β (regression coefficients) if the increase in the penalty term is offset by enough of a decrease in the sum of squares error (SSE). In order to ensure that results are generalizable, we pick λ (which determines how much the model will "care" about the penalty term) to maximize model performance on data that the model never saw during training (a hold-out set). This ensures that the model only includes predictor variables that robustly help it predict new data (the predictors that we can expect to generalize outside of our particular sample to the target population), setting the coefficients for all other predictors to zero. In exchange for performing near-optimal variable selection for us, Lasso regression does not provide a p-value for each remaining regression coefficient, but we can derive a p-value for the full model by comparing model performance on a test set (more data points the model did not see during training) to chance. This p-value, arguably, is more meaningful than those traditionally reported since it is derived from a measure of how well a model generalizes to new data, while p-values for ordinary linear regression are more prone to reach significance just because of noise within the sample. For more detail on the theory and practical implementation of the Lasso, see James et al.39.
Each time we fit a model we are actually fitting many models. First, we divide the data randomly into a training set (2/3 of the data) and a test set (the remaining 1/3 of the data). Next, we train models using many different values of λ (from 0.01 to \({10}^{10}\)) and select the model that minimizes the leave-one-out cross-validation score over the training set. We then compute the performance of this model on the test set (picking the metric of our choosing as a "cross-validation score," in our case \(r = {\text{corr}}\left( {\hat{y},~y} \right)\)) as a measure of how well the model predicts new data.
If using the cross-validation score for inference, one has to be concerned about whether performance on the test set may have been good (or bad) by mere chance, and as it happens, the random choice of test set can result in dramatically variable cross-validation scores (see Figs. 3, 4, 5). To account for this variability, we repeat this whole cross-validation procedure 1000 times for each model, each with a new, random training-test split, and report the full distribution of r values generated.
Predictive Performance of Behavioral Tests, Piano FFR, and a Combined Model on Pitch Labelling Performance. (a, b) Correlation between the predicted pitch labelling performances and the true pitch labelling performances on a test set are shown for 1000 cross-validation runs. The FFR to the piano tone predicts pitch labelling performance better than the behavioral tests as well as the combined model. (c) Predicted pitch labelling performance based on the behavioral tests plotted against actual, observed pitch labelling performance. Red dots represent subjects who self-reported as AP possessors. (d) Predicted pitch labelling performance based on a combined model of both behavioral tests and the piano FFR plotted against observed pitch labelling performance. Red dots represent subjects who self-reported as AP possessors. (e) Predicted pitch labelling performance based on the piano FFR plotted against observed pitch labelling performance. Red dots represent subjects who self-reported as AP possessors.
The analysis code is available at https://github.com/apex-lab/ap-ffr, and the data used in our analyses is available on Open Science Framework with https://doi.org/10.17605/OSF.IO/HRCVS.
Rauschecker, J. P. & Scott, S. K. Maps and streams in the auditory cortex: Nonhuman primates illuminate human speech processing. Nat. Neurosci. 12, 718–724 (2009).
Deutsch, D. Auditory Pattern Recognition (Wiley, 1986).
Huang, Y., Matysiak, A., Heil, P., König, R. & Brosch, M. Persistent neural activity in auditory cortex is related to auditory working memory in humans and nonhuman primates. Elife 5, e15441 (2016).
Kraus, N. & Chandrasekaran, B. Music training for the development of auditory skills. Nat. Rev. Neurosci. 11(8), 599–605 (2010).
Krishnan, A., Xu, Y., Gandour, J. & Cariani, P. Encoding of pitch in the human brainstem is sensitive to language experience. Cogn. Brain Res. 25(1), 161–168 (2005).
Song, J. H., Skoe, E., Wong, P. C. & Kraus, N. Plasticity in the adult human auditory brainstem following short-term linguistic training. J. Cogn. Neurosci. 20(10), 1892–1902 (2008).
Kraus, N., Skoe, E., Parbery-Clark, A. & Ashley, R. Experience-induced malleability in neural encoding of pitch, timbre, and timing: Implications for language and music. Ann. N. Y. Acad. Sci. 1169, 543 (2009).
Tzounopoulos, T. & Kraus, N. Learning to encode timing: Mechanisms of plasticity in the auditory brainstem. Neuron 62(4), 463–469 (2009).
Carcagno, S. & Plack, C. J. Subcortical plasticity following perceptual learning in a pitch discrimination task. J. Assoc. Res. Otolaryngol. 12(1), 89–100 (2011).
Musacchia, G., Strait, D. & Kraus, N. Relationships between behavior, brainstem and cortical encoding of seen and heard speech in musicians and non-musicians. Hear. Res. 241(1–2), 34–42 (2008).
Marmel, F. et al. Subcortical neural synchrony and absolute thresholds predict frequency discrimination independently. J. Assoc. Res. Otolaryngol. 14(5), 757–766 (2013).
Coffey, E. B., Colagrosso, E. M., Lehmann, A., Schönwiesner, M. & Zatorre, R. J. Individual differences in the frequency-following response: Relation to pitch perception. PLoS ONE 11(3), e0152374 (2016).
Takeuchi, A. H. & Hulse, S. H. Absolute pitch. Psychol. Bull. 113(2), 345 (1993).
Gockel, H. E., Carlyon, R. P., Mehta, A. & Plack, C. J. The frequency following response (FFR) may reflect pitch-bearing information but is not a direct representation of pitch. J. Assoc. Res. Otolaryngol. 12(6), 767–782 (2011).
Hedger, S. C., Heald, S. L. & Nusbaum, H. C. Absolute pitch may not be so absolute. Psychol. Sci. 24(8), 1496–1502 (2013).
Van Hedger, S. C., Heald, S. L., Uddin, S. & Nusbaum, H. C. A note by any other name: Intonation context rapidly changes absolute note judgments. J. Exp. Psychol. Hum. Percept. Perform. 44(8), 1268 (2018).
Zatorre, R. J. Absolute pitch: A model for understanding the influence of genes and development on neural and cognitive function. Nat. Neurosci. 6(7), 692–695 (2003).
Athos, E. A. et al. Dichotomy and perceptual distortions in absolute pitch ability. Proc. Natl. Acad. Sci. 104(37), 14795–14800 (2007).
Bermudez, P. & Zatorre, R. J. A distribution of absolute pitch ability as revealed by computerized testing. Music Percept. 27(2), 89–101 (2009).
Van Hedger, S. C., Veillette, J., Heald, S. L. M. & Nusbaum, H. C. Revisiting discrete versus continuous models of human behavior: The case of absolute pitch. PLoS ONE 15(12), e0244308 (2020).
Van Hedger, S. C., Heald, S. L., Koch, R. & Nusbaum, H. C. Auditory working memory predicts individual differences in absolute pitch learning. Cognition 140, 95–110 (2015).
Deutsch, D., Henthorn, T. & Dolson, M. Absolute pitch, speech, and tone language: Some experiments and a proposed framework. Music Percept. 21(3), 339–356 (2004).
Vanzella, P. & Schellenberg, E. G. Absolute pitch: Effects of timbre on note-naming ability. PLoS ONE 5(11), e15449 (2010).
Bahr, N., Christensen, C. A. & Bahr, M. Diversity of accuracy profiles for absolute pitch recognition. Psychol. Music 33(1), 58–93 (2005).
Wong, P. C., Skoe, E., Russo, N. M., Dees, T. & Kraus, N. Musical experience shapes human brainstem encoding of linguistic pitch patterns. Nat. Neurosci. 10(4), 420–422 (2007).
Levitin, D. J. & Rogers, S. E. Absolute pitch: Perception, coding, and controversies. Trends Cogn. Sci. 9(1), 26–33 (2005).
Keenan, J. P., Thangaraj, V., Halpern, A. R. & Schlaug, G. Absolute pitch and planum temporale. Neuroimage 14(6), 1402–1408 (2001).
Coffey, E. B. et al. Evolving perspectives on the sources of the frequency-following response. Nat. Commun. 10(1), 1–10 (2019).
Article MathSciNet CAS Google Scholar
Bidelman, G. M. Multichannel recordings of the human brainstem frequency-following response: scalp topography, source generators, and distinctions from the transient ABR. Hear. Res. 323, 68–80 (2015).
Ross, D. A., Gore, J. C. & Marks, L. E. Absolute pitch: Music and beyond. Epilepsy Behav. 7(4), 578–601 (2005).
Parbery-Clark, A., Tierney, A., Strait, D. L. & Kraus, N. Musicians have fine-tuned neural distinction of speech syllables. Neuroscience 219, 111–119 (2012).
Hood, L. Clinical Applications of the Auditory Brainstem Response (Singular, 1998).
Heald, S. L., Van Hedger, S. C. & Nusbaum, H. C. Auditory category knowledge in experts and novices. Front. Neurosci. 8, 260 (2014).
Kumar, S. et al. Resource allocation and prioritization in auditory working memory. Cogn. Neurosci. 4(1), 12–20 (2013).
Skoe, E. & Kraus, N. Auditory brainstem response to complex sounds: A tutorial. Ear Hear. 31(3), 302 (2010).
Friedman, J. H. On bias, variance, 0/1—loss, and the curse-of-dimensionality. Data Min. Knowl. Discov. 1(1), 55–77 (1997).
Baharloo, S., Johnston, P. A., Service, S. K., Gitschier, J. & Freimer, N. B. Absolute pitch: an approach for identification of genetic and nongenetic components. Am. J. Hum. Genet. 62(2), 224–231 (1998).
Baharloo, S., Service, S. K., Risch, N., Gitschier, J. & Freimer, N. B. Familial aggregation of absolute pitch. Am. J. Hum. Genet. 67(3), 755–758 (2000).
James, G., Witten, D., Hastie, T. & Tibshirani, R. An Introduction to Statistical Learning (Springer, 2013).
Department of Psychology, The University of Chicago, Chicago, IL, USA
Katherine S. Reis, Shannon L. M. Heald, John P. Veillette & Howard C. Nusbaum
Department of Psychology, Huron University College, London, ON, Canada
Stephen C. Van Hedger
Brain and Mind Institute, Western University, London, ON, Canada
Katherine S. Reis
Shannon L. M. Heald
John P. Veillette
Howard C. Nusbaum
S.M.L.H., S.C.V., and H.C.N. conceived of the presented idea; K.S.R. processed the experimental data; K.S.R. and J.P.V. analyzed the data; S.M.L.H. and H.C.N. supervised the project; K.S.R. and J.P.V. drafted the manuscript; all authors contributed to and approved the final manuscript.
Correspondence to Katherine S. Reis.
Reis, K.S., Heald, S.L.M., Veillette, J.P. et al. Individual differences in human frequency-following response predict pitch labeling ability. Sci Rep 11, 14290 (2021). https://doi.org/10.1038/s41598-021-93312-7 | CommonCrawl |
\begin{definition}[Definition:Ambivalent Group]
Let $G$ be a group.
Then $G$ is '''ambivalent''' {{iff}} every element of $G$ is conjugate to its inverse:
:$\forall g \in G : \exists h \in G : h g h^{-1} = g^{-1}$
That is, {{iff}} every element of $G$ is real.
\end{definition} | ProofWiki |
A web server for analysis, comparison and prediction of protein ligand binding sites
Harinder Singh1,
Hemant Kumar Srivastava1 and
Gajendra P. S. Raghava1, 2Email author
© Singh et al. 2016
One of the major challenges in the field of system biology is to understand the interaction between a wide range of proteins and ligands. In the past, methods have been developed for predicting binding sites in a protein for a limited number of ligands.
In order to address this problem, we developed a web server named 'LPIcom' to facilitate users in understanding protein-ligand interaction. Analysis, comparison and prediction modules are available in the "LPIcom' server to predict protein-ligand interacting residues for 824 ligands. Each ligand must have at least 30 protein binding sites in PDB. Analysis module of the server can identify residues preferred in interaction and binding motif for a given ligand; for example residues glycine, lysine and arginine are preferred in ATP binding sites. Comparison module of the server allows comparing protein-binding sites of multiple ligands to understand the similarity between ligands based on their binding site. This module indicates that ATP, ADP and GTP ligands are in the same cluster and thus their binding sites or interacting residues exhibit a high level of similarity. Propensity-based prediction module has been developed for predicting ligand-interacting residues in a protein for more than 800 ligands. In addition, a number of web-based tools have been integrated to facilitate users in creating web logo and two-sample between ligand interacting and non-interacting residues.
In summary, this manuscript presents a web-server for analysis of ligand interacting residue. This server is available for public use from URL http://crdd.osdd.net/raghava/lpicom.
This article was reviewed by Prof Michael Gromiha, Prof Vladimir Poroikov and Prof Zlatko Trajanoski.
Ligand-amino acid interaction analysis
Two-sample logo
Propensity-based analysis
Amino acid composition based analysis
Physicochemical property-based analysis
Ligands play a variety of roles in the regulation and expression of proteins. Currently, PDB has thousands of ligands and the majority of them bound non-covalently to various proteins. The non-covalent ligand binding occurs by intermolecular forces like hydrogen bonds, ionic bonds, hydrophobic-hydrophobic interaction, van der Waals forces, etc. 3D shape of the protein gets altered as a result of the ligand binding. These changes in the conformational state of the protein may activate or inhibit some specific function of the protein. Various methods have been developed to predict the binding affinity of ligands [1–9]. Many databases are also developed to summarize binding affinity of a diverse class of ligands [10, 11] or specific class of ligands [12, 13].
Ligands have high or low binding with specific amino acids depending on various factors (e.g. shape, charge, surface area). ATP has significantly higher interaction with glycine and least interaction with leucine [14]. Various studies have been performed to understand the binding behaviour of ligands with the amino acids in a protein. Many machine learning methods have also been developed to predict the preference of interacting and non-interacting amino acids with various ligands [15–24].
However, binding preference analysis between different ligands and protein was not carried out on a large dataset. Considering this, we performed a rigorous study to understand the binding behaviour of various ligands with different amino acids. This information can be used to either enhance or diminish the binding strength of the given ligand by mutating unfavourable residue with preferred residue at the site of binding. In addition, we developed a web-based platform for the analysis of amino acid preference for all the ligand present in PDB.
Clustering of nucleotides based on their binding sites
The nucleotides are clustered to understand similarity or dissimilarity in their binding sites. In this study only major nucleotides (e.g., adenine, guanine, cytosine, uracil, thymine monophosphate) are clustered based on residues preferred in their binding sites.
Propensity-based clustering
The propensity score of nucleotides interacting residues (\( {\mathbf{RP}}_{\mathbf{i}}\Big) \) is calculated using equation 3 and the propensity-based Euclidean distance (PED p,q ) between nucleotides is calculated using equation 5. The PED between each pair of nucleotides is used to construct a distance matrix, which was further used for clustering these nucleotides. Figure 1 depicts the propensity score based clustering of all the nucleotides. The propensity score of GMP, CDP and UTP ligands are negligible and hence these ligands are not included in the analysis. We defined preference of a residue in ligand binding site based on its propensity score, if score of a residue is lower than 5 than we called it low preferred residue. Similarly, we called a residue moderate if it has propensity score between 5 to 12; high if score is more than 12. As shown in Fig. 1, for most of nucleotides propensity score for different type of residues is low or moderate. It is clear from Fig. 1 that ATP and ADP nucleotides are highly similar in term of residue preferred in their binding sites as Euclidean distance is minimum. Similarly, AMP and UDP nucleotides are clustered together; while CMP nucleotide is out of cluster. The NAD and FAD nucleotides binding/interacting residues are also similar as they fall in same cluster.
Shows residue-wise propensity score for different nucleotides (left) and clustering of nucleotides based on propensity score (right)
Figure 1 indicates the low propensity score of W,Y,H,Q,N,K,D,L and F amino acids for CMP nucleotide. Based on propensity score one may conclude that CMP has a strong preference for amino acid S,C,R,I and G amino acids. Similarly, UDP binding sites are dominated by C,R,H,I,E and V amino acids, as the propensity of these residues is high for UDP. The interaction of CTP with most of the amino acids also falls in the low category. The interaction of GTP ligand with H,K and G amino acids falls in the moderate category while amino acid M shows negligible interaction with this ligand. Rest 16 amino acids show low interaction with GTP ligand. Interestingly ADP ligand shows low interaction with most of the amino acids except G and T where moderate interaction is observed. W,H,K,D,G and T amino acids show moderate interaction with ATP ligand and rest 14 amino acids show low interaction with this ligand. Amino acids show similarity generally on the basis of their category e.g. charged amino acids (K and D), hydrophobic amino acids (L, A and V) and polar amino acids (T and S) show similarity up to some extent. Clearly, there is no similarity between C & I, W & V etc. amino acids.
Clustering of nucleotides using physicochemical property-based
The physicochemical property based composition of nucleotides binding/interacting residues, \( {\mathbf{PC}}_{\mathbf{i}} \) is calculated using equation 6. The physicochemical composition based Euclidean distance (PCED p,q ) between nucleotides is calculated using equation 7. The PCED p,q of each nucleotides is used to construct a distance matrix, which was further used for clustering. Figure 2 shows the clustering of different nucleotides based on physicochemical properties of residues in their binding sites or interacting residues. In this ATP and ADP nucleotides falls in same cluster, it means ATP and ADP interacting residues have similar physicochemical properties. Similar trend was observed for GTP and GDP binding or interacting residues. The next group of nucleotides consist of UDP, AMP and CTP. The UMP nucleotide again occurs as isolated in the overall cluster of nucleotides. As expected the NAD and FAD nucleotides occur in the same group and are most similar with CMP and AVR (average of all ligand interacting amino acids in PDB) ligands. Aromatic and acidic amino acids show similar interactions and form a single group. The interaction of polar and charged amino acids is also similar to each other followed by basic group of amino acids. Non-polar and small amino acids show similarity in the interaction up to some extent. Aliphatic amino acids have similarity with the group of non-polar, small amino acids and are least similar to all other groups. UMP ligand shows low interaction with aromatic and aliphatic amino acids, strong interaction with acidic amino acids and moderate interaction with other groups.
Show property based residue composition of different nucleotides binding sites (left) and clustering of nucleotides based on the physicochemical properties of ligand interacting amino acids
Clustering of carbohydrates based on their binding sites
The propensity score of carbohydrates binding or interacting residues (\( {\mathbf{RP}}_{\mathbf{i}}\Big) \) is calculated using equation 3. In order to compute similarity/distance between different carbohydrates, we compute propensity based Euclidean distance (PED p,q ) between carbohydrates using equation 5. The PED between all pair of carbohydrate was compute for distance matrix, which was further used for clustering. Figure 3 shows the clustering of carbohydrates based on the propensity score of interacting amino acids. The interacting region of GLC and GLA carbohydrates are similar and occur in the same group. Both of these carbohydrates interacting regions are similar to the MAL carbohydrate. The MAN and FRU carbohydrates also show similar interactions and form the same group. Interactions of TRE carbohydrate with the group of FRU, MAN group are also similar up to some extent. The remaining two carbohydrates RIB and XLS occur as isolated in the overall cluster of carbohydrates. The F and T amino acids occur in the same group (F-T). C-Q, V-T, A-S-L-P, Y-D and M-G-K amino acids show similarity in the interaction with various carbohydrates. On the other hand, the interaction behaviour of W and C amino acids are different as clear from Fig. 3.
Shows residue-wise propensity score for different carbohydrates (left) and clustering of carbohydrates based on propensity score (right)
Similarly, GLA-GLC-MAL ligands and FRU-MAN-TRE ligands show very similar interactions with various amino acids. There is no similarity between the interaction behaviour of XLS and MAN ligands. XLS shows negligible interaction with A,S,L,P,M and G amino acids and strong interaction with W,Q,Y and I amino acids. Other amino acids show low or moderate interaction with XLS ligand. RIB shows a strong interaction with V,T,S and L-amino acids and negligible interaction with C,Q and P amino acids. TRE shows a strong interaction with C,L,A and I amino acids while other amino acids show low or moderate interaction with this ligand. MAN and FRU ligands show low interaction with most of the amino acids while MAL, GLC and GLA ligands show strong interaction. Additional file 1: Figure S1 displays the percentile of the interaction of graphical representation of these interactions in detail.
Description of web based tools of LPIcom
LPIcom has three different modules namely 'analysis of binding sites', 'comparison of multiple binding sites' and 'propensity based prediction' implemented in the LPIcom website for the analysis and prediction of interacting amino acids for various ligands. We consider a case study of various ligands e.g. ATP, ADP, GTP, NAD and FAD etc. for illustrating these modules.
Analysis of binding sites
This module calculates the amino acid composition of ligand interacting and non-interacting residues using equation 1. This server compute residue composition of ligand interacting and non-interacting residues. It shows the composition of interacting and non-interacting residues by a bar graph. In order to understand residue preference in ligand interaction, the server also shows the average amino acid composition of residues in proteins. This help user to understand whether the residue is preferred or not based on its composition, whether it is higher or lower than its average composition. In order to demonstrate the utility of this module of LPIcom server, we analyzed ATP binding sites (Fig. 4). It was observed that G, L and R amino acids are frequent binders followed by S and T amino acids. These results are in accordance with walker motif where G, L, S and T amino acids are dominated. Figure 4 also contains a pie chart in the left corner, which depicts the total number of ATP interacting and non-interacting residues.
Example output of analysis module of server LPIcom for ATP with composition option; bar graph shows the composition of ATP interacting and non-interacting residues along with the average amino acid composition of proteins
Analysis module of the server also allows the user to generate two sample logo for ligands based on their interaction. We created all possible overlapping pattern of length 21 residues in proteins; these patterns were classified as interacting and non-interacting patterns based on their central residue whether it is ligand interacting or non-interacting. The web logo of ATP interacting proteins (Additional file 1: Figure S2) shows the dominance of G, R, K, T and S amino acids. The most frequent neighbouring residues are G, T, and S amino acids. On the other hand, L and V amino acids occur as distant neighbouring residues. A further investigation seems necessary to understand the role of L-amino acid as a neighbour of ATP interacting residues. The two-sample logo is created for ATP using two-sample logo package to understand the preference of neighbour (Fig. 5). To detect potential motif in ATP interacting patterns, we used the meme program available in MEME suite [25]. Since motif detection is based on the frequency of the interacting pattern in a dataset, their equal representation is necessary for finding a motif. A 30 % non-redundant dataset of ATP interacting proteins is created, and meme program is used to find any potential motif. Motif depicted in Fig. 6 has a pattern of XXXXXGXXG[SVT]GK[TS] [TV][LIV][AL][RA]X[LI][AL]. The central part of the pattern (GXXG[SVT]GK[TS]) represents the walker motif (GXXXXGK(T/S). The motif is known in the case of ATP; however there are several ligand interacting motifs that need to be discovered. This tool helps in analyses of interacting and non-interacting residues based on the column chart and web logo, two sample logo and discovery of any potential motifs.
Two sample logo of ATP interacting and non-interacting residues using a window length of 21 amino acids. It is an example output of analysis module of LPIcom for ATP with logo option
The motif detected in ATP interacting proteins, which resembles the walker motif
Comparison of multiple ligands based on binding sites
As discussed in the above section, multiple ligands can be compared based on the interacting amino acids. The ligands can be compared based on either the amino acid composition of interacting amino acids or the propensity score of the interacting amino acids as described in methods section. We also implemented a third module for comparing ligands on the basis of the composition of physicochemical properties of interacting amino acids. Each module displays four charts namely column chart, area chart, pie chart and hierarchical cluster and the user can download the data as a text file. Next three sections briefly describe these three modules.
Composition based on comparison of ligands
We compared ten ligands based on the interacting amino acid composition computed using equation 1. The average amino acids composition of all the ligand is used for the reference dataset. Figure 7 depicts the percentage of composition of interacting amino acids with different ligands. It is clear that G amino acid has the highest frequency interaction with most of the ligands followed by R, S, T and V amino acids. On the other hand, C, M, P, Q and W amino acids show the least frequency. A, D, E, F, H, I, K, L, N and Y amino acids show moderate interaction with most of the ligands. UMP ligand with R amino acid shows unusually high frequency and, in general, the interaction of most of the amino acids is higher than average for this ligand as clear from Fig. 7.
Amino acid composition of interacting residues of various ligands
Propensity based on comparison of multiple ligands
We compared all these nucleotides based on their propensity score obtained from PDB using equation 3. In this case, the average propensity score of all amino acids cannot be calculated. Figure 8 shows the propensity score of different ligands as bar plot consists of all 20 amino acids. ATP has the least preference for C, A, V, Q, P, L, I and E amino acids while having a high preference or propensity score for G, D, W, T, S, K and H amino acids. Different ligands have different propensity score for the 20 amino acids. The analysis suggests that the propensity score is not dependent on the chemical property or size of the amino acid.
The propensity score of residues interacting with various nucleotides
Comparison ligands based on physicochemical composition of interacting amino acid
First, amino acids were grouped into different categories on the basis of their physicochemical properties (e. g. charged, acidic, basic, small, polar, non-polar, aliphatic, and aromatic) than the composition of each class of amino acids is computed using equation 6. The average physicochemical property of all ligand interacting amino acids in PDB is also calculated as a reference. A column chart is created for ATP, ADP, GTP, NAD and FAD interacting amino acid. Charged amino acids, especially basic amino acids, are more favoured in ATP, ADP and GTP interaction compared to NAD and FAD interaction as clear from Fig. 9. Small and polar amino acids are equally represented in all five ligands. ATP, ADP and GTP ligands show lower than the average interaction for non-polar, aromatic and aliphatic amino acids. On the other hand, NAD and FAD ligands show a higher than average interaction. Thus, ligands can be divided into two groups 1) ATP, ADP and GTP and 2) NAD and FAD on the basis of their physicochemical properties.
A column chart of various ligands based upon the physicochemical property of amino acids
Propensity-based prediction
The propensity based prediction method assign the propensity score of the residue according to the selected ligand using equation 3. These propensity scores are normalized in the range of 0–9 and depicted below the sequence (Fig. 10) using equation 4. The high probability region of ATP, ADP and GTP ligands are similar in the test sequence. Probability region of NAD and FAD ligands are also analogous to each other. We tested the performance of propensity based prediction on an independent dataset of 1301 PDB chains interacting with 50 different ligands. Each PDB chain is submitted to the prediction module with relevant ligand information and the predicted propensity scores are compared with the actual data to validate the performance. We achieved an average accuracy of 70 % (minimum 24.65 % to maximum 97.61 %, Additional file 1: Table S1). The region of amino acids having highest propensity score has a higher probability to interact with the given ligand. The similarity-based approach is recommended to use if machine learning methods are not available, and propensities based method is recommended for analysis/understanding of interacting residues in a given sequence.
Propensity-based prediction of ATP, ADP, GTP, NAD and FAD interacting residues in test examples
Binding preference analysis of a series of ligands with interacting amino acids is performed using a dataset of 824 ligands. Ligands are clustered based on the preference of interacting amino acids. The ligands having a similar preference for interacting residues have a higher probability of interaction with similar pockets if the difference in their size is not significantly bigger. Clustering the ligands based upon residue preference will help in better understanding of various ligand interactions. A web-based method named LPIcom is also developed for identification of favoured interacting residue with specific ligands. Three different approaches are used from LPIcom for analysis of interacting and non-interacting residues. 1) Comparison based on the amino acid composition of the specific ligand interacting and non-interacting residues. 2) Generation of 'Two sample logo' for comparison of interacting and non-interacting amino acids based upon t-test. 3) Detection of any potential motif in the interacting protein sequences using MEME suite. In addition, three modules are developed for comparison of interacting amino acids of multiple ligands. The first module compares the interacting amino acid composition of multiple ligands, the second module calculates the propensity score of interacting residue for each ligand and compares these propensity scores. The third module compares the physicochemical properties of amino acids for different ligands.
The propensity scores of various ligands are calculated, and the regions of all the protein sequences are highlighted on the basis of propensity scores. The propensity-based method can predict the probable interacting region for every ligand. These results may help biologist in better understanding the ligand interacting regions. Simply, the regions having highest propensity scores have the highest probability to interact with the ligand. The single ligand module helps to understand the interacting and non-interacting residues preference and to detect any potential motif in interacting PDB chains. It also provides a complete and non-redundant dataset for analysis and development of prediction methods. These comparison tools assist in analysing the amino acid preference of various ligands simultaneously. It is important for readers or users to understand limitation of web-server LPIcom describes in this study. As shown in Additional file 1: Table S3, median resolution of PDB chains for around 50 % ligands is poorer than 2.0 Å, even median resolution of 7 % ligands is poorer than 3.0 Å. It means prediction reliability of large number of ligands will be poor, as median of resolution of PDB chains is poor. In addition, user cannot use our server LPIcom for new ligands not included Additional file 1: Table S3.
The ligand interacting data was obtained from the ccPDB database [26] updated up to September 2015 release of PDB. The ligand-interacting amino acids information is extracted from PDB using the LPC software [27]. This software stores the information of each ligand interacting residue with the ligand name, number and chain id and the residue name, number and chain id. In this study, we consider ligand amino acid distance less than or equal to 4 Å for performing the analysis. In this study we only consider 824 ligands, having more than 30 binding sites in the PDB on the basis of the data release up to September 2015. The list of these 824 ligands is given in the Additional file 1: Table S2 and the respective PDBs resolution details are given in Additional file 1: Table S3.
Ligand-specific amino acid composition
The percent amino acids composition of interacting residues (or residues in ligand binding sites) of each ligand is calculated using equation 1.
$$ \mathbf{R}{\mathbf{C}}_{\mathbf{i}}=\frac{{\mathbf{R}}_{\mathbf{i}}}{\mathbf{N}} \times 100 $$
Where RC i is the percent composition of a residue of type i, R i is the number of residues of type i, and N is the total the number of all twenty interacting residues.
Similarity between two ligands based on their amino acid composition
In order to compute similarity or distance between two ligands, we compute similarity using amino acid composition of the ligand binding or interacting residues. The similarity between two ligands based on their amino acid composition is calculated using the Euclidean distance (ED). The composition based ED between two ligands p and q is calculated equation 2:
$$ \mathbf{C}\mathbf{E}{\mathbf{D}}_{\mathbf{p},\mathbf{q}}=\sqrt{{\displaystyle {\sum}_{\mathbf{i}=1}^{20}}\Big(\mathbf{R}{\mathbf{C}}_{\mathbf{i}}^{\mathbf{p}} - \mathbf{R}{\mathbf{C}}_{\mathbf{i}}^{\mathbf{q}}}\Big){}^2 $$
Where CED p,q is distance between two ligands p and q, RC i p is amino acid composition of residue type i for ligand p and, RC i p is amino acid composition of residue type i for ligand q.
Residues propensity for ligands
It is important to understand which residue is preferred or not preferred in binding sites of a ligand. In order to compute preference of a residue in binding site or interaction of a ligand, we compute residues propensity for each ligand. The ligand propensity for each type of residue for a given ligand is computed using equation 3.
$$ \mathbf{R}{\mathbf{P}}_{\mathbf{i}}=\frac{{\mathbf{R}}_{\mathbf{i}}}{{\mathbf{N}}_{\mathbf{i}}} \times 100 $$
Where RP i is ligand propensity score for residue type i, R i is number of interacting residues of type i and N i is the total number of residues (interacting and non-interacting) of type i. The propensities were further normalized between 0 to 9 using equation 4.
$$ \mathbf{N}{\mathbf{P}}_{\mathbf{i}} = \frac{{\mathbf{P}}_{\mathbf{i}} - {\mathbf{P}}_{\mathbf{min}}}{{\mathbf{P}}_{\mathbf{max}} - {\mathbf{P}}_{\mathbf{min}}} \times 9 $$
Where NP i is the normalized propensity of the residue type i, P min is the minimum propensity score out of twenty amino acids and Pmax is the maximum propensity score out of twenty amino acids.
Similarity between two ligands based on their residues propensity
In order to compute the similarity between two ligands based on residues preferred in their interaction, we compute ED between their residues propensity. The propensity based ED between two ligands p and q is calculated using equation 5.
$$ \mathbf{P}\mathbf{E}{\mathbf{D}}_{\mathbf{p},\mathbf{q}}=\sqrt{{\displaystyle {\sum}_{\mathbf{i}=1}^{20}}\Big(\mathbf{R}{\mathbf{P}}_{\mathbf{i}}^{\mathbf{p}} - \mathbf{R}{\mathbf{P}}_{\mathbf{i}}^{\mathbf{q}}}\Big){}^2 $$
Where PED p,q is distance between two ligands p and q, RP i p is residue composition of residue type i for ligand p and RP i p is residue composition of residue type i for ligand q.
Physicochemical property based composition
In order to understand physicochemical property (e.g., charge, polar, hydrophobic) of residues involved a ligand binding residues. We group all twenty types of residues in eight class based on their physicochemical property. As shown in Table 1, we group amino acids based on their major characteristics that include their size, charge, polarity and hydrophobicity. We compute composition of each class of residues of these eight classes (or composition of physicochemical property) using following equation 6.
The different physicochemical property of amino acids with the respective amino involved
Physicochemical property
Amino acid involved
ASP,GLU,LYS,HIS,ARG
Acidic amino acids
ASP,GLU
Basic amino acids
LYS,ARG,HIS
Small amino acids
PRO,ALA,CYS,GLY,SER,ASN,ASP,THR,VAL
Polar amino acids
SER,THR,TYR,ASN,GLN
Non polar amino acids
ALA,VAL,LEU,ILE,PRO,PHE,TRP,MET,CYS,GLY
Aromatic amino acids
PHE,TYR,TRP
Aliphatic amino acids
LEU,ILE,VAL,ALA,GLY
$$ \mathbf{P}{\mathbf{C}}_{\mathbf{i}}=\frac{{\mathbf{P}}_{\mathbf{i}}}{\mathbf{N}} \times 100 $$
Where PC i is the percent composition of a physicochemical property of type i, P i is the number of interacting residues having physicochemical property of type i and N is the total the number of interacting residues.
Ligand similarity based on physicochemical property of residues
In order to compute the similarity between two ligands based on their physicochemical property of residues involved in their binding sites, we compute ED between compositions of their physicochemical properties. The composition based ED between two ligands p and q is calculated using equation 7.
$$ \mathbf{P}\mathbf{C}\mathbf{E}{\mathbf{D}}_{\mathbf{p},\mathbf{q}}=\sqrt{{\displaystyle {\sum}_{\mathbf{i}=1}^{20}}\Big(\mathbf{P}{\mathbf{C}}_{\mathbf{i}}^{\mathbf{p}} - \mathbf{P}{\mathbf{C}}_{\mathbf{i}}^{\mathbf{q}}}\Big){}^2 $$
Where PCED p,q is distance between two ligands p and q, PC i p is composition of physicochemical property of type i for ligand p and, PC i p is residue composition of composition of physicochemical property i for ligand q.
Clustering of Ligands
In this study, we have used the 'dist function' available in 'R' package to obtain the distance matrix between multiple ligands. The distance matrix is used for generating clusters based on hierarchical clustering algorithm embedded in 'Hclust function' available in 'R' package. The cluster information along with distance matrix is used to generate the heat map using 'Heatmap function' also available in 'R' package.
Generation of the dynamic graph
High-charts library was used to display graph according to selected features. The generated charts can also be exported to various image formats. For creating web logo and two sample logo, we generated a pattern of window length 21 for a specific ligand interacting proteins with the central residue as the ligand-interacting residues. The web logo standalone package is used for displaying the logo of interacting amino acids [28]. A two sample logo is generated on the basis of interacting and non-interacting patterns using the default parameters [29]. Meme program from MEME suite [25] is used for motif identification in the non-redundant dataset of interacting proteins.
Validation dataset
The LPIcom database was generated from PDB complexes released up to September 2015. In order to validate the performance of our prediction module, we created a validation dataset. A 1301 PDB chains, for 50 commonly found ligands, were selected from PDB complexes released between October-December 2015 (Additional file 1: Table S1). Thus, PDB chains in validation dataset are entirely different from PDB chains used for prediction in LPIcom.
Reviewer 1: Response to Prof Michael Gromiha
In this work, the authors developed a web server for predicting ligand binding sites in proteins. They have analyzed the binding propensity of more than 700 ligands and the topmost ones are presented in the manuscript. Further, the ligands/amino acid residues have been clustered to understand the preference of binding. The details about the binding sites and other details are provided in the Additional file 1 and on the web. It is an interesting manuscript with several ligands together.
The manuscript could be improved by incorporating the following suggestions.
1. Propensity analysis has been carried out based on high, moderate and low. The plausible reasons could be discussed.
Response: We are thankful to the reviewer for the suggestion, in revised manuscript we clearly described propensity score in detail including modification of equations used for calculation. We defined preference of a residue in ligand binding site based on its propensity score if the score of a residue is lower than 5 then we called it low preference residue. Similarly, we called a residue moderate if it has propensity score between 5 to 12 and high if the score is more than 12. In revised manuscript, we incorporate suggestion of reviewer.
2. Analysis on statistical significance would validate the specific preference of residues/ligands.
Response: We agree with the reviewer that analysis should show whether the preference is really significant or it is by chance. In order to facilitate users to understand whether propensity or composition of ligand interacting residue is significant or not, we also compute and compare it with an average of each type of amino acids. This help user to understand whether a given residue is preferred in the binding site of a ligand. In revised version, we emphasize this point.
3. Several examples are given on the binding site prediction of ligands using example proteins and ligands produced no binding site results. It is better to provide examples with binding site residues. Also, these results should be checked.
Response: We are thankful to the reviewer for pointing the error. We have fixed all the errors.
4. In the Additional file 1 prediction performance of specific ligands are given. It will be beneficial if the data for all ligands are given although some of them would be poor due to their less occurrence in proteins-ligand complexes.
Response: We calculated the prediction performance of some of the ligands, which have significantly high frequency in the PDB. After getting comment of the reviewer, we also compute prediction accuracy for more ligands (50 ligands). It is not feasible to compute performance to all ligands (~800 ligands).
5. Several methods are available for ligand binding site prediction. A comparison with other existing prediction methods could be useful.
Response: Ideally one should compare newly developed prediction method with existing methods as suggested by a reviewer. In past our group also developed a number of methods for predicting ligand interacting residues (e.g., ATPint, NADbinder, GTPbinder, FADpred) where we compare their performance with existing methods. Development of prediction method even for a single ligand is time-consuming as one need to create clean datasets (e.g., non-redundant) and should evaluate using cross-validation techniques (internal and external validations). This is the reason, so far methods have been developed only for limited ligands. In this study, we described simple propensity based method for a large number of ligands. Though we also compute performance of our method on limited ligands but comparing performance with existing method will be unfair as we have not used clean dataset for training and cross-validation techniques. The objective of our method is to assist biologist in understanding the propensity scores of various amino acid and propensity based prediction of those ligands for which no specific method is available.
Reviewer 2: Response to Prof Vladimir Poroikov
In this paper freely available via Internet web-server LPIcom (Ligand-Protein Interactions Comparison and Analysis), which provides the possibility to study protein-ligand interactions, is described. The authors extracted from PDB the information about protein-ligand complexes for 724 ligands, which have 50 protein binding sites in PDB. This information was analyzed, to estimate the propensity of participating in protein-ligand interactions for each of twenty amino acid residues. Web-server consists of three modules provided Analysis, Comparison and Prediction functionalities. It provides the following facilities: a) assigning of ligand-interacting residues in a protein from the structure of protein-ligand complex; b) analysis of composition of ligand-specific interacting residues; c) comparison of binding sites of different ligands; d) generation of two sample logo of ligand binding sites; e) searching of ligand binding motifs; f) propensity-based prediction of ligand-interacting residues.
1 From the technical point of view, everything is well-done, except some misprints in the text at this web-site (e.g., "How to save and pritn the graph" - it should be "print").
Response: We are thankful to the reviewer for indicating the error and we have corrected the typing and grammatical mistakes in the revised manuscript.
Also, according to my knowledge, this is the first analytics of massive data on protein-ligand interactions from PDB, where information about at least 50 binding sites is available for the ligands. However, some questions arose regarding the possibility of application of the obtained results in a prospective mode. The authors declare that "This information can be used to either enhance or diminish the binding strength of the given ligand" (page 3, lines 37–38 of the manuscript).
1. It is unclear if and how the developed web-server could be applied to the new ligands, which are not included into the "training set" (724 ligands).
Response: The web-server cannot be applied to new ligands; we have increased the number of ligands from 724 to 824 which have a minimum number of ligand binding sites greater than 30. In future, we will update this database to include new ligands.
2. It is necessary to explain how the user might use the information provided by this web-server, to "enhance or diminish the binding energy of the given ligand." Since such application is of great importance in the field of computer-aided drug design, it would be great if the authors could present at least one case study with such application in the manuscript. Such example(s) could be based on the retrospective data for already studied set of ligands belonging to the same chemical series.
Response: In revised manuscript, we explain how this server can be used to enhance or to diminish a ligand binding site in a protein. This server provides propensity score or preference for each type of amino acid for a given ligand. Experimentalist may enhance ligand binding by mutating low propensity residue with high propensity residue in the binding site having similar physicochemical property. Similarly, one may also diminish ligand binding by mutating high propensity residue with low propensity residue. Every ligand has a specific preference towards a different type of residues, nucleotides-ligand prefer aliphatic residues and less preference for acidic residues. On the other hand, carbohydrates have more preference for acidic residues than aliphatic residues. Experimental researchers may use above information for increasing or decreasing binding affinity based propensity score. The server only suggests the residues based on the information available in the PDB. Multiple factors influence the binding strength of a residue in a given binding site apart from its affinity to interact with a particular ligand. The purpose of LPIcom is to provide the affinity information of residues toward different ligands as observed in PDB.
Minor: It would be great if in the Additional file 1 the authors present the estimates of the quality of the X-ray data in the protein-ligand complexes analyzed for each studied ligand (median, minimum and maximum values characterized the resolution for all binding-sites under consideration).
Response: The X-ray data of all PDB present in the LPIcom database are given in Additional file 1 : Table S3. We have provided the median, minimum and maximum X-ray resolution for each ligand as shown in Additional file 1 : Table S3.
Second Revision
Major comments: The authors have provided the responses on my major comments, and now the contents of their work is more clear for a scientific community. There is still some minor issues, which should be fixed prior to the publication. 1. It is necessary to provide the units for values presented in the Additional file 1: Table S3. 2. As one may see from the Additional file 1: Table S3, in some cases the median resolution in X-ray data is quite low (exceeded 2.00). The authors should comment in the manuscript if the obtained results are reliable enough in such cases. 3. It should be explicitely mentioned in the manuscript that the web-server cannot be applied for new ligands.
Response: We are thankful to reviewer for appreciating our efforts. 1. In Additional file 1: Table S3 units of resolution (angstrom) has been stated. 2. Yes, median resolution of ~50 % ligands exceed 2.0 Å, even median resolution of ~7 % ligands exceed 3.0 Å. In revised manuscript, we clearly mentioned limitations of our study as number of ligands have PDB chains of poor resolution. In addition, we also mentioned in last paragraph of 'Conclusion section' that our web-server couldn't by applied for new ligands.
Minor: Despite the correction of grammatical errors and misprints, the authors added new errors/misprints in the novel part of the manuscript; e.g., Page 10, Line 57: "twnety" it should be "twenty". The whole manuscript should be carefully checked, and all errors/misprints should be corrected. Despite the correction of grammatical errors and misprints, the authors added new errors/misprints in the novel part of the manuscript; e.g., Page 10, Line 57: "twnety" it should be "twenty". The whole manuscript should be carefully checked, and all errors/misprints should be corrected.
Response: We are grateful to the reviewer for indicating the grammatical errors. The manuscript has been carefully checked and corrected.
Reviewer 3: Response to Prof Zlatko Trajanoski
General comments The manuscript describes a web server for analysis of protein ligand binding sites. Although the topic is potentially of interest to a broader community, I don' see any considerable contribution neither from manuscript nor from the web server. The manuscript is difficult to read and the presented results seems to show simple statistical analysis of the amino acids which are binding ligands. What is the major contribution and how does this work add additional information compared to other papers?
Response: Best of our knowledge this is a unique server which allows users to analyse, compare and predict potential binding sites for a large number of ligands based on information in PDB.
Specifically, the work should be compared to the web servers already available (References 10 and 11) and the advantages/disadvantages highlighted.
Response: Ideally one should compare newly developed prediction method with existing methods as suggested by a reviewer. In past our group also developed a number of methods for predicting ligand interacting residues (e.g., ATPint, NADbinder, GTPbinder, FADpred) where we compare their performance with existing methods. Development of prediction method even for a single ligand is a time consuming as one need to create clean datasets (e.g., non-redundant) and should evaluate cross-validation techniques (internal and external validations). This is the reason, so far methods have been developed only for limited ligands. In this study, we described simple propensity-based method for a large number of ligands. Though we also compute performance of our method on limited ligands but comparing performance with existing method will be unfair as we have not used clean dataset for training and cross-validation techniques. The objective of our method is to assist biologist in understanding the propensity scores of various amino acid and propensity based prediction of those ligands for which no specific method is available.
Moreover, the web server itself was not thoroughly tested as evident by a number of issues raised bellow. Specific comments The implementation of the web server has several limitations some of which are provided below:
1) Typos and grammatical errors: For instance, from the input form and output of "analysis of binding sites" (http://crdd.osdd.net/raghava/lpicom/mut.php): - "User are required"; - "Click to Cutomize plot"; - "red color bars shows"; - "amino acid composition of all ligand"; - "High Resolutione".
Response: We check and removed all the errors from the revised manuscript and web-server.
2) Inconsistencies in the descriptions: - The page of "ligand statistics" (http://crdd.osdd.net/raghava/lpicom/ligand-data.php) is once referred as "the complete list of ligands" (http://crdd.osdd.net/raghava/lpicom/mut.php) and once as the list of "highly frequent ligand" (http://crdd.osdd.net/raghava/lpicom/predict.php). The second option is probably the correct one, since the web server provides results also for ligands that are not present in the list. However, the full sentence is difficult to understand: "Detail of highly frequent ligand in PDB is available from and view ligands having highest occurence in PDB HERE". - The description of results from "analysis of binding sites" (http://crdd.osdd.net/raghava/lpicom/mut.php), says: "blue color bars show ATP interacting and red color bars shows not interacting residues […]". But there are no red bars, only blue or back ones.
3) Inconsistencies in the web pages and broken links If "Click to Cutomize plot" is selected on the "analysis of binding sites" results page (http://crdd.osdd.net/raghava/lpicom/mut.php), a different web page is shown. Some links, such as "interacting PDB" are broken.
Response: We fixed these issues and now they are working fine.
4) Not-working modules (?) - The example of the module "Comparison of Ligands Binding Sites (Amino acid Composition)" (http://crdd.osdd.net/raghava/lpicom/compare.php) gives 0.00 % result on all amino acids and ligands. - The "Prediction of Ligand Interacting Residues" module (http://crdd.osdd.net/raghava/lpicom/predict.php) predicts 0 propensity score for all positions (there are no regions highlighted in red or green.
5) Finally, it would be useful to have descriptions of acronyms and link to external references (e.g. PDB), as well as a description of the full name of the ligand(s) for which the analysis was run, to have a confirmation of the selection.
Response: The information is already provided on the web-server and in the Additional file 1. We have updated the web server language for better understanding of the terminology.
ADP:
adenosine diphosphate
adenosine monophosphate
guanosine triphosphate
GDP:
guanosine diphosphate
CTP:
cytidine triphosphate
CMP:
cytidine monophosphate
UDP:
uridine diphosphate
UMP:
uridine monophosphate
NAD:
nicotinamide adenine dinucleotide
flavin adenine dinucleotide
GLA:
Alpha D-galactose
GLC:
Alpha-D-glucose
MAL:
FRU:
Alpha-D-mannose
TRE:
RIB:
XLS:
D-xylose (linear form)
Avr:
Authors are thankful to funding agencies, Council of Scientific and Industrial Research (project OSDD and GENESIS BSC0121), Govt. of India. Authors declare no conflict of interest.
Additional file 1: Figure S1. The propensity score of residues interacting with various carbohydrates. Figure S2. A web logo of ATP interacting patterns of 21-window length. Table S1. Performance of our propensity based prediction models on 50 major ligands, evaluated on independent datasets. Table S2. List of 824 ligands having more than 30 binding sites in the PDB. Table S3. Minimum, Maximum and Median Resolution of PDBs interacting with 824 ligands. (DOCX 276 kb)
Authors declare there are no financial conflict and no non-financial conflicts of interest.
HS carried out the primary work including the development of web server. HS collected the data used for performing the analysis. HKS performed the statistical analysis. HS and HKS prepared the manuscript. GPSR conceived the study, and participated in its design and coordination and finalized the manuscript. All authors read and approved the final manuscript.
Bioinformatics Centre, CSIR-Institute of Microbial Technology, Chandigarh, 160036, India
http://www.imtech.res.in/raghava/
Gonzales-Diaz H, Gia O, Uriarte E, Hernadez I, Ramos R, Chaviano M, Seijo S, Castillo JA, Morales L, Santana L, et al. Markovian chemicals "in silico" design (MARCH-INSIDE), a promising approach for computer-aided molecular design I: discovery of anticancer compounds. J Mol Model. 2003;9(6):395–407.View ArticlePubMedGoogle Scholar
Stumpf SH. Pathways to success: training for independent living. Monogr Am Assoc Ment Retard. 1990;15:1–111.PubMedGoogle Scholar
Speck-Planche A, Kleandrova VV, Luan F, Cordeiro MN. Unified multi-target approach for the rational in silico design of anti-bladder cancer agents. Anticancer Agents Med Chem. 2013;13(5):791–800.View ArticlePubMedGoogle Scholar
Speck-Planche A, Kleandrova VV, Luan F, Cordeiro MN. Chemoinformatics in anti-cancer chemotherapy: multi-target QSAR model for the in silico discovery of anti-breast cancer agents. Eur J Pharm Sci. 2012;47(1):273–9.View ArticlePubMedGoogle Scholar
Speck-Planche A, Kleandrova VV, Luan F, Cordeiro MN. Chemoinformatics in multi-target drug discovery for anti-cancer therapy: in silico design of potent and versatile anti-brain tumor agents. Anticancer Agents Med Chem. 2012;12(6):678–85.View ArticlePubMedGoogle Scholar
Estrada E, Uriarte E, Montero A, Teijeira M, Santana L, De Clercq E. A novel approach for the virtual screening and rational design of anticancer compounds. J Med Chem. 2000;43(10):1975–85.View ArticlePubMedGoogle Scholar
Gonzalez-Diaz H, Vina D, Santana L, de Clercq E, Uriarte E. Stochastic entropy QSAR for the in silico discovery of anticancer compounds: prediction, synthesis, and in vitro assay of new purine carbanucleosides. Bioorg Med Chem. 2006;14(4):1095–107.View ArticlePubMedGoogle Scholar
Gonzalez-Diaz H, Bonet I, Teran C, De Clercq E, Bello R, Garcia MM, Santana L, Uriarte E. ANN-QSAR model for selection of anticancer leads from structurally heterogeneous series of compounds. Eur J Med Chem. 2007;42(5):580–5.View ArticlePubMedGoogle Scholar
Singla D, Tewari R, Kumar A, Raghava GP. Designing of inhibitors against drug tolerant Mycobacterium tuberculosis (H37Rv). Chem Cent J. 2013;7(1):49.View ArticlePubMedPubMed CentralGoogle Scholar
Liu T, Lin Y, Wen X, Jorissen RN, Gilson MK. BindingDB: a web-accessible database of experimentally determined protein-ligand binding affinities. Nucleic Acids Res. 2007;35(Database issue):D198–201.View ArticlePubMedPubMed CentralGoogle Scholar
Yang J, Roy A, Zhang Y. BioLiP: a semi-manually curated database for biologically relevant ligand-protein interactions. Nucleic Acids Res. 2013;41(Database issue):D1096–1103.View ArticlePubMedPubMed CentralGoogle Scholar
Mangal M, Sagar P, Singh H, Raghava GP, Agarwal SM. NPACT: naturally occurring plant-based anti-cancer compound-activity-target database. Nucleic Acids Res. 2013;41(Database issue):D1124–1129.View ArticlePubMedPubMed CentralGoogle Scholar
Yadav IS, Singh H, Imran Khan M, Chaudhury A, Raghava GP, Agarwal SM. EGFRIndb: Epidermal Growth Factor Receptor Inhibitor Database. Anticancer Agents Med Chem. 2014;14(7):928–35.View ArticlePubMedGoogle Scholar
Chauhan J, Mishra N, Raghava G. Identification of ATP binding residues of a protein from its primary sequence. BMC Bioinformatics. 2009;10(1):434.View ArticlePubMedPubMed CentralGoogle Scholar
Agarwal S, Mishra NK, Singh H, Raghava GP. Identification of mannose interacting residues using local composition. PLoS One. 2011;6(9):e24039.View ArticlePubMedPubMed CentralGoogle Scholar
Ansari HR, Raghava GP. Identification of NAD interacting residues in proteins. BMC Bioinformatics. 2010;11:160.View ArticlePubMedPubMed CentralGoogle Scholar
Brylinski M, Skolnick J. Comparison of structure-based and threading-based approaches to protein functional annotation. Proteins. 2010;78(1):118–34.View ArticlePubMedPubMed CentralGoogle Scholar
Chauhan JS, Mishra NK, Raghava GP. Identification of ATP binding residues of a protein from its primary sequence. BMC Bioinformatics. 2009;10:434.View ArticlePubMedPubMed CentralGoogle Scholar
Chupakhin V, Marcou G, Baskin I, Varnek A, Rognan D. Predicting ligand binding modes from neural networks trained on protein–ligand interaction fingerprints. J Chem Inf Model. 2013;53(4):763–72.View ArticlePubMedGoogle Scholar
Jacob L, Vert JP. Protein-ligand interaction prediction: an improved chemogenomics approach. Bioinformatics. 2008;24(19):2149–56.View ArticlePubMedPubMed CentralGoogle Scholar
Mishra NK, Raghava GP. Prediction of FAD interacting residues in a protein from its primary sequence using evolutionary information. BMC Bioinformatics. 2010;11 Suppl 1:S48.View ArticlePubMedPubMed CentralGoogle Scholar
Roche DB, Buenavista MT, McGuffin LJ. The FunFOLD2 server for the prediction of protein-ligand interactions. Nucleic Acids Res. 2013;41(Web Server issue):W303–307.View ArticlePubMedPubMed CentralGoogle Scholar
Wass MN, Kelley LA, Sternberg MJ. 3DLigandSite: predicting ligand-binding sites using similar structures. Nucleic Acids Res. 2010;38(Web Server issue):W469–473.View ArticlePubMedPubMed CentralGoogle Scholar
Yang J, Roy A, Zhang Y. Protein-ligand binding site recognition using complementary binding-specific substructure comparison and sequence profile alignment. Bioinformatics. 2013;29(20):2588–95.View ArticlePubMedPubMed CentralGoogle Scholar
Bailey TL, Williams N, Misleh C, Li WW. MEME: discovering and analyzing DNA and protein sequence motifs. Nucleic Acids Res. 2006;34(Web Server issue):W369–373.View ArticlePubMedPubMed CentralGoogle Scholar
Singh H, Chauhan JS, Gromiha MM, Raghava G. ccPDB: compilation and creation of data sets from Protein Data Bank. Nucleic Acids Res. 2012;40(Database issue):D486–489.View ArticlePubMedPubMed CentralGoogle Scholar
Sobolev V, Sorokine A, Prilusky J, Abola EE, Edelman M. Automated analysis of interatomic contacts in proteins. Bioinformatics. 1999;15(4):327–32.View ArticlePubMedGoogle Scholar
Crooks GE, Hon G, Chandonia JM, Brenner SE. WebLogo: a sequence logo generator. Genome Res. 2004;14(6):1188–90.View ArticlePubMedPubMed CentralGoogle Scholar
Vacic V, Iakoucheva LM, Radivojac P. Two Sample Logo: a graphical representation of the differences between two sets of sequence alignments. Bioinformatics. 2006;22(12):1536–7.View ArticlePubMedGoogle Scholar | CommonCrawl |
\begin{document}
\baselineskip=18pt
\title{Dispersion as a survival strategy}
\author[Valdivino Vargas Junior]{Valdivino Vargas Junior} \address[Valdivino Vargas Junior]{Institute of Mathematics and Statistics, Federal University of Goias, Campus Samambaia, CEP 74001-970, Goi\^ania, GO, Brazil} \email{[email protected]} \thanks{Valdivino Vargas was supported by PNPD-CAPES (1536114), F\'abio Machado by CNPq (310829/2014-3) and Fapesp (09/52379-8) and Alejandro Roldan by CNPq (141046/2013-9).}
\author[F\'abio P. Machado]{F\'abio~Prates~Machado} \address[F\'abio P. Machado]{Statistics Department, Institute of Mathematics and Statistics, University of S\~ao Paulo, CEP 05508-090, S\~ao Paulo, SP, Brazil.} \email{[email protected]}
\author[Alejandro Rold\'an]{Alejandro~Rold\'an-Correa} \address[Alejandro Rold\'an]{Instituto de Matem\'aticas, Universidad de Antioquia, Calle 67, no 53-108, Medellin, Colombia} \email{[email protected]}
\keywords{Branching processes, catastrophes, population dynamics} \subjclass[2010]{60J80, 60J85, 92D25} \date{\today}
\begin{abstract} We consider stochastic growth models to represent population subject to catastrophes. We analyze the subject from different set ups considering or not spatial restrictions, whether dispersion is a good strategy to increase the population viability. We find out it strongly depends on the effect of a catastrophic event, the spatial constraints of the environment and the probability that each exposed individual survives when a disaster strikes. \end{abstract}
\maketitle
\section{Introduction} \label{S: Introduction}
Biological populations are often exposed to catastrophic events that cause mass extinction: Epidemics, natural disasters, etc. When mild versions of these disasters occur, survivors may develop strategies to improve the odds of their species survival. Some populations adopt dispersion as a strategy. Individuals of these populations disperse, trying to make new colonies that may succeed settling down depending on the new environment they encounter. Recently, Schinazi~\cite{S2014} and Machado \textit{et al.}~\cite{MRS2015} proposed stochastic models for this kind of population dynamics. For these models they concluded that dispersion is a good survival strategy. Earlier, Lanchier~\cite{Lanchier} considered the basic contact process on the lattice modified so that large sets of individuals are simultaneously removed, which also models catastrophes. In this work there are qualitative results about the effect of the shape of those sets on the survival of the process, with interesting non-monotonic results, and dispersion is proved to be a better strategy in some contexts.
Moreover, Brockwell \textit{et al.}~\cite{BGR1982} and later Artalejo \textit{et al.}~\cite{AEL2007} considered a model for the growth of a population (a single colony) subject to collapse. In their model, two types of effects when a disaster strikes were analyzed separately, \textit{binomial effect} and \textit{geometric effect}. After the collapse, the survivors remain together in the same colony (there is no dispersion). They carried out an extensive analysis including first extinction time, number of individuals removed, survival time of a tagged individual, and maximum population size reached between two consecutive extinctions. For a nice literature overview and motivation see Kapodistria \textit{et al.}~\cite{KPR2016}.
Based on the model proposed by Artalejo \textit{et al.}~\cite{AEL2007}, and adapting some ideas from Schinazi~\cite{S2014} and Machado~\textit{et al.}~\cite{MRS2015}, we analyze growth models of populations subject to disasters, where after the collapse species adopt dispersion as a survival strategy. We show that dispersion is not always a good strategy to avoid the population extinction. It strongly depends on the effect of a catastrophic event, the spatial constraints of the environment and the probability that each exposed individual survives when a disaster strikes.
This paper is divided into four sections. In Section 2 we define and characterize three models for the growth of populations subject to collapses. In Section 3 we compare the three models introduced in Section 2 and determine under what conditions the dispersion is a good strategy for survival, due to space restrictions and the effects when a disaster strikes. Finally, in Section 4 we prove the results from Sections 2 and 3.
\section{Growth models}
First we describe a model presented in Artalejo \textit{et al.}~\cite{AEL2007}. This is a model for a population which sticks together in one colony, without dispersion. The colony gives birth to a new individual at rate $\lambda>0$, while collapses happen at rate $\mu$. If at a collapse time the size of the population is $i$, it is reduced to $j$ with probability $\mu_{ij}$. The parameters $\mu_{ij}$ are determinated by how the collapse affects the population size. Next we describe two types of effects.
$\bullet$ \textit{Binomial effect:} Disasters reach the individuals simultaneously and independently of everything else. Each individual survives with probability $p<1$ (dies with probability $q=1-p$), meaning that \[ \mu_{ij}^B ={ i \choose j} p^j q^{i-j}, \ 0\leq j\leq i.\]
$\bullet$ \textit{Geometric effect:} Disasters reach the individuals sequentially and the effects of a disaster stop as soon as the first individual survives, if there are any survivor. The probability of next individual to survive given that everyone fails up to that point is $p<1,$ which means that \[ \mu_ {ij}^G = \left\{\begin{array}{ll} q^i, & j=0\\ pq^{i-j}, & 1\leq j \leq i.\end{array}\right.\]
The binomial effect is appropriate when the catastrophe affects the individuals in a independent and even way. The geometric effect would correspond to cases where the decline in the population is halted as soon as any individual survives the catastrophic event. This may be appropriate for some forms of catastrophic epidemics or when the catastrophe has a sequential propagation effect like in the predator-prey models - the predator kills prey until it becomes satisfied. More examples can be found in Artalejo \textit{et al.}~\cite{AEL2007} and in Cairns and Pollett~\cite{CairnsPollett}.
\subsection{Growth model without dispersion}
In Artalejo \textit{et al.}~\cite{AEL2007} the authors consider the binomial and the geometric effect separately as alternatives to the total catastrophe rule which instantaneously removes the whole population whenever a catastrophic event occurs.
Here we consider a mixture of both effects, that is, with probability $r$ the group is striken sequentially (geometric effect) and with probability $1-r$ the group is striken simultaneously (binomial effect). More precisely, \[ \mu_{ij}:=r\mu_{ij}^G+(1-r)\mu_{ij}^B.\]
We assume that the collapse rate $\mu$ equals 1. The size of the population (number of individuals in the colony) at time $t$ is a continuous time Markov process $\left\{X(t):t\geq 0\right\}$ whose infinitesimal generator $(q_{ij})_{i,j\geq 0}$ is given by \[q_{ij}=\left\{\begin{array}{ll}\lambda, & j=i+1, \ i\geq 0, \\ \mu_{ij}, & 0\leq j <i, \\ -(\lambda+\sum_{j=0}^{i-1}\mu_{ij}), & i=j, \\ 0& \text{otherwise.} \end{array}\right.\]
We also assume $X(0)=1$ and denote by $C^1(p,r,\lambda)$ the process described by $\left\{X(t):t\geq 0\right\}$. When $r=0$ and $r=1$, we obtain the models considered in Artalejo \textit{et al.}~\cite{AEL2007}.
\begin{theorem}[Artalejo \textit{et al.}~\cite{AEL2007}] \label{th:semdisp} Let $X(t)$ a process $C^1(p,r,\lambda)$, with $\lambda>0$ and $0<p<1$. Then, extinction (which means $X(t)=0$ for some $t>0$) occurs with probability $$\rho_1(r)=\left\{\begin{array}{ll} 1& \text{, when } r<1\\ \min\left\{\frac{1-p}{\lambda p}, 1\right\} & \text{, when} r=1. \end{array}\right.$$ Moreover, if $r<1$, or $r=1$ and $\lambda p < 1-p,$ the time it takes until extinction has finite expectation. \end{theorem}
\begin{obs} The result of Theorem~\ref{th:semdisp} has been shown by Artalejo \textit{et al.}~\cite{AEL2007} for the cases $r=0$ and $r=1$. They use the word \textit{extinction} to describe the event that $X(t) = 0$, for some $t>0$, for a process where state 0 is not an absorbing state. In fact the extinction time here is the first hitting time to the state 0. We keep using the word extinction for this model trough the paper.
From their result one can see that survival is only possible when the effect is purely geometric ($r=1$). The reason for that is quite clear: If $r<1$ the binomial effect strikes at rate $(1-r)>0$ so even if one considers $p=1$ when the geometric effect strikes, the population will die out as proved in Artalejo \textit{et al.}~\cite{AEL2007} for the case $r=0$. \end{obs}
\subsection{Growth model with dispersion but no spatial restriction.} Consider a population of individuals divided into separate colonies. Each colony begins with an individual. The number of individuals in each colony increases independently according to a Poisson process of rate $\lambda > 0 $. Every time an exponential time of mean 1 occurs, the colony collapses through a binomial or a geometric effect and each of the collapse survivors begins a new colony independently of everything else. We denote this process by $C^2(p,r,\lambda)$ and consider it starting from a single colony with just one individual.
The following theorem establishes necessary and sufficient conditions for survival in $C^2(p,r,\lambda).$
\begin{theorem}\label{th:disp1} The process $C^2(p,r,\lambda)$ survives with positive probability if and only if \begin{equation}\label{eqthdisp1}
\frac{p(\lambda+1)^2r}{\lambda p+1} +p(\lambda+1)(1-r)>1. \end{equation} \end{theorem}
Theorem~\ref{th:disp1} shows that, contrary to what happens in $C^1(p,r,\lambda)$, in $C^2(p,r,\lambda)$ the population is able to survive even when the binomial effect may occur $(r<1)$. See example~\ref{ex:bin}. In particular, if $r=0$ (pure binomial effect) the process survives with positive probability whenever $p(\lambda+1)>1$.
The next result shows how to compute the probability of extinction, which means, the probability that eventually the system becomes empty.
\begin{theorem}\label{th:disp2} Let $\rho_2(r)$ be the probability of extinction in $C^2(p,r,\lambda)$. Then $\rho_2(r)$ is the smallest non-negative solution of \begin{equation}\label{probext}\phi(s):=\frac{1}{1+\lambda p}\left[q+\frac{r(\lambda +1)ps}{1+\lambda -\lambda s}+\frac{(1-r)(\lambda +1)ps}{1+\lambda p - \lambda p s}\right]=s \end{equation} \end{theorem}
\begin{exa}\label{ex:bin} For $C^2(2/5,r,1)$ \[\phi(s)=\frac{3}{7}+\frac{4rs}{14-7s}+\frac{20(1-r)s}{49-14s}.\] The smallest non-negative solution for the equation $\phi(s)=s,$ is given by \[\rho_2(r)=\left\{\begin{array}{cl}1,& \ r\leq7/12\\ \displaystyle\frac{12r+49-\sqrt{144 r^2+1176 r+49}}{28},& \ r>7/12. \end{array}\right.\] \end{exa}
\begin{obs} For $r=0$ (pure binomial effect) and $r=1$ (pure geometric effect) the smallest non-negative solution for (\ref{probext}) is: $$\rho_2(0)=\min\left\{\frac{q}{\lambda p},1\right\} \hspace{0.5cm} \text{and}\hspace{0.5cm}\rho_2(1)=\min\left\{\frac{q(\lambda+1)}{\lambda(1+\lambda p)},1\right\}.$$
Observe that $\rho_2(0)\geq \rho_2(1)$ where the strict inequality holds provided $(1+\lambda+\lambda^2)^{-1}<p<1.$ Moreover, \begin{itemize} \item[$\bullet$] If $p<\displaystyle\frac{1}{1+\lambda+\lambda^2}$ then $\rho_2(0)=\rho_2(1)=1.$ \item[$\bullet$] If $\displaystyle\frac{1}{1+\lambda+\lambda^2}<p<\frac{1}{1+\lambda}$ then $\rho_2(0)=1$ and $\rho_2(1)=\displaystyle\frac{q(\lambda+1)}{\lambda(1+\lambda p)}.$ \item[$\bullet$] If $p>\displaystyle\frac{1}{1+\lambda}$ then $\rho_2(0)=\displaystyle\frac{q}{\lambda p}$ and $\rho_2(1)=\displaystyle\frac{q(\lambda+1)}{\lambda(1+\lambda p)}.$ \end{itemize}
Note that likewise as occurs in $C^1(p,r,\lambda)$, the binomial effect is a worst scenary than the geometric effect for the population survival in $C^2(p,r,\lambda)$. \end{obs}
\begin{obs} Observe that $\rho_1(r) \geq \rho_2(r)$ for $ r \in [0,1].$ In addition, if $r<1$ the inequality is strict provided (\ref{eqthdisp1}) holds. Moreover, $\rho_1(1) > \rho_2(1)$ for $\lambda(1+\lambda p)>q(\lambda+1).$ That means when there are no spatial restrictions, dispersion is a good strategy for population survival. That coincides with the results for the models presented and analyzed by Schinazi~\cite{S2014} and Machado \textit{et al.}~\cite{MRS2015}. \end{obs}
\subsection{Growth with dispersion and spatial restriction.}
Let $\mathcal{G}_m$ be a graph (finite or infinite) such that every vertex has $m$ neighbours, what is known as a $m-$regular graph. Let us define a process with dispersion and spatial restrictions on $\mathcal{G}_m$, starting from a single colony placed at one vertex of $\mathcal{G}_m$, with just one individual. The number of individuals in a colony grows following a Poisson process of rate $\lambda>0$. To each colony we associate an exponential time of mean 1 that indicates when the colony collapses. Each one of the individuals that survived the collapse (either a binomial or a geometric effect) picks randomly a neighbor vertex and tries to create a new colony at it. Among the survivors leaping to the same vertex trying to create a new colony at it, only one succeeds (disregarding the number of colonies already present at that vertex), the others die. So in this case when a colony collapses, it is replaced by 0,1, ... or $ m $ colonies. Finaly, every vertex can have any number of independent colonies. We denote this process by $C^3(p,r,\lambda,m)$.
The next result presents a necessary and sufficient condition for population survival in $C^3(p,r,\lambda,m)$.
\begin{theorem} \label{th:dispesp1} The process $C^3(p,r,\lambda,m)$ survives with positive probability if and only if $$\frac{mp(1+\lambda)^2r}{(m+\lambda)(\lambda p +1)}+\frac{mp(1+\lambda)(1-r)}{m+ \lambda p} > 1.$$ \end{theorem} The following result shows that the extinction probability for the process $C^3(p,r,\lambda,m)$ can be computed as the root of a polynomial of degree $m$.
\begin{theorem}\label{th:dispesp2} Let $\rho_3(r)$ be the probability of population extinction in $C^3(p,r,\lambda,m)$. Then $\rho_3(r)$ is the smallest non-negative solution of $$\psi(s):=r\psi _G(s)+(1-r)\psi _B(s)=s,$$ where {\small \[ \psi _B(s):=\frac{q}{1+\lambda p}+\frac{m(1+\lambda)}{\lambda}\sum_{k=1}^m {m \choose k}\left[\frac{-\lambda p s}{m(1+\lambda p)}\right]^k\sum_{j=0}^k {k \choose j}\frac{(-1)^j j^k}{m(1+\lambda p)-\lambda p j}, \]} {\small \[ \psi_G(s):=\frac{q}{1+\lambda p}+\frac{(1+\lambda)ps}{\lambda p +1}\sum_{k=1}^m {m \choose k}\left[\frac{-\lambda s}{m(1+\lambda )}\right]^{k-1}\sum_{j=0}^k {k \choose j}\frac{(-1)^{j-1}j^k}{m(1+\lambda )-\lambda j}.\]} \end{theorem}
\begin{exa} Consider $C^3(2/3,r,1,3).$ Then $$\psi(s)=\left(\frac{126 r}{3575}+\frac{32}{715}\right)s^3 +\left(\frac{138r}{3575}+\frac{144}{715}\right)s^2 +\left(\frac{36}{65}-\frac{24r}{325}\right)s +\frac{1}{5}.$$ Therefore, the smallest non-negative solution for $\psi(s)=s$ is given by $$\rho_3(r)=\frac{-440-132 r+\sqrt{22(14000+9375 r+792 r^2)}}{2 (80+63 r)}.$$ \end{exa}
\section{Dispersion as a survival strategy}
Towards being able to evaluate dispersion as a survival strategy we define $$\lambda^i(p,r):=\inf\{\lambda: \mathbb{P}[ C^i(p,r,\lambda) \text{ survives}]>0 \}, \quad \text{for } i=1,2$$ $$\text{and} \quad \lambda^3(p,r,m):=\inf\{\lambda: \mathbb{P}[ C^3(p,r,\lambda,m) \text{ survives}]>0 \}. $$
Observe that for $i=1,2$, when $0<\lambda^i(p,r)<\infty$ for $0<p<1,$ the graph of $\lambda^i(p,r)$ splits the parametric space $\lambda \times p$ into two regions. For those values of $(\lambda,p)$ above the curve $\lambda^i(p,r)$ there is survival in $C^i(p,r,\lambda)$ with positive probability, and for those values of $(\lambda,p)$ below the curve $\lambda^i(p,r)$ extinction occurs in $C^i(p,r,\lambda)$ with probability 1. The analogous happens also for i=3 and any $m$.
Next we establish some properties of $\lambda^2(p,r)$ and $\lambda^3(p,r,m).$ \begin{prop}\label{prop-disp-est} Let $0\leq r \leq 1$ and $0<p<1.$ Then, \begin{itemize} \item[$(i)$] $0 < \lambda^2(p,r) < \lambda^3(p,r,m+1) < \lambda^3(p,r,m) < \infty,$ for all $ m\geq 2.$ Besides $\lambda^3(p,r,1)=\infty.$ \item[$(ii)$] $\displaystyle\lim_{m\rightarrow\infty}\lambda^3(p,r,m)=\lambda^2(p,r).$ \end{itemize} \end{prop}
\begin{obs} From standard coupling arguments one can show the expected monotonocity relationship.
\noindent If $p_1 > p_2$ then \begin{align*} \lambda^i(p_1, r) & \leq \lambda^i(p_2, r), \ i=1,2 \\ \lambda^3(p_1, r,m) & \leq \lambda^3(p_2, r,m). \end{align*} If $r_1 > r_2$ then \begin{align*} \lambda^i(p, r_1) & \leq \lambda^i(p, r_2), \ i=1,2 \\ \lambda^3(p, r_1,m) & \leq \lambda^3(p, r_2,m). \end{align*} \end{obs}
For what follows $0<p<1.$ From Theorem \ref{th:semdisp} it follows that if $r<1$ then $\lambda^1(p,r)=\infty,$ and from Proposition \ref{prop-disp-est} we obtain that $$\lambda^2(p,r)<\lambda^3(p,r,m)<\lambda^1(p,r),$$ for all $m\geq2$. Then, provided binomial effect may strike $(r<1)$, dispersion is a good scenary for population survival either with or without spatial restrictions. \\
When binomial effect is not present $(r=1)$, which means, only geometric effect is present, it is simple to compute $\lambda^1(p,1),$ $\lambda^2(p,1)$ and $\lambda^3(p,1,m)$. From Theorems \ref{th:semdisp}, \ref{th:disp1} and \ref{th:dispesp1}, we have that
\begin{eqnarray*} \lambda^1(p,1)&=&\frac{1-p}{p},\\ \lambda^2(p,1)&=&\sqrt{\frac{1}{4}+\frac{1-p}{p}} -\frac{1}{2},\\ \lambda^3(p,1,m)&=&\frac{1-mp+\sqrt{(1-mp)^2+4m(m-1)p(1-p)}}{2p(m-1)}. \end{eqnarray*}
\noindent When $r=1$ (pure geometric effect) $\lambda^2(p,1) < \lambda^1(p,1).$ However, dispersion is not always a better scenary for population survival, as one can see in Figure~\ref{fig:sub1}. Observe that \[\lambda^3(p,1,m)\leq\lambda^1(p,1) \iff p\leq 1-\frac{1}{m-1}.\] Therefore, under a pure geometric effect, dispersion is an advantage or not for population survival depending on both $m$, the spatial restrictions, and $p$, the probability that an individual, when exposed to catastrophe, survives. See Figure \ref{fig:sub2}.
\begin{figure}
\caption{Graphics of $\lambda^1(p,1),\lambda^2(p,1), \lambda^3(p,1,5)$ }
\label{fig:sub1}
\end{figure}
\begin{figure}
\caption{Curve $p=1-(m-1)^{-1}.$ Best strategy for survival, when r=1, provided the spatial restrictions $(m)$ and the probability that an individual survives when facing a collapse $(p)$.}
\label{fig:sub2}
\end{figure}
\section{Proofs}
Theorem~\ref{th:semdisp} is part of Theorem 3.1 and Theorem 3.2 in Artalejo \textit{et al.}~\cite{AEL2007}. They work hard with the moment generating functions of the first excursion until 0 (the empty state) when the process (binomial and geometric catastrophes) starts from 1 individual. Here we present an alternative proof for $r<1$ by the use of Foster's theorem, enunciated next. For a proof of Foster's theorem see Fayolle~\textit{et. al. }\cite[Theorem 2.2.3]{FMM1995}.
\begin{theorem}[Foster's theorem] Let $\{W_n\}_{n\geq 0}$ be an irreducible and aperiodic Markov chain on countable state space $\mathcal{A}=\{\alpha_i,\ i\geq0\}.$ Then, $\{W_n\}_{n\geq 0}$ is ergodic if and only if there exists a positive function $f(\alpha), \ \alpha\in\mathcal{A},$ a number $\epsilon>0$ and a finite set $A\subset\mathcal{A}$ such that
$$\mathbb{E}[f(W_{n+1})-f(W_{n}) \ | \ W_n=\alpha_j]\leq -\epsilon, \quad \alpha_j\notin A,$$
$$\mathbb{E}[f(W_{n+1}) \ | \ W_n=\alpha_i] < \infty, \quad \alpha_i\in A.$$ \end{theorem}
Next we present the proof of Theorem~\ref{th:semdisp}.
\begin{proof}[Proof of Theorem \ref{th:semdisp}]
Let $\{Y_n\}_{n\geq 0}$ be a discrete-time Markov chain embedded on $C^1(p,r,\lambda),$ with transition probabilities given by $$\begin{array}{ll} P_{i,i+1}=\displaystyle\frac{\lambda}{\lambda +1},& \ i\geq 0, \\ \\ P_{i,j}=\displaystyle\frac{r\mu_{ij}^G+(1-r)\mu_{ij}^B}{\lambda +1}, & \ 0\leq j\leq i.\\ \end{array}$$
Ergodicity of $\{Y_n\}$ implies that the time until extintion of $C^1(p,r,\lambda)$ has finite mean.
Observe that $\{Y_n\}$ is irreducible and aperiodic. We use Foster's theorem to show that $\{Y_n\}_{n\geq 0}$ is ergodic for $0 \leq r<1$, $0<p<1$ and $\lambda>0$. Consider the function $f:\mathbb{N}\rightarrow \mathbb{R}^+$ defined by $f(i)=i+1$, $\epsilon>0$ and the set $$A:=\left\{i\in \mathbb{N}: \frac{\lambda-i(1-r)q}{1+\lambda}-\frac{rq(1-q^{i})}{p(1+\lambda)} >-\epsilon\right\}.$$
For $0 \leq r<1,$ $0<p<1$ and $\lambda>0,$ the set $A$ is finite. Moreover we have that
\noindent $\begin{array}{lll}
\bullet \ \mathbb{E}[f(Y_{n+1})&-&f(Y_{n}) \ | \ Y_n=i]=\displaystyle\sum_{j=0}^{i+1}[f(j)-f(i)]P_{i,j}\\ \\ &=& \displaystyle\frac{\lambda}{1+\lambda}+\sum_{j=0}^i (j-i)\left[ \frac{r\mu_{ij}^G+(1-r)\mu_{ij}^B}{1+\lambda}\right]\\ \\ &=&\displaystyle\frac{\lambda}{1+\lambda}+\frac{1}{1+\lambda}\left[-riq^i+r\sum_{j=1}^i (j-i)pq^{i-j} \right. \\ \\ && \ \left. + \ (1-r)\displaystyle\sum_{j=0}^i(j-i){i\choose j}p^jq^{i-j} \right]\\ \\ &=& \displaystyle\frac{\lambda-i(1-r)q}{1+\lambda}-\frac{rq(1-q^{i})}{p(1+\lambda)}\\ \\ &\leq & -\epsilon \quad \text{for } i\notin A. \end{array}$\\ \\
\noindent$\begin{array}{lll}
\bullet \ \mathbb{E}[f(Y_{n+1}) &\ |& \ Y_n=i] = \displaystyle\sum_{j=0}^{i+1}f(j)P_{i,j} \le (i+2)^2 < \infty \text{ for } i\in A.
\end{array}$\\
\noindent It follows from Foster's theorem that $\{Y_n\}$ is ergodic and that concludes the proof. \end{proof}
Seeking the proof of the other results we define the following auxiliary process. \\
\noindent \textbf{Auxiliary process $(Z_n^{r,i})_{n\geq 0}$}:
\noindent Consider $C^2(p,r,\lambda)$ and $C^3(p,r,\lambda,m)$. We define $Z_0^{r,i}=1$ for $i=2,3$, the number of colonies present at time 0 in each model. As soon as it collapses, $Z_1^{r,i}$, a random number of colonies will be created, the first generation. Each one of these colonies will give birth (at different times) to a random number of new colonies, the second generation. Let us define this quantity by $Z_2^{r,i}$. In general, for $n \geq 1$, if $Z_{n-1}^{r,i} = 0$ then $Z_n^{r,i}=0$. On the other hand, if $Z_{n-1}^{r,i} \geq 1$ then $Z_n^{r,i}$ is the number of colonies generated by the $(n-1)-th$ generation of colonies.
From the fact that the numbers of descendants of different colonies are independent and have the same distribution, we claim that $\{Z_n^{r,i}\}_{n \in \mathbb{N}}$ is a Galton-Watson process.
\begin{obs}\label{AuxPro} For $i=2,3,$ observe that $C^i(p,r,\lambda)$ dies out if and only if $\{Z_n^{r,i}\}_{n \in \mathbb{N}}$ dies out, which in turn happens almost surely if and only if $\mathbb{E}[Z_1^{r,i}] \leq 1$. The probability of extinction for $\{Z_n^{r,i}\}_{n \in \mathbb{N}}$ is the smallest non-negative solution of $\phi_{r,i}(s)=s,$ where $\phi_{r,i}(s)$ is the probability generating function of $Z_1^{r,i}$. \end{obs}
\begin{lem}\label{L:disp} The probability generating function of $Z_1^{r,2}$ is given by: \[ \phi_{r,2}(s)=\frac{1}{1+\lambda p}\left[q+\frac{r(\lambda +1)ps}{1+\lambda -\lambda s}+\frac{(1-r)(\lambda +1)ps}{1+\lambda p - \lambda p s}\right] \] and \[ \mathbb{E}[Z_1^{r,2}]=\frac{p(\lambda+1)^2r}{\lambda p+1} +p(\lambda+1)(1-r).\] \end{lem}
\begin{proof} $Z_1^{r,2}$ is the number of colonies in the first generation of $C^2(p,r,\lambda).$ Denote $Z_B:=Z_1^{0,2}$ and $Z_G:=Z_1^{1,2}.$ Firstly we show that
\begin{eqnarray}\label{E1:lemaaux1} \mathbb{P}[Z_B=k]&=&\left\{\begin{array}{ll} \displaystyle\frac{1+\lambda}{\lambda (1+\lambda p)}\left(\frac{\lambda p}{1+\lambda p}\right)^ k,& k\geq 1
\\ \displaystyle\frac{q}{1+\lambda p}, & k=0.\end{array}\right. \end{eqnarray}
\begin{eqnarray}\label{E2:lemaaux1}
\mathbb{P}[Z_G=k]&=&\left\{\begin{array}{ll} \displaystyle\frac{p}{1+\lambda p}\left(\frac{\lambda }{1+\lambda }\right)^ {k-1},& k\geq 1
\\ \displaystyle\frac{q}{1+\lambda p}, & k=0.\\\end{array}\right. \end{eqnarray}
\begin{defn} Let us consider the following random variables
\begin{itemize} \item $T$ the lifetime of the collony until the collapse time; \item $f_T(t)$ the density of the random variable T; \item $X_T$ the amount of individuals created in a collony until it collapes. \end{itemize} \end{defn}
Observe that \begin{eqnarray} \label{eq: ZBEZG}
\mathbb{P}[Z_B=k]=\int_0^{\infty} f_T(t) \sum_{n= 0 \vee k-1}^{\infty} \mathbb{P}(X_T=n|T=t)\mathbb{P}(Z_B=k|X_T=n; T=t)dt. \end{eqnarray}
Then, for $k=0$, we have that \[ \mathbb{P}[Z_B=0]=\int_0^\infty e^{-t} \sum_{n=0}^\infty \frac{e^{-\lambda t}(\lambda t)^n}{n!} q^{n+1} dt=q\int_0^\infty e^{-(\lambda p+1)t} dt=\frac{q}{1+\lambda p}. \]
For $k\geq 1,$ \begin{eqnarray*} \mathbb{P}[Z_B=k]&=&\displaystyle\int_0^\infty e^{-t} \sum_{n=k-1}^\infty \frac{e^{-\lambda t} (\lambda t)^n}{n!}{n+1 \choose k} p^kq^{n+1-k} dt \\ &=&q\left(\displaystyle\frac{p}{q}\right)^k \displaystyle\sum_{n=k-1}^\infty {n+1 \choose k} \frac{(\lambda q)^n}{n!} \int_0^\infty e^{-(\lambda +1)t} \ t^n dt \\ &=&q\left(\displaystyle\frac{p}{q}\right)^k \displaystyle\sum_{n=k-1}^\infty {n+1 \choose k} \frac{(\lambda q)^n}{n!} \frac{\Gamma(n+1)}{(\lambda +1)^{n+1}} \\ &=&\displaystyle\frac{q}{\lambda +1} \left(\frac{p}{q}\right)^k \displaystyle\sum_{n=k-1}^\infty {n+1 \choose k} \left(\frac{\lambda q}{\lambda +1}\right)^{n} \\ &=&\displaystyle\frac{q}{\lambda +1} \left(\frac{p}{q}\right)^k \left(\frac{\lambda q}{\lambda +1}\right)^{k-1}\displaystyle\sum_{j=0}^\infty {j+k \choose k} \left(\frac{\lambda q}{\lambda +1}\right)^{j} \\ &=&\displaystyle\frac{q}{\lambda +1} \left(\frac{p}{q}\right)^k \left(\frac{\lambda q}{\lambda +1}\right)^{k-1} \left(1-\frac{\lambda q}{\lambda +1}\right)^{-(k+1)}\\ &=&\displaystyle\frac{1+\lambda}{\lambda (1+\lambda p)}\left(\frac{\lambda p}{1+\lambda p}\right)^ k. \end{eqnarray*}
Similarly to~(\ref{eq: ZBEZG}), we obtain the distribution of $Z_G$. First observe that $\mathbb{P}[Z_B=0]=\mathbb{P}[Z_G=0]$. Besides, for $k\geq 1,$ \begin{eqnarray*} \mathbb{P}[Z_G=k]&=&\displaystyle\int_0^\infty e^{-t} \sum_{n=k-1}^\infty \frac{e^{-\lambda t}(\lambda t)^n}{n!} pq^{n+1-k} dt \\ &=&pq^{1-k}\displaystyle\sum_{n=k-1}^\infty \frac{(q\lambda )^{n}}{n!} \int_0^\infty e^{-(\lambda +1)t} \ t^ndt \\ &=& pq^{1-k}\displaystyle\sum_{n=k-1}^\infty \frac{(q\lambda )^{n}}{n!} \frac{\Gamma (n+1)}{(\lambda +1)^{n+1}} \\ &=&\displaystyle\frac{pq^{1-k}}{\lambda +1} \displaystyle\sum_{n=k-1}^\infty \left(\frac{q\lambda }{\lambda +1}\right)^n \\ &=&\displaystyle\frac{pq^{1-k}}{\lambda +1} \left(\frac{q\lambda }{\lambda +1}\right)^{k-1} \displaystyle\sum_{j=0}^\infty \left(\frac{q\lambda }{\lambda +1}\right)^j \\ &=&\displaystyle\frac{p}{1+\lambda p}\left(\frac{\lambda }{\lambda +1 }\right)^ {k-1}. \end{eqnarray*}
By (\ref{E1:lemaaux1}) we obtain the probability generating function of $Z_B$, $$\begin{array}{lll} \phi_B(s)&=&\mathbb{E}[s^{Z_B}]=\displaystyle\sum_{k\geq 0} \mathbb{P}[Z_B=k] \ s^k \\ &=& \displaystyle\frac{q}{1+\lambda p} + \frac{1+\lambda}{\lambda(1+\lambda p)}\sum_{k\geq 1} \left(\frac{\lambda p s}{1+\lambda p}\right)^{k}
\\ &=&\displaystyle\frac{1}{1+\lambda p}\left[q+\frac{(\lambda +1)ps}{1+\lambda p -\lambda ps}\right]. \end{array}$$
Besides, from (\ref{E2:lemaaux1}), we obtain the probability generating function of $Z_G$, $$\begin{array}{lll} \phi_G(s)&=&\mathbb{E}[s^{Z_G}]=\displaystyle\sum_{k\geq 0} \mathbb{P}[Z_G=k] \ s^k \\ &=& \displaystyle\frac{q}{1+\lambda p} + \frac{sp}{1+\lambda p}\sum_{k\geq 1} \left(\frac{\lambda s}{1+\lambda }\right)^{k-1}
\\ &=&\displaystyle\frac{1}{1+\lambda p}\left[q+\frac{(\lambda +1)ps}{1+\lambda -\lambda s}\right]. \end{array}$$
Finaly, the desired result follows after we observe that $$\phi_{r,2}(s)=r\phi_G(s)+(1-r)\phi_B(s),$$ and computing $\mathbb{E}[Z_1^{r,2}]=\phi_{r,2}'(1).$\\
\end{proof}
\begin{lem} \label{L: disp2} The probability generating function of $Z_1^{r,3}$ is given by:
\[ \psi_{r,3}(s)=r\psi_{G}(s)+(1-r)\psi_{B}(s),\] where {\small $$\psi _B(s):=\frac{q}{1+\lambda p}+\frac{m(1+\lambda)}{\lambda}\sum_{k=1}^m {m \choose k}\left[\frac{-\lambda p s}{m(1+\lambda p)}\right]^k\sum_{j=0}^k {k \choose j}\frac{(-1)^j j^k}{m(1+\lambda p)-\lambda p j},$$} {\small $$\psi_G(s):=\frac{q}{1+\lambda p}+\frac{(1+\lambda)ps}{\lambda p +1}\sum_{k=1}^m {m \choose k}\left[\frac{-\lambda s}{m(1+\lambda )}\right]^{k-1}\sum_{j=0}^k {k \choose j}\frac{(-1)^{j-1}j^k}{m(1+\lambda )-\lambda j}.$$} Furthermore,
$$\mathbb{E}[Z_1^{r,3}]=\frac{mp(\lambda +1)^2r}{(m+\lambda)(\lambda p +1)}+\frac{mp(\lambda +1)(1-r)}{m+ \lambda p}. $$
\end{lem}
\begin{proof} Consider $C^3(p,r,\lambda,m)$ starting from one colony placed at some vertex $x \in \mathcal{G}_m$. Besides the quantity already defined $Z_1^{r,3},$ consider also $Z$ the number of individuals that survived right after the collapse, before they compete for space.
From the definition of $C^3(p,r,\lambda,m)$ it follows that \begin{equation}\label{E} \mathbb{P}[Z=j]=r\mathbb{P}[Z_G=j]+(1-r)\mathbb{P}[Z_B=j], \end{equation} where $Z_B$ and $Z_G$ are the random variables defined in (\ref{E1:lemaaux1}) and (\ref{E2:lemaaux1}), respectively. By other side, for $k\in\{1,\ldots,m\}$ and $j\geq k$, observe that
$$\mathbb{P}[Z_1^{r,3}=k|Z=j]={m \choose k}\frac{T(j,k)}{m^j}.$$ By the inclusion-exclusion principle, $T(j,k)=\sum_{i=0}^k{k \choose i}(-1)^i(k-i)^j$ is the number of surjective functions whose domain is a set with $j$ elements and whose codomain is a set with $k$ elements. See Tucker~\cite{Tucker} p. 319.
Then, for $k\in\{1,\ldots,m\},$ \begin{eqnarray}\label{E4: lemaaux2} \mathbb{P}[Z_1^{r,3}=k]&=&r\sum_{j=k}^\infty {m \choose k}\frac{T(j,k)}{m^j}\mathbb{P}[Z_G=j] \nonumber\\ &&+(1-r)\sum_{j=k}^\infty {m \choose k}\frac{T(j,k)}{m^j}\mathbb{P}[Z_B=j]. \end{eqnarray}
By (\ref{E1:lemaaux1}), we have that\\
{\small $\displaystyle\sum_{j=k}^\infty {m \choose k}\frac{T(j,k)}{m^j}\mathbb{P}[Z_B=j]$ \begin{eqnarray}\label{E5: lemaaux2} &=&{m \choose k} \frac{1+\lambda}{\lambda(\lambda p+1)}\sum_{j=k}^\infty \left[\frac{\lambda p}{m(\lambda p+1)}\right]^{j}T(j,k)\nonumber\\ &=&{m \choose k} \frac{1+\lambda}{\lambda(\lambda p+1)} \left[\frac{\lambda p}{m(\lambda p+1)}\right]^{k}\sum_{j=0}^\infty \left[\frac{\lambda p}{m(\lambda p+1)}\right]^{j}T(j+k,k)\nonumber\\ &=& {m \choose k} \frac{1+\lambda}{\lambda(\lambda p+1)} \left[\frac{\lambda p}{m(\lambda p+1)}\right]^{k}\sum_{j=0}^\infty \left[\frac{\lambda p}{m(\lambda p+1)}\right]^{j}\sum_{i=0}^k {k \choose i}(-1)^i(k-i)^{j+k}\nonumber\\ &=& {m \choose k} \frac{1+\lambda}{\lambda(\lambda p+1)} \left[\frac{\lambda p}{m(\lambda p+1)}\right]^{k}\sum_{i=0}^k {k \choose i}(-1)^i(k-i)^{k}\sum_{j=0}^\infty \left[\frac{\lambda p(k-i)}{m(\lambda p+1)}\right]^{j}\nonumber\\ &=&{m \choose k} \frac{m(1+\lambda)}{\lambda} \left[\frac{\lambda p}{m(\lambda p+1)}\right]^{k}\sum_{i=0}^k {k\choose i}\frac{(-1)^i(k-i)^k}{m(\lambda p+1)-\lambda p (k-i)}. \end{eqnarray}}
Similarly, by (\ref{E2:lemaaux1}), we have that\\
{\small $\displaystyle\sum_{j=k}^\infty {m \choose k}\frac{T(j,k)}{m^j}\mathbb{P}[Z_G=j]$ \begin{eqnarray}\label{E6: lemaaux2} &=&{m \choose k} \frac{p}{m(\lambda p+1)}\sum_{j=k}^\infty \left[\frac{\lambda}{m(\lambda+1)}\right]^{j-1}T(j,k)\nonumber\\ &=&{m \choose k} \frac{p}{m(\lambda p+1)}\left[\frac{\lambda}{m(\lambda+1)}\right]^{k-1}\sum_{j=0}^\infty \left[\frac{\lambda}{m(\lambda+1)}\right]^{j}T(j+k,k)\nonumber\\ &=&{m \choose k} \frac{p}{m(\lambda p+1)}\left[\frac{\lambda}{m(\lambda+1)}\right]^{k-1}\sum_{j=0}^\infty \left[\frac{\lambda}{m(\lambda+1)}\right]^{j}\sum_{i=0}^k {k \choose i}(-1)^i(k-i)^{j+k}\nonumber\\ &=&{m \choose k} \frac{p}{m(\lambda p+1)}\left[\frac{\lambda}{m(\lambda+1)}\right]^{k-1}\sum_{i=0}^k {k \choose i}(-1)^i(k-i)^{k}\sum_{j=0}^\infty \left[\frac{\lambda(k-i)}{m(\lambda+1)}\right]^{j}\nonumber\\ &=&{m \choose k}\frac{(1+\lambda)p}{\lambda p +1}\left[\frac{\lambda }{m(1+\lambda )}\right]^{k-1}\sum_{i=0}^k {k \choose i}\frac{(-1)^i (k-i)^k}{m(1+\lambda )-\lambda(k-i)}. \end{eqnarray}}
Finally, observe that $\mathbb{P}[Z_1^{r,3}=0]=\mathbb{P}[Z=0]=q/(1+\lambda p)$. With (\ref{E4: lemaaux2}),(\ref{E5: lemaaux2}) and (\ref{E6: lemaaux2}) we obtain the probability generating function of $Z_1^{r,3}$.\\
To compute $\mathbb{E}[Z_1^{r,3}],$ consider enumerating each neighbour of the initial vertex $x$, from 1 to $m$. Next we describe $Z_1^{r,3}=\sum_{i=1}^m I_{i},$ where $I_{i}$ is the indicator function of the event \{A new colony is created in the first generation at the $i-th$ neighbour vertex of $x$ \}. Therefore, \begin{eqnarray}\label{E1: lemaaux2} \mathbb{E}[Z_1^{r,3}]=\sum_{i=1}^m \mathbb{P}[I_{i}=1]=m\mathbb{P}[I_{1}=1]. \end{eqnarray}
Observe that \begin{eqnarray}
\mathbb{P}[I_{1}=1|Z=k] =1-\left(\frac{m-1}{m}\right)^k \nonumber \end{eqnarray} and that by using (\ref{E}) we have that \begin{eqnarray}\label{E2: lemaaux2} \mathbb{P}[I_{1}=1]&=&r\sum_{k=1}^\infty\left[1-\left(\frac{m-1}{m}\right)^k\right]\mathbb{P}[Z_G=k] \nonumber \\ &&+(1-r)\sum_{k=1}^\infty\left[1-\left(\frac{m-1}{m}\right)^k\right]\mathbb{P}[Z_B=k]. \end{eqnarray}
Substituting (\ref{E1:lemaaux1}) and (\ref{E2:lemaaux1}) in (\ref{E2: lemaaux2}) one can see that \begin{eqnarray}\label{E3: lemaaux2} \mathbb{P}[I_1=1]&=&\frac{p(\lambda +1)^2r}{(m+\lambda)(\lambda p +1)}+\frac{p(\lambda +1)(1-r)}{m+ \lambda p}. \end{eqnarray}
Finally, plugging (\ref{E3: lemaaux2}) into (\ref{E1: lemaaux2}) we obtain the desired result. \end{proof}
\begin{proof}[Proofs of Theorems \ref{th:disp1} and \ref{th:dispesp1}] From Remark~\ref{AuxPro} one can see that $C^i(p,r,\lambda)$ survives if and only if $\mathbb{E}[Z_n^{r,i}]>1.$ From Lemmas~\ref{L:disp} and \ref{L: disp2} the result follows. \end{proof}
\begin{proof}[Proofs of Theorems \ref{th:disp2} and \ref{th:dispesp2}] From Remark~\ref{AuxPro} we have that the probabilities of extinction, $\rho_2(r)$ and $\rho_3(r)$, of $C^2(p,r,\lambda)$ and $C^3(p,r,\lambda,m)$, are the smallest solution in $[0,1]$ of $\phi_{r,i}(s)=s$ for $i=2$ and $3$ respectively. The desired results follow from Lemmas~ \ref{L:disp} and \ref{L: disp2}. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop-disp-est} $(i)$] First we define the following functions \begin{eqnarray*} f_m(\lambda)&:=&\frac{mp(1+\lambda)^2r}{(m+\lambda)(\lambda p +1)}+\frac{mp(1+\lambda)(1-r)}{m+ \lambda p} ,\\ f(\lambda)&:=&\frac{p(\lambda+1)^2r}{\lambda p+1} +p(\lambda+1)(1-r). \end{eqnarray*} From Theorems \ref{th:disp1} and \ref{th:dispesp1} it follows that $$\lambda^2(p,r)=\inf\{\lambda: f(\lambda)>1\},$$ $$\lambda^3(p,r,m)=\inf\{\lambda: f_m(\lambda)>1\}.$$
Observe that $f_m$ and $f$ are continuous functions on $[0,\infty),$ such that $f_m(0)=f(0)=p<1,$ $\displaystyle\lim_{\lambda\rightarrow \infty}f(\lambda)=\infty$ and $\displaystyle\lim_{\lambda\rightarrow \infty}f_m(\lambda)=m.$\\
Moreover, $\{f_m\}_{m \geq 1}$ is a strictly increasing sequence of strictly increasing functions on $(0,\infty)$ such that $\displaystyle\lim_{m\rightarrow\infty} f_m(\lambda)=f(\lambda)$. Similarly, $f$ is a strictly increasing function.
Then, from the intermediate value theorem and the strict monotonicity of $f$ we have that there is a unique $\lambda_*\in(0,\infty)$ such that $f(\lambda_*)=1.$ Moreover, from the definition of $\lambda^2(p,r)$ and the continuity of $f$, we have that \begin{eqnarray}\label{valorcritico} f(\lambda)=1 &\iff& \lambda = \lambda^2(p,r). \end{eqnarray} Thus, $\lambda_*= \lambda^2(p,r).$ Similarly, for $m\geq 2,$ we obtain that \[\lambda^3(p,r,m)\in(0,\infty)\] and \begin{eqnarray}
f_m(\lambda)=1 &\iff& \lambda = \lambda^3(p,r,m). \nonumber \end{eqnarray} Besides, from the strict monotonicity of $f_1$, it follows that \[\lambda^3(p,r,1)=\infty.\]
In order to show that $\lambda^3(p,r,m) > \lambda^3(p,r,m+1)$ for all $m \geq 2$ let us assume that $\lambda^3(p,r,m) \leq \lambda^3(p,r,m+1)$ for some $m \geq 2$ and proceed by contradiction. Note that \[ 1=f_m(\lambda^3(p,r,m))\leq f_m(\lambda^3(p,r,m+1))<f_{m+1}(\lambda^3(p,r,m+1))=1 \] \noindent which is cleary a contradiction. Analogously one can show that \[\lambda^2(p,r)<\lambda^3(p,r,m) \textrm{ for all } m\geq 1.\] \end{proof}
\begin{proof}[Proof of Proposition \ref{prop-disp-est} $(ii)$] Let us restrict the domain of the functions $f_m$ and $f$ to $[0,\lambda^3(p,r,2)].$ Observe that $f_m$ and $f$ are continuous functions, that $\displaystyle\lim_{m\rightarrow\infty}f_m=f$ and that $f_m(\lambda)<f_{m+1}(\lambda)$ for all $\lambda \in [0,\lambda^3(p,r,2)].$ Then, from Theorem 7.13 in Rudin~\cite{Rudin} we have that $f_m$ converges uniformly to $f$ on $[0,\lambda^3(p,r,2)]$.
From $(i)$ it follows that $\lambda^3(p,r,m) \in [0,\lambda^3(p,r,2)]$ for all $m\geq 2$ and the existence of $\theta:=\displaystyle\lim_{m\rightarrow\infty} \lambda^3(p,r,m).$ Then, from the uniform convergence of $f_m$ to $f$, it follows that $f(\theta)=\displaystyle\lim_{m\rightarrow\infty} f_{m}(\lambda^3(p,r,m))=1,$ (see Rudin \cite[exercise 9, chapter 7]{Rudin}). Finaly the result follows from (\ref{valorcritico}). \end{proof}
\section{Acknowledgments} The authors are thankful to Rinaldo Schinazi and Elcio Lebensztayn for helpful discussions about the model. V. Junior and A. Rold\'an wish to thank the Instituto de Matem\'atica e Estat\'{\i}stica of Universidade de S\~ao Paulo for the warm hospitality during their scientific visits to that institute. The authors are thankful for the two anonymous referees for a careful reading and many suggestions and corrections that greatly helped to improve the paper.
\end{document} | arXiv |
Learn more. See our Privacy Policy and User Agreement for details. Maximum Likelihood (1) Likelihood is a conditional probability. Interval estimators, such as confidence intervals or prediction intervals, aim to give a range of plausible values for an unknown quantity. We define three main desirable properties for point estimators. Define bias; Define sampling variability This is a case where determining a parameter in the basic way is unreasonable. Now, suppose that we would like to estimate the variance of a distribution $\sigma^2$. Moreover, statistics concepts can help investors monitor, Hypothesis Testing is a method of statistical inference. These are: 1) Unbiasedness: the expected value of the estimator (or the mean of the estimator) is simply the figure being estimated. An estimate is a specific value provided by an estimator. There are four main properties associated with a "good" estimator. Asymtotic Properties of Estimators: Plims and Consistency (PPTX, Size: 1.1MB) Sufficient Condition for Consistency (PPTX, Size: 143KB) Asymptotic Properties of Estimators: The Use of Simulation (PPTX, Size: 331KB) The Central limit Theorem (PPTX, Size: 819KB) reset + A - A; About the book. Step 1 — Identify a Base Story. IGNOU MA ECONOMICS MICROECONOMICS MEC 001 // JUNE 2014 PAPER SOLUTIONS, No public clipboards found for this slide. Burt Gerstman\Dropbox\StatPrimer\estimation.docx, 5/8/2016). 2 Unbiased Estimator As shown in the breakdown of MSE, the bias of an estimator is defined as b(θb) = E Y[bθ(Y)] −θ. Properties of Point Estimators 2. properties from a statistical point of view: the seemingly random variations of asset prices do share some quite non-trivial statistical properties. You can also check if a point estimator is consistent by looking at its corresponding expected value and varianceVariance AnalysisVariance analysis can be summarized as an analysis of the difference between planned and actual numbers. Unbiasedness. Statistical inference is the act of generalizing from the data ("sample") to a larger phenomenon ("population") with calculated degree of certainty. of an unbiased estimator: We assume suitable smoothness conditions, including that • The region of positivity of f(x;θ) is constant in θ; • Integration and differentiation can be interchanged. sample from a population with mean and standard deviation ˙. 152 5. The act of generalizing and deriving statistical judgments is the process of inference. It produces a single value while the latter produces a range of values. Bayesian estimation 6.4. Instead, a statistician can use the point estimator to make an estimate of the population parameter. Suppose we do not know f(@), but do know (or assume that we know) that f(@) is a member of a family of densities G. The estimation problem is to use the data x to select a member of G which is some appropriate sense is close to the true f(@). Pre-Algebra 3-8 Squares and Square Roots 25 64 144 225 400 1. The method of moments of estimating parameters was introduced in 1887 by Russian mathematician Pafnuty Chebyshev. The method of Maximum likelihood (ML) ML is point estimation method with some stronger theoretical properties than OLS (Appendix 4.A on pages 110-114) The estimators of coefficients 's by OLS and ML are identical. Only once we've analyzed the sample minimum can we say for certain if it is a good estimator or not, but it is certainly a natural first choice. Density estimators aim to approximate a probability distribution. For example, in a normal distribution, the mean is considered more efficient than the median, but the same does not apply in asymmetrical distributions. For example, a researcher may be interested in knowing the average weight of babies born prematurely. Example: = σ2/n for a random sample from any population. (i.e. Slide 33 Properties of Point Estimators Consistency A point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as … Suppose we do not know f(@), but do know (or assume that we know) that f(@) is a member of a family of densities G. The estimation problem is to use the data x to select a member of G which A point estimator is a statistic used to estimate the value of an unknown parameter of a population. PERIODIC CLASSIFICATION OF ELEMENTS.ppt . If there is a function Y which is an UE of , then the ... – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow.com - id: 577274-NDFiN Introduction References Amemiya T. (1985), Advanced Econometrics. The equations derived in step one are then solved using the sample mean of the population moments. Consistency tells us how close the point estimator stays to the value of the parameter as it increases in size. Principles. The statistics estimate population values, e.g., An estimator is a method for producing a best guess about a population value. Qualities of Estimators…Statisticians have already determined the "best" way to estimate a population parameter. Several methods can be used to calculate the point estimators, and each method comes with different properties. They use the sample data of a population to calculate a point estimate or a statistic that serves as the best estimate of an unknown parameterParameterA parameter is a useful component of statistical analysis. So far, finite sample properties of OLS regression were discussed. Statistical Inferences A random sample is collected on a population to draw conclusions, or make statistical inferences, about the population. If you continue browsing the site, you agree to the use of cookies on this website. Since it would be impossible to measure all babies born prematurely in the population, the researcher can take a sample from one location. Characteristics of Estimators. Story Points in agile are a complex unit that includes three elements: risk, complexity and repetition. 7-4 Methods of Point Estimation σ2 Properties of the Maximum Likelihood Estimator 2 22 1 22 2 22 1 ˆ 1 ()ˆ ()ˆ n i i MLE of is XX n n E n bias E n σ σ σσ σ σσ = =− − = − =−= ∑ bias is negative. The two main types of estimators in statistics are point estimators and interval estimators. Estimators 3. Definition: Given two unbiased estimators ̂ and ̂ of , the efficiency of ̂ relative to ̂ As such it has a distribution. • Need to examine their statistical properties and develop some criteria for comparing estimators • For instance, an estimator should be close to the true value of the unknown parameter. Desirable Properties of an Estimator A point estimator (P.E) is a sample statistic used to estimate an unknown population parameter. The variance measures the level of dispersion from the estimate, and the smallest variance should vary the least from one sample to the other. Consistency: An estimator θˆ = θˆ(X When the estimated value of the parameter and the value of the parameter being estimated are equal, the estimator is considered unbiased. Note that for g(θ) = θ the lower bound is simply the 3a) Mendeleev's periodic … CFI is the official provider of the Financial Modeling and Valuation Analyst (FMVA)™FMVA® CertificationJoin 350,600+ students who work for companies like Amazon, J.P. Morgan, and Ferrari certification program, designed to transform anyone into a world-class financial analyst. See our User Agreement and Privacy Policy. It refers to the characteristics that are used to define a given population. This distribution of course is determined the distribution of X 1;:::;X n. If … Point Estimation & Estimators Sections 7-1 to 7-2 1/26. Method of moments estimators can be criticised because they are not uniquely defined, so that if the method is used it is necessary to choose amongst possible estimators to find ones that best suit the data being analysed. CHAPTER 6. Now, suppose that we would like to estimate the variance of a distribution $\sigma^2$. Generalized Method of Moments (GMM) refers to a class of estimators which are constructed from exploiting the sample moment counterparts of population moment conditions (some- times known as orthogonality conditions) of the data generating model. 2. The most common Bayesian point estimators are the mean, median, and mode of the posterior distribution. Qualities desirable in estimators include unbiasedness, consistency, and relative efficiency: • An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter. Rev.R.Acad. From a statistical standpoint, a given set of observations are a random sample from an unknown population.The goal of maximum likelihood estimation is to make inferences about the population that is most likely to have generated the sample, specifically the joint probability distribution of the random variables {,, …}, not necessarily independent and identically distributed. A function that is used to find an approximate value of a population parameter from random samples of the population, A parameter is a useful component of statistical analysis. Measures of Central Tendency, Variability, Introduction to Sampling Distributions, Sampling Distribution of the Mean, Introduction to Estimation, Degrees of Freedom Learning Objectives. A confidence interval is an estimate of an interval in statistics that may contain a population parameter. [Note: There is a distinction Section 6: Properties of maximum likelihood estimators Christophe Hurlin (University of OrlØans) Advanced Econometrics - HEC Lausanne December 9, 2013 5 / 207. There is a random sampling of observations.A3. 1 When we want to study the properties of the obtained estimators, it is convenient to distinguish between two categories of properties: i) the small (or finite) sample properties, which are valid whatever the sample size, and ii) the asymptotic properties, which are associated with large samples, i.e., when tends to . It refers to the characteristics that are used to define a given population. For example, when finding the average age of kids attending kindergarten, it will be impossible to collect the exact age of every kindergarten kid in the world. The act of generalizing and deriving statistical judgments is the process of inference. Properties of Point Estimators Estimators are evaluated depending on three important properties: unbiasedness consistency efficiency Chapter 7: Interval Estimation: One Population. Statistical inference . 4.2 The Sampling Properties of the Least Squares Estimators The means (expected values) and variances of random variables provide information about the location and spread of their probability distributions (see Chapter 2.3). The process of point estimation involves utilizing the value of a statistic that is obtained from sample data to get the best estimate of the corresponding unknown parameter of the population. The above discussion suggests the sample mean, $\overline{X}$, is often a reasonable point estimator for the mean. NOTATION: ^ = X (a 'hat' over a parameter represents an estimator, X is the estimator here) Prior to data collection, X is a random variable and it is the statistic of interest calculated from the data when estimating . Cienc. 14.3 Bayesian Estimation. The linear regression model is "linear in parameters."A2. 8.2.2 Point Estimators for Mean and Variance The above discussion suggests the sample mean, $\overline{X}$, is often a reasonable point estimator for the mean. A good example of an estimator is the sample mean x, which helps statisticians to estimate the population mean, μ. properties compared to other estimation procedures, yet survives as an effective tool, easily implemented and of wide generality'. The unknown population parameter is found through a sample parameter calculated from the sampled data. The maximum likelihood estimator method of point estimation attempts to find the unknown parameters that maximize the likelihood function. Properties of Estimators ME104: Linear Regression Analysis Kenneth Benoit August 13, 2012. Population distribution f(x;θ). We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. A point estimator is a statistic used to estimate the value of an unknown parameter of a population. MLE is a function of sufficient statistics. It takes a known model and uses the values to compare data sets and find the most suitable match for the data. 2.1. Point estimators are functions that are used to find an approximate value of a population parameter from random samples of the population. Again, this variation leads to uncertainty of those estimators which we … Our first choice of estimator for this parameter should prob-ably be the sample minimum. It is used to test if a statement regarding a population parameter is correct. These properties tried to study the behavior of the OLS estimator under the assumption that you can have several samples and, hence, several estimators of the same unknown population parameter. The two main types of estimators in statistics are point estimators and interval estimators. The confidence interval is used to indicate how reliable an estimate is, and it is calculated from the observed data. There are point and interval estimators. We want good estimates. It uses sample data when calculating a single statistic that will be the best estimate of the unknown parameter of the population. Statistical inference is the act of generalizing from the data ("sample") to a larger phenomenon ("population") with calculated degree of certainty. Introduction Point Estimators Interval Estimators Unbiasedness Definition: A point estimator is unbiased if its expected value is equal to the population parameter. The expected value also indicates, Variance analysis can be summarized as an analysis of the difference between planned and actual numbers. But the sample mean Y is also an estimator of the popu-lation minimum. Let's walk through each step of the estimation process with Story Points. Here the Central … The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other. I The validity and properties of least squares estimation depend very much on the validity of the classical assumptions underlying the regression model. For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. The numerical value of the sample mean is said to be an estimate of the population mean figure. Exact. What properties should it have? Assuming $0\sigma^2\infty$, by definition \begin{align}%\label{} \sigma^2=E[(X-\mu)^2]. We say that ^ is an unbiased estimator of if E( ^) = Examples: Let X 1;X 2; ;X nbe an i.i.d. Point estimation, in statistics, the process of finding an approximate value of some parameter—such as the mean (average)—of a population from random samples of the population. To keep learning and developing your knowledge of financial analysis, we highly recommend the additional CFI resources below: Become a certified Financial Modeling and Valuation Analyst (FMVA)®FMVA® CertificationJoin 350,600+ students who work for companies like Amazon, J.P. Morgan, and Ferrari by completing CFI's online financial modeling classes and training program! It starts by taking known facts about a population and then applying the facts to a sample of the population. Also, we would want our estimator to be such that, as. Prerequisites. If you wish to opt out, please close your SlideShare account. Then for any unbiased estimator T = t(X) of g(θ) it holds V(T) = V(ˆg(θ)) ≥ {g0(θ)}2/i(θ). Its quality is to be evaluated in terms of the following properties: 1. Statistics as Estimators We use sample data compute statistics. STATISTICAL INFERENCE PART II SOME PROPERTIES OF ESTIMATORS 1 SOME PROPERTIES 93, N." 2, pp 217-220, 1999 Matemáticas A CLASS OF PPS ESTIMATORS OF POPULATION VARIANCE USING Here are the reasons why. DESIRABLE PROPERTIES OF ESTIMATORS 6.1.1 Consider data x that comes from a data generation process (DGP) that has a density f( x). This produces the best estimate of the unknown population parameters. It produces a single value while the latter produces a range of values. The interval of the parameter is selected in a way that it falls within a 95% or higher probability, also known as the confidence intervalConfidence IntervalA confidence interval is an estimate of an interval in statistics that may contain a population parameter. € Point Estimator… A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point. The expected value also indicates of the estimator and the value of the parameter being estimated. For each individual item, companies assess its favorability by comparing actual costs. Recap • Population parameter θ. [Note: There is a distinction Sample means are used to estimate population means and sample proportions are used to estimate population proportions) • A population parameter can be conveyed in two ways 1. "ö ! " sa re ga ma pa da ni H LI Be B C N O F Na Mg Al Si P S Cl K Ca Cr Tl Mn Fe Co and Ni Cu Zn Y In As Se Br Rb Sr Ce and La Zr--5. 2.4.1 Finite Sample Properties of the OLS and ML Estimates of STATISTICAL INFERENCE PART II SOME PROPERTIES OF ESTIMATORS * * * LEHMANN-SCHEFFE THEOREM Let Y be a css for . These and other varied roles of estimators are discussed in other sections. Fis.Nat. 14.2.1, and it is widely used in physical science.. Estimation ¥Estimator: Statistic whose calculated value is used to estimate a population parameter, ¥Estimate: A particular realization of an estimator, ¥Types of Estimators:! Properties of estimators Unbiased estimators: Let ^ be an estimator of a parameter . The following are the main characteristics of point estimators: The bias of a point estimator is defined as the difference between the expected valueExpected ValueExpected value (also known as EV, expectation, average, or mean value) is a long-run average value of random variables. 2.1.1 Properties of Point Estimators An estimator ϑbof a parameter ϑ is a random variable (a function of rvs X1,...,Xn) and the estimate ϑbobs is a single value taken from the distribution of ϑb. An estimator is said to be unbiased if its expected value is identical with the population parameter being estimated. For the point estimator to be consistent, the expected value should move toward the true value of the parameter. (1) An estimator is said to be unbiased if b(bθ) = 0. More EXAMPLES - Physical size, shape, freezing point, boiling point, melting point, magnetism, viscosity, density, luster and many more. - interval estimate: a range of numbers, called a conÞdence It uses sample data when calculating a single statistic that will be the best estimate of the unknown parameter of the population. If you continue browsing the site, you agree to the use of cookies on this website. This video covers the properties which a 'good' estimator should have: consistency, unbiasedness & efficiency. Hence, we are only trying to generate a value that is close to the true value. The form of ... Properties of MLE MLE has the following nice properties under mild regularity conditions. What is a good estimator? Page 5.2 (C:\Users\B. The conditional mean should be zero.A4. MLE for is an asymptotically unbiased estimator … Properties of Point Estimators and Methods of Estimation Relative efficiency: If we have two unbiased estimators of a parameter, ̂ and ̂ , we say that ̂ is relatively more efficient than ̂ if ( ̂ ) ̂ . )Notations Of Estimators 4.) Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. MLE for tends to underestimate The bias approaches zero as n increases. As we shall see, many of these assumptions are rarely appropriate when dealing with data for business. 21 7-3 General Concepts of Point Estimation 7-3.1 Unbiased Estimators Definition ÎWhen an estimator is unbiased, the bias is zero. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. • Desirable properties of estimators ... 7.1 Point Estimation • Efficiency: V(Estimator) is smallest of all possible unbiased estimators. Point estimation is the opposite of interval estimation. 1. The accuracy of any particular approximation is not known precisely, though probabilistic statements concerning the accuracy of such numbers as found over many experiments can be constructed. When it exists, the posterior mode is the MAP estimator discussed in Sec. Also, the closer the expected value of a parameter is to the value of the parameter being measured, the lesser the bias is. As of this date, Scribd will manage your SlideShare account and any content you may have on SlideShare, and Scribd's General Terms of Use and Privacy Policy will apply. Or we can say that. Show that X and S2 are unbiased estimators of and ˙2 respectively. Apoint estimatordrawsinferencesaboutapopulation by estimating the value of an unknown parameter using a single value or point. Hence an estimator is a r.v. Statisticians often work with large. Estimation 2.) A good estimator, as common sense dictates, is close to the parameter being estimated. Linear regression models have several applications in real life. 6.5 The Distribution of the OLS Estimators in Multiple Regression. Point estimation can be a sample statistic. Note that Unbiasedness, Efficiency, Consistency and Sufficiency are the criteria (statistical properties of estimator) to identify that whether a statistic is "good" estimator. The sum of all variances gives a picture of the overall over-performance or under-performance for a particular reporting period. 1. In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean).More formally, it is the application of a point estimator to the data to obtain a point estimate. As such, the means and variances of b1 and b2 provide information about the range of values that b1 and b2 are likely to take. Indeed, any statistic is an estimator. $\overline{x}$ is a point estimate for $\mu$ and s is a point estimate for $\sigma$. V(Y) Y • "The sample mean is not always most efficient when the population distribution is not normal. A Point Estimate is a statistic (a statistical measure from sample) that gives a plausible estimate (or possible a best guess) for the value in question. The point estimators yield single-valued results, although this includes the possibility of single vector-valued results and results that can be expressed as a single function. 2.4.1 Finite Sample Properties of the OLS and ML Estimates of Join 350,600+ students who work for companies like Amazon, J.P. Morgan, and Ferrari, A solid understanding of statistics is crucially important in helping us better understand finance. Burt Gerstman\Dropbox\StatPrimer\estimation.docx, 5/8/2016). Scribd will begin operating the SlideShare business on December 1, 2020 The properties of OLS described below are asymptotic properties of OLS estimators. 8.2.2 Point Estimators for Mean and Variance. On the other hand, interval estimation uses sample data to calculate the interval of the possible values of an unknown parameter of a population. We saw earlier that point probabilities in continuous distributions were virtually zero. Sample Mean X , a Point Estimate for the population mean The sample mean X is a point estimate for the population mean . For example, if statisticians want to determine the mean, or average, age of the world's population, how would they collect the exact age of every person in the world to take an average? What properties should it have? Now customize the name of a clipboard to store your clips. Point Estimate vs. Interval Estimate • Statisticians use sample statistics to use estimate population parameters. It is used to of a population. A statistic used to estimate a parameter is called a point estimator or simply an estimator. It is a random variable and therefore varies from sample to sample. When we want to study the properties of the obtained estimators, it is convenient to distinguish between two categories of properties: i) the small (or finite) sample properties, which are valid whatever the sample size, and ii) the asymptotic properties, which are associated with large samples, i.e., when tends to . Such properties, common across a wide range of instruments, markets and time periods are called stylized empirical facts. Harvard University Press. Viscosity - The resistance of a liquid to flowing. The first step is to derive equations that relate the population moments to the unknown parameters. For example, the population mean μ is found using the sample mean x̅. PROPERTIES OF ESTIMATORS (BLUE) KSHITIZ GUPTA 2. Point estimation is the opposite of interval estimation. 6. Since we want our estimate to be close to ϑ, the random variable ϑbshould be centred close to ϑ and have a small variance. An estimate is a specific value provided by an estimator. Recall that for a continuous variable, the probability of assuming any particular value is zero. Statistical inference . Looks like you've clipped this slide to already. The sum of all variances gives a picture of the overall over-performance or under-performance for a particular reporting period. The endpoints of the intervals are referred to as the upper and lower confidence limits. - point estimate: single number that can be regarded as the most plausible value of! " Story points are extremely important for lean startup and Agile methodology. As in simple linear regression, different samples will produce different values of the OLS estimators in the multiple regression model. Parametric Estimation Properties 3 Estimators of a parameter are of the form ^ n= T(X 1;:::;X n) so it is a function of r.v.s X 1;:::;X n and is a statistic. A distinction is made between an estimate and an estimator. 122 4. We want good estimates. 52 2. • Obtaining a point estimate of a population parameter • Desirable properties of a point estimator: • Unbiasedness • Efficiency • Obtaining a confidence interval for a mean when population standard deviation is known • Obtaining a confidence interval for a mean when population standard deviation is … (Esp) Vol. unwieldy sets of data, and many times the basic methods for determining the parameters of these data sets are unrealistic. Since the weight of pre-term babies follows a normal distribution, the researcher can use the maximum likelihood estimator to find the average weight of the entire population of pre-term babies based on the sample data. For example, the population mean μ is found using the sample mean x̅.. Desirable properties of an estimator Consistency Unbiasedness Efficiency •However, unbiased and/or efficient estimators do not always exist •Practitioners are not particularly keen on unbiasedness. In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean).More formally, it is the application of a point estimator to the data to obtain a point estimate. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. We can build interval with confidence as we are not only interested in finding the point estimate for the mean, but also determining how accurate the point estimate is. This is in contrast to an interval estimator, where the result would be a range of plausible values (or vectors or functions). DESIRABLE PROPERTIES OF ESTIMATORS 6.1.1 Consider data x that comes from a data generation process (DGP) that has a density f( x). The statistics estimate population values, e.g., An estimator is a method for producing a best guess about a population value. STATISTICAL INFERENCE PART I POINT ESTIMATION * * * * * * * * * * P(X=0|n=2,p=1/2)=1/4 … * * * * * * * * * * * * * * * STATISTICAL INFERENCE Determining certain unknown properties of a probability distribution on the basis of a sample (usually, a r.s.) Properties of estimators (blue) 1. Most often, the existing methods of finding the parameters of large populations are unrealistic. Author(s) David M. Lane. 3-8 Squares and Square Roots Warm Up Problem of the Day Lesson Presentation Pre-Algebra Warm Up Simplify. Hypothesis testing, In statistics and probability theory, independent events are two events wherein the occurrence of one event does not affect the occurrence of another event, In statistical hypothesis testing, the p-value (probability value) is a probability measure of finding the observed, or more extreme, results, when the null, Certified Banking & Credit Analyst (CBCA)™, Capital Markets & Securities Analyst (CMSA)™, Financial Modeling and Valuation Analyst (FMVA)™, Financial Modeling and Valuation Analyst (FMVA)®, Financial Modeling & Valuation Analyst (FMVA)®. What is a good estimator? Statistical Inference has two Parts:- Estimation And Testing of Hypothesis Topics Covered In this Unit 1.) View Notes - 4.SOME PROPERTIES OF ESTIMATORS - 552.ppt from STATISTICS STAT552 at Casablanca American School. Generally, the efficiency of the estimator depends on the distribution of the population. Clipping is a handy way to collect important slides you want to go back to later. The point estimator requires a large sample size for it to be more consistent and accurate. Is the most efficient estimator of µ? On the other hand, interval estimation uses sample data to calcul… ESTIMATION 6.1. 202 Problem of the Day A Shakespearean sonnet is a poem made … Statistics as Estimators We use sample data compute statistics. So they often tend to favor estimators such that the mean square error, MSE= , is as low as possible independently of the bias. Bayesian approach to point estimation Example 6.2 Suppose that X 1;:::;X n are iid N( ;1), and that a priori ˘N(0;˝ 2) for known ˝ 2. You can change your ad preferences anytime. The unknown population parameter is found through a sample parameter calculated from the sampled data. The most efficient point estimator is the one with the smallest variance of all the unbiased and consistent estimators. For each individual item, companies assess its favorability by comparing actual costs. The next step is to draw a sample of the population to be used to estimate the population moments. 82 3. • In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data • Example- i. X follows a normal distribution, but we do not know the parameters of our distribution, namely mean (μ) and variance (σ2 ) ii. It is used to, Expected value (also known as EV, expectation, average, or mean value) is a long-run average value of random variables. A point estimation is a type of estimation that uses a single value, a sample statistic, to infer information about the population. WHAT IS AN ESTIMATOR? Properties of Point Estimators. The first one is related to the estimator's bias.The bias of an estimator $\hat{\Theta}$ tells us on average how far $\hat{\Theta}$ is from the real value of $\theta$. Page 5.2 (C:\Users\B. Application of Point Estimator Confidence Intervals. Harry F. Martz, Ray A. Waller, in Methods in Experimental Physics, 1994. ... Iron having properties similar to Cobalt and Nickel are placed in different rows. A liquid to flowing produce different values of the OLS estimators in statistics are estimators! Be summarized as an analysis of the population mean the sample mean x̅ with mean and standard ˙. Its expected value should move toward the true value of an unknown parameter of the population wide... Having properties similar to Cobalt and Nickel are placed in different rows and interval estimators, such as intervals! To uncertainty of those estimators which we harry F. Martz, Ray A. Waller, in methods in Experimental,. Periodic … our first choice of estimator for the population of cookies on this website one... Estimate the variance of a linear regression, different samples will produce different values the! About the population mean figure parameters was introduced in 1887 by Russian mathematician Pafnuty Chebyshev %. To other Estimation procedures, yet survives as an effective tool, easily implemented and wide. May contain a population value main properties associated with a `` good ''.... Its favorability by comparing actual costs act of generalizing and deriving statistical judgments the... Show you more relevant ads close to the population, the bias is zero statistic. For tends to underestimate the bias approaches zero as n increases basic methods for determining the of... Of Page 5.2 ( C: \Users\B also indicates of the parameter being estimated or an... Being estimated b ( bθ ) = 0 when the population functions that are used to how... Parameter and the value of the estimator and the value of the OLS and ML estimates of 5.2! An unknown parameter of a distribution $ \sigma^2 $ this website sets are unrealistic popu-lation minimum the data. Îwhen an estimator of the classical assumptions underlying the regression model an effective tool, easily and! Estimation depend very much on the distribution of the population mean figure that, as leads to of. Mec 001 // JUNE 2014 PAPER SOLUTIONS, No public clipboards found for this parameter prob-ably. Can help investors monitor, Hypothesis Testing is properties of point estimators ppt case where determining a parameter smallest. Below are asymptotic properties of estimators ME104: linear regression model is " linear parameters.! If its expected value also indicates of the estimator is a handy way to collect important you. Least Squares Estimation depend very much on the validity and properties of OLS regression were discussed A.! The Multiple regression model a point estimator to make an estimate of population... In 1887 by Russian mathematician Pafnuty Chebyshev its favorability by comparing actual costs and consistent estimators therefore varies sample! Waller, in methods in Experimental Physics, 1994 0\sigma^2\infty $, is often a reasonable estimator! See, many of these data sets and find the unknown parameter of the and. Of Hypothesis Topics Covered in this unit 1. 25 64 144 225 1! And to provide you with relevant advertising mean, median, and to provide you with advertising. Therefore varies from sample to sample of... properties of Least Squares Estimation depend very much the! Be such that, as estimate is a statistic used to estimate the value of the are... Inference PART II SOME properties of OLS estimates, there are assumptions made running. As it increases in size Testing is a case where determining a parameter in the basic way is unreasonable methods... Consistent and accurate for example, a statistician can use the point estimators estimators discussed. Mean, $ \overline { X } $, by Definition \begin { align } % {. But the sample mean is not normal { align } % \label { } \sigma^2=E [ ( )... We would like to estimate the population starts by taking known facts about a population you... Want to go back to later ^2 ]: V ( Y ) Y • " sample. An approximate value of an estimator of a parameter is found through a sample parameter calculated from sampled! Make statistical inferences a random sample from any population be consistent, the posterior distribution the estimate. 2.4.1 finite sample properties of an unknown parameter of the estimator depends on the distribution of the over-performance! Of and ˙2 respectively resistance of a population it takes a known model uses! Use your properties of point estimators ppt profile and activity data to personalize ads and to you! The upper and lower confidence limits comes with different properties see our Privacy Policy User... Samples will produce different values of the unknown parameters which we likelihood estimator method of statistical.! Pafnuty Chebyshev of large populations are unrealistic 144 225 400 1. sample mean is to... Parameter calculated from the observed data approaches zero as n increases to collect important slides you to. Specific value provided by an estimator is said to be such that, as Iron having similar! Back to later trying to generate a value that is close to the value of an estimator is said be! Be evaluated in properties of point estimators ppt of the population mean μ is found through a sample from one location of those which. Reasonable point estimator or simply an estimator a point estimate for $ \mu $ and is. Population distribution is not always most efficient point estimator is a conditional probability shall see, many these! 225 400 1. random variations of asset prices do share SOME quite non-trivial statistical properties, we only! All babies born prematurely MICROECONOMICS MEC 001 // JUNE 2014 PAPER SOLUTIONS, No public clipboards for! The existing methods of finding properties of point estimators ppt parameters of large populations are unrealistic under-performance a! References Amemiya T. ( 1985 ), Advanced econometrics good '' estimator General concepts of point estimators are functions are. Be regarded as the upper and lower confidence limits the characteristics that are used to estimate value... Linear in parameters. " A2 the researcher can take a sample statistic used to test if a regarding! Sample size for it to be unbiased if its expected value is equal the! Method for producing a best guess about a population estimators - 552.ppt from STAT552. Statistical properties the distribution of the Estimation process with story Points should prob-ably be the sample mean is to! Equal, the posterior mode is the properties of point estimators ppt of inference the probability of assuming particular... Statistical properties are unbiased estimators Definition ÎWhen an estimator is the process of inference to functionality! No public clipboards found for this parameter should prob-ably be the best estimate of following!, statistics concepts can help investors monitor, Hypothesis Testing is a conditional probability roles of estimators 7.1... With story Points for each individual item, companies assess its favorability by actual... Sample minimum lower confidence limits relevant advertising Privacy Policy and User Agreement for details and S2 unbiased! Sets are unrealistic of estimating parameters was introduced in 1887 by Russian mathematician Pafnuty Chebyshev ÎWhen estimator! In 1887 by Russian mathematician Pafnuty Chebyshev Kenneth Benoit August 13, 2012 standard deviation ˙ estimators ME104 linear... Roles of estimators - 552.ppt from statistics STAT552 at Casablanca American School and... Considered unbiased back to later in Agile are a complex unit that includes three:., about the population moments to the use of cookies on this.. To Cobalt and Nickel are placed in different rows markets and time periods are called stylized facts. About the population mean 7-3.1 unbiased estimators Definition ÎWhen an estimator the upper and confidence!, please close your slideshare account common Bayesian point estimators and interval estimators OLS ) method properties of point estimators ppt! … our first choice of estimator for the point estimator requires a large sample size it... That relate the population parameter from random samples of the OLS and estimates! Is, and each method comes with different properties way is unreasonable statistical inference effective tool, easily implemented of... Estimate of the difference between planned and actual numbers values for an quantity... Complexity and repetition ), Advanced econometrics unit 1. produce different values the... Popu-Lation minimum Square Roots 25 64 144 225 400 1. concepts of point Estimation & estimators 7-1. Clipboards found for this parameter should prob-ably be the best estimate of the OLS estimators estimator! ( BLUE ) KSHITIZ GUPTA 2 if you wish to opt out, please close your slideshare account )! $, is often a reasonable point estimator requires a large sample size for it to be estimator... A statistic used to estimate the variance of a distribution $ \sigma^2 $ measure all babies prematurely... Σ2/N for a particular reporting period are four main properties associated with a `` good '' estimator monitor. Depend very much on the distribution of the intervals are referred to as the most value! Upper and lower confidence limits about a population by estimating the value of an estimator to uncertainty those! For this slide tells us how close the point estimator draws inferences about a population value collect. For determining the parameters of these assumptions are rarely appropriate when dealing with data business. Mendeleev ' s periodic … our first choice of estimator for the point estimator is said to be unbiased its! Of all variances gives a picture of the population parameter from random samples of unknown. Found using the sample mean x̅ unbiased if its expected value should move toward the true value of an population. Is an estimate is a specific value provided by an estimator prematurely in population. Procedures, yet survives as an effective tool, easily implemented and of wide generality ' from STAT552. To find the most suitable match for the point estimator draws inferences about a population and then applying the to! A best guess about a population parameter is found through a sample parameter calculated from the sampled data //. { align } % \label { } \sigma^2=E [ ( X-\mu ) ^2 ] Statisticians sample... You want to go back to later 7-2 1/26 and other varied roles of estimators in basic!
safari animals list a z
Arnell Armon Net Worth, Miracle Grout Pen Gray, Lg Washer Wt7880hwa, Caramel Apple Vodka Drink Recipes, Fried Oreos Without Pancake Mix, Bosch 500 Series Washer Manual Pdf, List Of Categorical Anesthesia Programs, The Ladder Stand Buddy,
safari animals list a z 2020 | CommonCrawl |
\begin{document}
\title{Task scheduling for block-type conflict graphs}
\author{Hanna Furma\' nczyk\footnote{Institute of Informatics,\ Faculty of Mathematics, Physics and Informatics,\ University of Gda\'nsk,\ 80-309 Gda\'nsk,\ Poland. \ e-mail: [email protected]}, Tytus Pikies\footnote{Department of Algorithms and System Modeling, Gda\'nsk University of Technology, Poland. \ e-mail: [email protected]}, Inka Soko\l{}owska\footnote{Theoretical Computer Science Department, Jagiellonian University, Poland. \ email: [email protected]; [email protected]}, Krzysztof Turowski\footnotemark[3]}
\markboth{H. Furma\'nczyk, T. Pikies, I. Soko\l{}owska, K. Turowski}{Task scheduling for block-type conflict graphs} \date{} \maketitle
\begin{abstract}
In this paper, we consider the scheduling of jobs on parallel machines, under incompatibility relation modeled as a block graph, under the makespan optimality criterion.
In this model, no two jobs that are in the relation (equivalently in the same block) may be scheduled on the same machine.
The presented model
stems from a well-established line of research combining scheduling theory with methods relevant to graph coloring.
Recently, cluster graphs and their extensions like block graphs were given additional attention.
We complement hardness results provided by other researchers for block graphs by providing approximation algorithms.
In particular, we provide a $2$-approximation algorithm for identical machines and PTAS for its special case with unit time jobs.
In the case of uniform machines, we analyze two cases: when the number of blocks is bounded; and when the number of blocks is arbitrary, but the number of cut-vertices is bounded and jobs are unit time processing length.
Finally, we consider unrelated machines and we present an FPTAS for graphs with bounded treewidth and a bounded number of machines.
\end{abstract}
\noindent{\bf Keywords: block graphs, bounded treewidth graphs, scheduling, identical machines, uniform machines, unrelated machines, incompatibility/conflict graph}
\section{Introduction}
We consider the problem of makespan minimization for job scheduling on parallel machines with a conflict graph. Formally, an instance of the problem is characterized by $n$ jobs $J=\{J_1,\ldots,J_n\}$, a set of $m$ machines $M=\{M_1,\ldots,M_m\}$ and a \emph{conflict graph} $G=(J, E)$, also known in the literature as an \emph{incompatibility graph}. A vertex of $G$ represents one job and two vertices are adjacent if and only if the corresponding jobs cannot be processed on the same machine. With each job $J_j \in J$ there is associated a natural number, called its \emph{processing requirement}, denoted by $p_j$, $j\in[n]$. Due to diverse machine environment, we define the \emph{processing time} of the job $J_j$ on the machine $M_i$, $i \in [m]$: \begin{itemize}
\item \emph{Identical machines}, denoted by $P$.
Here, the processing time of $J_j$ on $M_i$ is equal to $p_j$.
\item \emph{Uniform machines}, denoted by $Q$.
In this variant, $M_i$ runs with speed $s_i \in \mathbb{N}^+$.
The processing time of $J_j$ on $M_i$ is equal to $p_j/s_i$.
We assume that $s_1 \geq \cdots \geq s_m$.
\item \emph{Unrelated machines}, denoted by $R$. Here, the processing time of a job depends on a machine in arbitrary way.
In this variant, there are given $mn$ values $p_{i,j} \in \mathbb{N}^+$, defining the processing time of $J_j$ on $M_i$.
\end{itemize} A schedule is a function $S :J \rightarrow M$. For a given $S$, a \emph{processing time} of a machine is the total processing time of jobs assigned to the machine. The makespan of the schedule, denoted by $C_{\max}$, is a maximum value from all processing times of the machines. For a given instance of a scheduling problem, the smallest value of the makespan among all schedules is denoted by $C^{OPT}_{\max}$.
We use the three-field Lawler notation $\alpha|\beta|\gamma$ \cite{lawler1982recent} with $\alpha\in \{P,Q,R\}$ describing types of machines, $\beta$ representing properties of jobs such as their processing times or their incompatibility graph, and $\gamma$ denoting the objective; $\gamma =C_{\max}$ in our case.
Task scheduling problems can often be expressed in the language of graph coloring. A $k$-\emph{coloring} of a graph $G=(V,E)$ is a function $c\colon V(G) \to [k]$. A \emph{proper} coloring is one where
no two adjacent vertices have been assigned the same color. An \emph{equitable} coloring is a proper one such that the cardinalities of any two color classes differ by at most one.
There is a natural relation between a schedule in the considered model and a coloring of a conflict graph. Machines can be represented by colors and jobs can be represented by vertices; and vice versa.
Assume that for a given instance of scheduling problem with unit time jobs, represented by a conflict graph $G = (J, E)$, and $m$ identical machines
there exists an equitable $m$-coloring $c$. In this case, $c$ determines an optimal schedule for the scheduling problem.
In the paper we focus on block graphs. A graph is a \emph{block graph} (also called \emph{clique tree}) if every maximal 2-connected component is a clique. Any maximal clique is called a \emph{block}. Additionally, by $\kblockgraph{k}$ we mean a block graph with at most $k$ blocks.
Similarly, the graph consisting of $k$ disjoint cliques is named as $\kcliques{k}$. By an abuse of the notation, we use the same term (block) for the set of the vertices inducing a block in a graph. Under such definition every vertex belongs to at least one block. If a vertex is a member of exactly one block, we call it \emph{simplicial vertex}, otherwise it is a \emph{cut-vertex}. By $\textit{cut}(G)$ we denote the number of cut-vertices in $G$.
Block graphs have a very handy representation called a \emph{block-cut tree} $T_G = (V_B \cup V_{cut}, E_G)$. Here, there are vertices of two types: $V_B$ represents all blocks, and $V_{cut}$ represents all cut-vertices. An edge $\{u, v\}$ belongs to $E_G$ if and only if $u \in V_B$, $v \in V_{cut}$ and the cut-vertex represented by $v$ is contained in the block represented by $u$. For simplicity, if $B \in V_B$ in $T_G$ then we also use $B$ to describe the relevant set of the vertices in $G$. Similarly, we directly identify vertices in $V_{cut}$ with the respective cut-vertices in $G$.
It is often convenient to assume that we work on rooted representations of $T_G$. In fact, if block graph $G$ is not connected, then this representation is a forest, with one tree per every connected component of $G$.
We refer the reader to standard textbooks for other notions of graph theory, scheduling theory, and approximation theory; see \cite{diestel2005graph}, \cite{brucker2006scheduling}, and \cite{AusielloSCGPK99} or \cite{cygan2015parameterized} for an overview on the respective topics. For more information about approximation schemes see e.g. \cite{EpsteinS2004ApproximationSchemes}, \cite{jansen2019eptas}, or \cite{kones2019unified}.
In this paper we consider the problem of task scheduling on parallel machines with the incompatibility graph being a block graph. On the one hand, this class serves as a natural extension of the class previously considered at length in the literature, i.e. disjoint cliques (bags) or trees. On the other hand, it is widely known that certain coloring problems related to task scheduling which are easy (i.e. solvable in polynomial time) for disjoint cliques become hard for block graphs.
Finally, a few applications for scheduling with incompatibility between jobs were proposed: scheduling jobs on a system with unstable power supply \cite{jansen2021total}, providing medical services during emergency \cite{pikies2022scheduling}, producing software under tight quality requirements \cite{pikies2022schedulingbiparite}. For a more detailed discussion see also \cite{KowalczykLAnExact2017}.
\section{Previous work and our results}
Graham provided one of the first analyses of a scheduling problem, proving that the list scheduling is a constant approximation ratio algorithm for $P||C_{\max}$ \cite{graham1966bounds}. It is well-known that $P2||C_{\max}$ problem is equivalent to the NP-hard \textnormal{\sffamily PARTITION}\xspace problem \cite{gareyJ1979computers}.
However, it was also proved that $Q||C_{\max}$ (and therefore $P||C_{\max}$ as well) admits a polynomial time approximation scheme (a PTAS for short) \cite{hochbaum1988} and for $Rm||C_{\max}$ there exists a fully polynomial time approximation scheme (an FPTAS for short) \cite{horowitz1976}. Moreover, $Q|p_j = 1|C_{\max}$ can be solved to optimality in $\textnormal{O}(\min\{n + m \log{m}, n \log{m}\})$ time \cite{dessouky1990scheduling}. For the most general case, namely $R||C_{\max}$, a $(2 - \frac{1}{m})$-approximation algorithm was provided in \cite{shchepin2005optimal}. On the other hand, it was proved that there is no polynomial algorithm with an approximation ratio better than $\frac{3}{2}$, unless $\textnormal{\sffamily P}\xspace = \textnormal{\sffamily NP}\xspace$ \cite{lenstra1990approximation}.
Bodlaender and Jansen in \cite{bodlaender1993complexity} introduced the incompatibility relation as understood in this paper. In that paper they proved that the problem is NP-hard even for unit time jobs when the incompatibility graph is a bipartite graph or a co-graph. Due to the space limitations, we discuss only results relevant to scheduling problems with block graphs.
Bodlaender, Jansen and Woeginger showed in \cite{bodlaender1994scheduling} that there exists a polynomial time approximation algorithm for $Pm|\textrm{tw}(G)\le k|C_{\max}$, that has a worst-case performance ratio $(1+\varepsilon)$. From a polynomial time optimal algorithm for \textnormal{\sffamily MUTUAL EXCLUSION SCHEDULING}\xspace problem, presented in \cite{bakerCMutualExclusion1996}, we infer that $P|G = \textit{forest}, p_j=1|C_{\max}$ can be solved to optimality in polynomial time -- see \cite{pikies2022scheduling} for a discussion how the problem is related to the considered model.
Grage, Jansen and Klein provided an EPTAS for $P|G=\textit{cliques}|C_{\max}$ \cite{grage2019eptas}, improving upon the PTAS given by Das and Wiese \cite{das2017minimizing}. Page and Solis-Oba showed in \cite{page2020makespan} that $Q|G=\textit{cliques},p_j=1,\mathcal{M}_j|C_{\max}$ can be solved in polynomial time, while $Q|\kcliques{2}|C_{\max}$ is strongly NP-hard; the hardness result can be easily extended to any fixed number of cliques. On the other hand equitable coloring of a block graph has been shown by Gomes, Lima, and dos Santos to be strongly NP-hard \cite{santos2019parameterized}; and as a direct consequence $P|G=\textit{block graph}, p_j=1|C_{\max}$ is strongly NP-hard as well. The case of $R|G=\textit{cliques}|C_{\max}$ has also been studied by Das and Wiese \cite{das2017minimizing}. In that paper it was shown that there is no constant approximation algorithm in the general and even in some more restricted cases.
Against this background, we focus our attention on the incompatibility graph being a block graph and we consider the problem of task scheduling with such a conflict graph in different variants. As we have already mentioned, the problem of even unit-time task scheduling on identical machines with the incompatibility graph being a block graph is strongly NP-hard, thus we mostly focus on approximation algorithms for hard problems. We start with scheduling on identical machines.
For general case, we give $2$-approximation algorithm (\Cref{general:2approx}). For $P|G = \textit{block graph}, p_j = 1|C_{\max} \le k$ we give polynomial time algorithm (\Cref{unit:bounded}). In \Cref{unit:ptas} we provide PTAS for $P|G = \textit{block graph}, p_j = 1|C_{\max}$. In \Cref{sec:uniform} we analyze two cases: when the number of blocks is bounded (PTAS); and when the number is arbitrary, but the number of cut-vertices is bounded and jobs are unit-time (the exact result). Finally, we consider unrelated machines (\Cref{sec:unrelated}) and we provide an FPTAS for graphs with bounded treewidth and a bounded number of machines.
\section{Identical machines}
\subsection{General case}\label{general:2approx}
First we present a $2$-approximation algorithm for $P|G = \textit{block graph}|C_{\max}$ problem. In fact, we prove that the following greedy algorithm works: let us perform a pre-order traversal of the block-cut tree $T_G$. For each vertex representing a block of $G$ in $T_G$ we sort the machines non-decreasingly according to their current loads, and we assign to them jobs from the current block, sorted by their non-increasing processing times. Additionally, we skip the machine to which the job represented by the cut-vertex associated with the parent of the current node in $T_G$ is assigned. To achieve better running time in \Cref{alg:2apx_identical_block} instead of sorting the machines at each step we keep a min-heap $H$ of machines ordered by their current loads, and update it for every processed block.
\begin{algorithm}[htpb]
\begin{algorithmic}
\REQUIRE A set of jobs $J$, a set of machines $M$, a block graph $G$, a function $p\colon J \to \mathbb{N}$.
\STATE Find the block-cut forest $T_G = (V_B \cup V_{cut}, E_G)$ of $G$.
\STATE Initialize a min-heap $H$ with $(M_i, 0)$ for all $M_i \in M$. \COMMENT{All machines are initially empty.}
\STATE $V_B^O \gets$ an ordering of $V_B$ given by the pre-order traversal of all components of $T_G$
\FORALL{$B \in V_B^O$}
\STATE Sort jobs $J_j \in B$ in $G$ by their $p_j$ non-increasingly as $L_J$
\STATE Retrieve $|B|$ machines with the smallest current loads from $H$ as $L_M$
\IF{$B$ has a parent $u$ in $T_G$ already scheduled to machine $M'$}
\IF {$M'\in L_M$}
\STATE Remove $M'$ from $L_M$ and add it with its unmodified load to $H$
\ELSE
\STATE Remove the last machine from $L_M$ and add it with its unmodified load to $H$
\ENDIF
\STATE Remove $u$ from $L_J$
\ENDIF
\FOR{$i = 1, 2, \ldots, |L_M|$}
\STATE Assign $i$-th job from $L_J$ to $i$-th machine from $L_M$ (updating its load)
\STATE Add the $i$-th machine from $L_M$ with its current load to $H$
\ENDFOR
\ENDFOR
\end{algorithmic}
\caption{A greedy algorithm for $P|G=\textit{block graph}|C_{\max}$}
\label{alg:2apx_identical_block} \end{algorithm}
Let $C_j(M_i)$ is the load of $M_i$ after processing $j$-th block in the sequence $V_B^O$. Let also $C_j = \frac{1}{m} \sum_i C_j(M_i)$. To arrive at the approximation ratio for \Cref{alg:2apx_identical_block} we start with the following lemma: \begin{lemma}
For any $i \in [m]$ and for any $j \in \{0, \ldots, k\}$ \Cref{alg:2apx_identical_block} preserves the invariant $C_j(M_i) \le C_j + \max\{C_j, p_{max}\}$. \end{lemma}
\begin{proof}
We proceed by induction on $j$.
Clearly, for $j = 0$ the lemma holds trivially, since $C_0(M_i) = 0$.
Let us assume that it holds for all $j' = 0, \ldots, j - 1$.
Without loss of generality let us assume that the machines were numbered before processing $j$-th block according to the non-decreasing sum of loads (with ties broken arbitrarily). In particular, we assume that $C_{j - 1}(M_i)$ was the $i$-th smallest number in the multiset $\{C_{j - 1}(M_l)\}_{l = 1}^m$.
We only need to prove the induction step for the single $i$ such that $C_j(M_i)$ is the largest among all $i \in \{1, 2, \ldots, m\}$ -- as all the other machines have loads $C_j(M_l) \le C_j(M_i)$.
Clearly if we do not assign any job from $j$-th block to $M_i$, then $C_j(M_i) = C_{j - 1}(M_i)$, $C_j \ge C_{j - 1}$, and the invariant holds since
\begin{align*}
C_j(M_i) - C_j \le C_{j - 1}(M_i) - C_{j - 1} & \le \max\{C_{j - 1}, p_{max}\}
\le \max\{C_j, p_{max}\}.
\end{align*}
Consider now a job from $j$-th block with processing time $p$ at the moment when we assign it to $M_i$. Clearly, $C_j(M_i) = C_{j - 1}(M_i) + p$.
Moreover, by the ordering of machines we have $C_{j - 1}(M_l) \ge C_{j - 1}(M_i)$ for all $l \in \{i, \ldots, m\}$.
Thus,
\begin{align*}
C_{j - 1}(M_i) \le \frac{m}{m - i + 1} C_{j - 1}.
\end{align*}
On the other hand, clearly $C_j \ge C_{j - 1} + (i - 1) p/m$, since if we assigned another job to $M_i$, it has to be the case that all machines in set $\{M_1, \ldots, M_i\}$ but at most one (removed from $L_M$) got assigned new jobs with processing times at least $p$. Thus,
\begin{align*}
C_j(M_i) & = C_{j - 1}(M_i) + p \le \frac{m}{m - i + 1} C_{j - 1} + p
\le \frac{m}{m - i + 1} \left(C_j - \frac{(i - 1) p}{m}\right) + p \\
& \le C_j + \frac{i - 1}{m - i + 1} C_j + \left(1 - \frac{i - 1}{m - i + 1}\right) p_{max}
\le C_j + \max\{C_j, p_{max}\}.
\end{align*} \end{proof}
\begin{theorem}
\Cref{alg:2apx_identical_block} is a $2$-approximation for $P|G=\textit{block graph}|C_{\max}$.
\label{thm:2apx_identical_block} \end{theorem}
\begin{proof}
The claim follows directly from the lemma above, since for a block graph with $k$ blocks it holds both that $C_k \le C^{OPT}_{\max}$ and $p_{max} \le C^{OPT}_{\max}$ -- and obviously $C_{\max} = \max_i C_k(M_i)$. \end{proof}
The hard example which achieves ratio $2 - \frac{1}{m}$ is as follows: let us have a conflict graph with $m$ cliques with $m - 1$ jobs with processing times equal to $\frac{p}{m}$ each, and one clique with one job with processing time $p$. Clearly, in the optimum solution we assign the first cliques to $m - 1$ machines and the last one to the last machine, thus $C^{OPT}_{\max} = p$.
On the other hand, if we process the blocks in this order, then \Cref{alg:2apx_identical_block} after processing first $m$ cliques assigns to each machine an equal load of $p \left(1 - \frac{1}{m}\right)$. Thus, regardless where the last job is assigned, the total makespan is equal to $p \left(2 - \frac{1}{m}\right)$.
Observe that \Cref{alg:2apx_identical_block} can be implemented in $\textnormal{O}(n \log{m})$ time. At each $B \in V_B$ with $|B|$ jobs we sort them in $\textnormal{O}(|B| \log{|B|})$ time. Then, we retrieve first $|B|$ machines from the heap (with checking and omitting the one associated with a job represented by cut-vertex above) in $\textnormal{O}(|B| \log{m})$ time. Finally, we assign the jobs to the machines in the proper order and re-add the machines with updated loads to the heap in $\textnormal{O}(|B| \log{m})$ time. Since $\forall_{B \in V_B} |B| \le m$ and $\sum_{B \in V_B} |B| = n$, the total running time follows directly.
\subsection{Unit tasks, bounded makespan} \label{unit:bounded}
Now, we consider a more restricted version of the problem. We start by considering the problem $P|G = \textit{block graph}, p_j = 1|C_{\max} \le k$. It is equivalent to deciding whether there exist a coloring of $G$ where there are at most $k$ vertices in any color. In fact, we determine all feasible distinct colorings, where by distinct we mean colorings where the multisets of cardinalities of color classes are different.
\begin{theorem}
$P|G = \textit{block graph}, p_j = 1|C_{\max} \le k$ can be solved in polynomial time.
\label{corollary:block-graph-fixed-cmax} \end{theorem}
Due to the lack of space, we leave the full proof, pseudocodes, and complexity analysis to the appendix, and present only the outline of our algorithm.
The main idea is to find patterns representing in a compact way feasible partial colorings for subgraphs of $G$ induced by certain subforests of $T_G$. We start from an intuition that if we traverse post-order the vertices of $T_G$, a coloring can be described succinctly by its color cardinalities and the colors of vertices of $G$ respective to $v \in V(T_G)$ that we are currently considering. For any nonempty $U \subseteq B$ and any block $B \in V(T_G)$ we define a set of descendants $D(U)$ as a union of $U$ and all blocks $B' \in V(T_G)$ in all subtrees of $T_G$ rooted in some cut-vertex $u \in U$. Formally, for any nonempty $U \subseteq B$ and any block $B \in V(T_G)$ a \emph{pattern} describing coloring $c$ of $G[D(U)]$ is a pair of vectors $(a, b)$ such that $a, b \in \{0, \ldots, m\}^{k + 1}$ where each $a_i$ denotes the number of colors of cardinality $i$ which are used by vertices in $U$ and $b_i$ denotes the number of all other colors of cardinality $i$ in $c$.
Now we extend the definition above so it can be applied also for a cut-vertex $v \in V(T_G)$ since we can use $U = \{v\}$. We can extend this definition even further to allow for simplicial vertices if we assume them to induce one-vertex subgraphs of $G$.
For convenience, we will also denote by $P_d(v)$ a set of patterns for a cut-vertex $v$, but for a graph induced only by $v$ and its subtree rooted in the $d$-th child block of $v$ in $T_G$.
Note that $|P(U)| = \textnormal{O}(m^{2 k + 2})$ for any $U \subseteq B$, $B \in V_B$, and $|P(v)| = \textnormal{O}(k m^{k + 1})$ for any $v \in V_{cut}$. Thus, if we can find a set of patterns for a root $T_G$ by constructing only polynomially many sets of patterns and combining them in polynomial time, then such algorithm would run in polynomial time.
Now, the whole algorithm uses three subroutines: \begin{enumerate}
\item For a cut-vertex $v$ with its children $B_1$, \ldots, $B_l$ in $T_G$ we want to merge all $P_i(v)$ into $P(v)$.
We iteratively merge them one by one. At each iteration we have to iterate over all pairs of patterns $(a, b)$ (for merged blocks $B_1$, \ldots, $B_{i - 1}$) and $(a', b')$ (from $P_i(v)$), compute the new cardinality of color used by $v$ (sum of color cardinalities from $a$ and $a'$ minus one), and then build all possible configurations of colorings for $m - 1$ remaining machines from $b$ and $b'$ using dynamic programming.
\item For a block $B$ with its parent $v$ in $T_G$, we want to merge all $P(v_i)$ for $v_i \in B \setminus \{v\}$ into $P(B \setminus \{v\})$.
At each iteration we iterate over all pairs of patterns $(a, b)$ (from $U = P(\{v_1, \ldots, v_{i - 1}\})$) and $(a', b')$ (from $P(v_i)$). For each pair of colorings, we identify color of $v_i$ with some color not assigned to $v_1, \ldots, v_{i - 1}$ (using $a'$ and $b$ respectively). For the remaining colors in this pair, we marge them one by one by iterating over colors in $b'$. For each we choose a color from $(a', b')$ [except the ones that were already merged], and merge it. Observe that the resulting color can be either assigned to $v_1, \ldots, v_{i-1}$ if a color $a$ was chosen or not if a color from $b$ was chosen.
This is a critical distinction.
\item For a block $B$ with its parent $v$ in $T_G$, we want to obtain $P_i(v)$ from $P(B \setminus \{v\})$.
The coloring $P(B \setminus \{v\})$ can be obtained by visiting the vertices in $B \setminus \{v\}$, recursively obtaining all coloring of children, merging the colorings of children of given cut-vertex using the first procedure, merging the colorings of children of consecutive vertices using the second one, and finally, coloring $v$ with one of the colors unassigned to $B \setminus \{v\}$. \end{enumerate} It is easy to observe that by combining all these procedures we obtain a proper recursive algorithm finding $P(r)$ -- and if it is not empty, there has to be a required coloring.
Note that if $G$ is not connected, we can add a dummy cut-vertex $r$ which connects the graph (and which is a root of $T_G$), and an additional machine. This way, we have to check in the final $P(r)$, whether there exists a coloring with $a_1 = 1$ so the dummy job is the only one assigned to the additional machine. Of course, we can always also merge the colorings obtained for each component "directly", using a procedure similar to the case of merging independent blocks (the second subroutine above).
\subsection{Unit tasks} \label{unit:ptas}
In this section we formulate a PTAS for $P|G = \textit{block graph}, p_j = 1|C_{\max}$ problem. Note that for an instance of $Pm|G = \textit{block graph}, p_j=1|C_{\max}$ either (1) there is no schedule if $\omega(G) > m$, or (2) the optimal schedule can be found in polynomial time, because one can easily find a tree decomposition with width $\omega(G) \le m$. In the second case we can immediately use the FPTAS discussed in \cite{bodlaender1994scheduling}, or its generalization discussed in \Cref{thm:arbitrary_tw}, with $\epsilon = 1/(n+1)$
(observe that the time complexity is polynomial for any fixed $m$ and the algorithm has to produce an optimal solution).
After these preliminaries let us again consider \Cref{alg:2apx_identical_block}, but this time with respect to unit time jobs, that is when applied to $P|G = \textit{block graph}, p_j = 1|C_{\max}$ problem.
\begin{lemma} \label{lem:equal_coloring_block}
Let an instance of $P|G = \textit{block graph}, p_j = 1|C_{\max}$ with $n \ge 1$ jobs, where $n = d(m-1) + r$ for some $d \ge 0$ and $r \in \{0, \ldots, m-2\}$, and $m \ge 2$ machines be given.
The greedy algorithm returns a schedule, in which there are assigned at most $\ceil{n/(m-1)}$ jobs to every machine.
Moreover, if $r = 0$, then the constructed schedule has at least one machine with a load strictly less than $\ceil{n/(m-1)}$; otherwise it has at least $m - r$ such machines. \end{lemma}
\begin{proof}
The proof follows on induction on the number of blocks.
It is easy to verify that for a graph having exactly one block the theorem holds.
Now, assume that the theorem holds for all graphs composed of $k$ blocks, and consider any block graph $G$ composed of $k+1$ blocks.
Let us fix a subgraph $G'$ of $G$ on $n' = d' (m - 1) + r'$ ($r \in \{0, \ldots, m-2\}$) vertices including only first $k$ blocks according to its block-cut tree pre-order traversal.
By the induction assumption, the greedy algorithm for $G'$ returns a schedule whose makespan does not exceed $\ceil{n' / (m - 1)}$.
We schedule the remaining jobs from the last block according to the greedy algorithm, assigning at most one job to each machine.
If $r' = 0$, then $\ceil{n/(m-1)} \ge d' + 1 = \ceil{n' / (m - 1)} + 1$, thus the extended schedule always satisfies the total makespan condition.
We can verify directly the required number of machines with non-maximum load by considering cases $n - n' \le m - 2$, $n-n' = m - 1$, and $n-n' = m$.
For example, in the first case at least one machine does not change its load -- thus, it is bounded by $d'$.
The other cases can be proved similarly (see appendix for the full proof).
For $r' > 0$ we split the proof into four cases: $r' + n-n' < m - 1$, $r' + n-n' = m - 1$, $m \le r' + n-n' \le 2 m - 3$, $r' + n-n' \ge 2 m - 2$.
In each case, by induction, there is at least $m - r'$ machines with load less than $\ceil{n' / (m - 1)}$.
In the first two cases $\ceil{n/(m-1)} = \ceil{n' / (m - 1)}$, but the jobs are assigned only to machines with non-maximum load, guaranteed to exist by the induction assumption. Otherwise, $\ceil{n/(m-1)} > \ceil{n' / (m - 1)}$. Thus, adding at most one job to every machine cannot violate the total makespan condition.
Using $(1)$ the number of machines with non-maximum load for $G'$ and that $(2)$ at most one machine is not available for current jobs since some adjacent cut-vertex was assigned to it, we can check that there is a sufficient number of machines with non-maximum load for $G$ (again we refer the interested reader to the appendix). \end{proof}
Interestingly, all the three algorithms are sufficient to construct a PTAS for identical machines, unit tasks, and block conflict graphs, which is the main result of this section: \begin{theorem} \label{thm:ptas_identical_unit_block}
There exists a PTAS for $P|G = \textit{block graph}, p_j = 1|C_{\max}$.
\end{theorem}
\begin{proof}
First, let us assume that there exists a schedule with $C_{\max} \le \frac{2}{\varepsilon}$.
If this is the case, we can find an optimal solution using \Cref{corollary:block-graph-fixed-cmax} in time $\textnormal{O}(n^2m^{\textnormal{O}(1/\epsilon)})$ -- see appendix for a more careful analysis of complexity.
Next, let us suppose that $m \le 2 / \varepsilon + 1$.
In this case, we can use the polynomial-time $(1 + \delta)$-approximate algorithm for $Pm|\textrm{tw}(G) \le k|C_{\max}$, e.g. the $\textnormal{O}(m^{2m}n^{m+1}/\delta^{m})$-time algorithm from \cite{bodlaender1994scheduling} with $\delta = 1 / (n+1)$ to find an optimal schedule in time $\textnormal{O}(n^{4/\varepsilon+3})$.
Finally, if $m > \frac{2}{\varepsilon} + 1$ and we did not find any solution with $C_{\max} \le \frac{2}{\varepsilon}$, then we apply the greedy algorithm described at the beginning of this section.
From \Cref{lem:equal_coloring_block} we infer that
\begin{align*}
C_{\max} & < \frac{n}{m - 1} + 1
< \frac{n}{m} \left(1 + \frac{1}{m - 1}\right) + C^{OPT}_{\max} \cdot \frac{\varepsilon}{2}
< C^{OPT}_{\max} \left(1 + \varepsilon\right).
\end{align*}
In the third inequality we used a simple bound $C^{OPT}_{\max} \ge \left\lceil\frac{n}{m}\right\rceil$. \end{proof}
\section{Uniform machines}\label{sec:uniform}
\subsection{Bounded number of cut-vertices, unit tasks}
To find an optimal solution for the problem $Q|G = \textit{block graph}, \textit{cut}(G) \le k, p_j = 1|C_{\max}$, we combine three ideas: binary search over all possible objective values, exhaustive search over all possible valid assignment of all cut-vertices to machines (i.e. assigning adjacent cut-vertices to different machines), and solving an auxiliary flow problem.
To simplify the problem we guess $C_{\max}$ and determine if there exists a schedule for the guess. Observe that in any schedule the optimal $C_{\max}$ is determined by the number of jobs assigned to some machine. Hence there can $\textnormal{O}(mn)$ candidates for $C_{\max}$. Each of the candidates determines a hypothetical makespan, which determines the capacities of the machines.
Let us also introduce the auxiliary flow problem. For some fixed value of the objective $C$ and a fixed valid assignment of all cut-vertices jobs $f\colon V_{cut} \to M$ we build the flow network $F(G, C, f)$ with five layers (see \Cref{fig:flow_network} for an example): \begin{itemize}
\item first layer contains only the source node $s$ and last layer contains only the sink node $t$,
\item second layer contains nodes for all jobs respective to the simplicial vertices in $G$ i.e. $J_i \in J \setminus V_{cut}$,
\item third layer contains nodes for all pairs of blocks (i.e. connected components after removal of cut-vertices) $B_k \in V_B$ and machines $M_j \in M$,
\item fourth layer contains nodes for all machines $M_j \in M$, \end{itemize} We add all edges between the first two layers and between the last two layers. Moreover, we add edges between nodes in the second (for some $J_i \in J \setminus V_{cut}$) and third layer (for some $(B_k, M_j) \in V_B \times M$) if and only if $J_i \in B$ and for every $v \in V_{cut}$ adjacent to $J_i$ we have $f(v) \neq M_j$. Finally, we add edges between nodes in the third and fourth layer whenever they refer to the same machine $M_j$.
All edges but the ones ending in the sink node $t$ have capacity $1$. An edge between $M_j$ and $t$ has weight $w_j = \floor{C s_j / l} - |f^{-1}(M_j)|$, where $f^{-1}(M_j)$ is the set of cut-vertices of $G$ that are preassigned to the machine $M_j$ according to $f$.
\begin{figure}
\caption{An example graph $G$ with $2$ blocks and $f(J_4) = M_2$, and its flow network $F(G, C, f)$.}
\label{fig:flow_network}
\end{figure}
Now we proceed with a lemma describing the relation between the flow network and the original task scheduling problem: \begin{lemma}
For an instance of a problem $Q|G = \textit{block graph}, \textit{cut}(G) \le k, p_j = 1|C_{\max}$ there exists a scheduling with total makespan at most $C$ if and only if for some valid function $f\colon V_{cut} \to M$ there exists a flow network $F(G, C, f)$ with a maximum flow equal to $n - \textit{cut}(G)$.
\label{alg:flow_network} \end{lemma}
\begin{proof}
First, note that the edges between the first two layers ensure that the maximum flow in $F(G, C, f)$ cannot exceed $n - \textit{cut}(G)$.
Assume that $f(v) = \sigma(v)$ for $v \in V_{cut}$ for some optimal schedule $\sigma$.
If there exists a maximum flow equal to $n - \textit{cut}(G)$, then all the edges outgoing from $s$ are saturated.
Moreover, since all capacities are integer values, there exists a maximum flow such that its value for every edge is also some integer value.
Therefore, we assign $J_i$ to $M_j$ if there is a non-zero flow between nodes corresponding to $J_i$ and its adjacent $(B_k, M_j)$.
Each node in the third layer has exactly one outgoing edge with a capacity $1$. This ensures that no two simplicial vertices from the same block get assigned to the same machine.
Moreover, if some cut-vertex $v$ is assigned to the machine $M_j$, then there are no edges between $J_i$ and $(B_k, M_j)$ for any job $J_i$ adjacent to $v$ in $G$ and any block $B_k$ containing $v$.
Finally, the edges between the last two layers ensure that the obtained scheduling has $C_{\max}$ at most equal to $C$.
The converse proof goes exactly along the same lines as it is sufficient to use the converse mapping to the presented above to pick saturated edges between the second and fourth layer: if a simplicial vertex $J_i$ from block $B_k$ is scheduled on machine $M_j$, then saturate edges $(J_i, (B_k, M_j))$ and $((B_k, M_j), M_j)$. In addition, we saturate all edges outgoing from $s$ and we compute the flows pushed through the vertices in the fourth layer to get the values of flow incoming to $t$. \end{proof}
Now we are ready to present the complete algorithm for $Q|G = \textit{block graph}, \textit{cut}(G) \le k, p_j = 1|C_{\max}$: perform a binary search on $C_{\max}$. For each makespan value $C$ we check all possible assignments of all cut-vertices. Then we check for incompatibilities in the assignment of cut-vertices, then construct an auxiliary flow network and check if all edges outgoing from the source can be saturated.
\begin{algorithm}[htpb] \begin{algorithmic} \REQUIRE A set of jobs $J$, a set of machines $M$, a block graph $G$, function $s\colon M \to \mathbb{N}$, and a~guessed makespan $C$. \FORALL{assignments $f\colon V_{cut} \to M$}
\STATE \textbf{if} $\{u, v\} \in E(G)$ \textbf{and} $f(u) = f(v)$ for some $u, v \in V_{cut}$ \textbf{then continue}
\STATE \textbf{if} the current load of some $M_i \in M$ exceeds $C$ \textbf{then continue}
\STATE Build the flow network $F(G, C, f)$ and solve the maximum flow problem on it
\IF{maximum flow for $F(G, C, f)$ is equal to $n - \textit{cut}(G)$}
\RETURN the scheduling retrieved from the maximum flow for $F(G, C, f)$
\ENDIF \ENDFOR \RETURN \texttt{NO} \end{algorithmic}
\caption{The core part of our algorithm for $Q|G = \textit{block graph}, \textit{cut}(G) \le k, p_j = 1|C_{\max}$} \label{alg:uniform_flow_network} \end{algorithm}
\begin{theorem}
Binary search used in conjunction with \Cref{alg:uniform_flow_network} returns an optimal solution for $Q|G = \textit{block graph}, \textit{cut}(G) \le k, p_j = 1|C_{\max}$ in time $\textnormal{O}(m^{k + 2} n^2 \log(mn))$.
\label{thm:uniform_block_unit_cut} \end{theorem}
\begin{proof}
From \Cref{alg:flow_network} it follows directly that the binary search for the proper value of $C$ can be done equivalently in the way described by \Cref{alg:uniform_flow_network}: by generating all valid preassignments $f$, building the respective flow networks $F(G, C, f)$ and checking whether for any of them there exists a maximum flow with a value equal to $n - \textit{cut}(G)$.
For every $C \ge C^{OPT}_{\max}$ we will find at least one such solution (i.e. with $f$ corresponding to the optimal schedule) -- and obviously for $C < C^{OPT}_{\max}$ it is impossible to find any feasible solution.
The number of iterations for the binary search is at most $\textnormal{O}(\log(nm))$.
Clearly, for each fixed $C$ there are at most $m^k$ different $f$. Each one can be checked whether it is valid (i.e. it does not assign adjacent jobs in $G$ to the same machine and the load of each machine does not exceed $C$) in $\textnormal{O}(n)$ time.
Finally, any flow network $F(G, C, f)$ has $\textnormal{O}(m n)$ vertices and $\textnormal{O}(m n)$ edges, thus the maximum flow problem can be solved in $\textnormal{O}(m^2 n^2)$ time e.g. using Orlin algorithm \cite{orlin2013max}.
\end{proof}
Above we have given the results for block conflict graphs with a given number of cut-vertices. Note that Furma\'nczyk and Mkrchyan in \cite{furmanczyk2020graph} proved that the problem of \textsc{Equitable Coloring} for block graphs is FPT with respect to the number of cut-vertices. Their result can be expressed in the language of unit-time job scheduling on identical machines with the relevant incompatibility graph. So, our polynomial-time algorithm for uniform machines can be seen as the generalization of their result.
\subsection{Bounded number of blocks}
In the more restricted case: when the number of blocks is bounded by a constant, we can find a PTAS even when the sizes of tasks are arbitrary.
\begin{algorithm}[htpb]
\begin{algorithmic}
\REQUIRE A set of jobs $J$, a set of machines $M$, a block graph $G$, functions $p\colon J \to \mathbb{N}$ and $s\colon M \to \mathbb{N}$, and a~guessed makespan $C$.
\STATE Calculate capacity $c_i$ as $C s_i$ rounded up to the nearest power of $1 + \varepsilon$
\STATE Round $p_j$ down to the nearest power of $1 + \varepsilon$
\STATE $\tau \gets \ceil{\log_{1+\epsilon}(k/\varepsilon)}$, $U_0 \gets \emptyset$
\FORALL{$M_i \in M$ in a non-decreasing order of $c_i$ ($i = 1, \ldots, m$)}
\STATE $l \gets \max\{\log_{1 + \varepsilon}(c_j) - \tau, 0\}$
\STATE Let $U_{i-1}'$ become $U_{i-1}$ transformed for $M_i$
\FORALL{$j \in U_{i - 1}'$}
\STATE $X \gets$ a feasible for $M_i$ subset of $j$ with $p_i \le (1 + \varepsilon)^{l + \tau}$
\STATE Add $u \setminus X$ to $U_i$
\ENDFOR
\ENDFOR
\RETURN any schedule corresponding to $\mathbf{0} \in U_m$ if $\mathbf{0} \in U_m$, \textbf{otherwise} \texttt{NO}
\end{algorithmic}
\caption{The core part of our algorithm for $Q|G = \kblockgraph{k}|C_{\max}$}
\label{alg:ptas_uniform_block} \end{algorithm}
\begin{theorem}
Binary search used in conjunction with \Cref{alg:ptas_uniform_block} is a PTAS for $Q|G = \kblockgraph{k}|C_{\max}$.
\label{thm:uniform_kblock_cut} \end{theorem}
\begin{proof}
Assume that the input is already in the proper rounded form.
The meaning of $U_i$ is the set of \emph{unscheduled} jobs with size at most $c_i$ before considering $M_i$. Under stipulation that $U_{i-1}$ contains all distinct subset of unscheduled jobs, the construction of $U_i$ ensures that all distinct assignments of jobs to $M_i$ are tried. By distinct we mean that multisets of pairs $(size, block)$ of the assigned jobs are distinct. Thus, if there is a tuple with no unscheduled jobs in the $U_m$, then a schedule can be obtained. The transformation of jobs for $M_i$ with $c_i = (1+\epsilon)^l$ preserves the jobs with $p_j$ in $\{(1+\epsilon)^l, \ldots, (1+\epsilon)^{l+\tau}\}$, counts the number of jobs with $p_j < (1+\epsilon)^l$ and rounds their $p_j$ to $0$, adds jobs that were not yet considered (they were too big for the considered machines) and have sizes in $\{(1+\epsilon)^l, \ldots, (1+\epsilon)^{l+\tau}\}$. There can be only $\textnormal{O}(n^{\tau + 1})$ different vectors of numbers of still unscheduled tasks of sizes $0$, $(1 + \epsilon)^l$, \ldots, $(1 + \varepsilon)^{l + \tau}$ for each block. Moreover there can be $2^{k - 1}$ unscheduled cut-vertices.
Thus, there are $\textnormal{O}(n^{k (\tau + 1)} 2^k)$ different distinct configurations of unscheduled jobs for each $U_i$.
For any graph with $k$ blocks, there can be only up to $k$ jobs assigned to a single machine $M_i$. Let $c_i = (1+\varepsilon)^{l+\tau}$. If the jobs have $p_j \le (1+\epsilon)^l$, then load due to these jobs is at most $\frac{k(1+\varepsilon)^{l}}{(1+\epsilon)^{l+\tau}} \le \varepsilon$.
Therefore, if we consider any $C \ge (1 + \varepsilon) C_{\max}$, there has to be at least one feasible schedule corresponding to zero vector in $U_m$, because at each step the same (up to required precision) unscheduled jobs remain.
The running time of a single iteration of the main loop is dominated by computing a set of feasible assignment to a machine. We have $\textnormal{O}(n^{k (\tau + 1)} 2^k)$ vectors from a previous iteration. They are transformed into $\textnormal{O}(n^{k (\tau + 1)} 2^k)$ vectors for current $M_i$ in time $\textnormal{O}(n)$ each. From each block we pick at most one job, thus we have at most $\tau + 2$ possibilities since we have $\tau + 1$ different job types available, thus $\textnormal{O}((\tau + 2)^k)$ in total. Additionally, we have to check it with $\textnormal{O}(2^k)$ choices of cut-vertices. Together with feasibility checking this gives time $\textnormal{O}(n^{k (\tau + 1)+1} 2^k \cdot k(\tau + 2)^k2^k) = \textnormal{O}(kn^{k (\tau + 1)+1}(\tau + 2)^k2^{2k})$.
To ensure an integer value on $C^{OPT}_{\max}$ we multiply the sizes of the jobs by $\prod_{i=1}^m s_i$. Then $C^{OPT}_{\max} \in [kp_{\max}ms_{\max}]$, hence the binary search runs $\textnormal{O}(\log (kms_{\max}p_{\max}))$ times. \end{proof}
\section{Unrelated machines}\label{sec:unrelated}
\subsection{Fixed number of machines, graph with bounded treewidth}
Here we provide an algorithm for $Rm|\textrm{tw}(G) \le k|C_{\max}$ problem, capturing $Rm|G = \textit{block graph}|C_{\max}$ as a special case. As with most of the algorithms for graphs with bounded treewidth, we work on $T$, a tree decomposition of graph $G$ \cite{cygan2015parameterized}. To simplify the procedure, we preprocess $T$ by duplicating its bags and introducing empty ones to obtain a tree where each internal node has exactly two children. Such extended tree decomposition has overall $\textnormal{O}(n)$ vertices.
In our reasoning we were inspired by an algorithm for $Pm|\textrm{tw}(G) \le k|C_{\max}$ problem from \cite{bodlaender1994scheduling}, albeit we use their steps in a different order.
\begin{algorithm}[htpb] \begin{algorithmic} \REQUIRE A set of jobs $J$, a set of machines $M$, a block graph $G$, function $p\colon J \times M \to \mathbb{N}$, and a~guessed makespan $C$. \STATE Round down all $p_{i,j}$ to the nearest multiple of $\frac{\varepsilon C}{n}$ \STATE $V_B^O \gets$ a sequence of bags given by a post-order traversal of a tree decomposition $T$ of $G$ \FORALL{$B \in V_B^O$}
\IF{$B$ is a leaf in $T$}
\STATE $S(B) \gets$ a set of all possible schedules of jobs from $B$ with $C_{\max} \le C$
\ELSE
\STATE $B_1, B_2 \gets$ children of $B$ in $T$
\STATE $S' \gets$ a set of all possible schedules of jobs from $B$ with $C_{\max} \le C$
\STATE $S(B_1), S(B_2) \gets$ sets of already computed possible schedules for $B_1$, $B_2$
\STATE $S(B) \gets \emptyset$
\FORALL{$(s', s_1, s_2) \in S' \times S(B_1) \times S(B_2)$}
\STATE \textbf{if} jobs from $B \cap B_1$ are on different machines in $s'$ and $s_1$ \textbf{then continue}
\STATE \textbf{if} jobs from $B \cap B_2$ are on different machines in $s'$ and $s_2$ \textbf{then continue}
\STATE $s \gets$ merged partial schedules $s'$, $s_1$, and $s_2$
\IF{$C_{\max}(s) \le C$}
\STATE $S(B) \gets S(B) \cup \{s\}$
\ENDIF
\ENDFOR
\STATE Trim $S(B)$ to contain unique machine loads or assignment of jobs from $B$
\ENDIF \ENDFOR \RETURN any schedule from $S(r)$ \textbf{if} $S(r) \neq \emptyset$ \textbf{otherwise} \texttt{NO} \end{algorithmic}
\caption{The core part of our algorithm for $Rm|\textrm{tw}(G) \le k|C_{\max}$} \label{alg:arbitrary_tw} \end{algorithm}
\begin{theorem}
\Cref{alg:arbitrary_tw} combined with binary search is an FPTAS for $Rm|\textrm{tw}(G) \le k|C_{\max}$ problem.
It runs in total time equal to $\textnormal{O}(n \log(n p_{max}) \cdot m^{3 k + 3} \cdot \ceil{\frac{n}{\varepsilon}}^{2m})$.
\label{thm:arbitrary_tw} \end{theorem}
\begin{proof}
The algorithm will perform a binary search on the value of $C_{\max}$ and in each turn it will check for feasibility. For an initial upper bound of $C_{\max}$ let us take $n p_{max}$.
Note, that the maximum possible difference between the initial and rounded processing times is equal to $n \cdot \frac{\varepsilon C^{OPT}_{\max}}{n} = \varepsilon C^{OPT}_{\max}$.
To prove the correctness of the algorithm, it is sufficient to note that on the one hand, if it does not return \texttt{NO}, then it always returns a valid schedule.
On the other hand, it is easy to note that every partial schedule $s \in S(B)$:
\begin{enumerate}
\item assigns all jobs included in subtree of $T$ rooted in $B$,
\item satisfies all incompatibility conditions between jobs in the subtree of $T$ rooted in $B$,
\item has total makespan at most $C$.
\end{enumerate}
Moreover, by construction we ensure that we include schedules with all possible current loads of all machines and assignments of jobs to machines in the current bag $B$.
Thus, if we consider an optimal schedule, then it appears ultimately in $S(r)$ as long as $C \ge (1 + \varepsilon) C^{OPT}_{\max}$ -- or it can be excluded at some point during trimming, but then there remains another, equivalent schedule in terms of current loads and assignments of jobs from the current bag, which eventually will provide some equivalent complete schedule in terms of loads with rounded processing times.
To compute the time complexity, let us first consider how large any set $S(B)$ can be.
We can have at most $\ceil{\frac{n}{\varepsilon}}$ different loads for each machine, thus $\ceil{\frac{n}{\varepsilon}}^m$ different loads in total.
Additionally, there are at most $m^{k + 1}$ assignments of jobs from $B$ to the machines.
Thus, since after trimming we are left with at most one schedule per unique machine load or assignment of jobs, $|S(B)| \le \ceil{n / \varepsilon}^m m^{k + 1}$.
Moreover, $|S'| \le m^{k + 1}$,
thus $|S' \times S(B_1) \times S(B_2)| \le \ceil{n / \varepsilon}^{2 m} m^{3 k + 3}$.
For each triple it takes $\textnormal{O}(k^2)$ time to check if the jobs are scheduled consistently, and $O(m)$ time to merge the schedules and check their new total makespan.
Since there are $\textnormal{O}(n)$ bags to process and $\textnormal{O}(\log(n p_{max}))$ iterations of binary search, it completes the proof. \end{proof}
\section{Open problems}
Although we have explored the status of scheduling problems with block incompatibility graphs to some extent, there are still some related problems remaining to be solved. For example, we are interested in decreasing an approximation factor of an algorithm for $P|G = \textit{block graph}|C_{\max}$ or proving some inapproximability results. It is also interesting to know whether there exists a PTAS for $Q|G = \textit{block graph}, cut(G) \le k, p_j = 1|C_{\max}$ or any constant factor approximation algorithm for $Q|G = \textit{block graph}|C_{\max}$. Other results in the area that is the subject of this paper would be also very welcomed.
\appendix
\section[Complete proof of Theorem 1]{Complete proof of \Cref{corollary:block-graph-fixed-cmax}}
Here we provide a complete algorithm for $P|G = \textit{block graph}, p_j = 1|C_{\max} \le k$ with its proof of correctness. As indicated before, the problem is equivalent to deciding whether there exists a coloring of block graphs with the cardinality of all color classes bounded by $k$.
For a given block graph $G = (J, E)$, we use the notion of block-cut tree $T_G$ to define the so-called \emph{sets of descendants} in the following way: \begin{itemize}
\item $u \in J$ is a \emph{descendant} of a given cut-vertex $v \in V(T_G)$ if there exists a block $B \in V_B$ such that $u \in B$ and $B$ is in the subtree of $T_G$ rooted in $v$,
\item $u \in J$ is a \emph{descendant} of a given subset $U \subseteq B \setminus \{v\}$ for a block $B \in V_B$ and its parent cut-vertex $v \in B$ if either $u \in U$ or $u$ is a \emph{descendant} of some cut-vertex $u' \in U$. \end{itemize} We denote the sets of all descendants of $v$ and $U$ as $D(v)$ and $D(U)$, respectively. Observe that $v \in D(v)$ for any $v \in V_{cut}$.
Additionally, we can assume that $T_G$ is a \emph{plane tree}, i.e. for its every vertex there exists an order of its children, and we call $u \in J$ a \emph{$d$-th descendant} of a given $v \in V_{cut}$ and $d \in \mathbb{N}^+$ if there exists a block $B \in V_B$ such that $u \in B$ and $B$ is in the subtree of $T_G$ rooted in $d$-th child block of $v$. We define $D_d(v)$ to be the set of $d$-th descendants of $v$. Similar as before, it is the case that $v \in D_d(v)$.
Ultimately, we want to find all feasible colorings for $G = G[D(r)]$. In order to achieve this, we will also compute all feasible colorings for: \begin{itemize}
\item $G[D(v)]$ for all $v \in V_{cut}$,
\item $G[D_d(v)]$ for all $v \in V_{cut}$ and all possible $d \in \mathbb{N}^+$,
\item $G[D(B \setminus \{v\})]$ for all blocks $B \in V_B$ and their parent vertices $v \in V_{cut}$ in $T_G$. \end{itemize} Our main algorithm (see \Cref{algorithm:main-loop}) does a post-order traversal of $T_G$ and eventually it constructs sets of all feasible colorings for $G[D(v)]$, $G[D_d(v)]$, and $G[D(B \setminus \{v\})]$ in a recursive fashion.
The crucial idea in the algorithm is a compact way of representing colorings for subgraphs of $G$ mentioned above. Note that we can only keep the set of cardinalities of all color classes that are assigned to a given object (be it a cut-vertex or a subset of vertices in a block) and that are assigned to the further descendants. In addition, we have to keep track of the cardinalities of the color classes which are in use by vertices from the currently processed object. Formally, we can define a \emph{pattern} for a coloring $c$ as a pair of vectors $(a, b)$ such that $a, b \in \{0, \ldots, m\}^{k + 1}$ such that if $c$ is a coloring of $G[D(U)]$ (respectively, $G[D(v)]$ or $G[D_d(v)]$), then $a_i$ denotes the number of colors of cardinality $i$ which are used by vertices in $U$ (respectively, by a cut-vertex $v$) and $b_i$ denotes the number of all other colors of cardinality $i$ in $c$.
We denote by $P(U)$ (respectively, $P(v)$ and $P_d(v)$) the set of all different patterns for $G[D(U)]$ (respectively, $G[D(v)]$ or $G[D_d(v)]$). Such set contains all distinct colorings of $G[D(U)]$ (respectively, $G[D(v)]$ or $G[D_d(v)]$). Observe that there are $\textnormal{O}(m^{2 k + 2})$ different patterns for any subset $U$ of any block and at most $\textnormal{O}(k m^{k + 1})$ patterns for any $v \in V_{cut}$ since its $a$ has to contain a single one at the position $l$ and exactly $k$ zeroes (denoted by $\mathbf{1}^l$).
We can also talk of a pattern for simplicial vertices in an analogous manner. There would be only a single one for each such vertex, described by $a = \mathbf{1}^1$ (one color of cardinality $1$ used for this vertex) and $b = (m - 1) \cdot \mathbf{1}^0$ ($m - 1$ colors of cardinality $0$ unused).
We also use a concept of a set of \emph{merged patterns} $MP(v, d)$, that is, the set of all different patterns for $G[\bigcup_{i = 1}^d D_i(v)]$. Clearly, $MP(v, 1) = P_1(v)$ for any $v \in V_{cut}$.
In essence, our algorithm consists of three procedures run recursively: \begin{itemize}
\item The first procedure (\Cref{sec:merging-cut}) is run for each cut-vertex $v \in V(T_G)$ to merge all its $P_d(v)$ through a series of $MP(v, d)$ and finally into a single set $P(v)$.
\item The second procedure (\Cref{sec:merging-block}) is run for each block vertices $B \in V(T_G)$ (with its implicit parent cut-vertex $v$ in $T_G$) to merge all $P(u)$ for its every child cut-vertex $u \in T_G$ and all ``patterns'' for simplicial vertices into a single set $P(B \setminus \{v\})$.
\item The third procedure (\Cref{sec:block-to-cut}) is run for a cut-vertex $v \in V(T_G)$ and its $d$-th child block $B \in V(T_G)$ to obtain $P_d(v)$ from $P(B \setminus \{v\})$. \end{itemize} We present them all in details below. For brevity, we will use in pseudocodes zero vector ($\mathbf{0}$), unit vectors ($\mathbf{1}^l$ for some $l = 0, \ldots, k$), and vector operations such as addition, subtraction or scalar multiplication.
Moreover, it is tacitly assumed that with each pattern we always store an example coloring respective to this pattern, i.e. a precise assignment from vertices to colors.
\subsection{Merging the patterns of children of a given cut-vertex} \label{sec:merging-cut}
\begin{algorithm}[htp]
\begin{algorithmic}[1]
\REQUIRE {A set $MP(v, d - 1)$ of merged patterns $P_i(v)$ for $i = 1, 2, \ldots, d - 1$ and a set of patterns $P_d(v)$}
\ENSURE {A set $MP(v, d)$ of merged patterns $P_i(v)$ for $i = 1, 2, \ldots, d$}
\STATE \textbf{if} $d = 1$ \textbf{then return} $P_d(v)$
\STATE $MP(v, d) \gets \emptyset$
\FORALL{$(a, b) \in MP(v, d - 1)$}
\FORALL{$(a', b') \in P_d(v)$}
\STATE $x, x' \gets$ the only non-zero coordinates of $a$ and $a'$, respectively
\STATE \textbf{if} $x + x' - 1 > k$ \textbf{then continue}
\FORALL{$b^* \in \{0, \ldots, m\}^{k + 1}$ such that $\sum_i b^*_i = m - 1$}
\STATE $s \gets $ any sequence of $m - 1$ color cardinalities consistent with $b^*$
\STATE $\mathcal{P}(0) \gets \{b\}$, $\mathcal{P}'(0) \gets \{b'\}$
\FOR{$i = 1, \ldots, m - 1$}
\STATE $\mathcal{P}(i) \gets \emptyset$, $\mathcal{P}'(i) \gets \emptyset$
\FORALL{$(f, f') \in \mathcal{P}(i - 1) \times \mathcal{P}'(i - 1)$}
\FOR{$j = 1, \ldots, k-s_i$}
\IF{$f_{j} > 0$ \textbf{and} $f'_{s_i - j} > 0$}
\STATE Add $f - \mathbf{1}^{j}$ to $\mathcal{P}(i)$
\STATE Add $f' - \mathbf{1}^{s_i - j}$ to $\mathcal{P}'(i)$
\ENDIF
\ENDFOR
\ENDFOR
\ENDFOR
\IF{$\mathbf{0} \in \mathcal{P}(m - 1)$ \textbf{and} $\mathbf{0} \in \mathcal{P}'(m - 1)$}
\STATE Add $(\mathbf{1}^{x + x' - 1}, b^*)$ to $MP(v, d)$
\ENDIF
\ENDFOR
\ENDFOR
\ENDFOR
\RETURN $MP(v, d)$
\end{algorithmic}
\caption
{
An algorithm for merging a set of already merged patterns $P_i(v)$ for $i = 1, 2, \ldots, d - 1$ and a set of patterns $P_d(v)$.
}
\label{algorithm:composing-subcolorings-for-subblockgraphs} \end{algorithm}
First, we present a lemma about combining colorings of children of a cut-vertex $v$, to form colorings of descendants of $v$. Keep in mind that a (merged) pattern $(a,b)$ for descendants of given cut-vertex are such that there is unique $1$ in $a$. \begin{lemma}
For any $v \in V_{cut}$ and $d \in \mathbb{N}^+$ with a given set of merged patterns $MP(v, d - 1)$ and a given $P_d(v)$ \Cref{algorithm:composing-subcolorings-for-subblockgraphs} returns the set $MP(v, d)$.
\label{lem:composing-subcolorings-for-subblockgraphs} \end{lemma}
\begin{proof}
In the algorithm we determine if using some pattern $(a, b)$ from a set of the first $d - 1$ merged patterns $MP(v, d - 1)$ and some pattern $(a', b') \in P_d(v)$ a given merged pattern $(a^*, b^*)$ of $MP(v, d)$ can be obtained.
First, it has to hold that $v$ has to be colored with the unique color given by $a$ and $a'$.
We have to add the respective cardinalities and subtract one -- because this vertex is common in both colors, thus $a^*$ is determined uniquely.
Other classes can be composed by merging pairs of colors.
For simplicity, we check exhaustively the cardinalities of target color classes and we check if a sequence of such cardinalities can be achieved.
We may do this in a dynamic programming fashion, one color at a time.
For $i \in 0, \ldots, m-1$, the sets $\mathcal{P}(i)$ and $\mathcal{P}'(i)$ represents colors still to be merged after using some colors to construct colors of cardinalities $s_1, \ldots, s_i$.
For each tuple representing colors to be merged, we can choose in up to $k$ ways colors that can be combined to obtain a merged color of the desired size.
Clearly, each $MP(v, d)$ set does preserve only unique patterns. \end{proof}
\begin{corollary}
By applying \Cref{algorithm:composing-subcolorings-for-subblockgraphs} in a loop to a sequence of children of $v \in V_{cut}$ in $T_G$ in any order, we can obtain a set of all patterns for a graph induced by $G[D(v)]$, that is, $P(v)$.
\label{cor:composing-subcolorings-for-subblockgraphs} \end{corollary}
\begin{lemma}
If $v \in V_{cut}$ has exactly $d$ children in $T_G$, then a loop over all children using \Cref{algorithm:composing-subcolorings-for-subblockgraphs} requires $\textnormal{O}(dnk^4 m^{5k+6})$ time.
\label{lem:composing-subcolorings-for-subblockgraphs-complexity} \end{lemma}
\begin{proof}
The complexity is bounded very roughly.
The loops can be proceeded up to $\textnormal{O}(k m^{k+1}\cdot k m^{k+1} \cdot km^{k+1} \cdot m \cdot m^{2k+2} \cdot k) = \textnormal{O}(k^4m^{5k+6})$ times.
Moreover, with each addition of a coloring there can be an operation of merging the colors.
For simplicity, we can assume that the colors are stored as a list of vertices.
Hence, merging the lists and coping the colorings can be done in time $\textnormal{O}(m + n)$, which is = $\textnormal{O}(n)$ by the natural assumption that $m \le n$.
Together this gives an algorithm of complexity $\textnormal{O}(nk^4m^{5k+6})$.
Hence, by calling it $d$ times we obtain the desired complexity. \end{proof}
\subsection{Merging the colorings of children for a given block} \label{sec:merging-block}
\begin{algorithm}[ht]
\begin{algorithmic}[1]
\REQUIRE A block $B \in V_B$, its parent cut-vertex $v \in V_{cut}$ in $T_G$. Sets of all patterns $P(U)$ and $P(u)$ for some $U \subseteq B \setminus \{v\}$ and a vertex $u \in B \setminus (U \cup \{v\})$
\ENSURE A set of all patterns $P(U \cup \{u\})$
\STATE \textbf{if} $U = \emptyset$ \textbf{then return} $P(u)$ \COMMENT{Only for conformity with \Cref{algorithm:main-loop}}
\STATE $P(U \cup \{u\}) \gets \emptyset$
\FORALL{$(a, b) \in P(U)$}
\FORALL{$(a', b') \in P(u)$}
\STATE $x \gets$ the only non-zero coordinate in $a'$
\FOR{$i = 0, \ldots, k - x$}
\IF{$b_i > 0$}
\STATE $\mathcal{P}(0) \gets \{(a, b - \mathbf{1}^i, \mathbf{1}^{x + i}, \mathbf{0})\}$
\STATE $s \gets $ any sequence of $m - 1$ color cardinalities consistent with $b'$
\FOR{$j = 1, \ldots, m - 1$}
\STATE $\mathcal{P}(j) \gets \emptyset$
\FORALL{$(a^*, b^*, a^{**}, b^{**}) \in \mathcal{P}(j - 1)$}
\FOR{$l = 0, \ldots, k - s_j$}
\IF{$a^*_l > 0$}
\STATE Add $(a^* - \mathbf{1}^l, b^*, a^{**} + \mathbf{1}^{l + s_j}, b^{**})$ to $\mathcal{P}(j)$
\COMMENT {Color used for $B$}
\ENDIF
\IF{$b^*_l > 0$}
\STATE Add $(a^*, b^* - \mathbf{1}^l, a^{**}, b^{**} + \mathbf{1}^{l + s_j})$ to $\mathcal{P}(j)$
\COMMENT {Color not used for $B$}
\ENDIF
\ENDFOR
\ENDFOR
\ENDFOR
\FORALL{$(a^*, b^*, a^{**}, b^{**}) \in \mathcal{P}(m - 1)$}
\STATE Add $(a^{**}, b^{**})$ to $P(U \cup \{u\})$
\ENDFOR
\ENDIF
\ENDFOR
\ENDFOR
\ENDFOR
\RETURN $P(U \cup \{u\})$
\end{algorithmic}
\caption
{
For a given block $B$ together with its parent cut-vertex $v$, the procedure merges sets of patterns $U \subseteq (B \setminus \{v\})$, and sets of patterns of another vertex from $B \setminus \{U \cup \{v\}\}$.
}
\label{algorithm:merger-of-earlier-cliques-and-this-clique} \end{algorithm}
Now we present a lemma about combining sets of patterns for cut-vertices and simplicial vertices for a given block $B$. \begin{lemma}
For any $B \in V_B$, its parent cut-vertex $v \in V_{cut}$ in $T_G$, and any $U \subseteq B \setminus \{v\}$ and $u \in B \setminus (U \cup \{v\})$ with their given sets of patters $P(U)$ and $P(u)$ \Cref{algorithm:merger-of-earlier-cliques-and-this-clique} computes $P(U \cup \{u\})$.
\label{lem:merger-of-earlier-cliques-and-this-clique} \end{lemma}
\begin{proof}
First, we have to merge the color assigned to $u$ with other colors.
Of course, it cannot be used to any other vertex in $B$, hence we can merge only with the colors that are not assigned to vertices from $U$.
The third {\bfseries for} loop proceeds over all the choices.
Second, we have to merge other colors assigned to $D(u)$.
Observe that they can be merged with colors of $D(U)$ in any way -- either with colors assigned to the vertices in $B$ or with other colors.
To find all distinct ways of merging the colors we use a dynamic programming approach when proceeding over the colors of $D(u)$.
The next loop does so.
Precisely, at each step (when merging a color of $D(u)$) we preserve the colors that are yet to be merged and that were already merged.
The state of the dynamic program is denoted by $(a^*, b^*, a^{**}, b^{**})$.
The first two elements of the tuple represent the colors that are not yet merged.
The last two elements of the tuple represent the colors that are already merged.
Observe that we are not going to merge a given color of $D(u)$ with many colors of $D(U)$.
Perhaps, in some cases it would be feasible, but by an assumption that we are operating on all feasible patterns of $D(U)$, we do not have to consider this case.
Hence, we have to know which colors were already used (and they will not change anymore) and which are still available to be merged with -- and a tuple provides exactly that information.
In fact, for a given color of $D(u)$ we have to decide with a color of $D(U)$ of which size we will merge -- this is done by the innermost loop.
Having such a decision we can merge with a color assigned to the vertices of $B$ or with another color -- the {\bfseries ifs} are differentiating between the cases.
Finally, if in $\mathcal{P}(m - 1)$ there is some tuple, then it means that the corresponding pattern can be obtained.
Observe that for $i \in \{0, \ldots, m-1\}$ the set $\mathcal{P}(i)$ contain all obtainable patterns after merging $i$ colors from $P(u)$.
Finally, the returned set represents all patterns for all possible colorings of $G[D(U \cup \{u\})]$, thus it is exactly $P(U \cup \{u\})$. \end{proof}
\begin{corollary}
For any block $B \in V_B$ with its parent cut-vertex $v \in V_{cut}$ in $T_G$, a set of all patterns $P(B \setminus \{v\})$ can be obtained by calling \Cref{algorithm:merger-of-earlier-cliques-and-this-clique} for the consecutive vertices of $B \setminus \{v\}$.
\label{cor:merger-of-earlier-cliques-and-this-clique} \end{corollary}
\begin{lemma}
For any $B \in V_B$ and its parent cut-vertex $v$, patterns for all descendants of $B \setminus \{v\}$ can be obtained using \Cref{algorithm:composing-subcolorings-for-subblockgraphs} in $\textnormal{O}(|B|nk^4m^{7k+8})$ time.
\label{lem:merger-of-earlier-cliques-and-this-clique-complexity} \end{lemma}
\begin{proof}
The complexity of the algorithm is bounded very roughly.
The loops iterate $\textnormal{O}(m^{2k+2} \cdot km^{k+1} \cdot k \cdot m \cdot m^{4k+4} \cdot k)= \textnormal{O}(k^4m^{7k +8})$.
Taking into account the need of coping with the colorings and sizes of tuples this gives the algorithm of complexity $\textnormal{O}(nk^4m^{7k+8})$. \end{proof}
\subsection{Converting the patterns of a given block to the patterns of all descendants of given parent cut-vertex} \label{sec:block-to-cut}
\begin{algorithm}[htpb]
\begin{algorithmic}[1]
\REQUIRE A cut-vertex $v \in V(T_G)$, $d \in \mathbb{N}^+$, and $P(B \setminus \{v\})$ for a $d$-th child block $B$ of $v$ in $T_G$.
\STATE $P_d(v) \gets \emptyset$
\FORALL{$(a, b) \in P(B \setminus \{v\})$}
\FOR{$l = 0, \ldots, k - 1$}
\IF {$b_l > 0$}
\STATE Add $(\mathbf{1}^{l+1}, a + b - \mathbf{1}^l)$ to $P_d(v)$
\ENDIF
\ENDFOR
\ENDFOR
\RETURN $P_d(v)$
\end{algorithmic}
\caption{Compute $P_d(v)$ from $P(B \setminus \{v\})$ for the respective $d$-th child block $B$ of $v$ in $T_G$.}
\label{algorithm:coloring-cardinalities-for-a-block-and-fixed-vertex} \end{algorithm}
\begin{lemma}
Given $v \in V(T_G)$, $d \in \mathbb{N}^+$, and a set of patterns $P(B \setminus \{v\})$ for a $d$-th child block $B$ of $v$ in $T_G$ \Cref{algorithm:coloring-cardinalities-for-a-block-and-fixed-vertex} computes the set $P_d(v)$.
\label{lem:coloring-cardinalities-for-a-block-and-fixed-vertex} \end{lemma}
\begin{proof}
The algorithm assigns one of the colors that are not assigned to $B \setminus \{v\}$, and color $v$ with it.
Other colors, both assigned to $B \setminus \{v\}$ and not can be added (not merged) into one set -- they will not interfere with the colorings of other blocks. \end{proof}
\begin{lemma}
\Cref{algorithm:coloring-cardinalities-for-a-block-and-fixed-vertex} runs in time $\textnormal{O}(k m^{2 k + 2})$. \end{lemma}
The proof follows directly from the fact that $|P(B \setminus \{v\})| = \textnormal{O}(m^{2 k + 2})$.
\subsection{Computing the total time complexity}
Finally, we outline a recursive algorithm for constructing all feasible patterns $P(v)$ for a given cut-vertex, that gathers all the subroutines listed above.
\begin{algorithm}[htpb]
\begin{algorithmic}[1]
\REQUIRE {$v \in V_{cut}$}
\STATE Let $P(v) \gets (\mathbf{1}^{1}, (m - 1) \cdot \mathbf{1}^{0})$ \COMMENT{Initially only $v$ is colored}
\STATE $d_{max} \gets $ number of children of $v$ in $T_G$
\FOR{$d = 1, 2, \ldots, d_{max}$}
\STATE $B \gets$ $d$-th child block of $v$ in $T_G$
\STATE $U \gets \emptyset$, $P(U) \gets (\mathbf{0}, m \cdot \mathbf{1}^0)$ \COMMENT{$P(\emptyset)$ abuses notation, but allows for a cleaner loop}
\FORALL{$v' \in B \setminus \{v\}$}
\IF{$v' \in V_{cut}$}
\STATE $P(v') \gets$ \Cref{algorithm:main-loop} $(v')$
\ELSE
\STATE $P(v') \gets (\mathbf{1}^1, (m - 1) \cdot \mathbf{1}^0)$ \COMMENT{$v'$ is a simplicial vertex}
\ENDIF
\STATE $P(U \cup \{v'\}) \gets$ \Cref{algorithm:merger-of-earlier-cliques-and-this-clique} $(P(U), P(v'))$
\STATE $U \gets U \cup \{v'\}$
\ENDFOR
\STATE $P_d(v) \gets$ \Cref{algorithm:coloring-cardinalities-for-a-block-and-fixed-vertex} $(v, d, P(B \setminus \{v\}))$
\STATE $P(v) \gets$ \Cref{algorithm:composing-subcolorings-for-subblockgraphs} $(P(v), P_d(v))$
\ENDFOR
\RETURN $P(v)$
\end{algorithmic}
\caption{The main recursive function}
\label{algorithm:main-loop} \end{algorithm}
\begin{theorem}
For any $v \in V_{cut}$
\Cref{algorithm:main-loop} returns a set of all patterns $P(v)$.
\label{lem:MergeColoringsOfVertices} \end{theorem} \begin{proof} By \Cref{lem:merger-of-earlier-cliques-and-this-clique} and \Cref{cor:merger-of-earlier-cliques-and-this-clique}, the innermost \textbf{for} loop computes for every child block $B_i \in V_B$ of $v$ in $T_G$ a set $P(B_i \setminus \{v\})$. By \Cref{lem:coloring-cardinalities-for-a-block-and-fixed-vertex}, in line 13 of \Cref{algorithm:main-loop} we build from them the set $P_d(v)$. By \Cref{lem:composing-subcolorings-for-subblockgraphs} we merge it (together with patterns for the previous block) into $MP(v, d)$ for $d = 1, 2, \ldots$ in the consecutive iterations. Observe, that \Cref{algorithm:coloring-cardinalities-for-a-block-and-fixed-vertex} colors $v$. Finally, \Cref{cor:composing-subcolorings-for-subblockgraphs} proves that $MP(v, d_{max}) = P(v)$, which finishes the proof. This is by the assumption that initially $v$ is colored with one color with cardinality $1$ and by an observation that \Cref{algorithm:composing-subcolorings-for-subblockgraphs} takes into account that $v$ is already colored. \end{proof}
\begin{corollary}
For a root $r$ of $T_G$ \Cref{algorithm:main-loop} finds $P(r)$, the set of all patterns for a block graph $G$ such that where cardinalities of color classes are bounded by a constant $k$.
\label{theorem:final-k-coloring-block-graphs} \end{corollary}
\begin{theorem}
\Cref{algorithm:main-loop} executed for a root $r$ of $T_G$ runs in $\textnormal{O}(n^2 m^{7k+1})$ time.
\label{theorem:complexity-k-coloring-block-graphs} \end{theorem}
\begin{proof} Let us calculate in total how many times the algorithms will be called. Observe that \Cref{algorithm:composing-subcolorings-for-subblockgraphs,algorithm:merger-of-earlier-cliques-and-this-clique,algorithm:coloring-cardinalities-for-a-block-and-fixed-vertex} will be called $\textnormal{O}(n)$ times in total. This is due to the fact that \Cref{algorithm:composing-subcolorings-for-subblockgraphs} is called for each block, \Cref{algorithm:merger-of-earlier-cliques-and-this-clique} for each block or simplicial vertex, and \Cref{algorithm:coloring-cardinalities-for-a-block-and-fixed-vertex} for each block again. \Cref{algorithm:main-loop} consists of these calls (counted already) and some simple initialization called $\textnormal{O}(n)$ times in total. Together this gives time complexity $\textnormal{O}(n^2k^4 m^{5k+6} + n^2k^4m^{7k+4} + n m^{2k+2})) = \textnormal{O}(n^2k^4m^{7k+4})$. Note, however, that we can exploit the fact that the complexity of the algorithms depends on the number of children, which sums up to $\textnormal{O}(n)$.
\end{proof}
Note that if $G$ is not connected, instead of constructing an additional algorithm for merging patterns for connected components, we can simply add a dummy cut-vertex $r$ which connects the graph (and which is a root of $T_G$), and additional color. This way, we have to check in the final $P(r)$, whether there exists a coloring of the block graph rooted in $r$ with pattern $(a, b)$, where $a = \mathbf{1}^{1}$ which means that the dummy vertex has a unique color.
Clearly, \Cref{theorem:final-k-coloring-block-graphs} and \Cref{theorem:complexity-k-coloring-block-graphs} taken together translate directly to the claim presented in \Cref{corollary:block-graph-fixed-cmax}.
\section[Complete proof of Lemma 2]{Complete proof of \Cref{lem:equal_coloring_block}}
The proof follows on induction on the number of blocks. For a graph having exactly one block the theorem holds: we assign at most one job to every machine, and clearly $\ceil{n/(m-1)} \ge 1$. It is also easy to verify that if $r = 0$, then it has to be the case $n = m - 1$ for the problem to be feasible, thus there is one machine with a load less than $1$. Similarly, the cases for $n \le m - 2$ and $n = m$ hold.
Now, assume that the theorem holds for all graphs composed of $k$ blocks, and consider any block graph $G$ composed of $k+1$ blocks. Let us fix a subgraph $G'$ of $G$ on $n'$ vertices including only the first $k$ blocks according to its block-cut tree pre-order traversal. Let $n' = d' (m - 1) + r'$ be the number of the vertices of $G'$. By induction assumption, the greedy algorithm for $G'$ returns a schedule whose total makespan does not exceed $\ceil{n' / (m - 1)}$.
We schedule the remaining jobs from the last block according to the greedy algorithm, assigning at most one job to each machine.
If $r' = 0$, then it is clear that the limit on the number of jobs increased by at least one since $\ceil{n/(m-1)} \ge d' + 1 = \ceil{n' / (m - 1)} + 1$, thus the extended schedule satisfies the total makespan condition. By considering separately cases $n - n' \le m - 2$, $n-n' = m - 1$, and $n-n' = m$ we can verify directly that we have the required number of machines that are processing less than $\ceil{n/(m-1)}$ jobs. \begin{itemize}
\item In the first case we have at most $r = n-n'$ machines where the new jobs are assigned.
Hence, only these machines can have $\ceil{n/(m-1)}$ jobs in the extended schedule, thus $m - r$ machines have to be loaded with less than $\ceil{n/(m-1)}$ jobs.
\item In the second case it holds $r = r' = 0$, and clearly there is at least one machine with load less than $\ceil{n/(m-1)}$.
\item The last case means that a clique on $m$ vertices is added -- the $(k+1)$-th block has no common vertex with other $k$ blocks.
However, this means that the new upper limits is $\ceil{n/(m-1)} = \ceil{n' / (m - 1)} + 2$, and hence there are $m$ machines with load less than $\ceil{n/(m-1)}$, in fact. \end{itemize}
For $r' > 0$ we can split the proof into four cases. In each case, by induction, there is at least $m - r'$ machines processing less than $\ceil{n' / (m - 1)}$ jobs. However, due to the fact that the blocks may have a common cut-vertex, perhaps one of them is not available for scheduling. \begin{itemize}
\item If $r' + n-n' < m - 1$, or equivalently, $n-n' < m - r - 1$, then all jobs of $G - G'$ can be scheduled on machines processing less than $\ceil{n' / (m - 1)}$ jobs.
In the end $m - r' - (n-n') = m - r$ machines process less than $\ceil{n/(m-1)}$ jobs,
\item If $r' + n-n' = m - 1$, then we still schedule the jobs of $G - G'$ on the machines processing less than $\ceil{n' / (m - 1)}$ jobs.
After extending the schedule there remains at least one machine with load less than $\ceil{n' / (m - 1)}$.
\item If $m \le r' + n-n' \le 2 m - 3$, then, $r = r' + (n-n')-(m-1)$, and, by induction, there were at most $r'$ machines processing $\ceil{n' / (m - 1)}$ jobs.
Consider the case that one of the machines processing less than $\ceil{n' / (m - 1)}$ cannot be scheduled upon.
Moreover, let $n-n' > m - r' - 1$
We skip other subcases, because they are similar and easier to prove.
Observe that no more than $r' - (n - n' - (m-r'-1))$ machines were processing $\ceil{n' / (m - 1)}$ jobs and are processing $\ceil{n/(m-1)}$ jobs now.
Hence, there will be at least $r' - (n - n' - (m-r'-1)) + (m-r'-1) + 1$ machines that are processing less than $\ceil{n/(m-1)}$.
By a simple transformation we obtain the induction thesis.
\item if $r' + n-n' \ge 2 m - 2$, then it has to be the case $r = 0$, $n-n' = m$, $r' = m - 2$.
Hence, again the added block is a clique on $m$ vertices.
By induction there were at least $1$ machine processing less than $\ceil{n' / (m - 1)}$ jobs.
Clearly, at least this machine is processing less than $\ceil{n/(m-1)}$ in the extended schedule. \end{itemize} In the first two cases $\ceil{n/(m-1)} = \ceil{n' / (m - 1)}$, but the jobs are assigned only to machines that are processing less than $\ceil{n' / (m - 1)}$ jobs. The machines are guaranteed to exist, by induction. In the other cases $\ceil{n/(m-1)} > \ceil{n' / (m - 1)}$, thus by assigning at most one additional job to every machine we cannot violate the total makespan condition.
We verified that both the total makespan condition and the condition on the number of machines processing less than $\ceil{n/(m-1)}$ jobs are fulfilled. Therefore, we see that extending the scheduling for the last block is always possible, which concludes the proof of the induction step.
\end{document} | arXiv |
Development and evaluation of a meat mitochondrial metagenomic (3MG) method for composition determination of meat from fifteen mammalian and avian species
Mei Jiang1,
Shu-Fei Xu2,
Tai-Shan Tang3,
Li Miao4,
Bao-Zheng Luo5,
Yang Ni6,
Fan-De Kong2 &
Chang Liu1
Bioassessment and biomonitoring of meat products are aimed at identifying and quantifying adulterants and contaminants, such as meat from unexpected sources and microbes. Several methods for determining the biological composition of mixed samples have been used, including metabarcoding, metagenomics and mitochondrial metagenomics. In this study, we aimed to develop a method based on next-generation DNA sequencing to estimate samples that might contain meat from 15 mammalian and avian species that are commonly related to meat bioassessment and biomonitoring.
In this project, we found the meat composition from 15 species could not be identified with the metabarcoding approach because of the lack of universal primers or insufficient discrimination power. Consequently, we developed and evaluated a meat mitochondrial metagenomics (3MG) method. The 3MG method has four steps: (1) extraction of sequencing reads from mitochondrial genomes (mitogenomes); (2) assembly of mitogenomes; (3) mapping of mitochondrial reads to the assembled mitogenomes; and (4) biomass estimation based on the number of uniquely mapped reads. The method was implemented in a python script called 3MG. The analysis of simulated datasets showed that the method can determine contaminant composition at a proportion of 2% and the relative error was < 5%. To evaluate the performance of 3MG, we constructed and analysed mixed samples derived from 15 animal species in equal mass. Then, we constructed and analysed mixed samples derived from two animal species (pork and chicken) in different ratios. DNAs were extracted and used in constructing 21 libraries for next-generation sequencing. The analysis of the 15 species mix with the method showed the successful identification of 12 of the 15 (80%) animal species tested. The analysis of the mixed samples of the two species revealed correlation coefficients of 0.98 for pork and 0.98 for chicken between the number of uniquely mapped reads and the mass proportion.
To the best of our knowledge, this study is the first to demonstrate the potential of the non-targeted 3MG method as a tool for accurately estimating biomass in meat mix samples. The method has potential broad applications in meat product safety.
Meat represents a significant portion of daily human consumption. However, meat adulteration has become a global issue. Valuable and expensive meat, such as beef and mutton, is often detected mixed with cheaper chicken, duck, pork, mink and animal meat [1, 2]. For instance, two of the nine beef samples examined by Erol et al. contained horse and deer meat [3]. Such adulteration harms consumers' rights and interests [4] and disrupts market order [5]. Therefore, identifying adulterated ingredients in meat and meat products is essential.
Based on next-generation DNA sequencing, many methods for determining the biological composition of mixed samples have been developed, including metabarcoding [6], metagenomics [7, 8] and mitochondrial metagenomics (MMG) [9]. The metabarcoding approach depends on the PCR amplification of a particular marker for species determination. The metagenomics approach consists of two steps for species determination and biomass quantification, namely, shotgun sequencing and mapping of read to whole nuclear genomes. MMG is essentially a metagenomic method using mitochondrial genomes (mitogenome) instead of nuclear genomes as references. The PCR amplification-dependent metabarcoding method is the workhorse for the molecular determination of biological composition.
Numerous markers have been tested on animals, including 18S rRNA genes from the nuclear genome, 16S rRNA gene and cytochrome c oxidase I (COX1, CO1 or COI) gene from the mitogenome [10]. However, these PCR-dependent methods have limitations. Firstly, they require universal primers targeting particular markers, usually lacking across all taxa [11]. Different sets of universal markers and primer pairs complicate data integration when different markers are used, and different primer pairs are used for the same markers. Secondly, even with universal primers, template DNA molecules with different sequences have different melting properties, leading to amplification bias [12]. Consequently, the direct quantification of template DNA molecules with different sequences is difficult.
All-Food-Seq (AFS) is a recently developed metagenomics method [8], in which the non-targeted deep sequencing of total genomic DNA from foodstuff, followed by bioinformatics analysis, can identify species from all kingdoms of life with high accuracy. It facilitates the quantitative measurement of the main ingredients and detection of unanticipated food components. Conceptually, the AFS method has set up a framework for ultimate bio-surveillance.
However, the AFS method has several practical limitations. Firstly, the method is probably extremely complex for bioassessment and biomonitoring because a whole genome has a high degree of complexity. Secondly, although whole-genome databases have expanded rapidly, obtaining high-quality whole-genome sequences for a species requires many years. The effect of genomic diversity on bioassessment and biomonitoring is unknown. Thirdly, this study used simulated data rather than experimental data.
MMG delimits closely related species from mixed samples [13, 14]. This method is desirable because of its advantages. Firstly, a mitogenome and its genes are common phylogenetic, DNA barcoding and metabarcoding markers. Secondly, the structures of mitogenomes are conserved, whereas sequences can be highly diverse. Thirdly, mitogenomes are small and easy to obtain and can be directly reconstructed using bioinformatics methods. Fourthly, large numbers of mitogenomes are available in public databases. More than 10,000 mitogenomes have been included in the GenBank in December 2020. The performance and accuracy of metabarcoding and MMG in biomass estimation in invertebrate community samples have been evaluated [15]. Overall, MMG yields more informative predictions of biomass content from bulk macroinvertebrate communities than metabarcoding. However, despite that MMG has been applied to ecological assessment [9, 16,17,18,19,20,21], the use of MMG in mammalian and avian meat mixed samples have not been examined to the best of our knowledge.
In this study, we intended to use either metabarcoding or MMG to detect the potential mixing of meat from 15 mammalian and avian species on the basis of a market survey. Preliminary studies suggested that the most commonly used metabarcoding markers, COI and 16S, are unsuitable for simultaneously detecting meat from these 15 species. Thus, we tested MMG in mixed meat samples. This approach, called 'meat mitochondrial mitogenome (3MG)', circumvent the problem of marker selection, PCR bias and sequencing bias. Additionally, this approach takes advantage of the availability of mitogenomes for many species. The results showed that it can accurately determine the biological composition of meat mix samples and accurately estimate biomass. The method has a wide range of applications in food and pharmaceutical industries involving animal products.
Meat samples and mock mixed meat samples
We prepared mock samples with meat from the legs of 15 mammalian and avian species: Anas platyrhynchos (duck), Bos taurus (cattle), Camelus bactrianus (camel), Canis lupus familiaris (dog), Equus caballus (horse), Gallus gallus (chicken), Mus musculus (mouse), Mustela putorius voucher (ferret), Myocastor coypus (nutria), Nyctereutes procyonoides (raccoon dog), Oryctolagus cuniculus (rabbit), Ovis aries (sheep), Rattus norvegicus (rat), Sus scrofa domesticus (pig) and Vulpes vulpes (fox). Efforts had been made to extract meat samples with homogenous compositions intraspecificly and interspecificly. We obtained camel, nutria, fox, donkey and deer meat from breeding farms. Nanjing Medical University provided the mouse, rabbit and rat samples. The Entry-exit Inspection and Quarantine Bureau provided other meat samples. The detailed information regarding sample origin, particularly cities and institutions, is provided in Table 1.
Table 1 Information for meat samples used in this study
Two methods were used in mixing the samples. One mix contained meat samples in equal amounts from 15 species. This mix was referred to as the 'mix containing meat from 15 species' or 'M1'. The other mix contained meat from S. scrofa domesticus (pig) and G. gallus (chicken) in the following proportions: 10:0 ('sample 1; mix containing two species' or 'M2-S1'), 8:2 (M2-S2), 6:4 (M2-S3), 4:6 (M2-S4), 2:8 (M2-S5) and 0:10 (M2-S6). Each M1 or M2 sample has three replicates.
Loop-mediated isothermal amplification (LAMP)
We performed loop-mediated isothermal amplification (LAMP) experiments to validate the composition of mock samples (M1 and M2). LAMP methods for detecting ingredients that contain cattle, sheep, pig, chickens and duck meat were developed by the Technology center of Xiamen Entry-Exit Inspection and Quarantine Bureau of the People's Republic of China [22,23,24]. The probe and primer sequences target cytB genes from the corresponding species were provided in Table S1. The PCR reaction mix contained isothermal master mix (15 μL), primer mix (FIP, 2 μL; BIP, 2 μL; F3, 1 μL; B3, 1 μL) and DNA (1 μL). We added RNase-free water to the final reaction of 50 μL. The experimental conditions were as follows: amplification at 60 °C for 90 min and annealing from 98 °C to 80 °C at a rate of 0.05 °C per second.
DNA extraction, library construction and next-generation sequencing (NGS)
We extracted genomic DNA with a modified sodium dodecyl sulfate (SDS)-based method [25]. The integrity and concentration of the extracted DNA were detected through electrophoresis in 1% (w/v) agarose gel and spectrophotometer (Nanodrop 2000; Thermo Fisher Scientific, USA). The extracted DNA samples (100 ng) were subjected to library construction using NEBNext® Ultra™ II DNA library prep kit for Illumina® (New England BioLabs, USA) according to the manufacturer's recommendations. Each library had an insert size of 500 bp. The quantity and quality of the libraries were analysed using Agilent 2100 Bioanalyser (Agilent Technologies, USA). We sequenced the libraries using the HiSeq X reagent kits (Illumina, USA) in an Illumina Hiseq X sequencer. We deposited the data generated in this study in GenBank. The accession numbers were SRR9107560 and SRR9140737.
Construction of mitogenome reference databases
We constructed a database (15MGDB), which had complete mitogenome sequences from the 15 species. The 15 mitogenome sequences were downloaded from GenBank, with the following accession numbers: A. platyrhynchos (NC_009684), B. taurus (NC_006853), C. bactrianus (NC_009628), C. lupus familiaris (NC_002008), E. caballus (NC_001640), G. gallus (NC_001323), M. musculus (NC_005089), M. putorius voucher (NC_020638), M. coypus (NC_035866), O. cuniculus (NC_001913), O. aries (NC_001941), N. procyonoides (NC_013700), R. norvegicus (NC_001665), S. scrofa domesticus (NC_012095), V. vulpes (NC_008434). The sequences in 15MGDB were used in constructing a searchable database with the makeblastdb command from the BLAST+ (v2.7.1) software package [26] and with the option '-dbtype nucl -parse_seqids'.
Development of the 3MG analysis pipeline
The 3MG pipeline was developed using Python 2.7.15 with the following third-party software applications: pandas module in python, BBMap (v35.66; https://sourceforge.net/projects/bbmap/), MITOBim (v1.9.1) [27], Blast+ (v2.7.1) [26], bowtie2 (v2.3.4) [28] and samtools (v1.9) [29]. The source code, sample data and instruction for using the locally installed copy of the 3 mg pipeline and a singularity container for running the 3 mg pipeline can be found using the following link: http://1kmpg.cn/3mg/.
Determination of 3MG detection errors using simulation
We generated 21 sets of data through simulation. Reads from an M2-S1 sample containing 100% pork was used as the background. Reads were then extracted from the reads of M2-S6 containing only chicken with the seqtk program (v1.3-r106) and with the option 'seqtk sample -s100'. The reads extracted from M2-S6 were mixed with those from M2-S1 in the following percentages: 0.01, 0.1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100%. We prepared five replicates of simulated data for each percentage level and used default seeds. The resulting sample sets were then analysed using the 3MG pipeline. We calculated relative detection errors with the following formula as described previously [8].
$$\mathrm{Relative}\ \mathrm{error}= \mid \left(\mathrm{Number}\ \mathrm{of}\ \mathrm{chicken}\ \mathrm{reads}\ \mathrm{detected}-\mathrm{Number}\ \mathrm{of}\ \mathrm{chicken}\ \mathrm{reads}\ \mathrm{in}\ \mathrm{the}\ \mathrm{sample}\right) \mid /\left(\mathrm{Number}\ \mathrm{of}\ \mathrm{chicken}\ \mathrm{reads}\ \mathrm{in}\ \mathrm{the}\ \mathrm{sample}\right)$$
Comparison of the reference and assembled mitogenomes
We aligned the assembled sequences with their reference sequences for each species using the CLUSTALW2 (v2.0.12) program [30] with option '-type = dna -output = phylip'. We used these aligned sequences in constructing phylogenetic trees with the maximum likelihood (ML) method implemented in RaxML (v8.2.4) [31]. We calculated the intra-specific and inter-specific distances among mitogenomes using the distmat program from EMBOSS (v6.3.1) [28] with the options '-nucmethod = 0'. Corrections for multiple substitutions cannot be made through this method. Finally, we calculated the p-distances among mitogenomes with MEGA (v7) [32].
Detection of other contaminating biological composition
Taxon content in reads unmapped to mitogenomes were analysed using the RDP classifier (v2.12) [33]. The unmapped reads were assigned to COX1 and 16S rRNA database with an assignment confidence cutoff of 0.8. The 16S rRNA database is part of the RDP Classifier package. The COX1 database was downloaded from https://github.com/terrimporter/CO1Classifier/releases/tag/v3.2 [34]. The results were visualised using MEGAN (v6) [35] with the following LCA parameters: 'minSupportPercent = 0.02, minSupport = 1, minScore = 50.0, maxExpected = 0.01, topPercent = 10.0 and readAssignmentMode = readCount'.
Evaluation of the metabarcoding method for the 15 mammalian and avian species
To determine whether the mixture containing 15 species can be identified using metabarcoding, we analysed the availability of universal primers and the ability of their amplified products (if applicable) to distinguish the 15 species. For the COX1 gene, no primer matched the sequences from all the species. For instance, the maximum number of matched species was five when the primer pair I-B1 and COI-C04 was used (Table S2). For the 16S rRNA gene, only one primer set, 16Sbr-H, matched the sequences of all the species, and the amplified products showed high degrees of variations that were sufficient for distinguishing the 15 species (Fig. S1). For the 18S rRNA, only the primer Uni18S was found in the sequences of all the species, but the amplified products were highly conserved and could not be used in distinguishing the 15 species (Fig. S2). Previously, the performance of COI metabarcoding and that of shotgun mitogenome sequencing were compared. Shotgun sequencing can provide highly significant correlations between read number and biomass [17]. As a result, we focused on developing the metagenomic approach for the direct biomass estimation of meat samples from the 15 species.
Development of 3MG method
The 3MG pipeline can be divided into four steps (Fig. 1). The first step is 'extracting mitochondrial reads'. We searched next-generation sequencing (NGS) reads against 15MGDB by using the BLASTN command with the following parameters: -evalue = 1e-5 and –outfmt = 6. We extracted the matched reads using the 'filterbyname.sh' command in the BBMap software package (v35.66). The extracted reads were called 'mitochondrial reads' and used in the subsequent procedures.
Flow chart of the 3MG pipeline. The 15 species used for the qualitative analyses and the setup of meat from the two species for the quantitative analyses are shown on the top. The four steps are labeled as S1, S2, S3 and S4. The results of each step are shown in the black rectangle. The third-party tools are shown on the right side of the corresponding process
The second step is assembling mitogenomes from mitochondrial reads. The mitogenomes in the public database might have originated from a particular individual or subspecies. Thus, the sequences from the samples might differ from those in the public database because of intra-specific variations. To ensure accurate qualitative and quantitative analyses, we assembled the mitogenomes according to the NGS reads and used MITOBim (v1.9.1) [27] with the default parameters. The mitogenome sequences downloaded from GenBank were used as references. They were called reference mitogenomes in the subsequent text.
The third step is mapping reads to the assembled mitogenomes. We mapped the reads to each assembled sequence of the species with bowtie2 (v2.3.4) [28], using default parameters. We then extracted the mapped reads using samtools (v1.9) [29] with the following command: 'samtools view -bF 4.'
The fourth step is identifying and counting reads uniquely mapped to the assembled mitogenomes. The mitogenome sequences were highly conserved. Some reads may be mapped to the mitogenomes of multiple species. We calculated the p-distances among these mitogenomes to determine how conserved they were. The p-distances among the 15 mitogenomes ranged from 0.14 to 0.49 (Fig. S3). These numbers indicated a high degree of mitogenome sequence conservation. All the mapped reads may have originated from multiple sources. To overcome this problem, we developed a custom python script to remove non-specific reads. Specifically, we obtained 15 files recording the mapped reads of each species. We compared the mapped reads of the target specie with those of other 14 species and deleted non-specific reads appeared multiple files. The proportions of unique reads mapped to the mitogenome of a particular species in all mitochondrial reads were calculated. When the proportion was greater than 2% (the cutoff of 2% was set according to the results of Determination of detection sensitivity for 3MG methods based on simulated datasets section), the species was called 'presence'.
Determination of detection sensitivity for 3MG methods based on simulated datasets
We constructed 21 simulated datasets (Table 2, SD01-SD21) mixed with 30,000 pork (major composition) and chicken mitochondrial reads (minor composition) to determine detection sensitivity. The percentages of chicken mitochondrial reads were 0.01, 0.1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20,30, 40, 50, 60, 70, 80, 90 and 100% of all the mitochondrial reads in the simulated dataset. We prepared five replicates at each proportion. We analysed the data using the 3MG analysis pipeline. We calculated the relative detection error at each proportion (Table 2). At a high proportion, the quantitative detected results were similar to the simulation results. At 2–100%, we detected the minor composition with a relative error of < 5%. However, the accuracy was significantly reduced at 1, 0.1 and 0.01%. These results indicated that the method can detect a contaminant at a proportion of 2% and has an error rate of < 5%.
Table 2 Relative detection errors determined using the simulated dataset
Qualitative determination of biological composition with the 3MG method
Sequencing results and characterisation
We constructed mixed samples containing meat from 15 animals (M1). The animals fall into several categories. For example, pork, cattle, chicken, lamb, duck and rabbit are primarily used for human consumption. Ferret, nutria, raccoon dog and fox are commonly used in the fur industry. Dogs are companion animals. Camel and horse are used for multiple purposes. Rat and mouse co-inhabit with humans and their meat can potentially contaminate other meat for human consumption. Adulteration of meat not meant for human consumption have been reported, particularly through the addition of meat of fur animals to pork or beef or substitution of pork with horse meat [2]. The motivation is primarily economic gain, as profits can be made when expensive meat is replaced with cheap meat. Some of these adulterations can be culturally offensive, for example, adding pork in food for Islamic consumers [36, 37].
We constructed three M1 samples labeled as 'R1', 'R2' and 'R3', respectively. We obtained 23.45, 24.10 and 28.56 GB for each of the three replicates (Table 3). The percentages of bases having Quality scores ≥30 were 89.51, 88.97 and 89.97%.
Table 3 Summary of sequencing data for samples M1 and M2
Qualitative analysis of M1 sample's biological composition
We analysed the NGS data generated from the M1 samples using the pipeline 3MG. In step one, 331,866 (0.43%), 222,702 (0.28%) and 267,495 (0.28%) reads were mapped to the mitogenomes and were extracted as mitochondrial reads (Table 4). In step two, we successfully assembled all 15 mitogenomes from mitochondrial reads. We constructed a phylogenetic tree using 15 pairs of assembled and reference mitogenomes (Fig. 2). The alignment of the 15 pairs of mitogenomes is shown in Fig. S4. The reference and assembled mitogenomes for the same species were clustered together (left part of Fig. 2).
Table 4 Summary of reads mapped to the mitogenomes of the 15 species
Phylogenetic analysis of the reference ('R') and assembled ('A') mitogenomes. The intra-specific and inter-specific nucleotide distances of mitogenomes are shown in the squares to the right of the tree. The intra-specific nucleotide distance was calculated between the reference and assembled mitogenomes for the same species. We calculated the inter-specific nucleotide distances between each of the 14 pairs of mitogenomes. Each pair consisted of one focal mitogenome and a mitogenome. The average inter-specific distances are shown
To compare the intra-specific and inter-specific distances, we calculated the distances, as shown in the right part of Fig. 2. Intra-specific distance was the distance between the assembled and reference mitogenomes for a particular species. By contrast, inter-specific distance was the average distance between the assembled mitogenomes of the focal species and those of the other 14 species. Intra-specific nucleotide distances were much smaller than the inter-specific distances. Thus, we assembled the mitogenomes of specific species from the metagenomic data with high accuracy. Our assembled mitogenomes unlikely contained chimeric sequences because the inter-specific distances were significantly larger than the intra-specific distances.
In step three, we mapped these mitochondrial reads to the 15 assembled mitogenomes. Approximately 10,000–90,000 reads were mapped to each mitogenome (Table 4). However, an average of 52.07% reads was mapped to multiple species. Thus, using the total number of mapped reads led to the overestimation of the meat content of a particular species. For example, 79.77% of the reads mapped to the B. taurus (cattle) mitogenome were non-specific, and 63.70% of the reads mapped to the O. aries (sheep) mitogenome were non-specific. Using the total number of reads in estimating the beef and lamb content led to overestimation. Hence, 3MG determines biological composition according to the number of reads uniquely mapped to a particular mitogenome.
In step four, we identified reads uniquely mapped to the mitogenome of each species. The proportion of unique reads to all mitochondrial reads was more than 2% for 12 species in at least one replicate sample (Table 4). The mapped read proportions for B. taurus, O. cuniculus and R. norvegicus were approximately 1%. In summary, through the analysis of the 15 species mix with this 3MG pipeline, 12 of 15 (80%) species were successfully identified with a confidence level of 95%.
Validation of 3MG analysis results for M1 samples by using LAMP experiments
LAMP is commonly used in detecting the biological composition of meat products. We used LAMP results to evaluate the accuracy of the 3MG results. Unfortunately, LAMP protocols are available for the meat of only five of 15 species (pig, sheep, cattle, duck and chicken). Thus, only these five species in the M1 samples were tested. The experiments were conducted separately for each target species (Fig. 3). The results confirmed the presence of meat from pig (Fig. 3A), sheep (Fig. 3B), cattle (Fig. 3C), duck (Fig. 3D) and chicken (Fig. 3E) in our experimental samples, consistent with the results obtained from the 3MG method.
Composition determination of the M1 samples with the LAMP method. The X-axis represents time, whereas the Y-axis represents relative fluorescence abundance. Each species was tested in three replicates shown in different colors
Quantitative determination of biological composition in mixed sample
Sequencing results and characteristics
To determine the performance of 3MG in estimating the biological composition in biomass, we prepared a series of samples by mixing meat from S. scrofa domesticus (pig) and G. gallus (chicken) in different proportions. We performed DNA extraction, library construction, DNA sequencing and DNA analyses in the same way as those for the M1 samples. The sequencing results are summarised in Table 3. We generated an average of 2.95 GB of data with 19,749,625 raw reads for each M2 sample. Approximately 90% of bases had quality scores greater than 30.
Quantitative analysis of M2 samples' biological composition
The number of reads mapped to the pig and chicken mitogenomes were shown in Table S2. The proportion of reads uniquely mapped to the pig mitogenomes was called meat content estimated with 3MG. They were plotted against the known meat content (Fig. 4A). Regression analyses showed that the pork's estimated and known meat content had a correlation coefficient of 0.98. Similarly, based on relative read counts, the estimated meat content for chicken were plotted against known meat content (Fig. 4B). Regression analyses showed that the estimated and known meat content had a correlation coefficient of 0.98. The high correlation coefficient between the estimated and known content suggested that the 3MG method can use the percentage of uniquely mapped reads in quantitatively determining biological composition in a meat mix.
Results of quantitative analysis using the 3MG and LAMP for two species. A and B Quantitative analysis results of the 3MG for a combination of two species. The X-axis shows the mass proportions of pork (A) and chicken (B) in the mix samples. The Y-axis shows the proportions of reads uniquely mapped to pork (A) and chicken (B) mitogenomes from mixed samples. The R-value represents the correlation coefficient between the proportions of uniquely mapped reads and the mass proportions of the samples. C and D Results of quantitative analysis using LAMP for the mixed samples of two species. The Y-axis shows the peak times for detecting pork (C) and chicken (D) components in the mix samples. The R-value represents the correlation coefficient between peak time and the mass proportion in the mixed samples
Validation of 3MG analysis results using LAMP experiments
We conducted a LAMP experiment to determine the quantity of pork and chicken in different ratios. We then compared the LAMP results with those obtained from the 3MG method. The peak time for detecting composition was called meat content estimated by LAMP and plotted against the known meat content of pig (Fig. 4C) and chicken (Fig. 4D). Regression analyses showed that the estimated and known meat content had correlation coefficients of 0.99 (pork) and 0.96 (chicken). Consequently, the 3MG results were consistent with the LAMP results. However, the variations in the LAMP results for chicken were significantly higher than those in the 3MG results. This observation suggested that the 3MG results were more stable than the LAMP results, at least for chicken meat.
Estimation of the relative correction factor for DNA–biomass ratio from different species
We mixed the meat of 15 species to construct M1 samples in equal mass ratios as described earlier. However, the number of reads mapped to each mitogenome of the 15 species varied significantly (Table 4). This discrepancy was likely due to the differences in mitogenome DNA content among the 15 species at the same meat biomass. As a result, a correction factor was needed when the meat mass was estimated from uniquely mapped read counts for a particular species. Using the number of reads uniquely mapped to the S. scrofa domesticus mitogenome as the baseline, we calculated the relative correction factors for the other species. The correction factors were 1.00 for A. platyrhynchos, 0.77 for C. bactrianus, 1.46 for C. lupus familiaris, 0.59 for E. caballus, 0.65 for G. gallus, 0.32 for M. musculus, 0.40 for M. putorius voucher, 1.24 for M. coypus, 0.43 for N. procyonoides, 0.31 for O. aries and 1.94 for V. vulpes. This set of relative correction factors might correlate with the relative copy numbers of mitogenome in the muscle tissues of each species. They can be used in estimating meat content for different species. Detailed discussions on the ratios of nuclear DNA to mitochondrial DNA and DNA to biomass are provided in the following text.
To determine the presence of unexpected biological composition in the samples, we classified the unmapped reads with the RDP classifier and analysed them using MEGAN6. The unmapped reads can be divided into four categories: bacteria, Archaea, Eukaryota and 'not assigned' (Fig. 5). Five genera belonging to Eukaryota were annotated: Myocastor, Canis, Sus, Anas and Gallus. These reads may have been extremely diverse and thus were not mapped to the mitogenomes in the 3MG process. Overall, we detected few contaminants from other mammals and bacteria in our mock mix samples.
Analyses of reads unused by the 3MG. The phylogram shows the taxa at the genus level at which reads were mapped. Each circle of the tree represents a taxon labeled by its name and the number of reads assigned to it. The size of the circle represents the proportion of reads aligned to the taxon and cannot be aligned to a lower level of the taxon
Meat adulteration and contamination can affect consumers' well-being, disrupt market order and insult religious beliefs. Hence, the development of qualitative, quantitative and unbiased methods for detecting the composition of meat products is of great importance. In the present study, we found that meat composition from 15 species cannot be identified with the metabarcoding approach because of the lack of universal primers or the needed discrimination power. Therefore, we developed a meat mitochondrial metagenomics (3MG) method to determine the composition of 15 meat most commonly found in food markets.
The evaluation of detection sensitivity for the 3MG methods based on simulated datasets indicated that the method can detect a contaminating composition at a proportion of 2% and has an error rate of < 5%. This method successfully identified the presence of 12 of 15 (80%) species with the threshold of detection sensitivity. This result showed that the method can simultaneously detect the presence of multiple species with high sensitivity. It can detect a wide variety of adulterated meat in the market. In addition, the analyses of the two species mixed samples revealed correlation coefficients of 0.98 for pork and 0.98 for chicken between the number of uniquely mapped reads and the mass proportion. The 3MG results were more stable than the LAMP results, at least for chicken meat, indicating that the method can use the percentage of uniquely mapped reads in quantitatively determining biological composition in a meat mix.
To the best of our knowledge, this study is the first to demonstrate the usefulness of the mitochondrial metagenomics method in detecting meat composition and estimating biomass. This method has several advantages over methods based on PCR amplification and particular markers. It is a non-targeted approach and does not need to assume the biological composition of samples. Consequently, it is likely to have a lower false-negative detection rate. Given that PCR-based methods require species primers, they often fail to amplify sequences not matched by primers. Furthermore, the 3MG method is not affected by problems in PCR reactions, such as the generation of multiple PCR bands resulting from non-specific amplification. The detection of multiple composition with PCR-based methods requires simultaneous PCR reactions specific to multiple biological composition. This approach can be quite expensive. By contrast, the 3MG methods can potentially reduce the cost in this case. The 3MG method may facilitate the analysis of high-value products, such as medical and health-promoting products. In general, the 3MG method is suitable for non-targeted biomonitoring and requires meat composition with an abundance above specific levels, whereas PCR-based method is suitable for targeted biomonitoring and can detect biological composition at considerably low abundance levels.
We showed that the mitogenome DNAs of the 15 mammalian and avian species represent 0.28–0.43% of the total DNA. The generation of 1 GB of data costs around US$ 10, and 1 GB of data can produce sufficient mitochondrial reads for determining biological composition qualitatively and quantitatively. Mitogenomes from animals are relatively small and easy to assemble. In December 2020, more than ten thousand animal mitogenomes had been deposited in the NCBI RefSeq database (https://www.ncbi.nlm.nih.gov/genome/browse#!/organelles/). Thus, expecting that all species used in food and medicine will have their mitogenomes available soon is reasonable. Owing to the rapid drop in sequencing costs, fast accumulation of mitogenomes and the integrated bioinformatics software tools, we can expect the broad application of the 3MG method in the near term.
One problem encountered in this study was that beef was successfully detected using LAMP but was not detected in the mock sample when the 3MG method was used. One explanation is that the cattle mitogenome sequence has relatively small percentages of unique sequences. Our data showed that only 17.12–23.93% of reads mapped to the cattle mitogenome were unique to the cattle mitogenome (Table 4). LAMP primers were unique enough to amplify the cattle sequence successfully. Hence, the proportion of unique regions on a mitogenome is essential for its successful detection with the 3MG method. In addition, rabbit or rat meat was not successfully detected with the 3MG method. One explanation is that the mitochondrial DNA has a low proportion of all cellular DNA. Our data showed that the total number of reads mapped to the mitochondrial genomes of these two species was significantly lower than those mapped to the other species (Table 4). Additional studies are needed to optimise the 3MG for the detection of such species mixed samples. Several improvements can be made for the 3MG method. Firstly, internal controls can be added to the samples for the accurate determination of the amount of mito-DNA for particular animal species. As described previously [14, 38], the internal controls can be commonly used as metabarcoding markers, particularly COI and 18S. As the lack of universal primers prevent these markers in metabarcoding analysis, they should represent perfect sequences serving as internal controls for 3MG analysis. Secondly, correction factors should be estimated for biomass estimation based on read counts. For meat biomonitoring, biomass is more commonly used than read counts. Thus, an appropriate conversion rate from read count to the biomass for each meat type is needed. It should be emphasized that the sampling locations may affect the results of biomass estimation. For example, meat from different locations of the legs might have different ratios of fat and fibers, resulting in the variations in the DNA extraction rate and the nuclear to mitochondrial DNA ratio. In this study, we tried to extract samples with homogeneous compositions intraspecificly and interspecificly to minimize this effect. The sample heterogeneity problem is difficult to solve not only for the 3MG method, but also for other traditional detection methods in general. Therefore, we need to estimate two types of correction factors. The first one is the nuclear to mitochondrial DNA ratio (also known as the nuclear–mito ratio). The DNA to biomass ratio (DNA-mass ratio) should be calculated as well. Given that meat might contain different proportions of fats, a high degree of variations in nuclear–mito ratio and DNA–mass ratio are expected among different species. Thirdly, many studies have focused on the meat from 15 mammalian and avian species used in food safety biomonitoring. Meat from many other animal species is commonly consumed but has not been tested in the current study. For instance, fish represents another large group of animal meat widely consumed. The 3MG methods developed in the current study can be applied to fish meat in theory. Parameter optimisation and validation of 3MG on the assessment of fish meat are interesting subjects. Lastly, we should build an extensive reference database for unique mitogenomic DNA sequences from different varieties of related animal species given that many animals, such as chickens, pigs, cattle and sheep, have many endemic species. Building an extensive database containing variety-specific mitochondrial genome sequences will facilitate the identification of the sources of particular animal species.
The current research has laid the foundation for developing accurate and standard procedures for detecting the composition of meat qualitative and quantitatively. The methods will be necessary for the bioassessment and biomonitoring of meat products worldwide and significantly contribute to meat safety management.
The Next Generation Sequencing data for this research is available on the SRA database (https://www.ncbi.nlm.nih.gov/sra). The accession number of a mixture of 15 commonly used animal meat raw sequence reads in Biosample is SAMN11812028. The raw data can be download with the SRA accessions number SRR9107560.
Cawthorn D-M, Steinman HA, Hoffman LC. A high incidence of species substitution and mislabelling detected in meat products sold in South Africa. Food Control. 2013;32(2):440–9.
Premanandh J. Horse meat scandal – a wake-up call for regulatory authorities. Food Control. 2013;34(2):568–9.
Ayaz Y, Ayaz ND, Erol I. Detection of species in meat and meat products using enzyme-linked immunosorbent assay. J Muscle Foods. 2006;17(2):214–20.
Zhang W, Xue J. Economically motivated food fraud and adulteration in China: an analysis based on 1553 media reports. Food Control. 2016;67:192–8.
Peng GJ, Chang MH, Fang M, Liao CD, Tsai CF, Tseng SH, et al. Incidents of major food adulteration in Taiwan between 2011 and 2015. Food Control. 2016;72:145–52.
Bik HM, Porazinska DL, Creer S, Caporaso JG, Knight R, Thomas WK. Sequencing our way towards understanding global eukaryotic biodiversity. Trends Ecol Evol. 2012;27(4):233–43.
Carvalho DC, Palhares RM, Drummond MG, Gadanho M. Food metagenomics: next generation sequencing identifies species mixtures and mislabeling within highly processed cod products. Food Control. 2017;80:183–6.
Ripp F, Krombholz CF, Liu Y, Weber M, Schafer A, Schmidt B, et al. All-food-Seq (AFS): a quantifiable screen for species in biological samples by deep DNA sequencing. BMC Genomics. 2014;15:639.
Crampton-Platt A, Yu DW, Zhou X, Vogler AP. Mitochondrial metagenomics: letting the genes out of the bottle. GigaScience. 2016;5:15.
Rennstam Rubbmark O, Sint D, Horngacher N, Traugott M. A broadly applicable COI primer pair and an efficient single-tube amplicon library preparation protocol for metabarcoding. Ecol Evol. 2018;8(24):12335–50.
Deagle BE, Jarman SN, Coissac E, Pompanon F, Taberlet P. DNA metabarcoding and the cytochrome c oxidase subunit I marker: not a perfect match. Biol Lett. 2014;10(9):20140562.
Polz MF, Cavanaugh CM. Bias in template-to-product ratios in multitemplate PCR. Appl Environ Microbiol. 1998;64(10):3724–30.
Tang M, Tan M, Meng G, Yang S, Su X, Liu S, et al. Multiplex sequencing of pooled mitochondrial genomes-a crucial step toward biodiversity analysis using mito-metagenomics. Nucleic Acids Res. 2014;42(22):e166.
Ji Y, Huotari T, Roslin T, Schmidt NM, Wang J, Yu DW, et al. SPIKEPIPE: a metagenomic pipeline for the accurate quantification of eukaryotic species occurrences and intraspecific abundance change using DNA barcodes or mitogenomes. Mol Ecol Resour. 2020;20(1):256–67.
Bista I, Carvalho GR, Tang M, Walsh K, Zhou X, Hajibabaei M, et al. Performance of amplicon and shotgun sequencing for accurate biomass estimation in invertebrate community samples. Mol Ecol Resour. 2018;18(5):1020–34.
Choo LQ, Crampton-Platt A, Vogler AP. Shotgun mitogenomics across body size classes in a local assemblage of tropical Diptera: phylogeny, species diversity and mitochondrial abundance spectrum. Mol Ecol. 2017;26(19):5086–98.
Crampton-Platt A, Timmermans MJ, Gimmel ML, Kutty SN, Cockerill TD, Vun Khen C, et al. Soup to tree: the phylogeny of beetles inferred by mitochondrial metagenomics of a Bornean rainforest sample. Mol Biol Evol. 2015;32(9):2302–16.
Gillett CP, Crampton-Platt A, Timmermans MJ, Jordal BH, Emerson BC, Vogler AP. Bulk de novo mitogenome assembly from pooled total DNA elucidates the phylogeny of weevils (Coleoptera: Curculionoidea). Mol Biol Evol. 2014;31(8):2223–37.
Gómez-Rodríguez C, Crampton-Platt A, Timmermans MJTN, Baselga A, Vogler AP. Validating the power of mitochondrial metagenomics for community ecology and phylogenetics of complex assemblages. Methods Ecol Evol. 2015;6(8):883–94.
Tang M, Hardman CJ, Ji Y, Meng G, Liu S, Tan M, et al. High-throughput monitoring of wild bee diversity and abundance via mitogenomics. Methods Ecol Evol. 2015;6(9):1034–43.
Liu S, Wang X, Xie L, Tan M, Li Z, Su X, et al. Mitochondrial capture enriches mito-DNA 100 fold, enabling PCR-free mitogenomics biodiversity analysis. Mol Ecol Resour. 2016;16(2):470–9.
Xu S, Kong F, Miao L, Lin S. Establishment and application of fluorescent loop-mediated isothermal amplification for detecting chicken or duck derived ingredients. Chin J Anim Quarantine. 2018;35(2):77–81.
Xu S, Kong F, Miao L, Cai Z, Lin Z, Zhao R. Establishment of the loop-mediated isothermal amplification for the detection of bovine-derived materials. China Anim Health Inspect. 2017;33(12):94–9.
Xu S, Kong F, Miao L, Lin S, Lin Z. Establishment of loop-mediated isothermal amplification for detection of mutton-derived ingredients based on isothermal amplification platform. J Econ Anim. 2016;20(4):200–6.
Zhou J, Bruns MA, Tiedje JM. DNA recovery from soils of diverse composition. Appl Environ Microbiol. 1996;62(2):316–22.
Camacho C, Coulouris G, Avagyan V, Ma N, Papadopoulos J, Bealer K, et al. BLAST+: architecture and applications. BMC Bioinformatics. 2009;10:421.
Hahn C, Bachmann L, Chevreux B. Reconstructing mitochondrial genomes directly from genomic next-generation sequencing reads--a baiting and iterative mapping approach. Nucleic Acids Res. 2013;41(13):e129.
Rice P, Longden I, Bleasby A. EMBOSS: the European molecular biology open software suite. Trends Genet. 2000;16(6):276–7.
Li H, Handsaker B, Wysoker A, Fennell T, Ruan J, Homer N, et al. The sequence alignment/map (SAM) format and SAMtools. Bioinformatics. 2009;25(1 Pt 2):1653–4.
Larkin MA, Blackshields G, Brown NP, Chenna R, McGettigan PA, McWilliam H, et al. Clustal W and Clustal X version 2.0. Bioinformatics. 2007;23(21):2947–8.
Stamatakis A. RAxML version 8: a tool for phylogenetic analysis and post-analysis of large phylogenies. Bioinformatics. 2014;30(9):1312–3.
Kumar S, Stecher G, Tamura K. MEGA7: molecular evolutionary genetics analysis version 7.0 for bigger datasets. Mol Biol Evol. 2016;33(7):1870–4.
Wang Q, Garrity GM, Tiedje JM, Cole JR. Naive Bayesian classifier for rapid assignment of rRNA sequences into the new bacterial taxonomy. Appl Environ Microbiol. 2007;73(16):5261–7.
Porter TM, Hajibabaei M. Automated high throughput animal CO1 metabarcode classification. Sci Rep. 2018;8(1):4226.
Huson DH, Auch AF, Qi J, Schuster SC. MEGAN analysis of metagenomic data. Genome Res. 2007;17(3):377–86.
Aida AA, Che Man YB, Wong CMVL, Raha AR, Son R. Analysis of raw meats and fats of pigs using polymerase chain reaction for Halal authentication. Meat Sci. 2005;69(1):47–52.
Nakyinsige K, Man YBC, Sazili AQ. Halal authenticity issues in meat and meat products. Meat Sci. 2012;91(3):207–14.
Harrison JG, John Calder W, Shuman B, Alex Buerkle C. The quest for absolute abundance: the use of internal standards for DNA-based community ecology. Mol Ecol Resour. 2021;21(1):30–43.
We would like to thank the following scientific institutions for providing meat samples:
Jiangsu Entry-exit Inspection and Quarantine Bureau, Nanjing City, Jiangsu Province, China
The Breeding Farm of Camel, Alxa League, Inner Mongolia, China
Xinjiang Entry-exit Inspection and Quarantine Bureau, Urumqi City, Xinjiang, China
Nanjing Medical University, Nanjing City, Jiangsu Province, China
The breeding farm of nutria, Chongqing City, China
The breeding farm of fox, Suihua City, Heilongjiang Province, China
This work was supported by funds from CAMS Innovation Fund for Medical Sciences (CIFMS) [2021-I2M-1-022], The National Mega-Project for Innovative Drugs of China [2019ZX09735-002], National Science & Technology Fundamental Resources Investigation Program of China [2018FY100705], National Natural Science Foundation of China [81872966], Guiding Projects of the Bureau of Science and Technology of the Fujian Province (2015Y0031), Xiamen Science and Technology Program project (3502Z20154079), the Science and Technology Program Project of General Administration of Quality Supervision, Inspection and Quarantine of People's Republic of China (2014IK089, 2014IK234) and the Fifth Phase of '333 Project' in Jiangsu Province (No.BRA2016498). The benerfactors were not involved in the study design, data collection and analysis, decision to publish or manuscript preparation.
Institute of Medicinal Plant Development, Chinese Academy of Medical Sciences & Peking Union Medical College, 100193, Beijing, PR China
Mei Jiang & Chang Liu
Technology Center of Xiamen Entry-exit Inspection and Quarantine Bureau, Xiamen, Fujian, 361026, PR China
Shu-Fei Xu & Fan-De Kong
Technology Center of Jiangsu Entry-exit Inspection and Quarantine Bureau, Nanjing, Jiangsu, 210009, PR China
Tai-Shan Tang
Technology Center of Henan Entry-exit Inspection and Quarantine Bureau, Zhengzhou, Henan, 450003, PR China
Li Miao
Technology Center of Zhuhai Entry-exit Inspection and Quarantine Bureau, Zhuhai, Guangdong, 519000, PR China
Bao-Zheng Luo
College of Agriculture, Fujian Agriculture and Forestry University, Fuzhou, Fujian Province, 350002, PR China
Yang Ni
Mei Jiang
Shu-Fei Xu
Fan-De Kong
Chang Liu
CL and FDK conceived the study; MJ extracted DNA for next-generation sequencing and performed data analysis; SFX, LM and BZL conducted LAMP validation; TST collected the meat materials; YN helped in genome assembly; MJ and CL wrote the paper. All authors have read and agreed to the contents of the manuscript.
Correspondence to Fan-De Kong or Chang Liu.
The study including meat samples complies with relevant institutional, national and international guidelines and legislation. No specific permits were required for meats collection.
Additional file 1: Table S1.
The primer sequences used for the LAMP experiments. Table S2. Analysis of universal primers for COX1, 16S rRNA, and 18S rRNA genes. Table S3. Summary of reads mapped to the mitogenomes of S. scrofa domesticus and G. gallus. Fig. S1. The distribution of universal primers on 16 s rRNA sequences. Fig. S2. The distribution of universal primers on 18 s rRNA sequences. Fig. S3. The pairwise p-distance of the 15 mitogenomes sequences. Fig. S4. Alignment of reassembled sequences of mitogenomes and those downloaded from GenBank for 15 species. The prefix "A" and "R" represent the assembled and reference mitogenomes, respectively.
Jiang, M., Xu, SF., Tang, TS. et al. Development and evaluation of a meat mitochondrial metagenomic (3MG) method for composition determination of meat from fifteen mammalian and avian species. BMC Genomics 23, 36 (2022). https://doi.org/10.1186/s12864-021-08263-0
Composition determination
Biomass estimation
Meat mix
Mitogenome
Mitochondrial metagenomics
Analysis pipeline | CommonCrawl |
\begin{definition}[Definition:Triangle (Geometry)/Right-Angled/Adjacent]
:400px
In a right-angled triangle, for a given non-right angled vertex, the adjacent side which is ''not'' the hypotenuse is referred to as '''the adjacent'''.
In the above figure:
: the '''adjacent''' to vertex $A$ is side $c$
: the '''adjacent''' to vertex $C$ is side $a$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Insertion of Generators]
Let $S$ be a set, and let $S^*$ be its Kleene closure.
The '''insertion of generators (into $S^*$)''' is the mapping $i: S \to S^*$ defined by:
:$i \left({s}\right) := \left\langle{s}\right\rangle$
that is, it sends any element $s$ of $S$ to the one-term sequence containing only $s$.
\end{definition} | ProofWiki |
The influence of arm composition on the self-assembly of low-functionality telechelic star polymers in dilute solutions
Invited Article
Esmaeel Moghimi1,2,
Iurii Chubak3,
Dimitra Founta1,2,
Konstantinos Ntetsikas4,
George Polymeropoulos4,
Nikos Hadjichristidis4,
Christos N. Likos3 &
Dimitris Vlassopoulos1,2
Colloid and Polymer Science volume 299, pages 497–507 (2021)Cite this article
We combine synthesis, physical experiments, and computer simulations to investigate self-assembly patterns of low-functionality telechelic star polymers (TSPs) in dilute solutions. In particular, in this work, we focus on the effect of the arm composition and length on the static and dynamic properties of TSPs, whose terminal blocks are subject to worsening solvent quality upon reducing the temperature. We find two populations, single stars and clusters, that emerge upon worsening the solvent quality of the outer block. For both types of populations, their spatial extent decreases with temperature, with the specific details (such as temperature at which the minimal size is reached) depending on the coupling between inter- and intra-molecular associations as well as their strength. The experimental results are in very good qualitative agreement with coarse-grained simulations, which offer insights into the mechanism of thermoresponsive behavior of this class of materials.
Self-organization of building blocks due to external stimuli is ubiquitous in most materials and all living organisms in nature. Inspired by this, a remarkable body of work has been performed to understand and emulate their response to temperature [1,2,3], pH [4,5,6] and light [7, 8], enabling the controlled design of their structure assembly. Recent advances in polymer chemistry have led to the synthesis of various building blocks with complex architectures and functionalized properties. Such responsive building blocks that can self-organize into higher-order structures may form soft patchy particles, which have directional interactions and varying softness. Furthermore, topological effects that arise in systems with complex architecture can alone lead to a range of interesting phenomena in and out of thermodynamic equilibrium for both low [9,10,11,12,13,14,15] and high system densities [16,17,18,19,20,21]. Functionalized biomolecules such as DNA-grafted colloidal particles represent a typical example where patchiness reflects the competition between inter- and intra-particle associations [22,23,24,25]. However, despite its significance, DNA-based research is very specialized and yields limited amounts of samples. An alternative design of patchy particles is based on the so-called Telechelic Star Polymers (TSPs), that is star polymers with functionalized end groups [26,27,28,29]. A TSP consists of f amphiphilic AB-block copolymer arms grafted on a common center. The solvophilic A-block is attached at the center of the star, whereas the solvophobic B-block is exposed to the exterior of the star. On changing the solvent quality through temperature variation, the outer blocks become attractive and form patches on the surface of the particle [30,31,32]. With such a TSP system, it is thus possible to cover the entire range of inter-particle interactions, from purely repulsive to attractive soft colloids simply by changing the solvent quality. The self-assembly of TSPs at the single molecule level and in concentrated solutions depends mainly on three parameters: (i) the functionality f of the stars, (ii) the outer block size ratio α = NB/(NA + NB) (NA and NB denote the length of the respective block), and (iii) the attraction strength between the outer solvophobic blocks which is enhanced upon worsening solvent quality.
TSPs with low functionality (f ≤ 5) collapse into a watermelon structure with one single patch on the surface of the particle [30]. However, more complex structures with a richer distribution of patches are formed in TSPs with higher functionality [31]. Such soft patchy particles can preserve their properties such as the size, number, and arrangement of patches upon increasing the TSP concentration [33]. The inherent flexibility of such soft-particles leads to formation of ordered structures in the case of high functionality [33]. On the other hand, low-functionally TSPs tend to form micellar aggregates [34,35,36,37], which at relatively high concentrations self-assemble into long worm-like micelles [35, 38]. In previous work, we have examined the effects of temperature (or attraction strength) on the self-assembly of these low-f TSPs in dilute solution [32]. In the present work, we extend these investigations by addressing the effects of the block size ratio and arm length on the self-organization of TSPs with f = 3 in dilute solutions. The new results provide insight into the responsive behavior of TSPs, paving the way for the design of functional materials with tunable properties.
Telechelic star polymers (TSPs) with three arms made of 1,4-polybutadiene (PB) as the inner A-block and polystyrene (PS) as the outer B-block were synthesized by anionic polymerization and chlorosilane chemistry using high-vacuum techniques. Detailed information on the synthesis procedure can be found in Ref. [32]. Three different TSP samples have been used in the present study. Two of them have a similar total molar mass of about 40000 g/mol, albeit a different PS weight fraction of fPS = 0.14 and 0.33. The third sample, which was used in our previous work [32], has a lower molar mass of MW = 26700 g/mol and fPS = 0.23. The molar mass distribution in all three TSP samples is rather narrow with the corresponding polydispersity being around Ð = 1.03. The detailed molecular characteristics of the samples are listed in Table 1.
Table 1 Molecular characteristics of investigated star diblock copolymers (PS-b-PB)3
We have used 1-phenyldodecane as the solvent. It has a cloud-point at 53 ∘C for PS [39] and 22 ∘C for PB (the corresponding 𝜃-temperatures are expected to be slightly higher). Solutions were prepared by mixing an appropriate amount of the TSP with the solvent to reach the desired concentration. The sample degradation was inhibited by adding 0.1 wt% of the TSP of the antioxidant BHT (2,6-Di-tert-butyl-4-methylphenol). In order to fully dissolve TSPs, methylene chloride was used as the cosolvent. Then, the cosolvent was evaporated under ambient conditions until a constant weight was achieved.
Dynamic Light Scattering (DLS) was used to investigate the dynamics and self-assembly of TSPs in dilute solutions. In DLS, the normalized autocorrelation function \(G(q, t) = \left \langle I(q, 0) I(q, t)\right \rangle / \left \langle I(q, 0) \right \rangle ^{2}\) of the total scattered light intensity I(q) at the wave vector \(q = \left (4 \pi n_{0} / \lambda \right ) {\sin \limits } \left (\theta / 2\right )\) (n0 is the refractive index, 𝜃 denotes the scattering angle, and λ is the wavelength of the incident laser beam) is related to the normalized time correlation function of the scattered electric field E(q,t) by the Siegert relation:
$$ G(q,t) = 1 + f^{*}\left|\tilde{\alpha} g(q,t)\right|^{2} = 1 + f^{*}\left|C(q,t)\right|^{2}, $$
where f∗ stands for the coherence instrumental factor, \(\tilde {\alpha }\) is the fraction of I(q) associated with fluctuations relaxing with times longer than 0.1 µs [40,41,42], and C(q,t) denotes the intermediate scattering function (ISF). The inverse Laplace transformation using the constrained regularization method was applied to compute the relaxation spectrum \(H(\ln \tau )\). This method assumes that C(q,t) can be expressed as the superposition of exponentials:
$$ C(q, t) = {\int}_{-\infty}^{+\infty} H(\ln \tau) \exp{\left( - t / \tau \right)} \mathrm{d}\left( \ln \tau \right) $$
The characteristic relaxation times correspond to the peak positions of \(H(\ln \tau )\), whereas the area under the peak defines the value of \(\tilde {\alpha }\) in Eq. 1 and hence the intensity \(\tilde {\alpha } I(q)\) associated with the particular dynamic process. The transformation was performed with the program CONTIN [43] that yielded the relaxation time and intensity of the partitioning modes.
DLS experiments were performed on an ALV-5000 goniometer/correlator setup (ALV-GmbH, Germany). The light source was a Nd:YAG dye-pumped, air-cooled laser (100 mW) with the wavelength λ = 532 nm. The refractive index of 1-phenyldodecane is n0 = 1.482. Before each DLS experiment, the samples were equilibrated at T = 60 ∘C, which is above the cloud point of the outer PS-block, for 10 min to erase thermal history. Then, the sample was quenched to the desired temperature and equilibrated. The equilibration process was probed by measuring the ISF until it reached steady values over time. The duration of equilibration depended on the temperature and ranged from 10 min for T = 60 ∘C to 10 h for T = 20 ∘C.
Simulation details
To model TSP dynamics under worsening solvent conditions for its outer block, we have employed a coarse-grained dissipative particle dynamics (DPD) model with explicit solvent. In what follows, the inner blocks of a TSP are labeled "A," the outer ones—"B," whereas solvent particles—"S." In DPD, the total force Fi acting on the i th particle is composed of the conservative FC, dissipative FD, and random FR contributions [44]:
$$ \mathbf{F}_{i} = \sum\limits_{j \neq i} \left( \mathbf{F}_{ij}^{\text{C}} + \mathbf{F}_{ij}^{\text{D}} + \mathbf{F}_{ij}^{\text{R}} \right). $$
In Eq. 3 above, \(\mathbf {F}_{ij}^{\text {C}}\) is the conservative force acting between the i th and j th particle separated by a distance rij (here and in what follows, rij = ri −rj, rij = |rij|, \(\hat {\mathbf {r}}_{ij} = \mathbf {r}_{ij} / r_{ij}\), and vij = vi −vj):
$$ \mathbf{F}_{ij}^{\text{C}} = A_{ij} w(r_{ij}), $$
where Aij is the maximal repulsion between the particles and w(rij) is given by
$$ w(r_{ij}) = (1 - r_{ij}/r_{\text{cut}}) \theta(r_{\text{cut}}-r), $$
with 𝜃(x) denoting the Heaviside step function and the cutoff distance rcut being chosen as the unit of length (rcut = 1). Furthermore, \(\mathbf {F}_{ij}^{\text {D}}\) is the pairwise dissipative force
$$ \mathbf{F}_{ij}^{\text{D}} = - \gamma w(r_{ij})^{2} \left( \hat{\mathbf{r}}_{ij} \cdot \mathbf{v}_{ij} \right), $$
and \(\mathbf {F}_{ij}^{\text {R}}\) is the pairwise random force
$$ \mathbf{F}_{ij}^{\text{R}} = -\sqrt{2 \gamma k_{\text{B}} T/ {\Delta} t} \cdot \eta_{ij} w(r_{ij}), $$
where ηij is a Gaussian random number with zero mean and unit variance. The unit of mass is set by the (same) mass of every particle m, whereas the unit of energy was chosen to be kBT (kBT = 1). The simulation were performed using the HOOMD-blue simulation package [45,46,47,48] using friction coefficient γ = 4.5mτ− 1 and the equations of motion were integrated using the Velocity-Verlet algorithm [49] with time step Δt = 0.04τ, where \(\tau = r_{\text {cut}} \sqrt {m/k_{\text {B}}T}\) is the DPD unit of time.
To obtain the total polymerization degree N of the star arms in the experimental samples considered, we first estimated the molar volumes of PB and PS, given by vPS/PB = MW/ρPS/PB, where ρPS/PB is the corresponding molar density (ρPS = 1.05 g/mL and ρPB = 0.892 g/mL). N and αPS/PB were then computed on the basis of the PS reference segment volume 99.2 mL/mol and are listed in Table 2. In general, we are interested in the behavior of experimental systems in a rather narrow temperature range 20 ∘C < T < 60 ∘C, where the Flory-Huggins incompatibility parameter χPS-PB = 18.78/T − 9.6⋅10− 4 [50] does not change substantially (experimental values of χPS-PBN for the three samples in such temperature range are \(\chi _{\text {PS-PB}}N \lesssim 10\)), implying that the self-assembly is mainly controlled by the solvent selectivity towards the outer block.
Table 2 Composition of investigated star diblock copolymers (PS-b-PB)3
Given the computational cost of simulations with explicit solvent particles, we focused on a star polymer model with f = 3 arms containing N = 64 monomers, and systematically varied the outer block ratio α by changing NA and NB. All our simulations were performed at total particle density \(\rho r_{\text {cut}}^{3} = 3\). Bonded interactions were given by \(V_{\text {bond}}(r) = \tfrac {K}{2} (r - r_{\text {cut}})^{2}\) with K = 50kBT. The central particle, to which all arms were connected to, was treated as a monomer of type A. In DPD, repulsion amplitudes Aij can be directly related to the Flory-Huggins χij parameters [51]:
$$ A_{ij} \approx A_{ii} + \kappa(\rho) \chi_{ij}, $$
where κ(ρ) depends on the DPD density such that κ(3) = 3.49. In all simulations, we fixed the inter-block incompatibility parameter χAB = 0.23, which corresponds to AAB = 25.8 at \(\rho r_{\text {cut}}^{3} = 3\). Such value of χAB was obtained from a conservative experimental value \((\chi _{\text {AB}}N)_{\exp } = 10\) by taking into account finite polymer chain length corrections: \(\chi _{\text {AB}} = (\chi _{\text {AB}}N)_{\exp } \cdot (1 + 3.9 N^{2/3-2\nu }) / N\) with N = 64 and ν = 0.588 [51]. The incompatibility parameter χAS for the outer block and effective solvent particles was always set to χAS = 0, which corresponds to AAS = 25. Moreover, χBS was systematically varied between 0 and 7 with step ≈ 0.72, corresponding to ABS in the range between 25 and 50. Finally, note that the main goal of our simulations using such a coarse-grained model is not to quantitatively reproduce the change of star properties with increasing χBS, for which atomistic simulations with realistic solvent interactions would be necessary, but to qualitatively assess the effect of self-associations on the change of TSPs static and dynamic behavior.
We first focus on the effect of the outer PS-block fraction on the dynamic relaxation of the TSP system at low densities. In Fig. 1, we show the experimental ISFs at a fixed wave vector and various temperatures for the two samples with comparable total molar masses but with distinct PS weight fractions. The ISFs show two distinct trends upon changing temperature. At high temperatures, the ISFs show a single exponential decay that demonstrates the existence of individual stars in solution. However, when temperature is reduced below the cloud temperature of the outer-block, the ISF features a two-step decay which indicates the coexistence of two distinct populations in the system. The first decay (fast process) in the ISF is similar to the one observed at high temperatures and hence represents the individual stars in solution. On the other hand, the second decay (slow process) taking place at longer times suggests the presence of larger aggregates (clusters of TSPs). The slow process becomes more pronounced as the temperature is decreased. Interestingly, the two-step decay in the ISF appears at a slightly higher temperature for the TSP with a larger PS-fraction.
Experimental ISFs at constant wave vector q = 0.02475 nm− 1 and different temperatures for TSPs with the outer PS-block fractions of fPS = 0.14 (a) and fPS = 0.33 (b). Note that the plateau values of the ISF at short times are well below one. This is due to the fact that a part of the scattered intensity originates from density fluctuations of solvent molecules
To extract hydrodynamic sizes associated with the two processes in the solution, that is individual TSPs and clusters, the relaxation spectrum is calculated from the inverse Laplace transformation of the ISF using the constraint regularized method [43] discussed in Section 4. Typical results of such analysis for the TSPs with two different PS fractions at T = 60 ∘C and 30 ∘C are shown in Fig. 2. The relaxation spectrum at 60 ∘C shows a single peak, which is rather sharp, reflecting single exponential decay of the ISF mode. The position of the peak shifts to a slightly longer time for the TSP with a smaller PS fraction, indicating a larger hydrodynamic radius. At the lower temperature of 30 ∘C, the relaxation spectrum exhibits two well-separated peaks, as seen in Fig. 2b. Similarly to high temperatures, the position of the first peak shifts to a slightly longer time for the TSP with a smaller PS fraction, whereas the position of the second peak in both TSP samples is located at a similar time. These two peaks represent the relaxation times associated with individual TSPs and clusters, respectively. Subsequently, the two relaxation times are used to calculate the diffusion coefficients associated with each component. The diffusion coefficient for the fast mode (where qR < 1) is q-independent, whereas D for the slow mode (where qR ≥ 1) shows some q-dependence. In the latter case, D extrapolated to q = 0 is used to calculate Rh. Then, the hydrodynamic sizes of TSPs and clusters are obtained using the Stokes-Einstein-Sutherland relation.
ISF (open symbols, left axis) and its corresponding relaxation times spectrum (closed symbols, right axis) deduced from the constrained regularization method for the TSPs with PS fractions of fPS = 0.14 (black squares) and 0.33 (red circles) at q = 0.02475nm− 1 for T = 60 ∘C (a) and T = 30 ∘C (b)
We first examine the effect of the block size ratio α on the single TSP size upon cooling. In experiments, the radius of gyration was too small to be probed by DLS. Instead, we focused on the hydrodynamic radius Rh of individual TSPs in dilute solution, as calculated from the fast process in the ISF using the Stokes-Einstein-Sutherland relation. The temperature dependence of Rh for the three studied TSP samples is shown in Fig. 3. The single star size exhibits a two-step shrinkage upon reducing temperature or equivalently worsening the solvent quality. The first decay in size takes place at temperatures well below the cloud-point of outer PS-blocks, whereas the second drop is seen when temperature is reduced further below the cloud-point of the inner PB-block. Hence, the first decrease in size is associated with the collapse of outer blocks, whereas the second decay corresponds to the case when inner blocks start to collapse. At high temperatures, for the TSP with fPS = 0.14, we find Rh ≈ 5 nm and for fPS = 0.33, Rh ≈ 4.6 nm (see Fig. 3a). Although both TSPs have almost the same molar mass of about 40000 g/mol, the difference in their size originates from the difference in the fraction of PS. The radius of gyration of a star homopolymer in good solvent conditions is given by Rg ≈ κ(f)bNν, where ν = 0.588, N is the number of Kuhn segments in a star arm, b is the size of a Kuhn segment, and κ(f), which depends on the number of star arms f, is a numeric constant that takes into account the star functionality [52,53,54]. Using the latter relation, it can be found that the size of a star made of purely PB is about 65% larger compared with that made of purely PS, \(R_{\text {g}}^{\text {PB}} \approx 1.65 R_{\text {g}}^{\text {PS}}\), given that their molar mass is the same [55]. Hence, it is expected that the increase in PS fraction reduces the size of a TSP. In order to compare the collapse process for stars of different size, we have normalized the TSP size by the plateau value of Rh at high temperatures.
a The hydrodynamic radius, Rh, of individual TSPs in the dilute solution calculated from the fast process in the experimentally determined ISFs. bRh normalized by its plateau value at high temperatures. The dashed lines serve as a guide to the eye. The black arrows indicate the cloud-points of inner PB and outer PS blocks
The TSP with a larger fraction of outer PS-blocks (fPS = 0.33) shows the first-step reduction in size at higher temperatures and the second-step drop at slightly lower temperatures compared with the TSP with a smaller PS fraction (fPS = 0.14). In both stars, the decrease in size is about 15%, as seen in Fig. 3b. In addition, in Fig. 3, we show the results for a TSP with a smaller molar mass (26700 g/mol) with the outer PS-block fraction fPS = 0.23, which is between the other two higher molar mass TSPs with fPS = 0.14 and 0.33. The main difference is that the TSP with the smaller molar mass exhibits the decay in size at a much lower temperature compared with the other two, which originates from a smaller value of the incompatibility parameter \(\sim \chi _{\text {PS-S}}N\). Moreover, the decrease in size is also rather weaker (about 10%), which can be attributed to a shorter length of its arms.
In Fig. 4, we present the temperature dependence of clusters' Rh for the systems of TSPs with the same molar mass but two different PS fractions (MW = 40000 g/mol, fPS = 0.14 and 0.33). In both TSPs, the cluster size shrinks on cooling. However, a slight but consistent increase in the cluster size is observed when temperature is further reduced below the cloud-point of the inner PB-block. Moreover, the hydrodynamic cluster size does not show change with the fraction of outer PS block. In Fig. 4a, we additionally show the results for the TSP with smaller molar mass (MW = 26700 g/mol) and fPS = 0.23. In this case, the temperature dependence of the cluster size is the same as for the other two TSPs (fPS = 0.14,0.33). However, the smaller molar mass TSP shows a cluster size that is nearly three times larger. This could be due to a higher concentration of these TSPs. To rule out the effect of concentration, we have normalized the cluster size by the number density of TSPs in solution (Fig. 4b). The number density takes into account for the number of stickers available in the solution. With such normalization, the differences in cluster size between different TSPs are reduced to a great extent. The minor differences could originate from the complex nature of self-organization of TSPs due to differences in their molecular characteristics.
a The hydrodynamic radius, Rh, of clusters extracted from the slow process in the experimental ISFs. b The ratio of Rh to the number density n (see Table 1) of TSPs in the solution. The black arrow indicates the cloud-point of inner PB blocks
Simulation results
We now focus on static and dynamic properties of single TSPs under worsening solvent conditions for the outer B block, that is under increasing χBS. To do so, we simulated single stars with f = 3 arms of length N = 64 using a coarse-grained DPD model with explicit solvent particles, as described in detail in Section 4. The outer block ratio was systematically varied from 0.1 to 0.5 with step 0.1. The exact number of A- and B-type monomers in an arm was NA = 58, 52, 45, 39, 32 and NB = 6, 12, 19, 25, 32, respectively (the corresponding α = 0.1, 0.2, 0.3, 0.4, 0.5). For each state point (α, χBS), we performed 10–12 independent simulation runs of length 105τ, followed after a shorter equilibration period of 104τ. Single TSPs were simulated in a box of size L = 30rcut at the total particle density \(\rho r_{\text {cut}}^{3} = 3\). To check if such box size is sufficient to accommodate a TSP, we initially simulated the same star in good solvent conditions (χBS = 0) in a larger box with L = 35rcut, and we did not observe any substantial changes in its properties. In selective solvents, the TSP size is even smaller due to the formation of patches, which justifies the use of the same box size L = 30rcut in this case.
To assess single star shape properties, we computed the eigenvalues λi (i = 1,2,3, λ1 ≥ λ2 ≥ λ3) of the star's gyration tensor
$$ G_{ij} = \frac{1}{fN+1} \sum\limits_{k=1}^{fN + 1} {\Delta} r_{i}^{(k)} {\Delta} r_{j}^{(k)}, $$
where \({\Delta } r_{i}^{(k)}\) is the i th component of the k th monomer's position in the star's center of mass frame. In Fig. 5, we report the TSP's mean radius of gyration \(R_{\text {g}} = {\langle R_{\text {g}}^{2} \rangle }^{1/2}\) (\(R_{\text {g}}^{2} = \lambda _{1} + \lambda _{2} + \lambda _{3}\)) as well as the mean asphericity parameter \(\langle \lambda _{1} - \tfrac {1}{2}(\lambda _{2} + \lambda _{3}) \rangle \), which is positive and can vanish only for a completely symmetric configuration, as a function of χBS for different block length ratios α. The angles 〈⋯ 〉 denote an ensemble and time average. We find that the behavior of a single TSP size is generally very similar to the experimental one (see Fig. 3): upon increasing χBS, we first observe a rather small decrease in Rg, followed by a major drop at higher χBS. Such behavior of Rg is associated with the formation of a single patch, where all three arms of a TSP clump together (see Fig. 6). We find that the transition point shifts towards a higher χBS, that is a lower temperature because \(\chi \sim 1/T\), with decreasing α, which is in full accordance with the experimental behavior of the two samples with fPS = 0.14 and 0.33 that have a very similar total molar mass (see Fig. 3b). Afterwards, only a small reduction of Rg is observed upon increasing χBS, as seen in Fig. 3a. We also note that simulations do not capture the second drop in size which is observed in experiments for temperatures below the cloud-point of inner-block. The reason for this discrepancy is that in simulation, for simplicity reasons, the inner-block is assumed to be always in a good solvency condition. Hence, it only captures single step shrinkage process due to collapse of outer-block monomers.
a The mean radius of gyration of a TSP Rg as a function of χBS for different fractions of the outer block α = NB/N. b The mean asphericity of a TSP, computed as \(\langle \lambda _{1} - \tfrac {1}{2}(\lambda _{2} + \lambda _{3}) \rangle \), scaled with its mean radius of gyration Rg as a function of χBS for different α
Characteristic TSP conformations with α = 0.1 (left) and α = 0.3 (right) at a high χBS ≈ 7. B-monomers are blue, A-monomers—red, and star centers are black. Solvent particles are not shown for clarity
We furthermore find that the final TSP size decreases with increasing α, also in accordance with the experimental findings for the samples with fPS = 0.14 and 0.33 (see Fig. 3b). This behavior is associated with generally more open configuration of collapsed TSPs with small α that permit solvent flow through the TSP's interior. On the other hand, in the case of larger α, the solvophobic B-blocks form a single large patch that expels the solvent from its interior, resulting in more compact and symmetric configurations (see Figs. 6 and 5b). Interestingly, as can be seen in Fig. 5b, upon slightly increasing χBS from 0, the stars first become more aspherical, which can be attributed to the formation of transient patches between two out of three star arms. This is confirmed in Fig. 7 that reports the average number of patches formed by the star, the average number of arms in a patch, and the fraction of free arms as a function of χBS for different arm compositions. In agreement with earlier results [30], we find that such TSP with f = 3 forms only one patch for all α with all three arms contributing to it at high enough χBS. In addition, as seen from Fig. 7a and 7b, the point when all arms start to form a single patch corresponds to the point when Rg drops significantly (see Fig. 5a). Finally, to assess the influence of arm length N on the transition point for the watermelon-like structure formation, for α = 0.3, we additionally simulated stars with arm length N = 32,48,80 for different values of χBS. The comparison between the behavior of the radius of gyration of TSPs with different N for α = 0.3 is shown in Fig. 8. We find that the TSP with shorter arms features the star collapse at higher values of χBS, which therefore corresponds to lower temperatures in the experiments, being in line with the trend observed for the experimental sample with fPS = 0.23 that has a lower molar mass, see Fig. 3.
a The total number of formed patches as a function of χBS for different α. Two arms are defined as being in a common patch if there is at least one pair of monomers from the two distinct arms lying at a distance r ≤ rcut. The average number of arms in a patch (b) and the average fraction of free arms (c) as a function of χBS for different α. a, b, and c share the same legend shown in (a)
The mean radius of gyration Rg of a TSP as a function of χBS for the same fraction of the outer block α = 0.3 but different arm lengths N
The presence of faithful hydrodynamic interactions in DPD allows us to assess the influence of patch formation on the dynamics of single stars in solution. We did this by considering the mean-square displacement of TSP's center of mass, computed as:
$$ {\text{MSD}}(t) = \frac{1}{T-t} {\int}_{0}^{T-t} \left\langle [ \mathbf{R}(t^{\prime} + t) - \mathbf{R}(t^{\prime})]^{2} \right\rangle \text{d}t^{\prime}, $$
where R(t) is the position of the star's center of mass at time t, T is the total simulation time, and 〈⋯〉 stands for the average over independent simulations runs. Typical behavior of the MSD for different α as well as the extracted diffusion coefficients D is shown in Fig. 9. We find that the tendency to form patches, causing more compact watermelon-like structures, increases the diffusivity of the TSP. In the experiments, this behavior corresponds to a reduction in the hydrodynamic radius Rh, which is in good agreement with results shown in Fig. 3. Furthermore, this effect is especially significant for the case of high α, where D at high χBS can become about two times bigger compared with athermal conditions with χBS = 0, as seen in Fig. 9a, again in agreement with the experiments, where a larger reduction in Rh is seen for the TSP with a higher PS fraction. Finally, more open conformations of collapsed TSPs with low α make the increase in diffusivity less pronounced (for example, about 30% increase for α = 0.10).
a The diffusion coefficient D of the TSP's center of mass as a function of χBS for different α. D was extracted from the long-time behavior of the MSD of the star's center of mass, MSD = 6Dt. Typical MSD as a function of χBS are shown for α = 0.5 (b) and α = 0.1 (c)
In addition, we have considered the dynamics of internal patch reorganizations at the single-star level. In Fig. 10, we show the times tfree for an arm spent in the free state, that is not forming an association with other arms, as a function of χBS for different α. We find that for all α, the mean value of tfree initially decreases exponentially fast with increasing χBS (Fig. 10a), up to a point where a single patch forms. At this point, 〈tfree〉 drops to 0, indicating that the single patch is stable over the course of the whole simulation. The value of χBS where it happens compares with the point where a significant reduction of Rg occurs (Fig. 5a). Furthermore, in Fig. 10b, we show the distribution for tfree for α = 0.1 at various χBS, featuring tails that decay exponentially fast with increasing tfree in all cases.
a The mean time of an arm not belonging to a patch 〈tfree〉 as a function of χBS for different α. b The normalized distribution of tfree for α = 0.1 and various χBS
Finally, we consider the formation of inter-star aggregates in the dilute solution. To asses such behavior, it is necessary to simulate a sufficiently large number of stars, which becomes computationally restrictive if using the model with N = 64 that was employed for single star behavior discussed previously. We therefore resort to an even coarser model with, similarly, f = 3 but with N = 10, and in what follows we focus on the case with α = 0.3. We simulated 2000 such stars in a box with L = 70rcut at particle density \(\rho r_{\text {cut}}^{3} = 3\). In this model, the corresponding star concentration is c ≈ 0.4c∗, which is higher than the one used in the experiments (to reach an equivalent experimental concentration of c ≈ 0.04c∗, it would be necessary to simulate about a ten times bigger system containing \(\sim 10^{7}\) particles). Nevertheless, even in this regime, we remain at concentrations considerably below c∗. Initially, TSPs were initialized uniformly in the box and subsequently equilibrated in athermal solvent conditions for both blocks with χAS = χBS = 0 (χAB was set to χAB = 28.6 to match the experimental value (χABN)eff = 10, as explained in Section 4). Afterward, the incompatibility parameter for the outer block was increased to 4.3 (ABS = 40) over 2⋅106 integration time steps and then further equilibrated for another 2⋅106 steps. During the latter stages, the TSPs initially began to form small micelles that subsequently merged into worm-like structures, which then again merged into a single giant cylindrical aggregate, shown in the left column of Fig. 11. Note that such cylindrical architecture is rather a consequence of periodic boundary conditions. It is likely that symmetric spherical aggregates would form in a more dilute system with a larger simulation box size, as, for instance, recently shown in Ref. [56]. Nonetheless, this illustrates the tendency of TSPs to form large aggregates even at dilute conditions, as previously shown in the experimental cluster sizes in Fig. 4. We also considered the effect of lowering temperature on the structure of such aggregate by quenching χBS to 10 (ABS = 60) and equilibrating the system for another 106 integration time steps. As shown in the right column of Fig. 11, as a result of the χBS increase, the aggregate shrinks in the two transverse directions. This occurs because the solvophobic TSP blocks that lie in the aggregate's interior become more ordered and thus push away the remaining solvent (see the bottom row of Fig. 11). This further agrees with the experimental trend of decreasing cluster size with decreasing temperature (Fig. 4).
Left column: self-assembly of three-arm TSPs with N = 10 and α = 0.3 into a giant cylindrical aggregate at c ≈ 0.4c∗. The bottom row shows only the solvophobic B-blocks. Right column: the aggregate shrinks in the transverse directions upon increasing the solvophobicity of the B-monomers that become more ordered (bottom image)
In summary, we have investigated the self-assembly of TSPs with a variable size of the outer block as well as the arm length, which are subject to worsening solvent conditions, in dilute solutions. We find that two distinct modes in the experimental ISF appear upon lowering the temperature below a critical value: the fast-relaxing mode that corresponds to free stars in solutions as well as a slow-relaxing mode that indicates the presence of large aggregates. We find that the size of both populations decreases upon cooling. For single TSPs, the decay is associated with the formation of a single patch, where all three arms come together. From both experiments and simulations, we find that the temperature that corresponds to such transition increases with growing fraction of solvophobic monomers. However, we find that the transition temperature increases with the polymerization degree of TSP arms, when keeping the fraction of solvophobic monomers constant. The formed aggregates in solution are much bigger than single stars (\(\sim 100\) nm versus \(\sim 5\) nm). In simulations, albeit at a higher concentration of TSPs, we have found the formation of micellar aggregates with complex internal structure. Upon worsening the solvent quality for the outer block, the solvent is becoming more strongly expelled from the aggregate's interior, which causes the reduction of its size, similarly to the experimental behavior. We therefore speculate that similar objects also form at concentrations similar to the experimental ones, in line with assembled structures that have been recently observed in large-scale DPD simulations of linear diblock co-polymers in dilute conditions [56].
Pavlovic M, Antonietti M, Schmidt BV, Zeininger L (2020) J Colloid Interface Sci 575:88
Salonen A, Langevin D, Perrin P (2010) Soft Matter 6(21):5308
Vandenhaute M, Schelfhout J, Van Vlierberghe S, Mendes E, Dubruel P (2014) Eur Polym J 53:126
Dong J, Wang Y, Zhang J, Zhan X, Zhu S, Yang H, Wang G (2013) Soft Matter 9(2):370
Deng X, Livingston JL, Spear NJ, Jennings GK (2020) Langmuir 36(3):715
Fan TF, Park S, Shi Q, Zhang X, Liu Q, Song Y, Chin H, Ibrahim MSB, Mokrzecka N, Yang Y et al (2020) Nat Commun 11(1):1
Han P, Li S, Wang C, Xu H, Wang Z, Zhang X, Thomas J, Smet M (2011) Langmuir 27(23):14108
Balkenende DW, Monnier CA, Fiore GL, Weder C (2016) Nat Commun 7(1):1
Weiss LB, Nikoubashman A, Likos CN (2017) ACS Macro Lett 6(12):1426
Chubak I, Locatelli E, Likos CN (2018) Mol Phys 116(21-22):2911
Ahmadian Dehaghani Z, Chubak I, Likos CN, Ejtehadi MR (2020) Soft Matter 16:3029
Liebetreu M, Likos CN (2020) Comms Mater 1(1):4
Rauscher PM, Rowan SJ, de Pablo JJ (2018) ACS Macro Lett 7(8):938
Wu Q, Rauscher PM, Lang X, Wojtecki RJ, de Pablo JJ, Hore MJA, Rowan SJ (2017) Science 358(6369):1434
Moore NT, Lua RC, Grosberg AY (2004) Proc Natl Acad Sci U.S.A 101(37):13431
Slimani MZ, Bacova P, Bernabei M, Narros A, Likos CN, Moreno AJ (2014) ACS Macro Lett 3(7):611
Weiss LB, Likos CN, Nikoubashman A (2019) Macromolecules 52(20):7858
Rauscher PM, Schweizer KS, Rowan SJ, de Pablo JJ (2020) Macromolecules 53(9):3390
Smrek J, Chubak I, Likos CN, Kremer K (2020) Nat Commun 11(1):26
Klotz AR, Soh BW, Doyle PS (2020) Proc Natl Acad Sci U.S.A 117(1):121
Rosa A, Smrek J, Turner MS, Michieletto D (2020) ACS Macro Lett 9(5):743
Biffi S, Cerbino R, Nava G, Bomboi F, Sciortino F, Bellini T (2015) Soft Matter 11(16):3132
Mirkin CA, Letsinger RL, Mucic RC, Storhoff JJ (1996) Nature 382(6592):607
Dreyfus R, Leunissen ME, Sha R, Tkachenko AV, Seeman NC, Pine DJ, Chaikin PM (2009) Phys Rev Lett 102(4):048301
Angioletti-Uberti S, Mognetti BM, Frenkel D (2012) Nat Mater 11(6):518
Pitsikalis M, Hadjichristidis N, Mays JW (1996) Macromolecules 29(1):179
Broze G, Jérôme R, Teyssié P (1982) Macromolecules 15(3):920
Van Ruymbeke E, Vlassopoulos D, Mierzwa M, Pakula T, Charalabidis D, Pitsikalis M, Hadjichristidis N (2010) Macromolecules 43(9):4401
Vlassopoulos D, Pakula T, Fytas G, Pitsikalis M, Hadjichristidis N (1999) J Chem Phys 111(4):1760
Lo Verso F, Likos CN, Mayer C, Löwen H (2006) Phys Rev Lett 96:187802
Rovigatti L, Capone B, Likos CN (2016) Nanoscale 8:3288
Moghimi E, Chubak I, Statt A, Howard MP, Founta D, Polymeropoulos G, Ntetsikas K, Hadjichristidis N, Panagiotopoulos AZ, Likos CN, Vlassopoulos D (2019) ACS Macro Lett 8(7):766
Capone B, Coluzza I, LoVerso F, Likos CN, Blaak R (2012) Phys Rev Lett 109:238301
Koch C, Likos CN, Panagiotopoulos AZ, Lo Verso F (2011) Mol Phys 109(23-24):3049
Lo Verso F, Panagiotopoulos AZ, Likos CN (2009) Phys Rev E 79:010401
Lo Verso F, Panagiotopoulos AZ, Likos CN (2010) Faraday Discuss 144:143
Koch C, Panagiotopoulos AZ, Lo Verso F, Likos CN (2013) Soft Matter 9:7424
Koch C, Panagiotopoulos AZ, Lo Verso F, Likos CN (2015) Soft Matter 11:3530
Geerissen H, Wolf BA (1982) Chem Rapid Commun 3(1):17
Berne BJ, Pecora R (2000) Dynamic Light Scattering: with Applications to Chemistry, Biology and Physics (Courier Corporation)
Kroeger A, Belack J, Larsen A, Fytas G, Wegner G (2006) Macromolecules 39 (20):7098
Bockstaller M, Köhler W, Wegner G, Vlassopoulos D, Fytas G (2001) Macromolecules 34(18):6359
Provencher SW (1982) Comput Phys Commun 27(3):229
Groot RD, Warren PB (1997) J Chem Phys 107(11):4423
Anderson JA, Lorenz CD, Travesset A (2008) J Comput Phys 227(10):5342
Glaser J, Nguyen TD, Anderson JA, Lui P, Spiga F, Millan JA, Morse DC, Glotzer SC (2015) Comput Phys Commun 192:97
Howard MP, Panagiotopoulos AZ, Nikoubashman A (2018) Comput Phys Commun 230:10
Phillips CL, Anderson JA, Glotzer SC (2011) J Comput Phys 230(19):7191
Verlet L (1967) Phys Rev 159:98
Wolff T, Burger C, Ruland W (1993) Macromolecules 26(7):1707
Groot RD, Madden TJ (1998) J Chem Phys 108(20):8713
Likos CN (2001) Phys Rep 348(4):267
Daoud M, Cotton J (1982) J Phys France 43(3):531
Hsu HP, Nadler W, Grassberger P (2004) Macromolecules 37(12):4658
Rubinstein M, Colby RH (2003) Polymer physics. Oxford University Press, New York
Ye X, Khomami B (2020) Soft Matter 16:6056
I. C. acknowledges useful discussions with B. Capone. We are also grateful for a generous computational time at the Vienna Scientific Cluster.
Open access funding provided by University of Vienna. We would like to acknowledge financial support from KAUST under grant OSR-2016-CRG5-3073-03. We also acknowledge the additional support provided by a STSM Grant from COST Action CA17139.
Institute of Electronic Structure and Laser, FORTH, 71110, Heraklion, Crete, Greece
Esmaeel Moghimi, Dimitra Founta & Dimitris Vlassopoulos
Department of Materials Science, Technology, University of Crete, 71003, Heraklion, Crete, Greece
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria
Iurii Chubak & Christos N. Likos
Physical Sciences and Engineering Division, KAUST Catalysis Center, Polymer Synthesis Laboratory, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955, Kingdom of Saudi Arabia
Konstantinos Ntetsikas, George Polymeropoulos & Nikos Hadjichristidis
Esmaeel Moghimi
Iurii Chubak
Dimitra Founta
Konstantinos Ntetsikas
George Polymeropoulos
Nikos Hadjichristidis
Christos N. Likos
Dimitris Vlassopoulos
Correspondence to Christos N. Likos or Dimitris Vlassopoulos.
The authors declare that they have no conflict of interest.
E. M. and I. C. contributed equally.
Moghimi, E., Chubak, I., Founta, D. et al. The influence of arm composition on the self-assembly of low-functionality telechelic star polymers in dilute solutions. Colloid Polym Sci 299, 497–507 (2021). https://doi.org/10.1007/s00396-020-04742-0
Revised: 15 August 2020
Accepted: 23 August 2020
Micelles
Polymer brushes
Polymer synthesis | CommonCrawl |
\begin{document}
\jmlrheading{1}{2000}{1-48}{4/00}{10/00}{Aryan Mokhtari, Hamed Hassani, and Amin Karbasi}
\ShortHeadings{Decentralized Submodular Maximization}{Mokhtari, Hassani, and Karbasi} \firstpageno{1}
\title{Decentralized Submodular Maximization:\\ Bridging Discrete and Continuous Settings}
\name{Aryan Mokhtari$^{\S}$, Hamed Hassani$^{\dag}$, and Amin Karbasi$^{ \ddagger}$ \thanks{This work was done while A. Mokhtari was a Research Fellow at the Simons Institute for the Theory of Computing.}
\address{\normalsize $^\S$Laboratory for Information and Decision Systems, Massachusetts Institute of Technology \\ \normalsize$^{\dag}$Department of Electrical and Systems Engineering, University of Pennsylvania\\ \normalsize$^{\ddagger}$Department of Electrical Engineering and Computer Science, Yale University\\ \normalsize [email protected], [email protected], [email protected]}}
\maketitle
\begin{abstract} In this paper, we showcase the interplay between discrete and continuous optimization in network-structured settings. We propose the first fully decentralized optimization method for a wide class of non-convex objective functions that possess a diminishing returns property. More specifically, given an arbitrary connected network and a global \textit{continuous} submodular function, formed by a sum of local functions, we develop \alg (DCG), a message passing algorithm that converges to the tight $(1-1/e)$ approximation factor of the optimum global solution using only local computation and communication. We also provide strong convergence bounds as a function of network size and spectral characteristics of the underlying topology. Interestingly, DCG readily provides a simple recipe for decentralized \textit{discrete} submodular maximization through the means of continuous relaxations. Formally, we demonstrate that by lifting the local discrete functions to continuous domains and using DCG as an interface we can develop a consensus algorithm that also achieves the tight $(1-1/e)$ approximation guarantee of the global discrete solution once a proper rounding scheme is applied.
\end{abstract}
\section{Introduction} In recent years, we have reached unprecedented data volumes that are high dimensional and sit over (clouds of) networked machines. As a result, \textit{decentralized} collection of these data sets along with accompanying distributed optimization methods are not only desirable but very often necessary \citep{BoydEtalADMM11}.
The focus of this paper is on decentralized optimization, the goal of which is to maximize/minimize a global objective function --distributed over a network of computing units-- through local computation and communications among nodes. A canonical example in machine learning is fitting models using M-estimators where given a set of data points the parameters of the model are estimated through an empirical risk minimization \citep{DBLP:books/daglib/0097035}. Here, the global objective function is defined as an average of local loss functions associated with each data point. Such local loss functions can be convex (e.g., logistic regression, SVM, etc) or non-convex (e.g., non-linear square loss, robust regression, mixture of Gaussians, deep neural nets, etc) \citep{mei2016landscape}. Due to the sheer volume of data points, these optimization tasks cannot be fulfilled on a single computing cluster node. Instead, we need to opt for decentralized solutions that can efficiently exploit dispersed (and often distant) computational resources linked through a tightly connected network. Furthermore, local computations should be light so that they can be done on single machines. In particular, when the data is high dimensional, extra care should be given to any optimization procedure that relies on projections over the feasibility domain.
In addition to large scale machine learning applications, decentralized optimization is a method of choice in many other domains such as Internet of Things (IoT) \citep{abu2013data}, remote sensing \citep{ma2015remote}, multi-robot systems \citep{tanner2005towards}, and sensor networks \citep{rabbat2004distributed}. In such scenarios, individual entities can communicate over a network and interact with the environment by exchanging the data generated through sensing. At the same time they can react to events and trigger actions to control the physical world. These applications highlight another important aspect of decentralized optimization where \textit{private} data is collected by different sensing units \citep{yang2017survey}. Here again, we aim to optimize a global objective function while avoiding to share the private data among computing units. Thus, by design, one cannot solve such private optimization problems in a centralized manner and should rely on decentralized solutions where local private computation is done where the data is collected.
Continuous submodular functions, a broad subclass of non-convex functions with diminishing returns property, have recently received considerable attention \citep{bach2015submodular, bian16guaranteed}. Due to their interesting structures that allow strong approximation guarantees \citep{mokhtari2017conditional, bian16guaranteed}, they have found various applications, including robust budget allocation \citep{staib2017robust,soma2014optimal}, online resource allocation \citep{eghbali2016designing}, learning assignments \citep{golovin2014online}, as well as Adwords for e-commerce and advertising \citep{devanur2012online, mehta2007adwords}. However, all the existing work suffer from centralized computing. Given that many information gathering, data summarization, and non-parametric learning problems are inherently related to large-scale submodular maximization, the demand for a fully decentralized solution is immediate. In this paper, we develop the first decentralized framework for both continuous and discrete submodular functions. Our contributions are as follows: \begin{itemize}
\item \textit{Continuous submodular maximization:} For any global objective function that is monotone and continuous DR-submodular and subject to any down-closed and bounded convex body, we develop \alg, a decentralized and \textit{projection-free} algorithm that achieves the tight $(1 - 1/e- \epsilon)$ approximation guarantee in $O(1/\epsilon^2)$ rounds of local communication.
\item \textit{Discrete submodular maximization:} For any global objective function that is monotone and submodular and subject to any matroid constraint, we develop a discrete variant of the \alg algorithm that achieves the tight $(1-1/e-\epsilon)$ approximation ratio in $O(1/\epsilon^3)$ rounds of communication. \end{itemize}
\section{Related Work}
Decentralized optimization is a challenging problem as nodes only have access to separate components of the global objective function, while they aim to collectively reach the global optimum point. Indeed, one naive approach to tackle this problem is to broadcast local objective functions to all the nodes in the network and then solve the problem locally. However, this scheme requires high communication overhead and disregards the privacy associated with the data of each node. An alternative approach is the master-slave setting \citep{bekkerman2011scaling,shamir2014communication,zhang2015disco} where at each iteration, nodes use their local data to compute the information needed by the master node. Once the master node receives all the local information, it updates its decision and broadcasts the decision to all the nodes. Although this scheme protects the privacy of nodes it is not robust to machine failures and is prone to high overall communication time. In decentralized methods, these issues are overcame by removing the master node and considering each node as an independent unit that is allowed to exchange information with its neighbors.
{Convex decentralized consensus optimization} is a relatively mature area with a myriad of primal and dual algorithms \citep{bertsekas1989parallel}. Among primal methods, decentralized (sub)gradient descent is perhaps the most well known algorithm which is a mix of local gradient descent and successive averaging \citep{nedic2009distributed,yuan2016convergence}. It also can be interpreted as a penalty method that encourages agreement among neighboring nodes. This latter interpretation has been exploited to solve the penalized objective function using accelerated gradient descent \citep{jakovetic2014fast,qu2017accelerated}, Newton's method \citep{mokhtari2017network,bajovic2017newton}, or quasi-Newton algorithms \citep{eisen2017decentralized}. The methods that operate in the dual domain consider a constraint that enforces equality between nodes' variables and solve the problem by ascending on the dual function to find optimal Lagrange multipliers. A short list of dual methods are the alternating directions method of multipliers (ADMM) \citep{DBLP:journals/tsp/SchizasRG08,BoydEtalADMM11}, dual ascent algorithm \citep{rabbat2005generalized}, and augmented Lagrangian methods \citep{jakovetic2015linear,chatzipanagiotis2015convergence,mokhtari2016dsa}. Recently, there have been many attempts to extend the tools in decentralized consensus optimization to the case that the objective function is non-convex \citep{di2016next,sun2016distributed,hajinezhad2016nestt,tatarenko2017non}. However, such works are mainly concerned with reaching a stationary point and naturally cannot provide any optimality guarantee.
In this paper, our focus is to provide the first decentralized algorithms for both discrete and continuous submodular functions. Indeed, it is known that the centralized greedy approach of \citep{nemhauser1978analysis}, and its many variants \citep{feige2011maximizing, buchbinder2015tight, buchbinder2014submodular, feldman2017greed,DBLP:conf/icml/MirzasoleimanBK16}, reach tight approximation guarantees in various scenarios. Since such algorithms are sequential in nature, they do not scale to massive datasets. To partially resolve this issue, MapReduce style algorithms, with a master-slave architecture, have been proposed \citep{DBLP:conf/nips/MirzasoleimanKSK13,kumar2015fast,DBLP:conf/icml/BarbosaENW15,DBLP:conf/stoc/MirrokniZ15, qu2015distributed}.
One can extend the notion of diminishing returns to continuous domains \citep{wolsey1982analysis,bach2015submodular}.
Even though continuous submodular functions are not generally convex (nor concave) \citet{hassani2017gradient} showed that in the monotone setting and subject to a general bounded convex body constraint, stochastic gradient methods can achieve a $1/2$ approximation guarantee. The approximation guarantee can be tightened to $(1-1/e)$ by using Frank-Wolfe \citep{bian16guaranteed} or stochastic Frank-Wolfe \citep{mokhtari2017conditional}.
\section{Notation and Background} In this section, we review the notation that we use throughout the paper. We then give the precise definition of submodularity in discrete and continuous domains.\\ \\ \textbf{Notation.}
Lowercase boldface $\bbv$ denotes a vector and uppercase boldface $\bbW$ a matrix. The $i$-th element of $\bbv$ is written as $v_i$ and the element on the $i$-th row and $j$-th column of $\bbW$ is denoted by $w_{i,j}$. We use $\|\bbv\|$ to denote the Euclidean norm of vector $\bbv$ and $\|\bbW\|$ to denote the spectral norm of matrix $\bbW$. The null space of matrix $\bbW$ is denoted by $\rm{null}(\bbW)$. The inner product of vectors $\bbx,\bby$ is indicated by $\langle \bbx,\bby\rangle$, and the transpose of a vector $\bbv$ or matrix $\bbW$ are denoted by $\bbv^\dag$ and $\bbW^\dag$, respectively. The vector $\bbone_n\in \reals^n$ is the vector of all ones with $n$ components, and the vector $\bb0_p\in \reals^p$ is the vector of all zeros with $p$ components. \\ \\ \textbf{Submodulary.} A \textit{set} function $f:2^V\rightarrow \reals_+$, defined on the ground set $V$, is called submodular if for all $A,B\subseteq V$, we have $$f(A)+f(B)\geq f(A\cap B) + f(A\cup B).$$
We often need to maximize submodular functions subject to a down-closed set family $\mathcal{I}$. In particular, we say $\mathcal{I}\subset 2^V$ is a matroid if 1) for any $A\subset B\subset V$, if $B\in \mathcal{I}$, then $A\in \mathcal{I}$ and 2) for any $A,B\in \mathcal{I}$ if $|A|<|B|$, then there is an element $e\in B$ such that $A\cup\{e\}\in \mathcal{I}$.
The notion of submodularity goes beyond the discrete domain \citep{wolsey1982analysis, vondrak2007submodularity, bach2015submodular}. Consider a continuous function $F: \ccalX \to \reals_{+}$ where the set $\ccalX \subseteq \mathbb{R}^p$ is of the form $\ccalX=\prod_{i=1}^p\ccalX_i$ and each $\ccalX_i$ is a compact subset of $\reals_+$. We call the \textit{continuous} function $F$ submodular if for all $\bbx,\bby\in \ccalX$ we have
\begin{align}\label{eq:submodular_def} F(\bbx) + F(\bby) \geq F(\bbx \vee \bby) + F(\bbx \wedge \bby) , \end{align}
where $\bbx \vee \bby := \max (\bbx ,\bby )$ (component-wise) and $\bbx \wedge \bby := \min (\bbx ,\bby )$ (component-wise). In this paper, our focus is on differentiable continuous submodular functions with two additional properties: monotonicity and diminishing returns. Formally, a submodular function $F$ is monotone if
\begin{align}\label{eq:monotone_def} \bbx \leq \bby \quad \Longrightarrow \quad F(\bbx) \leq F(\bby), \end{align}
for all $\bbx,\bby\in \ccalX$. Note that $\bbx \leq \bby$ in \eqref{eq:monotone_def} means that $x_i\leq y_i$ for all $i=1,\dots,p$. Furthermore, a differentiable submodular function $F$ is called \textit{DR-submodular} (i.e., shows diminishing returns) if the gradients are antitone, namely, for all $\bbx,\bby\in \ccalX$ we have
\begin{align}\label{eq:antitone_def} \bbx \leq \bby \quad \Longrightarrow \quad \nabla F(\bbx) \geq \nabla F(\bby). \end{align} When the function $F$ is twice differentiable, submodularity implies that all cross-second-derivatives are non-positive \citep{bach2015submodular},
and DR-submodularity implies that all second-derivatives are non-positive \citep{bian16guaranteed}
In this work, we consider the maximization of continuous submodular functions subject to \textit{down-closed convex bodies} $\ccalC \subset \reals_{+}^p$ defined as follows. For any two vectors $\bbx, \bby\in \reals_{+}^p$, where $\bbx\leq \bby$, down-closedness means that if $\bby\in \ccalC$, then so is $\bbx\in \ccalC$. Note that for a down-closed set we have $\mathbf{0}_p \in \ccalC$.
\section{Decentralized Submodular Maximization} In this section, we state the problem of decentralized submodular maximization in continuous and discrete settings.\\ \\ \noindent\textbf{Continuous Case.} We consider a set of $n$ computing machines/sensors that communicate over a graph to maximize a global objective function. Each machine can be viewed as a node $i \in \mathcal{N} \triangleq \{1,\cdots,n\}$. We further assume that the possible communication links among nodes are given by a bidirectional connected \emph{communication graph} $\mathcal{G} = (\mathcal{N},\mathcal{E})$ where each node can only communicate with its neighbors in $\mathcal{G}$. We formally use $\ccalN_i$ to denote node $i$'s neighbors.
In our setting, we assume that each node $i \in \mathcal{N}$ has access to a local function $F_i:\ccalX\to\reals_{+}$. The nodes cooperate in order to maximize the aggregate monotone and continuous DR-submodular function $F:\ccalX\to\reals_{+}$ subject to a down-closed convex body $\ccalC\subset \ccalX\subset \reals_{+}^p$, i.e.,
\begin{equation}\label{original_optimization_problem1}
\ \max_{\bbx\in \ccalC} F(\bbx)
\ =\ \max_{\bbx\in \ccalC} \frac{1}{n}\sum_{i=1}^{n} F_i(\bbx). \end{equation}
The goal is to design a message passing algorithm to solve \eqref{original_optimization_problem1} such that: (i) at each iteration $t$, the nodes send their messages (and share their information) to their neighbors in $\mathcal{G}$, and (ii) as $t$ grows, all the nodes reach to a point $\bbx \in \ccalC$ that provides a (near-) optimal solution for \eqref{original_optimization_problem1}. \\ \\ \noindent \textbf{Discrete Case.} Let us now consider the discrete counterpart of problem~\eqref{original_optimization_problem1}. In this setting, each node $i \in \mathcal{N}$ has access to a local \emph{set} function $f_i:2^V \to \mathbb{R}_+$. The nodes cooperate in maximizing the aggregate monotone submodular function $f:2^V \to\reals_{+}$ subject to a matroid constraint $\ccalI$, i.e., \begin{equation}\label{original_optimization_problem2} \ \max_{S \in \mathcal{I}} f(S)
\ =\ \max_{S\in \mathcal{I}} \frac{1}{n}\sum_{i=1}^{n} f_i(S). \end{equation} Note that even in the centralized case, and under reasonable complexity-theoretic assumptions, the best approximation guarantee we can achieve for Problems~\eqref{original_optimization_problem1} and \eqref{original_optimization_problem2} is $(1-1/e)$ \citep{feige1998threshold}. In the following, we show that it is possible to achieve the same approximation guarantee in a decentralized setting.
\section{Decentralized Continuous Greedy Method}
In this section, we introduce the \alg (DCG) algorithm for solving Problem \eqref{original_optimization_problem1}. Recall that in a decentralized setting, the nodes have to cooperate (i.e., send messages to their neighbors) in order to solve the global optimization problem. We will explain how such messages are designed and communicated in DCG. Each node $i$ in the network keeps track of two local variables $\bbx_i, \bbd_i \in \reals^p$ which are iteratively updated at each round $t$ using the information gathered from the neighboring nodes. The vector $\bbx_i^t$ is the local decision variable of node $i$ at step $t$ whose value we expect to eventually converge to the $(1-1/e)$ fraction of the optimal solution of Problem \eqref{original_optimization_problem1}. The vector $\bbd_i^t$ is the estimate of the gradient of the global objective function that node $i$ keeps at step $t$.
To properly incorporate the received information from their neighbors, nodes should assign nonnegative weights to their neighbors. Define $w_{ij}\geq0$ to be the weight that node $i$ assigns to node $j$. These weights indicate the effect of (variable or gradient) information nodes received from their neighbors in order to update their local (variable or gradient) information. Indeed, the weights $w_{ij}$ must fulfill some requirements (later described in Assumption \ref{ass:weights}), but they are design parameters of DCG and can be properly chosen by the nodes prior to the implementation of the algorithm.
The first step at each round $t$ of DCG is updating the local gradient approximation vectors $\bbd_i^t$ using local and neighboring gradient information. In particular, node $i$ computes its vector $\bbd_i^t$ according to the update rule
\begin{align}\label{eq:gradient_update} \bbd_i^t = (1-\alpha) \!\!\!\sum_{j\in \ccalN_i \cup\{i\}}\!\! w_{ij}\bbd_{j}^{t-1}\ +\ \alpha \nabla F_i(\bbx_i^t), \end{align}
where $\alpha\in[0,1]$ is an averaging coefficient. Note that the sum $\sum_{j\in \ccalN_i\cup\{i\}} w_{ij}\bbd_{j}^{t-1}$ in \eqref{eq:gradient_update} is a weighted average of node $i$'s vector $\bbd_i^{t-1}$ and its neighbors $\bbd_j^{t-1}$, evaluated at step $t-1$. Hence, node $i$ computes the vector $\bbd_i^t$ by evaluating a weighted average of its current local gradient $\nabla F_i(\bbx_i^t)$ and the local and neighboring gradient information at step $t-1$, i.e., $\sum_{j\in \ccalN_i\cup\{i\}} w_{ij}\bbd_{j}^{t-1}$. Since the vector $\bbd_i^t$ is evaluated by aggregating gradient information from neighboring nodes, it is reasonable to expect that $\bbd_i^t$ becomes a proper approximation for the global objective function gradient $(1/n) \sum_{k=1}^n\nabla f_{k}(x)$ as time progresses. Note that to implement the update in \eqref{eq:gradient_update} nodes should exchange their local vectors $\bbd_i^t$ with their neighbors.
Using the gradient approximation vector $\bbd_i^t$, each node $i$ evaluates its local ascent direction $\bbv_i^t$ by solving the following linear program
\begin{align}\label{eq:descent_update} \bbv_i^t = \argmax_{\bbv\in \ccalC}\ \langle \bbd_i^t, \bbv\rangle. \end{align}
The update in \eqref{eq:descent_update} is also known as \textit{conditional gradient} update. Ideally, in a conditional gradient method, we should choose the feasible direction $\bbv\in \ccalC$ that maximizes the inner product by the full gradient vector $\frac{1}{n} \sum_{k=1}^n\nabla F_{k}(\bbx_i^t)$. However, since in the decentralized setting the exact gradient $\frac{1}{n} \sum_{k=1}^n\nabla F_{k}(\bbx_i^t)$ is not available at the $i$-th node, we replace it by its current approximation $\bbd_i^t$ and hence we obtain the update rule \eqref{eq:descent_update}.
\begin{algorithm}[tb] \caption{DCG at node $i$}\label{algo_DCG} \begin{algorithmic}[1] {\REQUIRE Stepsize $\alpha$ and weights $w_{ij}$ for $j\in\ccalN_{i}\cup\{i\}$
\STATE Initialize local vectors as $\bbx_i^0=\bbd_i^0=\bb0_p$
\STATE Initialize neighbor's vectors as $x_j^0=\bbd_j^0=\bb0_p$ if $j\in \ccalN_i$
\FOR {$t=1,2,\ldots, T$}
\STATE Compute $\displaystyle{\bbd_i^t = (1-\alpha) \!\!\!\! \sum_{j\in \ccalN_i \cup\{i\}} \!\!\!\! w_{ij}\bbd_{j}^{t-1}+\alpha \nabla F_i(\bbx_i^t)}$;
\STATE Exchange $\bbd_i^t$ with neighboring nodes ${j\in \ccalN_i }$
\STATE Evaluate $\bbv_i^t = \argmax_{\bbv\in \ccalC}\ \langle \bbd_i^t, \bbv\rangle$;
\STATE Update the variable $\displaystyle{\bbx_i^{t+1} = \!\!\!\! \sum_{j\in \ccalN_i\cup\{i\}} \!\!\!\! w_{ij}\bbx_{j}^{t}+\frac{1}{T} \bbv_i^t}$;
\STATE Exchange $\bbx_i^{t+1}$ with neighboring nodes ${j\in \ccalN_i }$
\ENDFOR} \end{algorithmic}\end{algorithm}
After computing the local ascent directions $\bbv_i^t$, the nodes update their local variables $x_{i}^{t}$ by averaging their local and neighboring iterates and ascend in the direction $\bbv_i^t$ with stepsize $1/T$ where $T$ is the total number of iterations, i.e.,
\begin{align}\label{eq:variable_update} \bbx_i^{t+1} =\! \sum_{j\in \ccalN_i\cup\{i\}} \!\!\! w_{ij}\bbx_{j}^{t}\ +\ \frac{1}{T} \bbv_i^t. \end{align}
The update rule \eqref{eq:variable_update} ensures that the neighboring iterates are not far from each other via the averaging term $\sum_{j\in \ccalN_i\cup\{i\}} w_{ij}\bbx_{j}^{t}$, while the iterates approach the optimal maximizer of the global objective function by ascending in the conditional gradient direction $\bbv_i^t$. The update in \eqref{eq:variable_update} requires a round of local communication among neighbors to exchange their local variables $\bbx_i^t$. The steps of the DCG method are summarized in Algorithm \ref{algo_DCG}.
Indeed, the weights $w_{ij}$ that nodes assign to each other cannot be arbitrary. In the following, we formalize the conditions that they should satisfy \citep{yuan2016convergence}.
\begin{assumption}\label{ass:weights} The weights that nodes assign to each other are nonegative, i.e., $w_{ij}\geq 0$ for all $i,j\in\ccalN$, and if node $j$ is not a neighbor of node $i$ then the corresponding weight is zero, i.e., $w_{ij}=0$ if $j\notin \ccalN_i$. Further, the weight matrix $\bbW\in\reals^{n\times n}$ with entries $w_{ij}$ satisfies
\begin{equation}\label{eqn_conditions_on_weights}
\bbW^{\dag}=\bbW, \quad
\bbW\bbone_n=\bbone_n, \quad
{\rm{null}}(\bbI-\bbW)={\rm{span}}(\bbone_n). \end{equation}
\end{assumption}
The first condition in \eqref{eqn_conditions_on_weights} ensures that the weights are symmetric, i.e., $w_{ij}=w_{ji}$. The second condition guarantees the weights that each node assigns to itself and its neighbors sum up to 1, i.e., $\sum_{j=1}^{n} w_{ij}=1$ for all $i$. Note that the condition $\bbW\bbone_n=\bbone_n$ implies that $\bbI-\bbW$ is rank deficient. Hence, the last condition in \eqref{eqn_conditions_on_weights} ensures that the rank of $\bbI-\bbW$ is exactly $n-1$. Indeed, it is possible to optimally design the weight matrix $\bbW$ to accelerate the averaging process as discussed in \citep{boyd2004fastest}, but this is not the focus of this paper. We should emphasize that $\bbW$ is not a problem parameter, and we design it prior to runnig DCG.
Notice that the stepsize $1/T$ and the conditions in Assumption~\ref{ass:weights} on the weights $w_{ij}$ are needed to ensure that the local variables $\bbx_i^t$ are in the feasible set $\ccalC$, as stated in the following proposition.
\begin{proposition} \label{prop_stay} Consider the proposed DCG method outlined in Algorithm \ref{algo_DCG}. If Assumption \ref{ass:weights} holds and nodes start from $\bbx_i^0=\bb0_p\in \ccalC$, then the local iterates $\bbx_i^t$ are always in the feasible set $\ccalC$, i.e., $\bbx_i^t\in \ccalC$ for all $i\in \ccalN$ and $t=1,\dots,T$. \end{proposition} \begin{myproof} Check Section \ref{app:prop_stay} in the Appendix. \end{myproof}
Let us now explain how DCG relates to and innovates beyond the exisiting work in submodular maximization as well as decentralized convex optimization. Note that in order to solve Problem~\eqref{original_optimization_problem1} in a \emph{centralized} fashion (i.e., when every node has access to \emph{all} the local functions) we can use the continuous greedy algorithm \citep{vondrak2008optimal}, a variant of the conditional gradient method. However, in decentralized settings, nodes have only access to their local gradients, and therefore, continuous greedy is not implementable. Similar to the decentralized convex optimization, we can address this issue via local information aggregation. Our proposed DCG method incorporates the idea of choosing the ascent direction according to a conditional gradient update as is done in the continuous greedy algorithm (i.e., the update rule \eqref{eq:descent_update}), while it aggregates the global objective function information through local communications with neighboring nodes (i.e., the update rule \eqref{eq:variable_update}). Unlike traditional consensus optimization methods that require exchanging nodes' local variables only \citep{nedic2009distributed,nedic2010constrained}, DCG also requires exchanging local gradient vectors to achieve a $(1-1/e)$ fraction of the optimal solution at each node (i.e., the update rule \eqref{eq:gradient_update}). This major difference is due to the fact that in conditional gradient methods, unlike proximal gradient algorithms, the local gradients can not be used instead of the global gradient. In other words, in the update rule \eqref{eq:descent_update}, we can not use the local gradients $\nabla F_i(\bbx_i^t)$ in lieu of $\bbd_i^t$. Indeed, there are settings for which such a replacement provides arbitrarily bad solutions. We formally characterize the convergence of DCG in Theorem \ref{theorem:main_theorem}.
\subsection{Extension to the Discrete Setting}
In this section we show how DCG can be used
for maximizing a decentralized submodular \emph{set} function $f$, namely Problem~\eqref{original_optimization_problem2}, through its continuous relaxation.
Formally, in lieu of solving Problem~\eqref{original_optimization_problem2}, we can form the following decentralized continuous optimization problem \begin{align}\label{eq:multilinear_program} \max_{\bbx \in \ccalC} \frac{1}{n} \sum_{i=1}^n F_i(\bbx), \end{align}
where $F_i$ is the multilinear extension of $f_i$ defined as
\begin{equation}\label{eq:def_multi_linear_extension} F_i(\bbx) = \sum_{S\subset V}f_i(S) \prod_{i\in S} x_i \prod_{j\notin S} (1-x_j) , \end{equation}
and the down-closed convex set $\ccalC= \text{conv}\{1_{I} : I\in \ccalI \}$ is the matroid polytope. Note that the discrete and continuous optimization formulations lead to the same optimal value \citep{calinescu2011maximizing}.
Based on the expression in \eqref{eq:def_multi_linear_extension}, computing the full gradient $\nabla F_i$ at each node $i$ will require an exponential computation in terms of $|V|$, since the number of summands in \eqref{eq:def_multi_linear_extension} is $2^{|V|}$. As a result, in the discrete setting, we will slightly modify the DCG algorithm and work with \emph{unbiased estimates} of the gradient that can be computed in time $O(|V|)$ (see Appendix~\ref{unbiased} for one such estimator). More precisely, in the discrete setting, each node $i \in \mathcal{N}$ updates three local variables $\bbx_i^t, \bbd_i^t, \bbg_i^t \in \reals^{|V|}$. The variables $\bbx_i^t, \bbd_i^t$ play the same role as in DCG and are updated using the messages received from the neighboring nodes. The variable $\bbg_i^t$ at node $i$ is defined to approximate the local gradient $\nabla F_i (\bbx_i^t)$. Consider the vector $\nabla \tilde{F}_i(\bbx_i^t)$ as an unbiased estimator of the local gradient $\nabla F_i (\bbx_i^t)$ at time $t$, and define the vector $\bbg_i^t$ as the outcome of the recursion
\begin{equation}\label{gradient_approx_update} \bbg_i^t = (1-\phi) \bbg_i^{t-1}+\phi \nabla \tilde{F}_i(\bbx_i^t), \end{equation}
where $\phi\in[0,1]$ is the averaging parameter. We initialize all vectors as $\bbg_i^{0}=\bb0\in \reals^{|V|}$. It was shown recently \citep{mokhtari2017conditional} that the averaging technique in \eqref{gradient_approx_update} reduces the noise of the gradient approximations. Therefore, the sequence of $\bbg_i^t$ approaches the true local gradient $ \nabla {F}_i(\bbx_i^t)$ as time progresses.
\begin{algorithm}[tb] \caption{Discrete DCG at node $i$}\label{algo_DDCG} \begin{algorithmic}[1] {\REQUIRE $\alpha,\phi\in[0,1]$ and weights $w_{ij}$ for $j\in\ccalN_{i}\cup\{i\}$;
\STATE Initialize local vectors as $\bbx_i^0=\bbd_i^0=\bbg_i^0=\bb0$ ;
\STATE Initialize neighbor's vectors as $\bbx_j^0=\bbd_j^0=\bb0$ if $j\in \ccalN_i$;
\FOR {$t=1,2,\ldots, T$}
\STATE Compute $\displaystyle{\bbg_i^t = (1-\phi) \bbg_i^{t-1}+\phi \nabla \tilde{F}_i(\bbx_i^t)}$;
\STATE Compute $\displaystyle{\bbd_i^t = (1-\alpha) \!\!\!\! \sum_{j\in \ccalN_i \cup\{i\}} \!\!\!\! w_{ij}\bbd_{j}^{t-1}+\alpha \bbg_i^t}$;
\STATE Exchange $\bbd_i^t$ with neighboring nodes ${j\in \ccalN_i }$;
\STATE Evaluate $\bbv_i^t = \argmax_{\bbv\in \ccalC}\ \langle \bbd_i^t, \bbv\rangle$;
\STATE Update the variable $\displaystyle{\bbx_i^{t+1} = \!\!\!\! \sum_{j\in \ccalN_i\cup\{i\}} \!\!\!\! w_{ij}\bbx_{j}^{t}+\frac{1}{T} \bbv_i^t}$;
\STATE Exchange $\bbx_i^{t+1}$ with neighboring nodes ${j\in \ccalN_i }$;
\ENDFOR} \STATE Apply proper rounding to obtain a solution for \eqref{original_optimization_problem2}; \end{algorithmic}\end{algorithm}
The steps of the \alg for the discrete setting is summarized in Algorithm~\ref{algo_DDCG}. Note that the major difference between the Discrete DCG method (Algorithm~\ref{algo_DDCG}) and the continuous DCG method (Algorithm~\ref{algo_DCG}) is in Step 5 in which the exact local gradient $\nabla F_i (\bbx_i^t)$ is replaced by the stochastic approximation $\bbg_i^t$ which only requires access to the computationally cheap unbiased gradient estimator $\nabla \tilde{F}_i(\bbx_i^t)$. The communication complexity of both the discrete and continuous versions of DCG are the same at each round. However, since we are using unbiased estimations of the local gradients $\nabla F_i(\bbx_i)$, the Discrete DCG takes more rounds to converge to a near-optimal solution compared to~continuous DCG. We characterize the convergence of Discrete DCG in Theorem \ref{theorem:main_theorem_discrete}. {Further, the implementation of Discrete DCG requires rounding the continuous solution to obtain a discrete solution for the original problem without any loss in terms of the objective function value. The provably lossless rounding schemes include the pipage rounding \citep{calinescu2011maximizing} and contention resolution~\citep{DBLP:journals/siamcomp/ChekuriVZ14}. }
\section{Convergence Analysis} \label{sec:convergence}
In this section, we study the convergence properties of DCG in both continuous and discrete settings. In this regard, we assume that the following conditions hold.
\begin{assumption}\label{ass:bounded_set} {Euclidean distance of the elements in the set $\ccalC$ are uniformly bounded, i.e., for all $\bbx,\bby \in \ccalC$ we have}
\begin{equation}
\|\bbx-\bby\|\leq D. \end{equation}
\end{assumption}
\begin{assumption}\label{ass:smoothness} The local objective functions $F_i(\bbx)$ are monotone and DR-submodular. Further, their gradients are $L$-Lipschitz continuous over the set $\ccalX$, i.e., for all $\bbx,\bby \in \ccalX$
\begin{equation}
\| \nabla F_i(\bbx) - \nabla F_i(\bby) \| \leq L \| \bbx-\bby \|. \end{equation}
\end{assumption}
\begin{assumption}\label{ass:smoothness2}
The norm of gradients $\|\nabla F_i(\bbx)\|$ are bounded over the convex set $\ccalC$, i.e., for all $\bbx \in \ccalC$, $i\in\ccalN$,
\begin{equation}
\| \nabla F_i(\bbx) \| \leq G. \end{equation}
\end{assumption}
The condition in Assumption \ref{ass:bounded_set} guarantees that the diameter of the convex set $\ccalC$ is bounded. Assumption \ref{ass:smoothness} is needed to ensure that the local objective functions $F_i$ are smooth. Finally, the condition in Assumption \ref{ass:smoothness2} enforces the gradients norm to be bounded over the convex set $\ccalC$. All these assumptions are customary and necessary in the analysis of decentralized algorithms. For more details, please check Section VII-B in \citet{jakovetic2014fast}.
We proceed to derive a constant factor approximation for DCG. Our main result is stated in Theorem~\ref{theorem:main_theorem}. However, to better illustrate the main result, we first need to provide several definitions and technical lemmas. Let us begin by defining the average variables $\bar{\bbx}^t$ as \begin{equation} \label{average_x} \bar{\bbx}^{t}= \frac{1}{n}\sum_{i=1}^n \bbx_i^t. \end{equation}
In the following lemma, we establish an upper bound on the variation in the sequence of average variables $\{\bar{\bbx}^t\}$.
\begin{lemma}\label{lemma:ar_in_avg_bound} Consider the proposed DCG algorithm defined in Algorithm \ref{algo_DCG}. Further, recall the definition of $\bar{\bbx}^{t}$ in \eqref{average_x}. If Assumptions \ref{ass:weights} and \ref{ass:bounded_set} hold, then the difference between two consecutive average vectors is upper bounded by \begin{align}\label{eq:var_in_avg_bound}
\|\bar{\bbx}^{t+1} - \bar{\bbx}^{t}\| \leq \frac{D}{T} . \end{align} \end{lemma}
\begin{myproof} Check Section \ref{app:lemma:ar_in_avg_bound} in the Appendix. \end{myproof}
Recall that at every node $i$, the messages are mixed using the coefficients $w_{ij}$, i.e., the $i$-th row of the matrix $\bbW$. It is thus not hard to see that the spectral properties of $\bbW$ (e.g. the spectral gap) play an important role in the
the speed of achieving consensus in decentralized methods.
\begin{definition} Consider the eigenvalues of $\bbW$ which can be sorted in a nonincreasing order as $1 = \lambda_{1}(\bbW) \geq \lambda_{2}(\bbW) \dots \geq \lambda_{n}(\bbW) > -1$. Define $\beta$ as the second largest magnitude of the eigenvalues of $\bbW$, i.e.,
\begin{align}\label{eq:def_beta}
\beta:= \max\{ | \lambda_{2}(\bbW) |, | \lambda_{n}(\bbW) | \}. \end{align} \end{definition}
As we will see, a mixing matrix $\bbW$ with smaller $\beta$ has a larger spectral gap $1-\beta$ which yields faster convergence \citep{boyd2004fastest,duchi2012dual}. In the following lemma, we derive an upper bound on the sum of the distances between the local iterates $\bbx_{i}^t$ and their average $\bar{\bbx}^{t}$, where the bound is a function of the graph spectral gap $1-\beta$, size of the network $n$, and the total number of iterations $T$.
\begin{lemma}\label{lemma:eq:bound_on_dif_from_avg} Consider the proposed DCG algorithm defined in Algorithm \ref{algo_DCG}. Further, recall the definition of $\bar{\bbx}^{t}$ in \eqref{average_x}. If Assumptions \ref{ass:weights} and \ref{ass:bounded_set} hold, then for all $t\leq T$ we have
\begin{align}\label{eq:bound_on_dif_from_avg}
\left( \sum_{i=1}^n \left\|\bbx_i^t-\bar{\bbx}^t\right\|^2\right)^{1/2} \leq \frac{\sqrt{n}D}{T(1-\beta)}. \end{align}
\end{lemma}
\begin{myproof} Check Section \ref{app:lemma:eq:bound_on_dif_from_avg} in the Appendix. \end{myproof}
Let us now define $\bar{\bbd}^t $ as the average of local gradient approximations $\bbd_i^t$ at step $t$, i.e., \begin{equation} \bar{\bbd}^t = \frac{1}{n} \sum_{i=1}^n \bbd_i^t. \end{equation} We will show in the following that the vectors $\bbd_i^t$ also become uniformly close to $\bar{\bbd}^t$.
\begin{lemma}\label{lemma:bound_on_gradient_consensus_error} Consider the proposed DCG algorithm defined in Algorithm \ref{algo_DCG}. If Assumptions \ref{ass:weights} and \ref{ass:smoothness} hold, then
\begin{align}\label{eq:bound_on_gradient_consensus_error}
\left(\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|^2\right)^{1/2} \leq \frac{\alpha \sqrt{n} G}{1-\beta(1-\alpha)}. \end{align}
\end{lemma}
\begin{myproof} Check Section \ref{proof:lemma:bound_on_gradient_consensus_error} in the Appendix. \end{myproof}
Lemma \ref{lemma:bound_on_gradient_consensus_error} guarantees that the individual local gradient approximation vectors $\bbd_i^t$ are close to the average vector $\bar{\bbd}^t$ if the parameter $\alpha$ is small. To show that the gradient vectors $\bbd_i^t$, generated by DCG, approximate the gradient of the global objective function, we further need to show that the average vector $\bar{\bbd}^t$ approaches the global objective function gradient $\nabla F$. We prove this claim in the following lemma.
\begin{lemma}\label{lemma:bound_on_gradient_error} Consider the proposed DCG algorithm defined in Algorithm \ref{algo_DCG}. If Assumptions \ref{ass:weights}-\ref{ass:smoothness2} hold, then
\begin{align}\label{eq:bound_on_gradient_error}
\left\|\bar{\bbd}^t - \frac{1}{n}\sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\| \leq (1-\alpha)^t G +\left(\frac{(1-\alpha)LD}{\alpha T} + \frac{ LD}{T(1-\beta)} \right). \end{align}
\end{lemma}
\begin{myproof} Check Section \ref{proof:lemma:bound_on_gradient_error} in the Appendix. \end{myproof}
By combining Lemmas \ref{lemma:bound_on_gradient_consensus_error} and \ref{lemma:bound_on_gradient_error} and setting $\alpha=1/\sqrt{T}$ we can conclude that the local gradient approximation vector $\bbd_i^t$ of each node $i$ is within $\mathcal{O}(1/\sqrt{T})$ distance of the global objective gradient $\nabla F(\bar{\bbx}^t)$ evaluated at $\bar{\bbx}^t$. We use this observation in the following theorem to show that the sequence of iterates generated by DCG achieves the tight $(1-1/e)$ approximation ratio of the optimum global solution.
\begin{theorem}\label{theorem:main_theorem} Consider the proposed DCG method outlined in Algorithm~\ref{algo_DCG}. Further, consider $\bbx^*$ as the global maximizer of Problem \eqref{original_optimization_problem1}. If Assumptions \ref{ass:weights}-\ref{ass:smoothness2} hold and we set $\alpha= 1/\sqrt{T}$,
for all nodes $j\in \ccalN$, the local variable $\bbx_j^T$ obtained after $T$ iterations satisfies \begin{align}\label{local_node_bound} F(\bbx_j^T)&\geq (1-e^{-1} )F(\bbx^*) - \frac{ LD^2+GD}{T^{1/2}}-\frac{GD}{T^{1/2}(1-\beta)}- \frac{ LD^2}{2T} - \frac{ GD+LD^2}{T(1-\beta)}. \end{align} \end{theorem}
\begin{myproof} Check Section \ref{proof:theorem:main_theorem} in the Appendix. \end{myproof}
Theorem \ref{theorem:main_theorem} shows that the sequence of the local variables ${\bbx}_j^t$, generated by DCG, is able to achieve the optimal approximation ratio $(1-1/e)$, while the error term vanishes at a sublinear rate of $\mathcal{O}(1/T^{1/2})$, i.e.,
\begin{align} F(\bbx_j^T) \geq (1-1/e ) F(\bbx^*)- \mathcal{O}\left(\frac{1}{(1-\beta)T^{1/2}}\right), \end{align}
which implies that the iterate of \textit{each node} reaches an objective value larger than $(1-1/e-\epsilon)OPT$ after $\mathcal{O}(1/\eps^2)$ rounds of communication. It is worth mentioning that the result in Theorem \ref{theorem:main_theorem} is consistent with classical results in decentralized optimization that the error term vanishes faster for the graphs with larger spectral gap $1-\beta$. We proceed to study the convergence properties of Discrete DCG in Algorithm \ref{algo_DDCG}. To do so, we first assume that the variance of the stochastic gradients $ \nabla \tilde{F}_i(\bbx)$ used in Discrete DCG is bounded. We justify this assumption in Remark~\ref{remdec}.
\begin{assumption}\label{ass:bounded_variance} The variance of the unbiased estimators $\nabla \tilde{F}(\bbx)$ is bounded above by $\sigma^2$ over the convex set $\ccalC$, i.e., for any $i \in \mathcal{N}$ and any vector $\bbx\in\ccalC$ we can write
\begin{equation}
\E{\| \nabla \tilde{F}_i(\bbx) - \nabla F_i(\bbx) \|^2} \leq \sigma^2, \end{equation} where the expectation is with respect to the randomness of the unbiased estimator. \end{assumption}
In the following theorem, we show that Discrete DCG achieves a $(1-1/e)$ approximation ration for Problem~\eqref{original_optimization_problem2}.
\begin{theorem}\label{theorem:main_theorem_discrete} Consider our proposed Discrete DCG algorithm outlined in Algorithm \ref{algo_DDCG}. Recall the definition of the multilinear extension function $F_i$ in \eqref{eq:def_multi_linear_extension}. If Assumptions \ref{ass:weights}-\ref{ass:bounded_variance} hold and we set $\alpha= T^{-1/2}$ and $\phi=T^{-2/3}$, then for all nodes $j\in \ccalN$ the local variables $\bbx_j^T$ obtained after running Discrete DCG for $T$ iterations satisfy
\begin{align}\label{eq:main_result_1} \E{F(\bbx_j^T)} &\geq (1-e^{-1} ) F(\bbx^*) -\frac{ GD+LD^2}{T(1-\beta)} -\frac{LD^2}{2T} -\frac{\sqrt{6} LD^2}{T^{2/3}}-\frac{\sqrt{12} LD^2}{(1-\beta)T^{2/3}}
\nonumber\\ & \quad
- \frac{ D(\sigma^2+G^2)^{1/2}}{T^{1/2}(1-\beta)}
- \frac{DG+LD^2}{T^{1/2}}
-\frac{\sqrt{2}\sigma+\sqrt{12} LD^2+4DG}{T^{1/3}}
-\frac{ \sqrt{24}LD^2}{(1-\beta)T^{1/3}}, \end{align}
where $\bbx^*$ is the global maximizer of Problem \eqref{eq:multilinear_program}.
\end{theorem}
\begin{myproof} Check Section \ref{proof:theorem:main_theorem_discrete} in the Appendix. \end{myproof}
Theorem \ref{theorem:main_theorem_discrete} states that the sequence of the local variables ${\bbx}_j^t$, generated by Discrete DCG, is able to achieve the optimal approximation ratio $(1-1/e)$ in expectation, while the error term vanishes at a sublinear rate of $\mathcal{O}(1/T^{1/3})$, i.e.,
\begin{equation} \label{eq:main_result_1000} \E{F(\bbx_j^T)}\geq (1-e^{-1} )F(\bbx^*) - \mathcal{O}\left(\frac{1}{(1\!-\!\beta)T^{1/3}}\right). \end{equation}
Hence, the iterate of \textit{each node} reaches an objective value larger than $(1-1/e-\epsilon)OPT$ after $\mathcal{O}(1/\eps^3)$ rounds of communication.
\begin{remark} \label{remdec} For any submodular set function $h: 2^V \to \mathbb{R}$ with associated multilinear extension $H$, it can be shown that its Lipschitz constant $L$ and the gradient norm $G$ are both bounded above by
$m_f \sqrt{|V|}$, where $m_f$ is the maximum marginal value of $f$, i.e., $m_f = \max_{i \in V} f(\{i\})$ (see, \citet{hassani2017gradient}). Similarly, it can be shown that for the unbiased estimator in Appendix~\ref{unbiased} we have $\sigma \leq m_f \sqrt{|V|}$. \end{remark}
\section{Numerical Experiments} We will consider a discrete setting for our experiments and use Algorithm~\ref{algo_DDCG} to find a decentralized solution. The main objective is to demonstrate how consensus is reached and~how the global objective increases depending on the topology of the network and the parameters of the algorithm.
For our experiments, we have used the MovieLens data set. It consists of 1 million ratings (from 1 to 5) by $M=6000$ users for $p=4000$ movies. We consider a network of $n = 100$ nodes. The data has been distributed equally between the nodes of the network, i.e., the set of users has been partitioned into $100$ equally-sized sets and each node in the network has access to only one chunk (partition) of the data. The global task is to find a set of $k$ movies that are most satisfactory to \emph{all} the users (the precise formulation will appear shortly). However, as each of the nodes in the network has access to the data of a small portion of the users, the nodes have to cooperate (exchange information) in order to fulfil the global task.
\begin{figure}
\caption{The logarithm of the distance-to-average at final round $T$ is plotted as a function of $T$. Note that when the underlying graph is complete or Erdos-Renyi (ER) with a good average degree, then consensus will be achieved even for small number of iterations $T$. However, for poor connected graphs such as the line graph, reaching consensus requires a large number of iterations.}
\label{fig-dist}
\end{figure}
We consider a well motivated objective function for the experiments. Let $r_{\ell,j}$ denote the rating of user $\ell$ for movie $j$ (if such a rating does not exist in the data we assign $r_{\ell,j}$ to 0). We associate to each user $\ell$ a ``facility location" objective function $g_\ell (S) = \max_{j\in S} r_{\ell,j}$, where $S$ is any subset of the movies (i.e. the ground set $V$ is the set of the movies). Such a function shows how much user $\ell$ will be ``satisfied" by a subset $S$ of the movies. Recall that each node $i$ in the network has access to the data of a (small) subset of users which we denote by $\mathcal{U}_i$. The objective function associated with node $i$ is given by $f_i(S) = \sum_{\ell \in \mathcal{U}_i} g_\ell(S)$. With such a choice of the local functions, our global task is hence to solve problem~\eqref{original_optimization_problem2} when the matroid $\mathcal{I}$ is the $k$-uniform matroid (a.k.a. the $k$-cardinality constraint).
We consider three different choices for the underlying communication graph between the $100$ nodes: A line graph (which looks like a simple path from node 1 to node 100), an Erdos-Renyi random graph (with average degree $5$), and a complete graph. The matrix $\bbW$ is chosen as follows (based on each of the three graphs). If $(i,j)$ is and edge of the graph, we let $w_{i,j} = 1/(1+\max(d_i, d_j))$. If $(i,j)$ is not an edge and $i,j$ are distinct integers, we have $w_{i,j} = 0$. Finally we let $w_{i,i} = 1 -\sum_{j \in \mathcal{N}} w_{i,j}$. It is not hard to show that the above choice for $\bbW$ satisfies Assumption~\ref{ass:weights}.
Figure~\ref{fig-dist} shows how consensus is reached w.r.t each of the three underlying networks. To measure consensus, we plot the (logarithm of) distance-to-average value $\frac{1}{n} \sum_{i=1}^n || \bbx_i^T - \bar{\bbx}^T ||$ as a function of the total number of iterations $T$ averaged over many trials (see \eqref{average_x} for the definition of $\bar{\bbx}^T$). It is easy to see that the distance to average is small if and only if all the local decisions $\bbx_i^T$ are close to the average decision $\bar{\bbx}^T$. As expected, it takes much less time to reach consensus when the underlying graph is fully connected (i.e. complete graph). For the line graph, the convergence is very slow as this graph has the least degree of connectivity.
\begin{figure}
\caption{The average objective value is plotted as a function of the cardinality constraint $k$ for different choices of the communication graph as well as number of iterations $T$. Note that ``ER" stands for the Erdos-Reny graph with average degree $5$, ``Line" stands for the line graph and ``Complete" is for the complete graph. We have run Algorithm~\ref{algo_DDCG} for $T = 50$ and $T=1000$. }
\label{fig-vals22}
\end{figure}
Figure~\ref{fig-vals22} depicts the obtained objective value of Discrete DCG (Algorithm~\ref{algo_DDCG}) for the three networks considered above. More precisely, we plot the value $\frac{1}{n} \sum_{i=1}^n f(\bbx_i^T)$ obtained at the end of Algorithm~\ref{algo_DDCG} as a function of the cardinality constraint $k$. We also compare these values with the value obtained by the centralized greedy algorithm (i.e. the centralized solution). A few comments are in order. The performance of Algorithm~\ref{algo_DDCG} is close to the centralized solution when the underlying graph is the Erdos-Renyi (with average degree 5) graph or the complete graphs. This is because for both such graphs consensus is achieved from the early stages of the algorithm. By increasing $T$, we see that the performance becomes closer to the centralized solution. However, when the underlying graph is the line graph, then consensus will not be achieved unless the number of iterations is significantly increased. Consequently, for small number of iterations (e.g., $T \leq 1000$) the performance of the algorithm will not be close to the centralized solution. Indeed, this is not surprising as the line graph is poorly~connected.
\section{Conclusion}\label{sec_conclusion}
In this paper, we proposed the first fully decentralized optimization method for maximizing a monotone and continuous DR-submodular function where its components are available at the different nodes of a connected graph. We developed Decentralized Continuous Greedy (DCG) that achieves a $(1-1/e-\eps)$ approximation guarantee with $\mathcal{O}(1/\eps^2)$ local rounds of communication. We also showed that our continuous algorithm can be used to provide the first $(1-1/e)$ tight approximation guarantee for maximizing a monotone submodular set function subject to a general matroid constraint in a decentralized fashion. In particular, we demonstrated that by lifting the local discrete functions to the continuous domain and using DCG as an interface, after $\mathcal{O}(1/\eps^3)$ rounds of communication, each node achieves a tight $(1-1/e-\eps)$ fractional approximate solution. Such solutions can be efficiently rounded in order to obtain discrete solutions with the same approximation guarantee.
\acks{The work of A. Mokhtari was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant CCF-1740425. The work of A. Karbasi was supported by DARPA Young Faculty Award (D16AP00046).}
\section{Appendix}
\subsection{Proof of Proposition~\ref{prop_stay}}\label{app:prop_stay} Define $\bbx_{con}=[\bbx_1;\dots;\bbx_n]\in \reals^{np}$ and $\bbv_{con}=[\bbv_1;\dots;\bbv_n]\in \reals^{np}$ as the concatenation of the local variables and descent directions, respectively. Using these definitions and the update in \eqref{eq:variable_update} we can write
\begin{align}\label{proof_2_100} \bbx_{con}^{t+1} = (\bbW\otimes\bbI)\bbx_{con}^{t}+\frac{1}{T} \bbv_{con}^t, \end{align}
where $ \bbW\otimes\bbI\in \reals^{np\times np}$ is the kronecker product of the matrices $\bbW\in\reals^{n\times n}$ and $\bbI\in\reals^{p\times p}$. If we set $\bbx_i^0=\bb0_p$ for all nodes $i$, it follows that $\bbx_{con}^0=\bb0_{np}$. Hence, by applying the update in \eqref{proof_2_100} recursively we obtain that the iterate $\bbx_{con}^t$ is equal to
\begin{align}\label{proof_2_200} \bbx_{con}^{t} =\frac{1}{T} \sum_{s=0}^{t-1}(\bbW\otimes\bbI)^{t-1-s} \bbv_{con}^s. \end{align}
We proceed by showing that if the local blocks of a vector $\bbv_{con}\in\reals^{np}$ belong to the feasible set $\ccalC$, i.e., $\bbv_i\in\ccalC$ for $i=1,\dots,n$, then the local vectors of $\bby_{con}=(\bbW\otimes\bbI)\bbv_{con}\in\reals^{np}$ also in the set $\ccalC$. Note that if the condition $\bby_{con}=(\bbW\otimes\bbI)\bbv_{con}$ holds, then the $i$-th block of $\bby_{con}=[\bby_1;\dots;\bby_n]$ can be written as
\begin{align}\label{proof_2_400} \bby_i=\sum_{j=1}^nw_{ij}\bbv_{j}. \end{align}
Since we assume that all $\{\bbv_{j}\}_{j=1}^n$ belong to the set $\ccalC$ and the set $\ccalC$ is convex, the weighted average of these vectors also is in the set $\ccalC$, i.e., $\bby_i\in \ccalC$. This argument indeed holds for all blocks $\bby_i$ and therefore $\bby_i\in \ccalC$ for $i=1,\dots,n$. This argument verifies that if we apply any power of the matrix $\bbW\otimes\bbI$ to a vector $\bbv_{con}\in\reals^{np}$ whose blocks belong to the set $\ccalC$, then the local components of the output vector also belong to the set $\ccalC$. Therefore, the local components of each of the terms $(\bbW\otimes\bbI)^{t-1-s} \bbv_{con}^s$ in \eqref{proof_2_200} belong to the set $\ccalC$. The fact that $\bbx_i$ which is the $i$-th block of the vector $\bbx_{con}^t$, is the average of $T$ terms that are in the set $\ccalC$ ($\bbx_{con}^t$ is the average of the vectors $(\bbW\otimes\bbI)^{t-1}\bbv_{con}^0, \dots, (\bbW\otimes\bbI)^{0} \bbv_{con}^{t-1} $ with weights $1/T$ and the vector $\bb0_{np}$ with weight $(T-t)/T$), implies that $\bbx_i^t\in \ccalC$. This result holds for all $i\in\{1,\dots,n\}$ and the proof is complete.
\subsection{Proof of Lemma~\ref{lemma:ar_in_avg_bound}}\label{app:lemma:ar_in_avg_bound} By averaging both sides of the update in \eqref{eq:variable_update} over the nodes in the network and using the fact $w_{ij}=0$ if $i$ and $j$ are not neighbors we can write
\begin{align}\label{proof_1_100} \frac{1}{n}\sum_{i=1}^n \bbx_i^{t+1} &= \frac{1}{n}\sum_{i=1}^n \sum_{j\in \ccalN_i\cup\{i\}} w_{ij}\bbx_{j}^{t}+\frac{1}{T} \frac{1}{n}\sum_{i=1}^n\bbv_i^t\nonumber\\ & = \frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n w_{ij}\bbx_{j}^{t}+\frac{1}{T} \frac{1}{n}\sum_{i=1}^n\bbv_i^t\nonumber\\ & = \frac{1}{n}\sum_{j=1}^n \bbx_{j}^{t} \sum_{i=1}^nw_{ij}+\frac{1}{T} \frac{1}{n}\sum_{i=1}^n\bbv_i^t\nonumber\\ & = \frac{1}{n}\sum_{j=1}^n \bbx_{j}^{t}+\frac{1}{T} \frac{1}{n}\sum_{i=1}^n\bbv_i^t, \end{align}
where the last equality holds since $\bbW^T\bbone_n=\bbone_n$ (i.e. $\bbW$ is a doubly stochastic matrix). By using the definition of the average iterate vector $\bar{\bbx}^t$ and the result in \eqref{proof_1_100} it follows that
\begin{align}\label{proof_1_200} \bar{\bbx}^{t+1} = \bar{\bbx}^{t} +\frac{1}{T} \frac{1}{n}\sum_{i=1}^n\bbv_i^t. \end{align}
Since $\bbv_i^t$ belongs to the convex set $\ccalC$ its Euclidean norm is bounded by $\|\bbv_i^t\|\leq D$ according to Assumption \ref{ass:bounded_set}. This inequality and the expression in \eqref{proof_1_200} yield
\begin{align}
\|\bar{\bbx}^{t+1} - \bar{\bbx}^{t}\| \leq \frac{D}{T}, \end{align}
and the claim in \eqref{eq:var_in_avg_bound} follows.
\subsection{Proof of Lemma~\ref{lemma:eq:bound_on_dif_from_avg}}\label{app:lemma:eq:bound_on_dif_from_avg}
Recall the definitions $\bbx_{con}=[\bbx_1;\dots;\bbx_n]\in \reals^{np}$ and $\bbv_{con}=[\bbv_1;\dots;\bbv_n]\in \reals^{np}$ for the concatenation of the local variables and descent directions, respectively. These definitions along with the update in \eqref{eq:variable_update} lead to the expression
\begin{align}\label{proof_3_100} \bbx_{con}^{t} =\frac{1}{T} \sum_{s=0}^{t-1}(\bbW\otimes\bbI)^{t-1-s} \bbv_{con}^s. \end{align}
If we premultiply both sides of \eqref{proof_3_100} by the matrix $(\frac{\bbone_n\bbone_n^{\dag}}{n}\otimes\bbI) $ which is the kronecker product of the matrices $(1/n)(\bbone_n\bbone_n^{\dag})\in\reals^{n\times n}$ and $\bbI\in\reals^{p\times p}$ we obtain
\begin{align}\label{proof_3_200} \left(\frac{\bbone_n\bbone_n^{\dag}}{n}\otimes \bbI\right) \bbx_{con}^t =\frac{1}{T} \sum_{s=0}^{t-1}\left(\left(\frac{\bbone_n\bbone_n^{\dag}}{n}\bbW^{t-1-s}\right)\otimes\bbI\right) \bbv_{con}^s. \end{align}
The left hand side of \eqref{proof_3_200} can be simplified to
\begin{align}\label{proof_3_300} \left(\frac{\bbone_n\bbone_n^{\dag}}{n}\otimes \bbI\right) \bbx_{con}^t=\bar{\bbx}_{con}^t , \end{align}
where $\bar{\bbx}_{con}^t=[\bar{\bbx}^t;\dots;\bar{\bbx}^t]$ is the concatenation of $n$ copies of the average vector $\bar{\bbx}^t$. Using the equality in \eqref{proof_3_300} and the simplification $\bbone_n\bbone_n^{\dag}\bbW=\bbone_n\bbone_n^{\dag}$, we can rewrite \eqref{proof_3_200} as
\begin{align}\label{proof_3_400} \bar{\bbx}_{con}^t= \frac{1}{T} \sum_{s=0}^{t-1}\left(\frac{\bbone_n\bbone_n^{\dag}}{n}\otimes\bbI\right) \bbv_{con}^s. \end{align}
Using the expressions in \eqref{proof_3_100} and \eqref{proof_3_400} we can derive an upper bound on the difference $ \|\bbx_{con}^{t}-\bar{\bbx}_{con}^t\|$ as
\begin{align}\label{proof_3_500}
\|\bbx_{con}^{t}-\bar{\bbx}_{con}^t\|
& =\frac{1}{T} \left\| \sum_{s=0}^{t-1}\left(\left[ \bbW^{t-1-s}-\frac{\bbone_n\bbone_n^{\dag}}{n} \right]\otimes\bbI\right) \bbv_{con}^s \right\|\nonumber\\
& \leq\frac{1}{T} \sum_{s=0}^{t-1}\left\| \bbW^{t-1-s}-\frac{\bbone_n\bbone_n^{\dag}}{n} \right\|\|\bbv_{con}^s\|\nonumber\\
& \leq\frac{\sqrt{n}D}{T} \sum_{s=0}^{t-1}\left\| \bbW^{t-1-s}-\frac{\bbone_n\bbone_n^{\dag}}{n} \right\|, \end{align}
where the first inequality follows from the Cauchy-Schwarz inequality and
the fact that the norm of a matrix does not change if we kronecker it by the identity matrix, the second inequality holds since $\|\bbv_i^t\|\leq D$ and therefore $\|\bbv_{con}^t\|\leq \sqrt{n}D$. Note that the eigenvectors of the matrices $\bbW$ and $\bbW^{t-s-1}$ are the same for all $s=0,\dots,t-1$. Therefore, the largest eigenvalue of $\bbW^{t-s-1}$ is 1 with eigenvector $\bbone_n$ and its second largest magnitude of the eigenvalues is $\beta^{t-1-s}$, where $\beta$ is the second largest magnitude of the eigenvalues of $\bbW$. Also, note that since $\bbW^{t-1-s}$ has $\bbone_n$ as one of its eigenvectors, then all the other eigenvectors of $\bbW$ are orthogonal to $\bbone_n$. Hence, we can bound the norm $\|\bbW^{t-1-s}-({\bbone_n\bbone_n^{\dag}})/(n)\|$ by $\beta^{t-1-s}$. Applying this substitution into the right hand side of \eqref{proof_3_500} yields
\begin{align}\label{proof_3_600}
\|\bbx_{con}^{t}-\bar{\bbx}_{con}^t\|
\leq\frac{\sqrt{n}D}{T} \sum_{s=0}^{t-1} \beta^{t-1-s} \leq\frac{\sqrt{n}D}{T(1-\beta)}. \end{align}
Since $ \|\bbx_{con}^{t}-\bar{\bbx}_{con}^t\|^2=\sum_{i=1}^n \left\|\bbx_i^t-\bar{\bbx}^t\right\|^2$, the claim in \eqref{eq:bound_on_dif_from_avg} follows.
\subsection{Proof of Lemma \ref{lemma:bound_on_gradient_consensus_error}}\label{proof:lemma:bound_on_gradient_consensus_error}
Recall the definition of the vector $\bbx_{con}=[\bbx_1;\dots;\bbx_n]\in \reals^{np}$ as the concatenation of the local variables, and define $\bbd_{con}=[\bbd_1;\dots;\bbd_n]\in \reals^{np}$ as the concatenation of the local approximate gradients. Further, consider the function $F_{con}:\ccalX^n\to\reals$ which is defined as $F_{con}(\bbx_{con})=F_{con}(\bbx_1,\dots,\bbx_n):=\sum_{i=1}^n F_i(\bbx_i)$. According to these definitions and the update in \eqref{eq:gradient_update}, we can show that
\begin{align}\label{eq:proof_10_1} \bbd_{con}^t = (1-\alpha) (\bbW\otimes\bbI) \bbd_{con}^{t-1}+\alpha \nabla F_{con}(\bbx_{con}^t), \end{align}
where $ \bbW\otimes\bbI\in \reals^{np\times np}$ is the kronecker product of the matrices $\bbW\in\reals^{n\times n}$ and $\bbI\in\reals^{p\times p}$. Considering the initialization $\bbd_{con}^0=\bb0_p$, applying the update in \eqref{eq:proof_10_1} recursively from step $1$ to $t$ leads to
\begin{align}\label{eq:proof_10_2} \bbd_{con}^t=\alpha \sum_{s=1}^{t} (((1-\alpha)\bbW)^{t-s}\otimes \bbI) \nabla F_{con}(\bbx_{con}^s). \end{align}
If we multiply both sides of \eqref{eq:proof_10_2} from left by the matrix $(\frac{\bbone_n\bbone_n^{\dag}}{n}\otimes\bbI) \in \reals^{np\times np}$ and use the properties of the weight matrix $\bbW$, i.e., $\bbone_n^{\dag}\bbW^{t-s}=\bbone_n^{\dag}$, we obtain that
\begin{align}\label{eq:proof_10_3} \bar{\bbd}_{con}^t=\alpha \sum_{s=1}^{t} (1-\alpha)^{t-s}\left(\frac{\bbone_n\bbone_n^{\dag}}{n}\otimes \bbI\right) \nabla F_{con}(\bbx_{con}^s), \end{align}
where $\bar{\bbd}_{con}^t=[\bar{\bbd}^t;\dots;\bar{\bbd}^t]$ is the concatenation of $n$ copies of the average vector $\bar{\bbd}^t$. Hence, the difference $ \|\bbd_{con}^t -\bar{\bbd}_{con}^t\|$ can be upper bounded by
\begin{align}\label{eq:proof_10_4}
\|\bbd_{con}^t -\bar{\bbd}_{con}^t\| &= \alpha \left\| \sum_{s=1}^{t} (1-\alpha)^{t-s}(\bbW^{t-s}\otimes \bbI) \nabla F_{con}(\bbx_{con}^s)- \sum_{s=1}^{t} (1-\alpha)^{t-s}\left[\frac{\bbone_n\bbone_n^{\dag}}{n}\otimes \bbI\right] \nabla F_{con}(\bbx_{con}^s) \right\| \nonumber\\
&= \alpha \left\| \sum_{s=1}^{t}(1-\alpha)^{t-s} \left[(\bbW^{t-s}-\frac{\bbone_n\bbone_n^{\dag}}{n})\otimes \bbI \right] \nabla F_{con}(\bbx_{con}^s) \right\| \nonumber\\ &\leq \alpha \sqrt{n}G \sum_{s=1}^{t}(1-\alpha)^{t-s} \beta^{t-s} \nonumber\\ &\leq \frac{\alpha \sqrt{n} G}{1-\beta(1-\alpha)}, \end{align}
where the first equality is implied by replacing $\bbd_{con}^t$ and $\bar{\bbd}_{con}^t$ with the expressions in \eqref{eq:proof_10_2} and \eqref{eq:proof_10_3}, respectively, the second equality is achieved by regrouping the terms,
the first inequality holds since $\|\nabla F_i(x_i^s)\|\leq G$ and $\|\bbW^{t-s-1}-(\bbone_n\bbone_n^{\dag})/n\|\leq \beta^{t-s-1}$, and finally the last inequality is valid since $\sum_{s=1}^{t}((1-\alpha)\beta)^{t-s}\leq \frac{1}{1-(\beta(1-\alpha))}$. Now considering the result in \eqref{eq:proof_10_4} and the expression $ \|\bbd_{con}^t -\bar{\bbd}_{con}^t\|^2=\sum_{i=1}^n \|\bbd_i^t-\bar{\bbd}^t\|^2$, the claim in \eqref{eq:bound_on_gradient_consensus_error} follows.
\subsection{Proof of Lemma \ref{lemma:bound_on_gradient_error}}\label{proof:lemma:bound_on_gradient_error}
Considering the update in \eqref{eq:gradient_update}, we can write the sum of local ascent directions $\bbd_i^t$ at step $t$ as
\begin{align}\label{app_proof_100} \sum_{i=1}^n \bbd_i^t &= (1-\alpha) \sum_{i=1}^n \sum_{j=1}^n w_{ij}\bbd_{j}^{t-1}+\alpha \sum_{i=1}^n \nabla F_i(\bbx_i^t) \nonumber\\ & = (1-\alpha) \sum_{j=1}^n \bbd_{j}^{t-1} \sum_{i=1}^n w_{ij}+\alpha \sum_{i=1}^n \nabla F_i(\bbx_i^t) \nonumber\\ &= (1-\alpha) \sum_{j=1}^n \bbd_{j}^{t-1} +\alpha \sum_{i=1}^n \nabla F_i(\bbx_i^t), \end{align}
where the last equality holds since $\sum_{i=1}^n w_{ij}=1$ which is the consequence of $\bbW^\dag \bbone_n=\bbone_n$. Now, we use the expression in \eqref{app_proof_100} to bound the difference $\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \|$. Hence,
\begin{align}\label{app_proof_200}
&\left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|\nonumber\\
&=\left\| (1-\alpha) \sum_{j=1}^n \bbd_{j}^{t-1} +\alpha \sum_{i=1}^n \nabla F_i(\bbx_i^t) - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\| \nonumber\\
&=\left\| (1-\alpha) \sum_{j=1}^n \bbd_{j}^{t-1} -(1-\alpha ) \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1}) + (1-\alpha)\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1})+\alpha \sum_{i=1}^n \nabla F_i(\bbx_i^t) - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\| \nonumber\\
&=\Bigg\| (1-\alpha) \!\left[ \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right]\!
+(1-\alpha )\! \left[\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1}) - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t}) \right]\! \nonumber\\ &\qquad +\alpha\! \left[\sum_{i=1}^n \nabla F_i(\bbx_i^t) - \nabla F_i(\bar{\bbx}^{t}) \right]\!
\Bigg\| \nonumber\\
&\leq (1-\alpha) \left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\|
+(1-\alpha ) \left\|\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1}) - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t}) \right\|
\nonumber\\
&\qquad +\alpha \left\|\sum_{i=1}^n \nabla F_i(\bbx_i^t) - \nabla F_i(\bar{\bbx}^{t}) \right\| . \end{align} The first equality is the outcome of replacing $\sum_{i=1}^n \bbd_i^t $ by the expression in \eqref{app_proof_100}, the second equality is obtained by adding and subtracting $(1-\alpha ) \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1})$, in the third equality we regroup the terms, and the inequality follows from applying the triangle inequality twice. Applying the Cauchy--Schwarz inequality to the second and third summands in \eqref{app_proof_200} and using the Lipschitz continuity of the gradients lead to
\begin{align}\label{app_proof_300}
&\left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\| \nonumber\\
&\leq (1-\alpha) \left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\|
+(1-\alpha ) L \sum_{i=1}^n \left\|\bar{\bbx}^{t-1} - \bar{\bbx}^{t}\right\|
+ \alpha L \sum_{i=1}^n \| \bbx_i^t - \bar{\bbx}^{t}\|. \end{align}
According to the result in Lemma \ref{lemma:ar_in_avg_bound}, we can bound the $\sum_{i=1}^n\|\bar{\bbx}^{t+1} - \bar{\bbx}^{t}\| $ by $ {nD}/{T}$. Further, the result in Lemma \ref{lemma:eq:bound_on_dif_from_avg} shows that $(\sum_{i=1}^n \|\bbx_i^t-\bar{\bbx}^t\|^2)^{1/2} \leq \frac{\sqrt{n}D}{T(1-\beta)}.$ Since by the Cauchy--Swartz inequality it holds that $(\sum_{i=1}^n \|\bbx_i^t-\bar{\bbx}^t\|^2)^{1/2}\geq \frac{1}{\sqrt{n}}\sum_{i=1}^n \|\bbx_i^t-\bar{\bbx}^t\|$, it follows that $\sum_{i=1}^n \|\bbx_i^t-\bar{\bbx}^t\|\leq (nD)/(T(1-\beta))$. Applying these substitutions into \eqref{app_proof_300} yields
\begin{align}\label{app_proof_400}
\left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|
\leq (1-\alpha) \left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\| +\frac{(1-\alpha )LnD}{T} + \frac{\alpha LnD}{T(1-\beta)} . \end{align}
By multiplying both of sides of \eqref{app_proof_400} by $1/n$ and applying the resulted inequality recessively for $t$ steps we obtain
\begin{align}\label{app_proof_500}
&\left\|\frac{1}{n}\sum_{i=1}^n \bbd_i^t - \frac{1}{n}\sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|\nonumber\\
&\leq (1-\alpha)^t \left\| \frac{1}{n} \sum_{j=1}^n \bbd_{j}^{0}-\frac{1}{n}\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{0}) \right\| +\left(\frac{(1-\alpha )LD}{T} + \frac{\alpha LD}{T(1-\beta)} \right)\sum_{s=0}^{t-1} (1-\alpha)^s\nonumber\\
&\leq (1-\alpha)^t \frac{1}{n}\sum_{i=1}^n\left\|\nabla F_i(\bar{\bbx}^{0}) \right\| +\frac{(1-\alpha)LD}{\alpha T} + \frac{ LD}{T(1-\beta)} \nonumber\\ &\leq (1-\alpha)^t G +\frac{(1-\alpha)LD}{\alpha T} + \frac{ LD}{T(1-\beta)} , \end{align}
where the second inequality holds since $ \sum_{j=1}^n \bbd_{j}^{0}=\bb0_p$ and $\sum_{s=0}^{t-1} (1-\alpha)^s\leq 1/\alpha$, and the last inequality follows from Assumption \ref{ass:smoothness2}.
\subsection{Proof of Theorem \ref{theorem:main_theorem}}\label{proof:theorem:main_theorem}
Recall the definition of $\bar{\bbx}^{t}=\frac{1}{n}\sum_{i=1}^n \bbx_i^t$ as the average of local variables at step $t$. Since the gradients of the global objective function are $L$-Lipschitz we can write
\begin{align}\label{final_thm_proof_100} \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t})
&\geq \frac{1}{n}\langle \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t}), \bar{\bbx}^{t+1}-\bar{\bbx}^{t} \rangle -\frac{L}{2}\| \bar{\bbx}^{t+1}-\bar{\bbx}^{t}\|^2 \nonumber\\
&= \frac{1}{T}\langle \frac{1}{n}\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t}), \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle -\frac{L}{2T^2}\left\|\frac{1}{n}\sum_{i=1}^n\bbv_i^t \right\|^2, \end{align}
where the equality holds due to the expression in \eqref{proof_1_200}. Note that the term $\|(1/n)\sum_{i=1}^n\bbv_i^t \|^2$ can be upper bounded by $D^2$ according to Assumption \ref{ass:bounded_set}, since $(1/n)\sum_{i=1}^n\bbv_i^t \in \ccalC$ . Apply this substition into \eqref{final_thm_proof_100} and add and subtract $(1/nT)\sum_{i=1}^n \bbd_i^t$ to obtain
\begin{align}\label{final_thm_proof_200} &\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) \nonumber\\ &\geq \frac{1}{T}\langle\frac{1}{n} \sum_{i=1}^n \bbd_i^t, \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle +\frac{1}{T}\langle \frac{1}{n}\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t})-\frac{1}{n}\sum_{i=1}^n \bbd_i^t, \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle -\frac{LD^2}{2T^2}. \end{align}
Now by rewriting the inner product $\langle \sum_{i=1}^n \bbd_i^t,\sum_{i=1}^n\bbv_i^t \rangle $ as $\sum_{i=1}^n\sum_{j=1}^n \langle \bbd_i^t, \bbv_j^t \rangle= \sum_{j=1}^n \langle \sum_{i=1}^n \bbd_i^t, \bbv_j^t \rangle$, we can rewrite the right hand side of \eqref{final_thm_proof_200} as
\begin{align}\label{final_thm_proof_300} &\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) \nonumber\\ &\geq \frac{1}{n^2T} \sum_{j=1}^n \langle \sum_{i=1}^n\bbd_i^t, \bbv_j^t \rangle +\frac{1}{T}\langle\frac{1}{n} \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t})-\frac{1}{n}\sum_{i=1}^n \bbd_i^t, \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle -\frac{LD^2}{2T^2}\nonumber\\ &= \frac{1}{nT} \sum_{j=1}^n \langle \bbd_j^t, \bbv_j^t \rangle +\frac{1}{nT} \sum_{j=1}^n \langle (\frac{1}{n} \sum_{i=1}^n\bbd_i^t-\bbd_j^t), \bbv_j^t \rangle +\frac{1}{T}\langle \sum_{i=1}^n \frac{1}{n}\nabla F_i(\bar{\bbx}^{t})-\frac{1}{n}\sum_{i=1}^n \bbd_i^t, \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle -\frac{LD^2}{2T^2}. \end{align}
Note that in the last step we added and and subtracted $ (1/nT) \sum_{j=1}^n \langle \bbd_j^t, \bbv_j^t \rangle$. Now according to the update in \eqref{eq:descent_update} we can write, $\langle \bbd_j^t, \bbv_j^t \rangle = \max_{\bbv\in \ccalC}\ \langle \bbd_j^t, \bbv\rangle \geq \langle \bbd_j^t, \bbx^*\rangle$. Hence, we can replace $\langle \bbd_j^t, \bbv_j^t \rangle$ by its lower bound $ \langle \bbd_j^t, \bbx^*\rangle$ to obtain
\begin{align} \label{final_thm_proof_400} &\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) \nonumber\\ &\geq \frac{1}{nT} \sum_{j=1}^n \langle \bbd_j^t, \bbx^* \rangle +\frac{1}{nT} \sum_{j=1}^n \langle ( \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t), \bbv_j^t \rangle +\frac{1}{T}\langle \frac{1}{n}\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t})-\frac{1}{n}\sum_{i=1}^n \bbd_i^t, \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle -\frac{LD^2}{2T^2}. \end{align}
Adding and subtracting $\frac{1}{n^2T} \sum_{j=1}^n \langle \sum_{i=1}^n \bbd_i^t, \bbx^* \rangle $ and regrouping the terms lead to
\begin{align} \label{final_thm_proof_500} \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})&-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) \geq \frac{1}{n^2T} \sum_{j=1}^n \langle \sum_{i=1}^n \bbd_i^t, \bbx^* \rangle +\frac{1}{nT} \sum_{j=1}^n \langle \bbd_j^t -\frac{1}{n}\sum_{i=1}^n \bbd_i^t, \bbx^* \rangle \nonumber\\ &\qquad +\frac{1}{nT} \sum_{j=1}^n \langle \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t, \bbv_j^t \rangle +\frac{1}{T}\langle \frac{1}{n}\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t})-\frac{1}{n}\sum_{i=1}^n \bbd_i^t, \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle -\frac{LD^2}{2T^2}. \end{align}
Further add and subtract the expression $\frac{1}{n^2T} \sum_{j=1}^n \langle \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t), \bbx^* \rangle $ and combine the terms to obtain
\begin{align} \label{final_thm_proof_600} &\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) \nonumber\\ &\geq \frac{1}{n^2T} \sum_{j=1}^n \langle \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t), \bbx^* \rangle +\frac{1}{nT} \sum_{j=1}^n \langle ( \frac{1}{n}\sum_{i=1}^n \bbd_i^t - \frac{1}{n} \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t), \bbx^* \rangle +\frac{1}{nT} \sum_{j=1}^n \langle (\bbd_j^t - \frac{1}{n}\sum_{i=1}^n \bbd_i^t, \bbx^* \rangle \nonumber\\ &\qquad +\frac{1}{nT} \sum_{j=1}^n \langle ( \frac{1}{n} \sum_{i=1}^n\bbd_i^t-\bbd_j^t), \bbv_j^t \rangle +\frac{1}{T}\langle \sum_{i=1}^n \frac{1}{n} \nabla F_i(\bar{\bbx}^{t})- \frac{1}{n}\sum_{i=1}^n \bbd_i^t, \frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle -\frac{LD^2}{2T^2}\nonumber\\ &= \frac{1}{nT} \langle \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t), \bbx^* \rangle +\frac{1}{nT} \langle \sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t), \bbx^*-\frac{1}{n}\sum_{i=1}^n\bbv_i^t \rangle \nonumber\\ &\qquad +\frac{1}{nT} \sum_{j=1}^n \langle ( \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t), \bbv_j^t-\bbx^* \rangle -\frac{LD^2}{2T^2}. \end{align}
The monotonicity of the average function $(1/n)\sum_{i=1}^n F_i(\bbx)$ and its concavity along positive directions imply that $\langle (1/n) \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t), \bbx^* \rangle \geq (1/n) \sum_{i=1}^n F_i(\bbx^*)- (1/n) \sum_{i=1}^n F_i(\bar{\bbx}^t)$. By applying this substitution into \eqref{final_thm_proof_600} and using the Cauchy-Schwarz inequality we obtain
\begin{align} \label{final_thm_proof_700} \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) &\geq \frac{1}{nT} \left[ \sum_{i=1}^n F_i(\bbx^*)- \sum_{i=1}^n F_i(\bar{\bbx}^t)\right]
\!-\!\frac{1}{nT} \sum_{j=1}^n \left\| \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t\right\| \| \bbv_j^t-\bbx^* \| \nonumber\\ &\qquad \quad
-\frac{1}{nT} \left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t) \right\| \left\| \bbx^*-\frac{1}{n}\sum_{i=1}^n\bbv_i^t \right\|-\frac{LD^2}{2T^2}. \end{align}
Now we proceed to derive lower bounds for the negative terms on the right hand side of \eqref{final_thm_proof_700}. Note that all $\bbv_i^t$ for $i=1,\dots,n$ belong to the convex set $\ccalC$ and therefore the average vector $\frac{1}{n}\sum_{i=1}^n\bbv_i^t $ is also in the set. Hence, we can bound the difference $\| \bbx^*-\frac{1}{n}\sum_{i=1}^n\bbv_i^t \|$ by $D$ according to Assumption \ref{ass:bounded_set}. Indeed, the norm $\| \bbv_j^t-\bbx^* \|$ is also upper bounded by $D$ and hence we can write
\begin{align} \label{final_thm_proof_800} &\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) \nonumber\\ &\geq \frac{1}{nT} \left[ \sum_{i=1}^n F_i(\bbx^*)- \sum_{i=1}^n F_i(\bar{\bbx}^t)\right]
-\frac{D}{nT} \left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t) \right\|
-\frac{D}{nT} \sum_{j=1}^n \left\| \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t\right\| -\frac{LD^2}{2T^2}. \end{align}
The result in Lemma \ref{lemma:bound_on_gradient_consensus_error} implies that
$(\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|^2)^{1/2} \leq \frac{\alpha \sqrt{n} G}{1-\beta(1-\alpha)}$. Note that based on the Cauchy--Swartz inequality it holds that $(\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|^2)^{1/2}\geq \frac{1}{\sqrt{n}}\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|$, and hence, $\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|\leq \frac{\alpha n G}{1-\beta(1-\alpha)}$. Using this result and recalling the definition $\bar{\bbd}^t:=(1/n)\sum_{i=1}^n\bbd_i^t$, we obtain that
\begin{equation}\label{final_thm_proof_900}
\frac{1}{n} \sum_{j=1}^n \left\| \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t\right\| \leq \frac{\alpha G}{1-\beta(1-\alpha)}. \end{equation}
Replace the term $\frac{1}{n} \sum_{j=1}^n \left\| \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t\right\|$ in \eqref{final_thm_proof_800} by its upper bound in \eqref{final_thm_proof_900} and use the result in Lemma \ref{lemma:bound_on_gradient_error} to replace $\frac{1}{n}\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \|$ by its upper bound in \eqref{eq:bound_on_gradient_error}. Applying these substitutions yields
\begin{align} \label{final_thm_proof_1000} \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) &\geq \frac{1}{T} \left[ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^t)\right] - \frac{(1-\alpha)^t GD}{T} -\frac{(1-\alpha)LD^2}{\alpha T^2} \nonumber\\ & \qquad - \frac{ LD^2}{(1-\beta)T^2} - \frac{\alpha GD}{(1-\beta(1-\alpha))T} -\frac{LD^2}{2T^2}. \end{align}
Set $\alpha=1/\sqrt{T}$ and regroup the terms to obtain
\begin{align} \label{final_thm_proof_1100}
\frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1}) &\leq\left(1- \frac{1}{T}\right) \left[ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^t)\right] + \frac{(1-(1/\sqrt{T}))^t GD}{T}
\nonumber\\ &\qquad +\frac{LD^2}{ T^{3/2}} + \frac{ LD^2}{(1-\beta)T^2} + \frac{ GD}{(1-\beta)T^{3/2}} +\frac{LD^2}{2T^2}. \end{align}
By applying the inequality in \eqref{final_thm_proof_1100} recursively for $t =0,\dots,T-1$ we obtain
\begin{align} \label{final_thm_proof_1200}
\frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{T}) &\leq\left(1- \frac{1}{T}\right)^T \left[ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^0)\right] + \sum_{t=0}^{T-1}\frac{\left(1- 1/\sqrt{T}\right)^t GD}{T}
\nonumber\\ &\qquad +\sum_{t=0}^{T-1}\frac{LD^2}{ T^{3/2}}+\sum_{t=0}^{T-1} \frac{ LD^2}{(1-\beta)T^2} +\sum_{t=0}^{T-1} \frac{ GD}{(1-\beta)T^{3/2}} +\sum_{t=0}^{T-1}\frac{LD^2}{2T^2}. \end{align}
By using the inequality $\sum_{t=0}^{T-1}(1- 1/\sqrt{T})^t\leq \sqrt{T}$ and simplifying the terms on the right hand side \eqref{final_thm_proof_1200} we obtain that to the expression
\begin{align} \label{final_thm_proof_1300} & \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{T}) \nonumber\\ &\leq\frac{1}{e} \left[ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^0)\right] +\frac{ GD}{T^{1/2}} +\frac{LD^2}{ T^{1/2}} + \frac{ LD^2}{(1-\beta)T} +\frac{ GD}{(1-\beta)T^{1/2}} +\frac{LD^2}{2T}\nonumber\\ &=\frac{1}{e} \left[ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^0)\right] +\frac{ LD^2+GD(1+(1-\beta)^{-1})}{T^{1/2}} + \frac{ LD^2(0.5+(1-\beta)^{-1})}{T}, \end{align}
where to derive the first inequality we used $(1-1/T)^T\leq 1/e$. Note that we set $\bbx_i^0=\bb0_p$ for all $i\in\ccalN$ and therefore $\bar{\bbx}^0=\bb0_p$. Since we assume that $F_i(\bb0_p)\geq 0$ for all $i\in\ccalN$, it implies that $\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^0)=\frac{1}{n}\sum_{i=1}^n F_i(\bb0_p)\geq 0$ and the expression in \eqref{final_thm_proof_1300} can be simplified to
\begin{align} \label{final_thm_proof_1400} \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^T)\geq (1-e^{-1} ) \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*) - \frac{ LD^2+GD(1+(1-\beta)^{-1})}{T^{1/2}} - \frac{ LD^2(0.5+(1-\beta)^{-1})}{T} . \end{align}
Also, since the norm of local gradients is uniformly bounded by $G$, the local functions $F_i$ are $G$-Lipschitz. This observation implies that \begin{align} \label{final_thm_proof_1500}
\left|\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^T)-\frac{1}{n}\sum_{i=1}^n F_i(\bbx_j^T)\right| \leq \frac{G}{n}\sum_{i=1}^n\|\bar{\bbx}^T-\bbx_j^T\| \leq \frac{GD}{T(1-\beta)}, \end{align}
where the second inequality holds by using the result in Lemma \ref{lemma:eq:bound_on_dif_from_avg} and the Cauchy-Schwartz inequality. Therefore, by combining the results in \eqref{final_thm_proof_1400} and \eqref{final_thm_proof_1500} we obtain that for all $j=\ccalN$ \begin{align} \frac{1}{n}\sum_{i=1}^n F_i(\bbx_j^T) &\geq (1-e^{-1} ) \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*) - \frac{ LD^2+GD(1+(1-\beta)^{-1})}{T^{1/2}}\nonumber\\ &\qquad - \frac{ GD(1-\beta)^{-1}+LD^2(0.5+(1-\beta)^{-1})}{T}, \end{align}
and the claim in \eqref{local_node_bound} follows.
\subsection{How to Construct an Unbiased Estimator of the Gradient in Multilinear Extensions}\label{unbiased}
In this section, we provide an unbiased estimator for the gradient of a multilinear extension. We thus consider an arbitrary submodular set function $h: 2^V \to \mathbb{R}$ with multilinear $H$. Our goal is to provide an unbiased estimator for $\nabla H(\bbx)$. We have $H(\bbx) = \sum_{S\subseteq V} \prod_{i\in S}x_i\prod_{j\not\in S} (1-x_j) h(S)$. Now, it can easily be shown that $$\frac{\partial H}{\partial x_i} = H(\bbx;x_i\leftarrow 1) - H(\bbx;x_i\leftarrow 0).$$ where for example by $(\bbx;x_i\leftarrow 1)$ we mean a vector which has value $1$ on its $i$-th coordinate and is equal to $\bbx$ elsewhere. To create an unbiased estimator for $\frac{\partial H}{\partial x_i} $ at a point $\bbx$ we can simply sample a set $S$ by including each element in it independently with probability $x_i$ and use $h(S \cup \{i\}) - h(S \setminus \{i\})$ as an unbiased estimator for the $i$-th partial derivative. We can sample one single set $S$ and use the above trick for all the coordinates. This involves $n$ function computations for $h$. Having a mini-batch size $B$ we can repeat this procedure $B$ times and then average.
Note that since every element of the unbiased estimator is of the form $h(S \cup \{i\}) - h(S \setminus \{i\})$ for some chosen set $S$, then due to submodularity of the function $h$ every element of the unbiased estimator is bounded above by the maximum marginal value of $h$ (i.e. $\max_{i \in V}$ h(\{i\})). As a result, the norm of the unbiased estimator (of the gradient of $H$) is bounded above by $\sqrt{|V|} \max_{i \in V} h(\{i\})$.
\subsection{Proof of Theorem \ref{theorem:main_theorem_discrete}}\label{proof:theorem:main_theorem_discrete}
The steps of the proof are similar to the one for Theorem 1. In particular, for the Discrete DCG method we can also show that the expressions in \eqref{final_thm_proof_100}-\eqref{final_thm_proof_800} hold and we can write
\begin{align} \label{final_thm_discrete_proof_100} &\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) \nonumber\\ &\geq \frac{1}{nT} \left[ \sum_{i=1}^n F_i(\bbx^*)- \sum_{i=1}^n F_i(\bar{\bbx}^t)\right]
-\frac{D}{nT} \left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t) \right\|
-\frac{D}{nT} \sum_{j=1}^n \left\| \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t\right\| -\frac{LD^2}{2T^2}. \end{align}
Now we proceed to derive upper bounds for the norms on the right hand side of \eqref{final_thm_discrete_proof_100}. To derive these bounds we use the results in Lemmata~\ref{lemma:ar_in_avg_bound} and \ref{lemma:eq:bound_on_dif_from_avg} which also hold for the Discrete DCG algorithm.
We first derive an upper bound for the sum $ \sum_{j=1}^n \| \frac{1}{n}\sum_{i=1}^n\bbd_i^t-\bbd_j^t\| $ in \eqref{final_thm_discrete_proof_100}. To achieve this goal the following lemma is needed.
\begin{lemma}\label{lemma:bound_on_g}
Consider the proposed Discrete DCG method defined in Algorithm \ref{algo_DDCG}. If Assumptions \ref{ass:smoothness2} and \ref{ass:bounded_variance} hold, then for all $i\in \ccalN$ and $t\geq0$ the expected squared norm $\E{\| \bbg_i^t\|^2}$ is bounded above by \begin{equation}\label{bound_on_g}
\E{\| \bbg_i^t\|^2} \leq K^2, \end{equation} where $K^2=\sigma^2+G^2$. \end{lemma}
\begin{myproof} Considering the condition in Assumption \ref{ass:bounded_variance} on the variance of stochastic gradients, we can define $K^2:=\sigma^2+G^2$ as an upper bound on the expected norm of stochastic gradients, i.e., for all $\bbx\in\ccalC$ and $i\in \ccalN$
\begin{equation} \label{final_thm_discrete_proof_300}
\E{\|\nabla \tilde{F}_i(\bbx_i^t)\|^2} \leq K^2. \end{equation}
Now we use an induction argument to show that the expected norm $\E{\|\bbg_{i}^t\|^2}\leq K^2$.
Since the iterates are initialized at $\bbg_{i}^0=\bb0$, the update in \eqref{gradient_approx_update} implies that $\E{\|\bbg_{i}^1\|^2\mid \bbx_i^1}= \phi^2 \E{\|\nabla \tilde{F}_i(\bbx_i^1)\|^2 \mid \bbx_i^1}\leq \phi^2 K^2\leq K^2$. Since $\E{\E{\|\bbg_{i}^1\|^2\mid \bbx_i^1}}=\E{\|\bbg_{i}^1\|^2}$ it follows that $\E{\|\bbg_{i}^1\|^2}\leq K^2$. Now we proceed to show that if $\E{\|\bbg_{i}^{t-1}\|^2}\leq K^2$ then $\E{\|\bbg_{i}^t\|^2}\leq K^2$.
Recall the update of $\bbg_i^t$ in \eqref{gradient_approx_update}. By computing the squared norm of both sides and using the Cauchy-Schwartz inequality we obtain that
\begin{align}
\|\bbg_{i}^t\|^2 \leq (1-\phi)^2 \|\bbg_{i}^{t-1}\|^2 + \phi^2 \| \nabla \tilde{F}_i(\bbx_i^t)\|^2 + 2\phi(1-\phi)\|\bbg_{i}^{t-1}\| \| \nabla \tilde{F}_i(\bbx_i^t)\|. \end{align}
Compute the expectation with respect to the random variable corresponding to the stochastic gradient $\nabla \tilde{F}_i(\bbx_i^t)$ to obtain
\begin{align}\label{sdaasdad}
\E{\|\bbg_{i}^t\|^2\mid \bbx_i^t} \leq (1-\phi)^2 \|\bbg_{i}^{t-1}\|^2 + \phi^2 \E{\| \nabla \tilde{F}_i(\bbx_i^t)\|^2 \mid \bbx_i^t}+ 2\phi(1-\phi)\|\bbg_{i}^{t-1}\|\E{ \| \nabla \tilde{F}_i(\bbx_i^t)\|\mid \bbx_i^t}. \end{align}
Note that according to Jensen's inequality $\E{\|\nabla \tilde{F}_i(\bbx_i^t)\|^2} \leq K^2$ implies that $\E{\|\nabla \tilde{F}_i(\bbx_i^t)\|} \leq K$. Replacing these bounds into \eqref{sdaasdad} yields
\begin{align}
\E{\|\bbg_{i}^t\|^2\mid \bbx_i^t} \leq (1-\phi)^2 \|\bbg_{i}^{t-1}\|^2 + \phi^2K^2+ 2K\phi(1-\phi)\|\bbg_{i}^{t-1}\|. \end{align}
Now by computing the expectation of both sides with respect to all sources of randomness from $t=0$ and using the simplification $\E{\E{\|\bbg_{i}^t\|^2\mid \bbx_i^t}}=\E{\|\bbg_{i}^t\|^2}$ we can write
\begin{align}
\E{\|\bbg_{i}^t\|^2\mid }
& \leq (1-\phi)^2 \E{\|\bbg_{i}^{t-1}\|^2} + \phi^2K^2+ 2K\phi(1-\phi)\E{\|\bbg_{i}^{t-1}\|}\nonumber\\ &\leq (1-\phi)^2 K^2+ \phi^2K^2+ 2K\phi(1-\phi)K \nonumber\\ &= K^2, \end{align}
and the claim in \eqref{bound_on_g} follows by induction. \end{myproof}
We use the result in Lemma \ref{lemma:bound_on_g} to find an upper bound for the sum $(1/n)\sum_{j=1}^n \left\| \bar{\bbd}^t-\bbd_j^t\right\| $ on the right hand side of \eqref{final_thm_discrete_proof_100}.
\begin{lemma}\label{lemma:bound_on_d} Consider the proposed Discrete DCG method defined in Algorithm \ref{algo_DDCG}. If Assumptions \ref{ass:weights}, \ref{ass:smoothness2} and \ref{ass:bounded_variance} hold, then for all $i\in \ccalN$ and $t\geq0$ we have \begin{equation}\label{bound_on_d}
\E{\frac{1}{n}\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|}\leq \frac{\alpha K}{1-\beta(1-\alpha)}, \end{equation} where $K=(\sigma^2+G^2)^{1/2}$. \end{lemma}
\begin{myproof} Define the vector $\bbg_{con}^t=[\bbg_1^t;\dots;\bbg_n^t]$ as the concatenation of the local vectors $\bbg_i^t$ at time $t$. Further, recall the definitions of the vectors $\bbx_{con}=[\bbx_1;\dots;\bbx_n]\in \reals^{np}$ and $\bbd_{con}=[\bbd_1;\dots;\bbd_n]\in \reals^{np}$ as the concatenation of the local variables and local approximate gradients, respectively, and the definition of $\bar{\bbd}_{con}^t=[\bar{\bbd}^t;\dots;\bar{\bbd}^t]$ as the concatenation of $n$ copies of the average vector $\bar{\bbd}^t$. By following the steps of the proof for Lemma~\ref{lemma:bound_on_gradient_consensus_error}, it can be shown that
\begin{align}\label{final_thm_discrete_proof_500}
\|\bbd_{con}^t -\bar{\bbd}_{con}^t\| &= \left\| \alpha \sum_{s=1}^{t}(1-\alpha)^{t-s} \left[(\bbW^{t-s}-\frac{\bbone_n\bbone_n^{\dag}}{n})\otimes \bbI \right] \bbg_{con}^t \right\| \nonumber\\
&\leq \alpha \sum_{s=1}^{t}(1-\alpha)^{t-s} \left\|(\bbW^{t-s}-\frac{\bbone_n\bbone_n^{\dag}}{n})\otimes \bbI \right\| \left\| \bbg_{con}^t\right\| \nonumber\\
&\leq \alpha \sum_{s=1}^{t}(1-\alpha)^{t-s} \beta^{t-s}\left\| \bbg_{con}^t\right\| . \end{align}
By computing the expected value of both sides and using the result in \eqref{bound_on_g} we obtain that
\begin{align}\label{final_thm_discrete_proof_500}
\E{\|\bbd_{con}^t -\bar{\bbd}_{con}^t\|} &\leq \alpha \sqrt{n}K \sum_{s=1}^{t}(1-\alpha)^{t-s} \beta^{t-s} \nonumber\\ &\leq \frac{\alpha \sqrt{n} K}{1-\beta(1-\alpha)}, \end{align}
where to derive the first inequality we use the fact that \begin{equation}
\E{\| \bbg_{con}^t\| }\leq(\E{\| \bbg_{con}^t\|^2})^{1/2}=(\E{(\sum_{i=1}^n \|\bbg_i^t\|^2)})^{1/2}=(\sum_{i=1}^n\E{ \|\bbg_i^t\|^2})^{1/2}\leq \sqrt{n}K. \end{equation} By combining the result in \eqref{final_thm_discrete_proof_500} with the inequality \begin{equation}
\frac{1}{n}\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|\leq \frac{1}{\sqrt{n}} \left[\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\|^2\right]^{1/2} =\frac{1}{\sqrt{n}} \|\bbd_{con}^t -\bar{\bbd}_{con}^t\|, \end{equation} the claim in \eqref{bound_on_d} follows. \end{myproof}
The result in Lemma \ref{lemma:bound_on_d} shows $\frac{1}{n}\sum_{i=1}^n\|\bbd_i^t-\bar{\bbd}^t\| $ is bounded above by $(\alpha K)/({1-\beta(1-\alpha)})$ in expectation. To bound the second sum in \eqref{final_thm_discrete_proof_100}, which is$ \left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^t) \right\| $, we first introduce the following lemma, which was presented in \citep{mokhtari2017conditional} in a slightly different form.
\begin{lemma}\label{lemma:bound_on_stoc} Consider the proposed Discrete DCG method defined in Algorithm \ref{algo_DDCG}. If Assumptions \ref{ass:weights}-\ref{ass:bounded_variance} hold and we set $\phi=1/T^{2/3}$, then for all $i\in \ccalN$ and $t\geq0$ we have \begin{align}\label{bound_on_stoch}
\E{\sum_{i=1}^n\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2}
&\leq
\left(1-\frac{1}{2T^{2/3}}\right)^tnG^2
+ \frac{6nL^2D^2C}{T^{4/3}}
+\frac{2n\sigma^2+12nL^2D^2C}{T^{2/3}}, \end{align} where $C:=1+({2}/{(1-\beta)^2})$. \end{lemma}
\begin{myproof}
Use the update $\bbg_i^t := (1-\phi) \bbg_i^{t-1} + \phi \nabla \tilde{F}(\bbx_i^t)$ to write the squared norm $\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2$ as
\begin{align}\label{proof:bound_on_grad_100}
\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2
=\|\nabla F_i(\bbx_i^t) -(1-\phi) \bbd_{t-1} - \phi \nabla \tilde{F}_i(\bbx_i^t)\|^2. \end{align}
Add and subtract the term $(1-\phi)\nabla F_i(\bbx_i^{t-1})$ to the right hand side of \eqref{proof:bound_on_grad_100} and regroup the terms to obtain
\begin{align}\label{proof:bound_on_grad_200}
&\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2 \nonumber\\
&=\|\phi(\nabla F_i(\bbx_i^t)-\nabla \tilde{F}_i(\bbx_i^t)) +(1-\phi)(\nabla F_i(\bbx_i^t)-\nabla F_i(\bbx_i^{t-1}))
+(1-\phi)(\nabla F_i(\bbx_i^{t-1}) - \bbg_i^{t-1} )\|^2. \end{align}
Define $\ccalF^t$ as a sigma algebra that measures the history of the system up until time $t$. Expanding the square and computing the conditional expectation $\E{\cdot\mid \ccalF^t}$ of the resulted expression yield
\begin{align}\label{proof:bound_on_grad_300}
&\E{\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2\mid\ccalF^t} =\phi^2\E{\|\nabla F_i(\bbx_i^t)-\nabla \tilde{F}_i(\bbx_i^t)\|^2\mid\ccalF^t}+ (1-\phi)^2\|\nabla F_i(\bbx_i^{t-1}) - \bbg_i^{t-1} \|^2
\nonumber\\
& \quad
+(1-\phi)^2\|\nabla F_i(\bbx_i^t)-\nabla F_i(\bbx_i^{t-1})\|^2
+ 2 (1-\phi)^2\langle \nabla F_i(\bbx_i^t)-\nabla F_i(\bbx_i^{t-1}) , \nabla F_i(\bbx_i^{t-1}) - \bbg_i^{t-1} \rangle,
\end{align}
where we have used the fact $\E{\nabla \tilde{F}_i(\bbx_i^t) \mid\ccalF^t}=\nabla F_i(\bbx_i^t) $.
The term $\E{\|\nabla F_i(\bbx_i^t)-\nabla \tilde{F}_i(\bbx_i^t)\|^2\mid\ccalF^t}$ can be bounded above by $\sigma^2$ according to Assumption \ref{ass:bounded_variance}. Based on Assumption~\ref{ass:smoothness}, we can also show that the squared norm $\|\nabla F_i(\bbx_i^t)-\nabla F_i(\bbx_i^{t-1})\|^2$ is upper bounded by $L^2\|\bbx_i^t-\bbx_i^{t-1}\|^2$. Moreover, the inner product $2\langle \nabla F_i(\bbx_i^t)\!-\!\nabla F_i(\bbx_i^{t-1}) , \nabla F_i(\bbx_i^{t-1}) - \bbd_{t-1} \rangle$ can be upper bounded by $\zeta \|\nabla F_i(\bbx_i^{t-1}) - \bbd_{t-1}\|^2+(1/\zeta) L^2\|\bbx_i^t-\bbx_i^{t-1}\|^2$ using Young's inequality (i.e., $2\langle \bba,\bbb\rangle \leq \zeta\|\bba\|^2+\|\bbb\|^2/\beta$ for any $\bba,\bbb\in \reals^n$ and $\zeta>0$) and the condition in Assumption \ref{ass:smoothness}, where $\zeta>0$ is a free scalar. Applying these substitutions into \eqref{proof:bound_on_grad_300} leads to
\begin{align}\label{proof:bound_on_grad_400}
&\E{\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2\mid\ccalF^t} \nonumber\\ & \leq \phi^2\sigma^2
+(1-\phi)^2 (1+\zeta^{-1})L^2\|\bbx_i^t-\bbx_i^{t-1}\|^2
+(1-\phi)^2(1+\zeta)\|\nabla F_i(\bbx_i^{t-1}) - \bbg_i^{t-1} \|^2.
\end{align}
By setting $\zeta=\phi/2$ we can replace $(1-\phi)^2 (1+\zeta^{-1})$ and $(1-\phi)^2(1+\zeta)$ by their upper bounds $(1+2\phi^{-1})$ and $(1-\phi/2)$, respectively. Applying theses substitutions and summing up both sides of the resulted inequality for $i=1,\dots,n$ lead to
\begin{align}\label{proof:bound_on_grad_401}
&\E{\sum_{i=1}^n\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2\mid\ccalF^t} \nonumber\\ & \leq n\phi^2\sigma^2
+ L^2(1+2\phi^{-1}) \sum_{i=1}^n\|\bbx_i^t-\bbx_i^{t-1}\|^2
+\left(1-\frac{\phi}{2}\right)\sum_{i=1}^n\|\nabla F_i(\bbx_i^{t-1}) - \bbg_i^{t-1}\|^2.
\end{align}
Now we proceed to derive an upper bound for the sum $\sum_{i=1}^n\|\bbx_i^t-\bbx_i^{t-1}\|^2$. Note that using the Cauchy-Schwartz inequality and the results in Lemmata~\ref{lemma:ar_in_avg_bound} and \ref{lemma:eq:bound_on_dif_from_avg} we can show that
\begin{align}\label{proof:bound_on_grad_402}
\sum_{i=1}^n\|\bbx_i^t-\bbx_i^{t-1}\|^2
&\leq \sum_{i=1}^n\left(3 \left\|\bbx_i^t-\bar{\bbx}^t\right\|^2 +3 \left\|\bar{\bbx}^t-\bar{\bbx}^{t-1}\right\|^2+3\left\|\bar{\bbx}^{t-1}-{\bbx}_i^{t-1}\right\|^2\right)\nonumber\\ &\leq \frac{3nD^2}{T^2(1-\beta)^2} + \frac{3nD^2}{T^2}+\frac{3nD^2}{T^2(1-\beta)^2}\nonumber\\ &= \frac{3nD^2}{T^2}\left(1+\frac{2}{(1-\beta)^2}\right). \end{align}
Replace the sum $ \sum_{i=1}^n\|\bbx_i^t-\bbx_i^{t-1}\|^2$ in \eqref{proof:bound_on_grad_401} by its upper bound in \eqref{proof:bound_on_grad_402} and compute the expectation with respect to $\ccalF_0$ to obtain
\begin{align}\label{proof:bound_on_grad_800}
&\E{\sum_{i=1}^n\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2}\nonumber\\ &\leq
\left(1-\frac{\phi}{2}\right)\E{\sum_{i=1}^n\|\nabla F_i(\bbx_i^{t-1}) - \bbg_i^{t-1} \|^2}
+n\phi^2\sigma^2
+ (1+2\phi^{-1})\frac{3nL^2D^2}{T^2}\left(1+\frac{2}{(1-\beta)^2}\right). \end{align}
Set $\phi=T^{-2/3}$ to obtain
\begin{align}\label{proof:bound_on_grad_900}
&\E{\sum_{i=1}^n\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2}\nonumber\\ &\leq
\left(1-\frac{1}{2T^{2/3}}\right)\E{\sum_{i=1}^n\|\nabla F_i(\bbx_i^{t-1}) - \bbg_i^{t-1} \|^2}
+\frac{n\sigma^2}{T^{4/3}}
+ \frac{3nL^2D^2C}{T^2}
+\frac{6nL^2D^2C}{T^{4/3}}, \end{align}
where $C:=\left(1+\frac{2}{(1-\beta)^2}\right)$. Applying the expression in \eqref{proof:bound_on_grad_900} recursively leads to
\begin{align}\label{proof:bound_on_grad_1000}
&\E{\sum_{i=1}^n\|\nabla F_i(\bbx_i^t) - \bbg_i^t\|^2}\nonumber\\ &\leq
\left(1-\frac{1}{2T^{2/3}}\right)^t\sum_{i=1}^n\|\nabla F_i(\bbx_i^{0}) - \bbd_{0} \|^2 +\left(\frac{n\sigma^2}{T^{4/3}}
+ \frac{3nL^2D^2C}{T^2}
+\frac{6nL^2D^2C}{T^{4/3}}\right) \sum_{s=0}^{t-1} \left(1-\frac{1}{2T^{2/3}}\right)^s\nonumber\\
&\leq
\left(1-\frac{1}{2T^{2/3}}\right)^t\sum_{i=1}^n\|\nabla F_i(\bbx_i^{0}) - \bbd_{0} \|^2 +\frac{2n\sigma^2}{T^{2/3}}
+ \frac{6nL^2D^2C}{T^{4/3}}
+\frac{12nL^2D^2C}{T^{2/3}}\nonumber\\
&\leq
\left(1-\frac{1}{2T^{2/3}}\right)^tnG^2 +\frac{2n\sigma^2}{T^{2/3}}
+ \frac{6nL^2D^2C}{T^{4/3}}
+\frac{12nL^2D^2C}{T^{2/3}}, \end{align}
and the claim in \eqref{bound_on_stoch} follows. \end{myproof}
We use the result in Lemma \ref{lemma:bound_on_stoc} to derive an upper bound for $\left\|\frac{1}{n}\sum_{i=1}^n \bbd_i^t - \frac{1}{n}\sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|$ in expectation.
\begin{lemma}\label{lemma:bound_on_whatever} Consider the proposed Discrete DCG method defined in Algorithm \ref{algo_DDCG}. If Assumptions \ref{ass:weights}-\ref{ass:bounded_variance} hold and we set $\alpha= 1/\sqrt{T}$ and $\phi=1/T^{2/3}$, then for all $i\in \ccalN$ and $t\geq0$ we have \begin{align}\label{bound_on_whatever}
\E{\left\|\frac{1}{n}\sum_{i=1}^n \bbd_i^t - \frac{1}{n}\sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|} & \leq G\left(1-\frac{1}{T^{1/2}}\right)^t
+G\left(1-\frac{1}{2T^{2/3}}\right)^{t/2}
+\frac{LD}{ T^{1/2}} \nonumber\\ &\qquad + \frac{ LD}{T(1-\beta)}
+ \frac{\sqrt{6} LDC^{1/2}}{T^{2/3}}
+\frac{\sqrt{2}\sigma+\sqrt{12} LDC^{1/2}}{T^{1/3}}, \end{align} where $C:=1+({2}/{(1-\beta)^2})$. \end{lemma}
\begin{myproof} The steps of this proof are similar to the ones in the proof of Lemma \ref{lemma:bound_on_gradient_error}. It can be shown that
\begin{align}\label{a}
&\left\|\sum_{i=1}^n \bbd_i^t -\sum_{i=1}^n\nabla F_i(\bar{\bbx}^t)\right\|\nonumber\\
&=\left\| (1-\alpha) \sum_{j=1}^n \bbd_{j}^{t-1} +\alpha \sum_{i=1}^n \bbg_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\| \nonumber\\
&=\left\| (1-\alpha) \sum_{j=1}^n \bbd_{j}^{t-1} -(1-\alpha ) \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1}) + (1-\alpha)\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1})+\alpha \sum_{i=1}^n \bbg_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\| \nonumber\\
&=\left\| (1-\alpha) \left[ \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right]
+(1-\alpha ) \left[\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1}) - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t}) \right]
+\alpha \left[\sum_{i=1}^n \bbg_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t}) \right]
\right\| \nonumber\\
&\leq (1-\alpha) \left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\|
+(1-\alpha ) \left\|\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1}) - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t}) \right\|
+\alpha \left\|\sum_{i=1}^n \bbg_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t}) \right\| \nonumber\\
&\leq (1-\alpha) \left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\|
+(1-\alpha ) \left\|\sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1}) - \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t}) \right\|
+\alpha \left\|\sum_{i=1}^n \bbg_i^t - \sum_{i=1}^n \nabla F_{i}(\bbx_i^t) \right\| \nonumber\\
&\qquad +\alpha \left\|\sum_{i=1}^n \nabla F_{i}(\bbx_i^t) - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t}) \right\|. \end{align} The first equality is the outcome of replacing $\sum_{i=1}^n \bbd_i^t $ by the expression in \eqref{app_proof_100}, the second equality is obtained by adding and subtracting $(1-\alpha ) \sum_{i=1}^n \nabla F_i(\bar{\bbx}^{t-1})$, in the third equality we regroup the terms, and the inequality follows from applying the triangle inequality twice. Applying the Cauchy--Schwarz inequality to the second and third summands in \eqref{app_proof_200} and using the Lipschitz continuity of the gradients lead to
\begin{align}\label{app}
\left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|
&\leq (1-\alpha) \left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\|
+(1-\alpha ) L \sum_{i=1}^n \left\|\bar{\bbx}^{t-1} - \bar{\bbx}^{t}\right\| \nonumber\\
&\qquad + \alpha L \sum_{i=1}^n \| \bbx_i^t - \bar{\bbx}^{t}\| +\alpha \left\|\sum_{i=1}^n \bbg_i^t -\sum_{i=1}^n \nabla F_{i}(\bbx_i^t) \right\|\nonumber\\
&\leq (1-\alpha) \left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\| +\frac{(1-\alpha )LnD}{T} \nonumber\\ &\qquad
+ \frac{\alpha LnD}{T(1-\beta)} +\alpha\sum_{i=1}^n \left\| \bbg_i^t - \nabla F_{i}(\bbx_i^t) \right\|, \end{align}
where the last inequality follows from Lemmata~\ref{lemma:ar_in_avg_bound} and \ref{lemma:eq:bound_on_dif_from_avg}. Using the inequality \begin{align}
\frac{1}{\sqrt{n}} \E{\sum_{i=1}^n \left\| \bbg_i^t - \nabla F_{i}(\bbx_i^t) \right\|}
\leq \E{\left(\sum_{i=1}^n \left\| \bbg_i^t - \nabla F_{i}(\bbx_i^t) \right\|^2\right)^{1/2}}
\leq \left(\E{\sum_{i=1}^n \left\| \bbg_i^t - \nabla F_{i}(\bbx_i^t) \right\|^2}\right)^{1/2}, \end{align}
and the result in Lemma \ref{lemma:bound_on_whatever} we obtain that
\begin{align}\label{appppp}
\E{\sum_{i=1}^n \left\| \bbg_i^t - \nabla F_{i}(\bbx_i^t) \right\|} &\leq \sqrt{n} \left[ \left(1-\frac{1}{2T^{2/3}}\right)^tnG^2 +\frac{2n\sigma^2}{T^{2/3}}
+ \frac{6nL^2D^2C}{T^{4/3}}
+\frac{12nL^2D^2C}{T^{2/3}}\right]^{1/2}\nonumber\\
&\leq nG\left(1-\frac{1}{2T^{2/3}}\right)^{t/2} +\frac{\sqrt{2}n\sigma}{T^{1/3}}
+ \frac{\sqrt{6}nLDC^{1/2}}{T^{2/3}}
+\frac{\sqrt{12}nLDC^{1/2}}{T^{1/3}}, \end{align}
where the second inequality holds since $\sum_{i}a_i^2 \leq(\sum_{i}a_i)^2 $ for $a_i\geq0$. Compute the expected value of both sides of \eqref{app} and replace $ \E{\sum_{i=1}^n \left\| \bbg_i^t - \nabla F_{i}(\bbx_i^t) \right\|}$ by its upper bound in \eqref{appppp} to obtain
\begin{align}\label{app2222}
\E{\left\|\sum_{i=1}^n \bbd_i^t - \sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|}
&\leq (1-\alpha) \E{\left\| \sum_{j=1}^n \bbd_{j}^{t-1}-\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{t-1}) \right\| } +\frac{(1-\alpha )LnD}{T} + \frac{\alpha LnD}{T(1-\beta)} \nonumber\\ & \quad +\alpha nG\left(1-\frac{1}{2T^{2/3}}\right)^{t/2} \!\!
+ \frac{\sqrt{6}\alpha nLDC^{1/2}}{T^{2/3}}
+\frac{\sqrt{2}n\alpha\sigma+\sqrt{12}\alpha nLDC^{1/2}}{T^{1/3}}. \end{align}
By multiplying both of sides of \eqref{app_proof_400} by $1/n$ and applying the resulted inequality recessively for $t$ steps we obtain
\begin{align}\label{app_pro}
&\E{\left\|\frac{1}{n}\sum_{i=1}^n \bbd_i^t - \frac{1}{n}\sum_{i=1}^n\nabla F_i(\bar{\bbx}^t) \right\|}\nonumber\\
&\leq (1-\alpha)^t \left\| \frac{1}{n} \sum_{j=1}^n \bbd_{j}^{0}-\frac{1}{n}\sum_{i=1}^n\nabla F_i(\bar{\bbx}^{0}) \right\| \nonumber\\ &\ + \!\left[\frac{(1\!-\!\alpha )LD}{T} + \frac{\alpha LD}{T(1\!-\!\beta)} + \alpha G\left[1-\frac{1}{2T^{2/3}}\right]^{t/2} \!\!\!
+ \frac{\sqrt{6}\alpha LDC^{1/2}}{T^{2/3}}
+\frac{\sqrt{2}\alpha \sigma\!+\!\sqrt{12}\alpha LDC^{1/2}}{T^{1/3}}\right]\sum_{s=0}^{t-1} (1-\alpha)^s\nonumber\\
&\leq (1-\alpha)^t \frac{1}{n}\sum_{i=1}^n\left\|\nabla F_i(\bar{\bbx}^{0}) \right\| +\frac{(1-\alpha)LD}{\alpha T} + \frac{ LD}{T(1-\beta)} +G\left(1-\frac{1}{2T^{2/3}}\right)^{t/2} \nonumber\\ &\qquad
+ \frac{\sqrt{6} LDC^{1/2}}{T^{2/3}}
+\frac{\sqrt{2}\sigma+\sqrt{12} LDC^{1/2}}{T^{1/3}}\nonumber\\ &\leq \left(1-\frac{1}{T^{1/2}}\right)^t G +\frac{(1-\alpha)LD}{\alpha T} + \frac{ LD}{T(1-\beta)} +G\left(1-\frac{1}{2T^{2/3}}\right)^{t/2} \nonumber\\ &\qquad + \frac{\sqrt{6} LDC^{1/2}}{T^{2/3}}
+\frac{\sqrt{2}\sigma+\sqrt{12} LDC^{1/2}}{T^{1/3}}, \end{align}
which yields the claim in \eqref{bound_on_whatever}. \end{myproof}
Now we can complete the proof of Theorem \ref{theorem:main_theorem_discrete} using the results in Lemmata \ref{lemma:bound_on_d} and \ref{lemma:bound_on_whatever} as well as the expression in \eqref{final_thm_discrete_proof_100}. Replace the terms on the right hand side of \eqref{final_thm_discrete_proof_100} by their upper bounds in Lemmata \ref{lemma:bound_on_d} and \ref{lemma:bound_on_whatever} to obtain
\begin{align} \label{final_thm_discrete_proof_1001} &\E{\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1})-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t}) } \nonumber\\ &\geq \E{ \frac{1}{nT} \left[ \sum_{i=1}^n F_i(\bbx^*)- \sum_{i=1}^n F_i(\bar{\bbx}^t)\right]} -\left(1-\frac{1}{T^{1/2}}\right)^t \frac{DG}{T} -\frac{(1-\alpha)LD^2}{\alpha T^2} -\frac{ LD^2}{T^2(1-\beta)} \nonumber\\ & \quad -\frac{DG}{T}\left(1-\frac{1}{2T^{2/3}}\right)^{t/2} - \frac{\sqrt{6} LD^2C^{1/2}}{T^{5/3}}
-\frac{\sqrt{2}\sigma+\sqrt{12} LD^2C^{1/2}}{T^{4/3}}- \frac{ D(\sigma^2+G^2)^{1/2}}{T^{3/2}(1-\beta(1-\alpha))}-\frac{LD^2}{2T^2}. \end{align}
Regrouping the terms implies that
\begin{align} \label{final_thm_discrete_proof_10011} &\E{ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{t+1}) } \nonumber\\ &\leq \left(1- \frac{1}{T}\right) \E{ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^t)} +\left(1-\frac{1}{T^{1/2}}\right)^t \frac{DG}{T} +\frac{(1-\alpha)LD^2}{\alpha T^2} +\frac{ LD^2}{T^2(1-\beta)} \nonumber\\ & \quad +\frac{DG}{T}\left(1-\frac{1}{2T^{2/3}}\right)^{t/2}+\frac{\sqrt{6} LD^2C^{1/2}}{T^{5/3}}
+\frac{\sqrt{2}\sigma+\sqrt{12} LD^2C^{1/2}}{T^{4/3}}+ \frac{ D(\sigma^2+G^2)^{1/2}}{T^{3/2}(1-\beta(1-\alpha))}+\frac{LD^2}{2T^2}. \end{align}
Now apply the expression in \eqref{final_thm_discrete_proof_10011} for $t=0,\dots,T-1$ to obtain
\begin{align} \label{final_thm_discrete_proof_10012} &\E{ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)-\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{T}) } \nonumber\\ &\leq \left(1- \frac{1}{T}\right)^T \E{ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^0)} +\frac{(1-\alpha)LD^2}{\alpha T} +\frac{ LD^2}{T(1-\beta)}
+\frac{\sqrt{2}\sigma+\sqrt{12} LD^2C^{1/2}}{T^{1/3}}
\nonumber\\ & \quad +\frac{\sqrt{6} LD^2C^{1/2}}{T^{2/3}}
+ \frac{ D(\sigma^2+G^2)^{1/2}}{T^{1/2}(1-\beta(1-\alpha))}+\frac{LD^2}{2T}
+\sum_{t=0}^T\left(1-\frac{1}{T^{1/2}}\right)^t \frac{DG}{T}+\sum_{t=0}^T\frac{DG}{T}\left(1-\frac{1}{2T^{2/3}}\right)^{t/2} \nonumber\\
&\leq \left(1- \frac{1}{T}\right)^T \E{ \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*)- \frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^0)} +\frac{(1-\alpha)LD^2}{\alpha T} +\frac{ LD^2}{T(1-\beta)} +\frac{\sqrt{6} LD^2C^{1/2}}{T^{2/3}}
\nonumber\\ & \quad
+ \frac{ D(\sigma^2+G^2)^{1/2}}{T^{1/2}(1-\beta(1-\alpha))}+\frac{LD^2}{2T}
+ \frac{DG}{T^{1/2}}+\frac{4DG}{T^{1/3}} +\frac{\sqrt{2}\sigma+\sqrt{12} LD^2C^{1/2}}{T^{1/3}} , \end{align}
where in the last inequality we use the inequalities $\sum_{t=0}^T\left(1-\frac{1}{2T^{2/3}}\right)^{t/2} \leq \frac{1}{1-(1-\frac{1}{2T^{2/3}})^{1/2}}\leq 4T^{2/3}$ and $\sum_{t=0}^T\left(1-\frac{1}{T^{1/2}}\right)^t \leq T^{1/2}$. Regrouping the terms and using the inequality $(1-1/T)^T\leq 1/e$ lead to
\begin{align} \label{final_thm_discrete_proof_1002} \E{\frac{1}{n}\sum_{i=1}^n F_i(\bar{\bbx}^{T})} &\geq (1-e^{-1} ) \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*) -\frac{LD^2}{T^{1/2}} -\frac{ LD^2}{T(1-\beta)} -\frac{\sqrt{6} LD^2C^{1/2}}{T^{2/3}}
\nonumber\\ & \quad
- \frac{ D(\sigma^2+G^2)^{1/2}}{T^{1/2}(1-\beta)}-\frac{LD^2}{2T}
- \frac{DG}{T^{1/2}}-\frac{4DG}{T^{1/3}} -\frac{\sqrt{2}\sigma+\sqrt{12} LD^2C^{1/2}}{T^{1/3}}. \end{align}
Now using the argument in \eqref{final_thm_proof_1500}, we can show that the result in \eqref{final_thm_discrete_proof_1002} implies that for all $j=\ccalN$ it holds
\begin{align} \label{final_thm_discrete_proof_10022222} \E{\frac{1}{n}\sum_{i=1}^n F_i({\bbx_j}^{T})} &\geq (1-e^{-1} ) \frac{1}{n}\sum_{i=1}^n F_i(\bbx^*) -\frac{LD^2}{T^{1/2}} -\frac{GD+ LD^2}{T(1-\beta)} -\frac{\sqrt{6} LD^2C^{1/2}}{T^{2/3}}
\nonumber\\ & \qquad
- \frac{ D(\sigma^2+G^2)^{1/2}}{T^{1/2}(1-\beta)}-\frac{LD^2}{2T}
- \frac{DG}{T^{1/2}}-\frac{4DG}{T^{1/3}} -\frac{\sqrt{2}\sigma+\sqrt{12} LD^2C^{1/2}}{T^{1/3}}. \end{align}
Since $C:=1+\frac{2}{(1-\beta)^2}$ it can be shown that $C^{1/2}=(1+\frac{2}{(1-\beta)^2})^{1/2}\leq 1+\frac{\sqrt{2}}{1-\beta}$. Applying this upper bound into \eqref{final_thm_discrete_proof_10022222} yields the claim in \eqref{eq:main_result_1}.
\end{document} | arXiv |
MATH FINANCE ENGINEERING FINANCE CHARTS MATH WORKSHEETS CURRENCY CONVERTER MULTIPLICATION TABLES
equations tricks history notes register login
3x3 Matrix Multiplication Calculator
add to notes
TIMES TABLES PRE-ALGEBRA ALGEBRA GEOMETRY MATRIX PROBABILITY & STATISTICS LOAN & MORTGAGE INTEREST INVESTMENT CREDIT & DEBIT PROFIT & LOSS CURRENCY CONVERTER DIGITAL COMPUTATION MECHANICAL ELECTRICAL ELECTRONICS METEOROLOGY ENVIRONMENTAL TIME & DATE UNIT CONVERSION
Matrix Determinant Matrix Inverse Transpose Matrix Matrix Addition & Subtraction Matrix Multiplication Cramers Rule Gauss Elimination
4 x 4 Matrix Multiplication 3 x 3 Matrix Multiplication 2 x 2 Matrix Multiplication Square Matrix
Matrix A
Matrix B
A x B
3x3 matrix multiplication calculator uses two matrices $A$ and $B$ and calculates the product $AB$. It is an online math tool specially programmed to perform multiplication operation between the two matrices $A$ and $B$. The matrix multiplication is not commutative operation. In the matrix multiplication $AB$, the number of columns in matrix $A$ must be equal to the number of rows in matrix $B$.
It is necessary to follow the next steps:
Enter two matrices in the box. Elements of matrices must be real numbers.
Press the "GENERATE WORK" button to make the computation;
3x3 matrix multiplication calculator will give the product of the first and second entered matrix.
Input: Two matrices. The number of columns in the first matrix must be equal to the number of rows in the second matrix;
Output: A matrix.
$3\times 3$ Matrix Multiplication Formula:
The product of two matrices $A=(a_{ij})_{3\times 3}$ and $B=(a_{ij})_{3\times 3}$ is determined by the following formula $$\begin{align}&\left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{array} \right)\cdot \left( \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} &b_{32} & b_{33} \\ \end{array} \right)\\&= \left(\begin{array}{ccc} a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31}& a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}& a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33} \\ a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31} &a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32}& a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}\\ a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31} &a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32} & a_{31}b_{13}+a_{32}b_{23}+a_{33}b_{33}\\ \end{array}\right)\end{align}$$
What is Matrix?
Matrices are a powerful tool in mathematics, science and life. Matrices are everywhere and they have significant applications. For example, spreadsheet such as Excel or written a table represents a matrix. The word "matrix" is the Latin word and it means "womb". This term was introduced by J. J. Sylvester (English mathematician) in 1850. The first need for matrices was in the studying of systems of simultaneous linear equations.
A matrix is a rectangular array of numbers, arranged in the following way $$A=\left( \begin{array}{cccc} a_{11} & a_{12} & \ldots&a_{1n} \\ a_{21} & a_{22} & \ldots& a_{2n} \\ \ldots &\ldots &\ldots&\ldots\\ a_{m1} & a_{m2} & \ldots&a_{mn} \\ \end{array} \right)=\left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots&a_{1n} \\ a_{21} & a_{22} & \ldots& a_{2n} \\ \ldots &\ldots &\ldots&\ldots\\ a_{m1} & a_{m2} & \ldots&a_{mn} \\ \end{array} \right]$$ There are two notation of matrix: in parentheses or box brackets. The terms in the matrix are called its entries or its elements. Matrices are most often denoted by upper-case letters, while the corresponding lower-case letters, with two subscript indices, are the elements of matrices. For examples, matrices are denoted by $A,B,\ldots Z$ and its elements by $a_{11}$ or $a_{1,1}$, etc. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.
The size of a matrix is a Descartes product of the number of rows and columns that it contains. A matrix with $m$ rows and $n$ columns is called an $m\times n$ matrix. In this case $m$ and $n$ are its dimensions. If a matrix consists of only one row, it is called a row matrix. If a matrix consists only one column is called a column matrix. A matrix which contains only zeros as elements is called a zero matrix. A square matrix is a matrix with the same number of rows and columns. A square matrix with all elements as zeros except for the main diagonal, which has only ones, is called an identity matrix. For instance, the following matrices $$I_1=(1),\; I_2=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right),\ldots ,I_n=\left( \begin{array}{cccc} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & 1 \\ \end{array} \right)$$ are identity matrices of size $1\times1$, $2\times 2, \ldots$ $n\times n$, respectively.
How to Find the Product of $n\times n$ Matrices?
Many operations with matrices make sense only if the matrices have suitable dimensions. In other words, they should be the same size, with the same number of rows and the same number of columns.
When we deal with matrix multiplication, matrices $A=(a_{ij})_{m\times p}$ with $m$ rows, $p$ columns and $B=(b_{ij})_{r\times n}$ with $r$ rows, $n$ columns can be multiplied if and only if $p=r$. This means, that the number of columns of the first matrix, $A$, must be equal to the number of rows of the second matrix, $B$. The product of these matrix is a new matrix that has the same number of rows as the first matrix, $A$, and the same number of columns as the second matrix, $B$. So, the corresponding product $C=A\cdot B$ is a matrix of size $m\times n$. Elements $c_{ij}$ of this matrix are $$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}\ldots+a_{ip}b_{pj}\quad\mbox{for}\;i=1,\ldots,m,\;j=1,\ldots,n.$$ For example, $3\times 3$ matrix multiplication is determined by the following formula $$\begin{align}&\left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{array} \right)\cdot \left( \begin{array}{ccc} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} &b_{32} & b_{33} \\ \end{array} \right)\\&= \left(\begin{array}{ccc} a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31}& a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}& a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33} \\ a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31} &a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32}& a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}\\ a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31} &a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32} & a_{31}b_{13}+a_{32}b_{23}+a_{33}b_{33}\\ \end{array}\right)\end{align}$$
Properties of Matrix Multiplication
Matrix multiplication is not commutative in general, $AB \not BA$. In some cases, it is possible that the product $AB$ exists, while the product $BA$ does not exist. Both products $AB$ and $BA$ are defined if and only if the matrices $A$ and $B$ are square matrices of a same size.
If $A=(a_{ij})_{mn}$, $B=(b_{ij})_{np}$ and $C=(c_{ij})_{pk}$, then matrix multiplication is associative, i.e. $$A(BC)=(AB)C$$
If $A=(a_{ij})_{mn}$, $B=(b_{ij})_{np}$, $C=(c_{ij})_{np}$ and $D=(d_{ij})_{pq}$, then the matrix multiplication is distributive with respect of matrix addition, i.e. $$\begin{align} A(B+C)&=AB+AC\\ (B+C)D&=BD+CD\end{align}$$
If $A_{n\times n}$ is a square matrix, it exists an identity matrix $I_{n\times n}$ such that $$AI=IA=A$$
For example, let us find the product $AB$ for $$A=\left( \begin{array}{ccc} 10 & 20 & 10 \\ 4 & 5 & 6 \\ 2 & 3 & 5 \\ \end{array} \right)\quad\mbox{and}\quad B=\left( \begin{array}{ccc} 3 & 2 & 4 \\ 3 & 3 & 9 \\ 4 & 4 & 2 \\ \end{array} \right)$$ Using the $3\times 3$ matrix multiplication formula, the product $AB$ is the matrix $$\begin{align} C&=\left( \begin{array}{ccc} 10\cdot3+20\cdot3+10\cdot4 & 10\cdot2+20\cdot3+10\cdot4 & 10\cdot4+20\cdot9+10\cdot2 \\ 4\cdot3+5\cdot3+6\cdot4 & 4\cdot2+5\cdot3+6\cdot4 & 4\cdot4+5\cdot9+6\cdot2\\ 2\cdot3+3\cdot3+5\cdot4 & 2\cdot2+3\cdot3+5\cdot4 & 2\cdot4+3\cdot9+5\cdot2\\ \end{array} \right)\\&=\left( \begin{array}{ccc} 130 & 120 & 240 \\ 51 & 47 & 73 \\ 35 & 33 & 45 \\ \end{array} \right)\end{align}$$
The matrix multiplication work with steps shows the complete step-by-step calculation for finding the product $AB$ of two $3\times 3$ matrices $A$ and $B$ using the matrix multiplication formula. For any other matrices, just supply elements of $2$ matrices whose elements are real numbers and click on the GENERATE WORK button. The grade school students and people who study math use this matrix multiplication calculator to generate the work, verify the results of multiplication matrices derived by hand, or do their homework problems efficiently. The grade school students can also use this calculator for solving linear equations.
Real World Problems Using 3x3 Matrix Multiplication
One of the main application of matrix multiplication is in solving systems of linear equations. Transformations in two or three dimensional Euclidean geometry can be represented by $2\times 2$ or $3\times 3$ matrices. Dilation, translation, axes reflections, reflection across the $x$-axis, reflection across the $y$-axis, reflection across the line $y=x$, rotation, rotation of $90^o$ counterclockwise around the origin, rotation of $180^o$ counterclockwise around the origin, etc, use $2\times 2$ and $3\times 3$ matrix multiplications.
3x3 Matrix Multiplication Practice Problems
Practice Problem 1 :
Find the product $AB$ for $$A=\left( \begin{array}{cc} 4& 20 \\ 5 & 5 \\ 2 &-6 \\ \end{array} \right)\quad\mbox{and}\quad B=\left( \begin{array}{cc} 3 & 2 \\ 3 & 3 \\ \end{array} \right)$$ Practice Problem 2 :
Find the image of a transformation of the vertex matrix $\left( \begin{array}{cc} 3 & 2 \\ 3 & 3 \\ \end{array} \right)$ when it is rotated $90^o$ counterclockwise around the origin.
The matrix multiplication calculator, formula, example calculation (work with steps), real world problems and practice problems would be very useful for grade school students (K-12 education) to understand the matrix multiplication of two or more matrices. Using this concept they can solve systems of linear equations and other linear algebra problems in physics, engineering and computer science.
4x4, 3x3 & 2x2 Matrix Determinant Calculator
Transpose Matrix Calculator
nxn Inverse Matrix Calculator
4x4 Matrix Addition & Subtraction Calculator
3x3 Matrix Addition Calculator
3x3 Matrix Subtraction Calculator
Squared Matrix Calculator
Gauss Elimination Calculator
Welcome to ncalculators!
By continuing with ncalculators.com, you acknowledge & agree to our terms of use & privacy policy
You must login to use this feature!
Privacy Terms Disclaimer Feedback
© All Rights Reserved 2011 - 2019 | CommonCrawl |
\begin{definition}[Definition:Dual Relation/Inverse of Complement]
Let $\RR \subseteq S \times T$ be a binary relation.
Then the '''dual''' of $\RR$ is denoted $\RR^d$ and is defined as:
:$\RR^d := \paren {\overline \RR}^{-1}$
where:
:$\overline \RR$ denotes the complement of $\RR$
:$\paren {\overline \RR}^{-1}$ denotes the inverse of the complement of $\RR$.
\end{definition} | ProofWiki |
Use the theory of group trisections to find invariants of $4$-manifold trisections.
Define a notion of group trisection for trisections with boundary.
Adapt group trisections to the setting of bridge trisections and use it to get invariants of knotted surfaces in $S^4$.
Accourding to a classical theorem by Wall, any two homotopoy equivalent, simply connected smooth closed $4$-manifolds $X$ and $Y$ become diffeomorphic after stabilizing by taking connected sums with some number of $S^1\times S^2$'s. What does it says about trisections of the trivial group?
Cite this as: AimPL: Trisections and low-dimensional topology, available at http://aimpl.org/trisections. | CommonCrawl |
Experimental Probability
Using Sample spaces to determine Probabilities
Probabilities of Games
Traffic Light Problems
Comparisons from Experiments (Investigation)
As we are beginning to see, probabilities occur in a wide variety of situations and we can visualise them through a wide variety of diagrams and tables.
Probabilities are also used in statistical analysis, usually as a way of predicting what might happen in the future based on past events that have been researched and recorded.
Obtaining probabilities from frequency tables
Our rule for probability is $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes
For example, the jump lengths of Long Jumpers in an athletics competition resulted in this data, we might then be interested in what the probability is that someone jumps more than $119$119cm.
and this will result in: total favourable = $8+4+1=13$8+4+1=13 and total possible is $31$31
$\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}=\frac{13}{31}$total favourable outcomes total possible outcomes =1331 ≈ $42%$42%
We can use frequency tables quite easily to answer a range of questions with regards to probabilities.
Be careful of the language more than, less than, at least or at most. They are subtly different. For example for numbers from 1 to 10, the set {at least 4} contains elements {4,5,6,7,8,9,10} but the set {more than 4} contains the elements {5,6,7,8,9,10}.
Obtaining probabilities from graphs
This graph depicts data resulting from watching the number of people in a queue at a concert ticket stand and how long they waited in line to be served.
Let's look at how we calculate the probability of the following $2$2 questions.
a) What is the probability that someone was served in less than $40$40 minutes?
b) What is the probability that someone was served in a time of at least $40$40 minutes?
From the graph we can see that less than $40$40 minutes includes the number of people served in $30$30 minutes ($30$30 people) and the number of people served in $35$35 minutes ($60$60 people).
We will need the total number of people served, we can find this by adding up all the values of the columns
$30+60+110+50+30+10+10=300$30+60+110+50+30+10+10=300
$\text{P(time to service < 40) }$P(time to service < 40) $=$= $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes
$=$= $\frac{60}{300}$60300
$=$= $\frac{1}{5}$15
$=$= $20%$20%
b) Now the question of "served in a time of at least $40$40 minutes" means find
$\text{P(time to service > 40) }$P(time to service > 40)
This is the complementary event for $\text{P(time to service < 40) }$P(time to service < 40) which we have already worked out. So the answer for this is
$\text{P(time to service }$P(time to service $>=$>= $\text{40) }$40) $=$= $1-\text{P(time to service < 40) }$1−P(time to service < 40)
$=$= $1-0.2$1−0.2
$=$= $0.8$0.8
We can use graphs quite easily to answer a range of questions with regards to probabilities, things to remember:
Take care when reading off information, check scales and check the axis labels.
The following table shows the frequency of the lengths jumped at a long jump competition. The class interval is a range of distances measured in centimeters.
Class Interval
$0-39$0−39 $6$6
$40-79$40−79 $9$9
$80-119$80−119 $7$7
$120-159$120−159 $8$8
Sum $33$33
According to the table, what is the probability that someone jumped more than 119 cm?
What is the probability that someone's jump measured between 80 and 159cm (inclusive)?
This frequency graph shows the number of people that were served at a furniture store, and the length of time it took to serve them.
Time (mins)Number of people10203040506070809010030 - 3435-3940-4445-4950-5455-5960-64
What is the probability that someone was served in under $40$40 minutes?
What is the probability that someone had to wait at least $50$50 minutes to be served?
S6-3
Investigate situations that involve elements of chance: A comparing discrete theoretical distributions and experimental distributions, appreciating the role of sample size B calculating probabilities in discrete situations.
Investigate a situation involving elements of chance | CommonCrawl |
www.springer.com The European Mathematical Society
Pages A-Z
StatProb Collection
Project talk
Weyl group
From Encyclopedia of Mathematics
The Weyl group of symmetries of a root system. Depending on the actual realization of the root system, different Weyl groups are considered: Weyl groups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.
Let $ G $ be a connected affine algebraic group defined over an algebraically closed field $ k $ . The Weyl group of $ G $ with respect to a torus $ T \subset G $ is the quotient group $$ W(T,\ G) = N _{G} (T) / Z _{G} (T), $$ considered as a group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ . Here $ N _{G} (T) $ is the normalizer (cf. Normalizer of a subset) and $ Z _{G} (T) $ is the centralizer of $ T $ in $ G $ . The group $ W(T,\ G) $ is finite. If $ T _{0} $ is a maximal torus, $ W( T _{0} ,\ G) $ is said to be the Weyl group $ W(G) $ of the algebraic group $ G $ . This definition does not depend on the choice of a maximal torus $ T _{0} $ ( up to isomorphism). The action by conjugation of $ N _{G} ( T _{0} ) $ on the set $ B ^ {T _{0}} $ of Borel subgroups (cf. Borel subgroup) in $ G $ containing $ T _{0} $ induces a simply transitive action of $ W( T _{0} ,\ G) $ on $ B ^ {T _{0}} $ . The action by conjugation of $ T $ on $ G $ induces an adjoint action of $ T $ on the Lie algebra $ \mathfrak g $ of $ G $ . Let $ \Phi (T,\ G) $ be the set of non-zero weights of the weight decomposition of $ \mathfrak g $ with respect to this action, which means that $ \Phi (T,\ G) $ is the root system of $ \mathfrak g $ with respect to $ T $ ( cf. Weight of a representation of a Lie algebra). $ \Phi (T,\ G) $ is a subset of the group $ X(T) $ of rational characters of the torus $ T $ , and $ \Phi (T,\ G) $ is invariant with respect to the action of $ W(T,\ G) $ on $ X(T) $ .
Let $ G $ be a reductive group, let $ Z(G) ^{0} $ be the connected component of the identity of its centre and let $ T _{0} $ be a maximal torus of $ G $ . The vector space $$ X(T _{0} /Z(G) ^{0} ) _ {\mathbf Q} = X(T _{0} /Z(G) ^{0} ) \otimes _ {\mathbf Z} \mathbf Q $$ is canonically identified with a subspace of the vector space $$ X(T _{0} ) _ {\mathbf Q} = X(T _{0} ) \otimes _ {\mathbf Z} \mathbf Q . $$ As a subset of $ X {( T _{0} )} _ {\mathbf Q} $ , the set $ \Phi ( T _{0} ,\ G) $ is a reduced root system in $ X( T _{0} /Z(G) ^{0} ) _ {\mathbf Q} $ , and the natural action of $ W( T _{0} ,\ G) $ on $ {X( T _{0} )} _ {\mathbf Q} $ defines an isomorphism between $ W( T _{0} ,\ G) $ and the Weyl group of the root system $ \Phi (T _{0} ,\ G) $ . Thus, $ W(T _{0} ,\ G) $ displays all the properties of a Weyl group of a reduced root system; e.g. it is generated by reflections (cf. Reflection).
The Weyl group of a Tits system is a generalization of this situation (for its exact definition see Tits system).
The Weyl group $ W $ of a finite-dimensional reductive Lie algebra $ \mathfrak g $ over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of $ W $ in the Cartan subalgebra $ \mathfrak p $ of $ \mathfrak g $ is a faithful representation of $ W $ . The group $ W $ is often identified with the image of this representation, being regarded as the corresponding linear group in $ \mathfrak p $ generated by the reflections. The concept of a "Weyl group" was first used by H. Weyl
in the special case of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple finite-dimensional Lie algebra, as the Weyl group of its root system. A relative Weyl group may be defined for an affine algebraic group $ G $ defined over an algebraically non-closed field. If $ T $ is a maximal $ k $ - split torus of $ G $ , then the quotient group $ N _{G} (T)/ Z _{G} (T) $ ( the normalizer of $ T $ over its centralizer in $ G $ ), regarded as the group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ , is said to be the relative Weyl group of $ G $ .
For the Weyl group of a symmetric space, see Symmetric space. The Weyl group of a real connected non-compact semi-simple algebraic group is identical with the Weyl group of the corresponding symmetric space. For the affine Weyl group see Root system.
[1a] H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen I" Math. Z. , 23 (1925) pp. 271–309 MR1544744
[1b] H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen II" Math. Z. , 24 (1925) pp. 328–395 MR1544744
[2] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[4] N. Bourbaki, "Lie groups and Lie algebras" , Elements of mathematics , Hermann (1975) (Translated from French) MR2109105 MR1890629 MR1728312 MR0979493 MR0682756 MR0524568 Zbl 0319.17002
[5a] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. I.H.E.S. , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[5b] A. Borel, J. Tits, "Complément à l'article "Groupes réductifs" " Publ. Math. I.H.E.S. , 41 (1972) pp. 253–276 MR0315007
[6] F. Bruhat, J. Tits, "Groupes algébriques simples sur un corps local" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 23–36 MR0230838 Zbl 0263.14016
[7] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101
The affine Weyl group is the Weyl group of an affine Kac–Moody algebra. One may define a Weyl group for an arbitrary Kac–Moody algebra.
The Weyl group as an abstract group is a Coxeter group.
Weyl groups play an important role in representation theory (see Character formula).
[a1] J. Tits, "Reductive groups over local fields" A. Borel (ed.) W. Casselman (ed.) , Automorphic forms, representations and -functions , Proc. Symp. Pure Math. , 33:1 , Amer. Math. Soc. (1979) pp. 29–69 MR0546588 Zbl 0415.20035
[a2] J.E. Humphreys, "Reflection groups and Coxeter groups" , Cambridge Univ. Press (1991) MR1066460 Zbl 0768.20016 Zbl 0725.20028
The Weyl group of a connected compact Lie group $ G $ is the quotient group $ W = N/T $ , where $ N $ is the normalizer in $ G $ of a maximal torus $ T $ of $ G $ . This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra $ \mathfrak t $ of $ T $ ( the isomorphism is realized by the adjoint representation of $ N $ in $ \mathfrak t $ ), and may be characterized with the aid of the root system $ \Delta $ of the Lie algebra $ \mathfrak g $ of $ G $ ( with respect to $ \mathfrak t $ ), as follows: If $ \alpha _{1} \dots \alpha _{r} $ is a system of simple roots of the algebra, which are linear forms on the real vector space $ \mathfrak t $ , the Weyl group is generated by the reflections in the hyperplanes $ \alpha _{i} (x) = 0 $ . Thus, $ W $ is the Weyl group of the system $ \Delta $ ( as a linear group in $ \mathfrak t $ ). $ W $ has a simple transitive action on the set of all chambers (cf. Chamber) of $ \Delta $ ( which, in this case, are referred to as Weyl chambers). It should be noted that, in general, $ N $ is not the semi-direct product of $ W $ and $ T $ ; all the cases in which it is have been studied. The Weyl group of $ G $ is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group $ G _{\mathbf C} $ ( cf. Complexification of a Lie group).
A.S. Fedenko
How to Cite This Entry:
Weyl group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_group&oldid=44291
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Weyl_group&oldid=44291"
TeX done
About Encyclopedia of Mathematics
Impressum-Legal | CommonCrawl |
If the sum of the squares of nonnegative real numbers $a,b,$ and $c$ is $39$, and $ab + bc + ca = 21$, then what is the sum of $a,b,$ and $c$?
Since $$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (39) + 2(21) = 81,$$ it follows that $a+b+c = \pm 9$. Since $a,b,c \ge 0$ we find $a+b+c=\boxed{9}$. | Math Dataset |
Home » Hot Topics: Kadison-Singer, Interlacing Polynomials, and Beyond
Hot Topics: Kadison-Singer, Interlacing Polynomials, and Beyond March 09, 2015 - March 13, 2015
To apply for Funding you must register by:
January 05, 2015 about 8 years ago
Hot Topic, Hot Topic
MSRI: Simons Auditorium, Atrium
Organizers Sorin Popa (University of California, Los Angeles), LEAD Daniel Spielman (Yale University), Nikhil Srivastava (University of California, Berkeley), Cynthia Vinzant (University of Washington)
Petter Branden (Royal Institute of Technology (KTH))
Chris Godsil (University of Waterloo)
Alice Guionnet (École Normale Supérieure de Lyon)
Osman Guler (University of Maryland Baltimore County)
Nicholas Harvey (University of British Columbia)
William Johnson (Texas A & M University)
Alexandra Kolla (University of Illinois at Urbana-Champaign)
Adam Marcus (Yale University)
Shayan Oveis Gharan (University of Washington)
Pablo Parrilo (Massachusetts Institute of Technology)
Sorin Popa (University of California, Los Angeles)
Gideon Schechtman (Weizmann Institute of Science)
Dimitri Shlyakhtenko (University of California, Los Angeles)
Daniel Spielman (Yale University)
Nikhil Srivastava (University of California, Berkeley)
Nicole Tomczak-Jaegermann (University of Alberta)
Stefaan Vaes (Katholieke Universiteit Leuven)
Alain Valette (Université de Neuchâtel)
Victor Vinnikov (Ben Gurion University of the Negev)
Cynthia Vinzant (University of Washington)
In a recent paper, Marcus, Spielman and Srivastava solve the Kadison-Singer Problem by proving Weaver's KS2 conjecture and the Paving Conjecture. Their proof involved a technique they called the "method of interlacing families of polynomials" and a "barrier function" approach to proving bounds on the locations of the zeros of real stable polynomials. Using these techniques, they have also proved that there are infinite families of Ramanujan graphs of every degree, and they have developed a very simple proof of Bourgain and Tzafriri's Restricted Invertibility Theorem. The goal of this workshop is to help build upon this recent development by bringing together researchers from the disparate areas related to these techniques, including Functional Analysis, Spectral Graph Theory, Free Probability, Convex Optimization, Discrepancy Theory, and Real Algebraic Geometry. Bibliography (PDF)
In a recent paper, Marcus, Spielman and Srivastava solve the Kadison-Singer Problem by proving Weaver's KS2 conjecture and the Paving Conjecture. Their proof involved a technique they called the "method of interlacing families of polynomials" and a "barrier function" approach to proving bounds on the locations of the zeros of real stable polynomials. Using these techniques, they have also proved that there are infinite families of Ramanujan graphs of every degree, and they have developed a very simple proof of Bourgain and Tzafriri's Restricted Invertibility Theorem. The goal of this workshop is to help build upon this recent development by bringing together researchers from the disparate areas related to these techniques, including Functional Analysis, Spectral Graph Theory, Free Probability, Convex Optimization, Discrepancy Theory, and Real Algebraic Geometry.
46-XX - Functional analysis {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx}
26-XX - Real functions [See also 54C30]
Show Funding
To apply for funding, you must register by the funding application deadline displayed above.
Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.
Show Lodging
MSRI does not hire an outside company to make hotel reservations for our workshop participants, or share the names and email addresses of our participants with an outside party. If you are contacted by a business that claims to represent MSRI and offers to book a hotel room for you, it is likely a scam. Please do not accept their services.
MSRI has preferred rates at the Hotel Shattuck Plaza, depending on room availability. Guests can call the hotel's main line at 510-845-7300 and ask for the MSRI- Mathematical Science Research Institute discount. To book online visit this page (the MSRI rate will automatically be applied).
MSRI has preferred rates at the Graduate Berkeley, depending on room availability. Reservations may be made by calling 510-845-8981. When making reservations, guests must request the MSRI preferred rate. Enter in the Promo Code MSRI123 (this code is not case sensitive).
MSRI has preferred rates at the Berkeley Lab Guest House, depending on room availability. Reservations may be made by calling 510-495-8000 or directly on their website. Select "Affiliated with the Space Sciences Lab, Lawrence Hall of Science or MSRI." When prompted for your UC Contact/Host, please list Chris Marshall ([email protected]).
MSRI has a preferred rates at Easton Hall and Gibbs Hall, depending on room availability. Guests can call the Reservations line at 510-204-0732 and ask for the MSRI- Mathematical Science Research Inst. rate. To book online visit this page, select "Request a Reservation" choose the dates you would like to stay and enter the code MSRI (this code is not case sensitive).
Additional lodging options may be found on our short term housing page.
Laws of non-commutative polynomials in $n$-tuples of free variables
Free probability, random matrices and transport maps
We will discuss transport maps techniques to construct isomorphisms of C^* algebras and study the local fluctuations of the eigenvalues of polynomials in several GUE matrices.
Polynomial convolutions and connections to free probability
Ramanujan graphs from finite free convolutions.
Towards Constructing Expanders via Lifts: Hopes and Limitations
In this talk, I will examine the spectrum of random k-lifts of d-regular graphs. We show that, for random shift k-lifts (which includes 2-lifts), if all the nontrivial eigenvalues of the base graph G are at most \lambda in absolute value, then with high probability depending only on the number n of nodes of G (and not on k), if k is *small enough*, the absolute value of every nontrivial eigenvalue of the lift is at most O(\lambda). While previous results on random lifts were asymptotically true with high probability in the degree of the lift k, our result is the first upperbound on spectra of lifts for bounded k. In particular, it implies that a typical small lift of a Ramanujan graph is almost Ramanujan. I will also discuss some impossibility results for large k, which, as one consequence, imply that there is no hope of constructing large Ramanujan graphs from abelian k-lifts. based on joint and ongoing work with Naman Agarwal Karthik Chandrasekaran and Vivek Madan
Expanders and box spaces
Box spaces of finitely generated groups are disjoint union of Cayley graphs of finite quotients associated with some decreasing sequence of finite index normal subgroups of the given group. In 1973 Margulis gave the first explicit construction of expanders by proving that box spaces of property (T) groups are expanders. In 2012 Mendel and Naor showed the existence of two expanders $F_1,F_2$ such that $F_1$ does not coarsely embed into $F_2$. In February 2015, Hume constructed a continuum of expanders with unbounded girth, not coarsely embedding into one another. In joint work with Ana Khukhro, we construct countably many expanders with bounded girth, as box spaces of groups with property $(\tau)$, and prove that they do not coarsely embed into one another.
Paving over arbitrary MASAs in von Neumann algebras
I will present some recent work with Stefaan Vaes, in which we consider a paving property for a MASA $A$
in a von Neumann algebra $M$, that we call \emph{\so-paving}, involving approximation in the {\so}-topology, rather
than in norm (as in classical Kadison-Singer paving).
If $A$ is the range of a normal conditional expectation, then {\so}-paving is equivalent to
norm paving in the ultrapower inclusion $A^\omega\subset M^\omega$.
We conjecture that any MASA in any von Neumann algebra satisfies {\so}-paving.
We use recent work of Marcus-Spielman-Srivastava to check this for all MASAs in $\mathcal B(\ell^2\mathbb N)$, all Cartan subalgebras in amenable von Neumann
algebras and in group measure space II$_1$ factors arising from profinite actions.
By work of mine from 2013, the conjecture also holds true for singular MASAs in II$_1$ factors, and we obtain an improved paving size
$C\varepsilon^{-2}$, which we show to be sharp.
A survey of discrepancy theory
Discrepancy theory has been an important research area in combinatorics and geometry for several decades. The Marcus-Spielman-Srivastava theorem can be viewed as a "spectral" discrepancy theorem. In this talk we will survey some of the classical results and recent progress in this area, as well as mentioning some open questions.
Contact: [email protected]
Scientific Description
Funding & Logistics
Visa/Immigration | CommonCrawl |
Operator space
In functional analysis, a discipline within mathematics, an operator space is a normed vector space (not necessarily a Banach space)[1] "given together with an isometric embedding into the space B(H) of all bounded operators on a Hilbert space H.".[2][3] The appropriate morphisms between operator spaces are completely bounded maps.
Equivalent formulations
Equivalently, an operator space is a subspace of a C*-algebra.
Category of operator spaces
The category of operator spaces includes operator systems and operator algebras. For operator systems, in addition to an induced matrix norm of an operator space, one also has an induced matrix order. For operator algebras, there is still the additional ring structure.
See also
• Gilles Pisier
• Operator system
References
1. Paulsen, Vern (2002). Completely Bounded Maps and Operator Algebras. Cambridge University Press. p. 26. ISBN 978-0-521-81669-4. Retrieved 2022-03-08.
2. Pisier, Gilles (2003). Introduction to Operator Space Theory. Cambridge University Press. p. 1. ISBN 978-0-521-81165-1. Retrieved 2008-12-18.
3. Blecher, David P.; Christian Le Merdy (2004). Operator Algebras and Their Modules: An Operator Space Approach. Oxford University Press. First page of Preface. ISBN 978-0-19-852659-9. Retrieved 2008-12-18.
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
• Bounded
• Compact
• Hilbert–Schmidt
• Normal
• Nuclear
• Trace class
• Transpose
• Unbounded
• Unitary
Algebras
• Banach algebra
• C*-algebra
• Spectrum of a C*-algebra
• Operator algebra
• Group algebra of a locally compact group
• Von Neumann algebra
Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
Advanced topics
• Approximation property
• Balanced set
• Choquet theory
• Weak topology
• Banach–Mazur distance
• Tomita–Takesaki theory
• Mathematics portal
• Category
• Commons
Spectral theory and *-algebras
Basic concepts
• Involution/*-algebra
• Banach algebra
• B*-algebra
• C*-algebra
• Noncommutative topology
• Projection-valued measure
• Spectrum
• Spectrum of a C*-algebra
• Spectral radius
• Operator space
Main results
• Gelfand–Mazur theorem
• Gelfand–Naimark theorem
• Gelfand representation
• Polar decomposition
• Singular value decomposition
• Spectral theorem
• Spectral theory of normal C*-algebras
Special Elements/Operators
• Isospectral
• Normal operator
• Hermitian/Self-adjoint operator
• Unitary operator
• Unit
Spectrum
• Krein–Rutman theorem
• Normal eigenvalue
• Spectrum of a C*-algebra
• Spectral radius
• Spectral asymmetry
• Spectral gap
Decomposition
• Decomposition of a spectrum
• Continuous
• Point
• Residual
• Approximate point
• Compression
• Direct integral
• Discrete
• Spectral abscissa
Spectral Theorem
• Borel functional calculus
• Min-max theorem
• Positive operator-valued measure
• Projection-valued measure
• Riesz projector
• Rigged Hilbert space
• Spectral theorem
• Spectral theory of compact operators
• Spectral theory of normal C*-algebras
Special algebras
• Amenable Banach algebra
• With an Approximate identity
• Banach function algebra
• Disk algebra
• Nuclear C*-algebra
• Uniform algebra
• Von Neumann algebra
• Tomita–Takesaki theory
Finite-Dimensional
• Alon–Boppana bound
• Bauer–Fike theorem
• Numerical range
• Schur–Horn theorem
Generalizations
• Dirac spectrum
• Essential spectrum
• Pseudospectrum
• Structure space (Shilov boundary)
Miscellaneous
• Abstract index group
• Banach algebra cohomology
• Cohen–Hewitt factorization theorem
• Extensions of symmetric operators
• Fredholm theory
• Limiting absorption principle
• Schröder–Bernstein theorems for operator algebras
• Sherman–Takeda theorem
• Unbounded operator
Examples
• Wiener algebra
Applications
• Almost Mathieu operator
• Corona theorem
• Hearing the shape of a drum (Dirichlet eigenvalue)
• Heat kernel
• Kuznetsov trace formula
• Lax pair
• Proto-value function
• Ramanujan graph
• Rayleigh–Faber–Krahn inequality
• Spectral geometry
• Spectral method
• Spectral theory of ordinary differential equations
• Sturm–Liouville theory
• Superstrong approximation
• Transfer operator
• Transform theory
• Weyl law
• Wiener–Khinchin theorem
| Wikipedia |
Leonhard Euler
Leonhard Euler (/ˈɔɪlər/ OY-lər,[lower-alpha 1] German: [ˈleːɔnhaʁt ˈɔʏlɐ] (listen);[lower-alpha 2] 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function.[6] He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.[7]
Leonhard Euler
Portrait by Jakob Emanuel Handmann, 1753
Born(1707-04-15)15 April 1707
Basel, Swiss Confederacy
Died18 September 1783(1783-09-18) (aged 76) [OS: 7 September 1783]
Saint Petersburg, Russian Empire
Alma materUniversity of Basel (MPhil)
Known for
• Contributions
• namesakes
Spouse
Katharina Gsell
(m. 1734; died 1773)
Salome Abigail Gsell
(m. 1776)
Scientific career
FieldsMathematics · Physics
Institutions
• Imperial Russian Academy of Sciences
• Berlin Academy
ThesisDissertatio physica de sono (Physical dissertation on sound) (1726)
Doctoral advisorJohann Bernoulli
Doctoral studentsJohann Hennert
Other notable students
• Nicolas Fuss
• Stepan Rumovsky
• Joseph-Louis Lagrange (epistolary correspondent)
• Anders Johan Lexell
Signature
Notes
• He is the father of the mathematician Johann Euler.
• He is listed by an academic genealogy as the equivalent to the doctoral advisor of Joseph Louis Lagrange.[1]
Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he is the master of us all."[8][lower-alpha 3] Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."[9][lower-alpha 4] Euler is also widely considered to be the most prolific; his 866 publications as well as his correspondences are collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quarto volumes.[11][12][13] He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
Euler is credited for popularizing the Greek letter $\pi $ (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation $f(x)$ for the value of a function, the letter $i$ to express the imaginary unit ${\sqrt {-1}}$, the Greek letter $\Sigma $ (capital sigma) to express summations, the Greek letter $\Delta $ (uppercase delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters.[14] He gave the current definition of the constant $e$, the base of the natural logarithm, now known as Euler's number.[15]
Euler was also the first practitioner of graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg). He became famous for, among many other accomplishments, solving the Basel problem, after proving that the sum of the infinite series of squared integer reciprocals equaled exactly π2/6, and for discovering that the sum of the numbers of vertices and faces minus edges of a polyhedron equals 2, a number now commonly known as the Euler characteristic. In the field of physics, Euler reformulated Newton's laws of physics into new laws in his two-volume work Mechanica to better explain the motion of rigid bodies. He also made substantial contributions to the study of elastic deformations of solid objects.
Early life
Leonhard Euler was born on 15 April 1707, in Basel, Switzerland, to Paul III Euler, a pastor of the Reformed Church, and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics.[16] He was the oldest of four children, having two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich.[17][16] Soon after the birth of Leonhard, the Euler family moved from Basel to the town of Riehen, Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood.[16]
From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some years earlier at the University of Basel. Around the age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics.[16]
In 1720, at thirteen years of age, Euler enrolled at the University of Basel.[7] Attending university at such a young age was not unusual at the time.[16] The course on elementary mathematics was given by Johann Bernoulli, the younger brother of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography:[18]
"the famous professor Johann Bernoulli [...] made it a special pleasure for himself to help me along in the mathematical sciences. Private lessons, however, he refused because of his busy schedule. However, he gave me a far more salutary advice, which consisted in myself getting a hold of some of the more difficult mathematical books and working through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when he resolved one of my objections, ten others at once disappeared, which certainly is the best method of making happy progress in the mathematical sciences."
It was during this time that Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor.[19][20]
In 1723, Euler received a Master of Philosophy with a dissertation that compared the philosophies of René Descartes and Isaac Newton.[16] Afterwards, he enrolled in the theological faculty of the University of Basel.[20]
In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono[21][22] with which he unsuccessfully attempted to obtain a position at the University of Basel.[23] In 1727, he entered the Paris Academy prize competition (offered annually and later biennially by the academy beginning in 1720)[24] for the first time. The problem posed that year was to find the best way to place the masts on a ship. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place.[25] Over the years, Euler entered this competition 15 times,[24] winning 12 of them.[25]
Career
Saint Petersburg
Johann Bernoulli's two sons, Daniel and Nicolaus, entered into service at the Imperial Russian Academy of Sciences in Saint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available.[23] On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia.[26][27] When Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler.[23] In November 1726, Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.[23]
Euler arrived in Saint Petersburg in May 1727.[23][20] He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.[28] Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as a medic in the Russian Navy.[29]
The academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler.[25] The academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg.[30] The Russian conservative nobility then gained power upon the ascension of the twelve-year-old Peter II.[30] The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and universities.[30]
Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia assumed power.[31] Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731.[31] He also left the Russian Navy, refusing a promotion to lieutenant.[31] Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department.[32] In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell.[33] Frederick II had made an attempt to recruit the services of Euler for his newly established Berlin Academy in 1740, but Euler initially preferred to stay in St Petersburg.[34] But after Emperor Anna died and Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave to Berlin, arguing he was in need of a milder climate for his eyesight.[34] The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members.[34]
Berlin
Concerned about the continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia.[35] He lived for 25 years in Berlin, where he wrote several hundred articles.[20] In 1748 his text on functions called the Introductio in analysin infinitorum was published and in 1755 a text on differential calculus called the Institutiones calculi differentialis was published.[36][37] In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences[38] and of the French Academy of Sciences.[39] Notable students of Euler in Berlin included Stepan Rumovsky, later considered as the first Russian astronomer.[40][41] In 1748 he declined an offer from the University of Basel to succeed the recently deceased Johann Bernoulli.[20] In 1753 he bought a house in Charlottenburg, in which he lived with his family and widowed mother.[42][43]
Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess.[44] This work contained Euler's exposition on various subjects pertaining to physics and mathematics and offered valuable insights into Euler's personality and religious beliefs. It was translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. The popularity of the Letters testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.[37]
Despite Euler's immense contribution to the academy's prestige and having been put forward as a candidate for its presidency by Jean le Rond d'Alembert, Frederick II named himself as its president.[43] The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways, the polar opposite of Voltaire, who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit.[37] Frederick also expressed disappointment with Euler's practical engineering abilities, stating:
I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry![45]
Throughout his stay in Berlin, Euler maintained a strong connection to the academy in St. Petersburg and also published 109 papers in Russia.[46] He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin.[46] In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops.[42] Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, with Empress Elizabeth of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time.[47] Euler decided to leave Berlin in 1766 and return to Russia.[48]
During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published.[49] This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to the St. Petersburg Academy, which had retained him as a member and paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in two parts in 1748. In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income.[50] He was even involved in the design of the water fountains at Sanssouci, the King's summer palace.[51]
Return to Russia
The political situation in Russia stabilized after Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he was assisted by his student Anders Johan Lexell.[52] While living in St. Petersburg, a fire in 1771 destroyed his home.[53]
Personal life
On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell, a painter from the Academy Gymnasium in Saint Petersburg.[33] The young couple bought a house by the Neva River.
Of their thirteen children, only five survived childhood,[54] three sons and two daughters.[55] Their first son was Johann Albrecht Euler, whose godfather was Christian Goldbach.[55]
Three years after his wife's death in 1773,[53] Euler married her half-sister, Salome Abigail Gsell (1723–1794).[56] This marriage lasted until his death in 1783.
His brother Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy.[34]
Eyesight deterioration
Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever,[57] he became almost blind in his right eye. Euler blamed the cartography he performed for the St. Petersburg Academy for his condition,[58] but the cause of his blindness remains the subject of speculation.[59][60] Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as "Cyclops". Euler remarked on his loss of vision, stating "Now I will have fewer distractions."[58] In 1766 a cataract in his left eye was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in the left eye as well.[39] However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased;[61] and, in 1775, he produced, on average, one mathematical paper every week.[39]
Death
In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with Anders Johan Lexell when he collapsed and died from a brain hemorrhage.[59] Jacob von Staehlin wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss, one of Euler's disciples, wrote a more detailed eulogy,[54] which he delivered at a memorial meeting. In his eulogy for the French Academy, French mathematician and philosopher Marquis de Condorcet, wrote:
il cessa de calculer et de vivre— ... he ceased to calculate and to live.[62]
Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island. In 1837, the Russian Academy of Sciences installed a new monument, replacing his overgrown grave plaque. To commemorate the 250th anniversary of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery.[63]
Contributions to mathematics and physics
Main article: Contributions of Leonhard Euler to mathematics
Part of a series of articles on the
mathematical constant e
Properties
• Natural logarithm
• Exponential function
Applications
• compound interest
• Euler's identity
• Euler's formula
• half-lives
• exponential growth and decay
Defining e
• proof that e is irrational
• representations of e
• Lindemann–Weierstrass theorem
People
• John Napier
• Leonhard Euler
Related topics
• Schanuel's conjecture
Euler worked in almost all areas of mathematics, including geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory, and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes.[39] Euler's name is associated with a large number of topics. Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonard Euler was the author of a quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century.[14]
Mathematical notation
Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function[6] and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the letter i to denote the imaginary unit.[64] The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones.[65]
Analysis
The development of infinitesimal calculus was at the forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour[66] (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms,[67] such as
$e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\lim _{n\to \infty }\left({\frac {1}{0!}}+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{n}}{n!}}\right).$
Euler's use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):[66]
$\sum _{n=1}^{\infty }{1 \over n^{2}}=\lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}.$
He introduced the constant
$\gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{n}}-\ln(n)\right)\approx 0.5772,$
now known as Euler's constant or the Euler–Mascheroni constant, and studied its relationship with the harmonic series, the gamma function, and values of the Riemann zeta function.[68]
Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms.[64] He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies
$e^{i\varphi }=\cos \varphi +i\sin \varphi $
which was called "the most remarkable formula in mathematics" by Richard P. Feynman.[69]
A special case of the above formula is known as Euler's identity,
$e^{i\pi }+1=0$
Euler elaborated the theory of higher transcendental functions by introducing the gamma function[70][71] and introduced a new method for solving quartic equations.[72] He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations.
Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions, and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.[73]
Number theory
Euler's interest in number theory can be traced to the influence of Christian Goldbach,[74] his friend in the St. Petersburg Academy.[57] Much of Euler's early work on number theory was based on the work of Pierre de Fermat. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form $ 2^{2^{n}}+1$ (Fermat numbers) are prime.[75]
Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and prime numbers; this is known as the Euler product formula for the Riemann zeta function.[76]
Euler invented the totient function φ(n), the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem.[77] He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one, a result otherwise known as the Euclid–Euler theorem.[78] Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of Carl Friedrich Gauss, particularly Disquisitiones Arithmeticae.[79] By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.[80]
Euler also contributed major developments to the theory of partitions of an integer.[81]
Graph theory
In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg.[82] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory.[82]
Euler also discovered the formula $V-E+F=2$ relating the number of vertices, edges, and faces of a convex polyhedron,[83] and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object.[84] The study and generalization of this formula, specifically by Cauchy[85] and L'Huilier,[86] is at the origin of topology.[83]
Physics, astronomy, and engineering
Part of a series on
Classical mechanics
${\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})$
Second law of motion
• History
• Timeline
• Textbooks
Branches
• Applied
• Celestial
• Continuum
• Dynamics
• Kinematics
• Kinetics
• Statics
• Statistical mechanics
Fundamentals
• Acceleration
• Angular momentum
• Couple
• D'Alembert's principle
• Energy
• kinetic
• potential
• Force
• Frame of reference
• Inertial frame of reference
• Impulse
• Inertia / Moment of inertia
• Mass
• Mechanical power
• Mechanical work
• Moment
• Momentum
• Space
• Speed
• Time
• Torque
• Velocity
• Virtual work
Formulations
• Newton's laws of motion
• Analytical mechanics
• Lagrangian mechanics
• Hamiltonian mechanics
• Routhian mechanics
• Hamilton–Jacobi equation
• Appell's equation of motion
• Koopman–von Neumann mechanics
Core topics
• Damping
• Displacement
• Equations of motion
• Euler's laws of motion
• Fictitious force
• Friction
• Harmonic oscillator
• Inertial / Non-inertial reference frame
• Mechanics of planar particle motion
• Motion (linear)
• Newton's law of universal gravitation
• Newton's laws of motion
• Relative velocity
• Rigid body
• dynamics
• Euler's equations
• Simple harmonic motion
• Vibration
Rotation
• Circular motion
• Rotating reference frame
• Centripetal force
• Centrifugal force
• reactive
• Coriolis force
• Pendulum
• Tangential speed
• Rotational frequency
• Angular acceleration / displacement / frequency / velocity
Scientists
• Kepler
• Galileo
• Huygens
• Newton
• Horrocks
• Halley
• Maupertuis
• Daniel Bernoulli
• Johann Bernoulli
• Euler
• d'Alembert
• Clairaut
• Lagrange
• Laplace
• Hamilton
• Poisson
• Cauchy
• Routh
• Liouville
• Appell
• Gibbs
• Koopman
• von Neumann
• Physics portal
• Category
Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers, the constants e and π, continued fractions, and integrals. He integrated Leibniz's differential calculus with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method[87] and the Euler–Maclaurin formula.[88][89][90]
Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering.[91] Besides successfully applying his analytic tools to problems in classical mechanics, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the Sun. His calculations contributed to the development of accurate longitude tables.[92]
Euler made important contributions in optics.[93] He disagreed with Newton's corpuscular theory of light,[94] which was the prevailing theory of the time. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.[95]
In fluid dynamics, Euler was the first to predict the phenomenon of cavitation, in 1754, long before its first observation in the late 19th century, and the Euler number used in fluid flow calculations comes from his related work on the efficiency of turbines.[96] In 1757 he published an important set of equations for inviscid flow in fluid dynamics, that are now known as the Euler equations.[97]
Euler is well known in structural engineering for his formula giving Euler's critical load, the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness.[98]
Logic
Euler is credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.[99]
An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset, and disjointness). Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it.
Euler diagrams (and their refinement to Venn diagrams) were incorporated as part of instruction in set theory as part of the new math movement in the 1960s.[100] Since then, they have come into wide use as a way of visualizing combinations of characteristics.[101]
Music
One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae (Attempt at a New Theory of Music), hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.[102] Even when dealing with music, Euler's approach is mainly mathematical,[103] for instance, his introduction of binary logarithms as a way of numerically describing the subdivision of octaves into fractional parts.[104] His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that remained with him throughout his life.[103]
A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2mA, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2m (where "m is an indefinite number, small or large, so long as the sounds are perceptible"[105]), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2m.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2m.5, major third + minor sixth (C–E–C); the fourth is 2m.32, two-fourths and a tone (C–F–B♭–C); the fifth is 2m.3.5 (C–E–G–B–C); etc. Genres 12 (2m.33.5), 13 (2m.32.52) and 14 (2m.3.53) are corrected versions of the diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2m.33.52) is the "diatonico-chromatic", "used generally in all compositions",[106] and which turns out to be identical with the system described by Johann Mattheson.[107] Euler later envisaged the possibility of describing genres including the prime number 7.[108]
Euler devised a specific graph, the Speculum musicum,[109][110] to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see above). The device drew renewed interest as the Tonnetz in Neo-Riemannian theory (see also Lattice (music)).[111]
Euler further used the principle of the "exponent" to propose a derivation of the gradus suavitatis (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only.[112] Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form
$ds=\sum _{i}(k_{i}p_{i}-k_{i})+1,$
where pi are prime numbers and ki their exponents.[113]
Personal philosophy and religious beliefs
Euler opposed the concepts of Leibniz's monadism and the philosophy of Christian Wolff.[114] Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler also labelled Wolff's ideas as "heathen and atheistic".[115]
Euler was a religious person throughout his life.[20] Much of what is known of Euler's religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of scripture.[116][117]
There is a famous legend[118] inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this non-sequitur: "Sir, ${\frac {a+b^{n}}{n}}=x$, hence God exists—reply!" Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is apocryphal, given that Diderot himself did research in mathematics.[119] The legend was apparently first told by Dieudonné Thiébault with embellishment by Augustus De Morgan.[118]
Commemorations
Main article: List of things named after Leonhard Euler
Euler was featured on both the sixth[120] and seventh[121] series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences.[122] The asteroid 2002 Euler was named in his honour.[123]
Selected bibliography
Euler has an extensive bibliography. His books include:
• Mechanica (1736)
• Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744)[124] (A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense)[125]
• Introductio in analysin infinitorum (1748)[126][127] (Introduction to Analysis of the Infinite)[128]
• Institutiones calculi differentialis (1755)[127][129] (Foundations of differential calculus)
• Vollständige Anleitung zur Algebra (1765)[127] (Elements of Algebra)
• Institutiones calculi integralis (1768–1770)[127] (Foundations of integral calculus)
• Letters to a German Princess (1768–1772)[37]
• Dioptrica, published in three volumes beginning in 1769[93]
It took until 1830 for the bulk of Euler's posthumous works to be individually published,[130] with an additional batch of 61 unpublished works discovered by Paul Heinrich von Fuss (Euler's great-grandson and Nicolas Fuss's son) and published as a collection in 1862.[130][131] A chronological catalog of Euler's works was compiled by Swedish mathematician Gustaf Eneström and published from 1910 to 1913.[132] The catalog, known as the Eneström index, numbers Euler's works from E1 to E866.[133] The Euler Archive was started at Dartmouth College[134] before moving to the Mathematical Association of America[135] and, most recently, to University of the Pacific in 2017.[136]
In 1907, the Swiss Academy of Sciences created the Euler Commission and charged it with the publication of Euler's complete works. After several delays in the 19th century,[130] the first volume of the Opera Omnia, was published in 1911.[137] However, the discovery of new manuscripts continued to increase the magnitude of this project. Fortunately, the publication of Euler's Opera Omnia has made steady progress, with over 70 volumes (averaging 426 pages each) published by 2006 and 80 volumes published by 2022.[138][12][14] These volumes are organized into four series. The first series compiles the works on analysis, algebra, and number theory; it consists of 29 volumes and numbers over 14,000 pages. The 31 volumes of Series II, amounting to 10,660 pages, contain the works on mechanics, astronomy, and engineering. Series III contains 12 volumes on physics. Series IV, which contains the massive amount of Euler's correspondences, unpublished manuscripts, and notes only began compilation in 1967. The series is projected to span 16 volumes, eight volumes of which have been released as of 2022.[12][137][14]
• Illustration from Solutio problematis... a. 1743 propositi published in Acta Eruditorum, 1744
• The title page of Euler's Methodus inveniendi lineas curvas.
• Euler's 1760 world map.
• Euler's 1753 map of Africa.
Notes
1. The pronunciation /ˈjuːlər/ YOO-lər is considered incorrect[2][3][4][5]
2. However, in the Swiss variety of Standard German with audible /r/: [ˈɔʏlər].
3. The quote appeared in Gugliemo Libri's review of a recently published collection of correspondence among eighteenth-century mathematicians: "... nous rappellerions que Laplace lui même, ... ne cessait de répéter aux jeunes mathématiciens ces paroles mémorables que nous avons entendues de sa propre bouche : 'Lisez Euler, lisez Euler, c'est notre maître à tous.' " [... we would recall that Laplace himself, ... never ceased to repeat to young mathematicians these memorable words that we heard from his own mouth: 'Read Euler, read Euler, he is our master in everything.][lower-alpha 5]
4. This quote appeared in a letter from Carl Friedrich Gauss to Paul Fuss dated September 11, 1849:[10] "Die besondere Herausgabe der kleinern Eulerschen Abhandlungen ist gewiß etwas höchst verdienstliches, [...] und das Studium aller Eulerschen Arbeiten doch stets die beste durch nichts anderes zu ersetzende Schule für die verschiedenen mathematischen Gebiete bleiben wird." [The special publication of the smaller Euler treatises is certainly something highly deserving, [...] and the study of all Euler's works will always remain the best school for the various mathematical fields, which cannot be replaced by anything else.]
5. Libri, Gugliemo (January 1846). "Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIe siècle, ..." [Mathematical and physical correspondence of some famous geometers of the eighteenth century, ...]. Journal des Savants (in French): 51. Archived from the original on 9 August 2018. Retrieved 7 April 2014.
References
1. Leonhard Euler at the Mathematics Genealogy Project Retrieved 2 July 2021; Archived
2. "Euler". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989.
3. "Euler". Merriam–Webster's Online Dictionary. 2009. Archived from the original on 25 April 2009. Retrieved 5 June 2009.
4. "Euler, Leonhard". The American Heritage Dictionary of the English Language (5th ed.). Boston: Houghton Mifflin Company. 2011. Archived from the original on 4 October 2013. Retrieved 30 May 2013.
5. Higgins, Peter M. (2007). Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections. Oxford University Press. p. 43. ISBN 978-0-19-921842-4.
6. Dunham 1999, p. 17.
7. Debnath, Lokenath (2010). The Legacy of Leonhard Euler : A Tricentennial Tribute. London: Imperial College Press. pp. vii. ISBN 978-1-84816-525-0.
8. Dunham 1999, p. xiii "Lisez Euler, lisez Euler, c'est notre maître à tous."
9. Grinstein, Louise; Lipsey, Sally I. (2001). "Euler, Leonhard (1707–1783)". Encyclopedia of Mathematics Education. Routledge. p. 235. ISBN 978-0-415-76368-4.
10. Fuß, Paul Heinrich; Gauß, Carl Friedrich (11 September 1849). "Carl Friedrich Gauß → Paul Heinrich Fuß, Göttingen, 1849 Sept. 11".
11. "Leonhardi Euleri Opera Omnia (LEOO)". Bernoulli Euler Center. Archived from the original on 11 September 2022. Retrieved 11 September 2022.
12. "The works". Bernoulli-Euler Society. Archived from the original on 11 September 2022. Retrieved 11 September 2022.
13. Gautschi 2008, p. 3.
14. Assad, Arjang A. (2007). "Leonhard Euler: A brief appreciation". Networks. 49 (3): 190–198. doi:10.1002/net.20158. S2CID 11298706.
15. Boyer, Carl B (1 June 2021). "Leonhard Euler". Encyclopedia Britannica. Archived from the original on 3 May 2021. Retrieved 27 May 2021.
16. Gautschi 2008, p. 4.
17. Calinger 2016, p. 11.
18. Gautschi 2008, p. 5.
19. Calinger 1996, p. 124.
20. Knobloch, Eberhard; Louhivaara, I. S.; Winkler, J., eds. (May 1983). Zum Werk Leonhard Eulers: Vorträge des Euler-Kolloquiums im Mai 1983 in Berlin (PDF). Birkhäuser Verlag. doi:10.1007/978-3-0348-7121-1. ISBN 978-3-0348-7122-8.
21. Calinger 2016, p. 32.
22. Euler, Leonhard (1727). Dissertatio physica de sono [Physical dissertation on sound] (in Latin). Basel: E. and J. R. Thurnisiorum. Archived from the original on 6 June 2021. Retrieved 6 June 2021 – via Euler archive.
Translated into English as
Bruce, Ian. "Euler's Dissertation De Sono : E002" (PDF). Some Mathematical Works of the 17th & 18th Centuries, including Newton's Principia, Euler's Mechanica, Introductio in Analysin, etc., translated mainly from Latin into English. Archived (PDF) from the original on 10 June 2016. Retrieved 12 June 2021.
23. Calinger 1996, p. 125.
24. "The Paris Academy". Euler Archive. Mathematical Association of America. Archived from the original on 30 July 2021. Retrieved 29 July 2021.
25. Calinger 1996, p. 156.
26. Calinger 1996, pp. 121–166.
27. O'Connor, John J.; Robertson, Edmund F. "Nicolaus (II) Bernoulli". MacTutor History of Mathematics Archive. University of St Andrews. Retrieved 2 July 2021.
28. Calinger 1996, pp. 126–127.
29. Calinger 1996, p. 127.
30. Calinger 1996, p. 126.
31. Calinger 1996, p. 128.
32. Calinger 1996, pp. 128–129.
33. Gekker & Euler 2007, p. 402.
34. Calinger 1996, pp. 157–158.
35. Gautschi 2008, p. 7.
36. Euler, Leonhard (1787). "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum" [Foundations of Differential Calculus, with Applications to Finite Analysis and Series]. Academiae Imperialis Scientiarum Petropolitanae (in Latin). Petri Galeatii. 1: 1–880. Archived from the original on 6 May 2021. Retrieved 8 June 2021 – via Euler Archive.
37. Dunham 1999, pp. xxiv–xxv.
38. Stén, Johan C.-E. (2014). "Academic events in Saint Petersburg". A Comet of the Enlightenment. Vita Mathematica. Vol. 17. Birkhäuser. pp. 119–135. doi:10.1007/978-3-319-00618-5_7. See in particular footnote 37, p. 131.
39. Finkel, B. F. (1897). "Biography – Leonard Euler". The American Mathematical Monthly. 4 (12): 297–302. doi:10.2307/2968971. JSTOR 2968971. MR 1514436.
40. Trimble, Virginia; Williams, Thomas; Bracher, Katherine; Jarrell, Richard; Marché, Jordan D.; Ragep, F. Jamil, eds. (2007). Biographical Encyclopedia of Astronomers. Springer Science+Business Media. p. 992. ISBN 978-0-387-30400-7. Available at Archive.org
41. Clark, William; Golinski, Jan; Schaffer, Simon (1999). The Sciences in Enlightened Europe. University of Chicago Press. p. 395. ISBN 978-0-226-10940-4. Archived from the original on 22 April 2021. Retrieved 15 June 2021.
42. Knobloch, Eberhard (2007). "Leonhard Euler 1707–1783. Zum 300. Geburtstag eines langjährigen Wahlberliners". Mitteilungen der Deutschen Mathematiker-Vereinigung. 15 (4): 276–288. doi:10.1515/dmvm-2007-0092. S2CID 122271644.
43. Gautschi 2008, pp. 8–9.
44. Euler, Leonhard (1802). Letters of Euler on Different Subjects of Physics and Philosophy, Addressed to a German Princess. Translated by Hunter, Henry (2nd ed.). London: Murray and Highley. Archived via Internet Archives
45. Frederick II of Prussia (1927). Letters of Voltaire and Frederick the Great, Letter H 7434, 25 January 1778. Richard Aldington. New York: Brentano's.
46. Vucinich, Alexander (1960). "Mathematics in Russian Culture". Journal of the History of Ideas. 21 (2): 164–165. doi:10.2307/2708192. ISSN 0022-5037. JSTOR 2708192. Archived from the original on 3 August 2021. Retrieved 3 August 2021 – via JSTOR.
47. Gindikin, Simon (2007). "Leonhard Euler". Tales of Mathematicians and Physicists. Springer Publishing. pp. 171–212. doi:10.1007/978-0-387-48811-0_7. ISBN 978-0-387-48811-0. See in particular p. 182 Archived 10 June 2021 at the Wayback Machine.
48. Gautschi 2008, p. 9.
49. Knobloch, Eberhard (1998). "Mathematics at the Prussian Academy of Sciences 1700–1810". In Begehr, Heinrich; Koch, Helmut; Kramer, Jürg; Schappacher, Norbert; Thiele, Ernst-Jochen (eds.). Mathematics in Berlin. Basel: Birkhäuser Basel. pp. 1–8. doi:10.1007/978-3-0348-8787-8_1. ISBN 978-3-7643-5943-0.
50. Thiele, Rüdiger (2005). "The Mathematics and Science of Leonhard Euler (1707–1783)". Mathematics and the Historian's Craft. CMS Books in Mathematics. New York: Springer Publishing. pp. 81–140. doi:10.1007/0-387-28272-6_6. ISBN 978-0-387-25284-1.
51. Eckert, Michael (2002). "Euler and the Fountains of Sanssouci". Archive for History of Exact Sciences. 56 (6): 451–468. doi:10.1007/s004070200054. ISSN 0003-9519. S2CID 121790508.
52. Maehara, Hiroshi; Martini, Horst (2017). "On Lexell's Theorem". The American Mathematical Monthly. 124 (4): 337–344. doi:10.4169/amer.math.monthly.124.4.337. ISSN 0002-9890. JSTOR 10.4169/amer.math.monthly.124.4.337. S2CID 125175471. Archived from the original on 20 August 2021. Retrieved 16 June 2021.
53. Thiele, Rüdiger (2005). "The mathematics and science of Leonhard Euler". In Kinyon, Michael; van Brummelen, Glen (eds.). Mathematics and the Historian's Craft: The Kenneth O. May Lectures. Springer Publishing. pp. 81–140. ISBN 978-0-387-25284-1.
54. Fuss, Nicolas (1783). "Éloge de M. Léonhard Euler" [Eulogy for Leonhard Euler]. Nova Acta Academiae Scientiarum Imperialis Petropolitanae (in French). 1: 159–212. Archived from the original on 20 August 2021. Retrieved 19 May 2018 – via Bioheritage Diversity Library. Translated into English as "Eulogy of Leonhard Euler by Nicolas Fuss". MacTutor History of Mathematics archive. Translated by Glaus, John S. D. University of St Andrews. Archived from the original on 26 December 2018. Retrieved 30 August 2006.
55. Calinger 1996, p. 129.
56. Gekker & Euler 2007, p. 405.
57. Gautschi 2008, p. 6.
58. Eves, Howard W. (1969). "Euler's blindness". In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes, Quadrants III and IV. Prindle, Weber, & Schmidt. p. 48. OCLC 260534353. Also quoted by Richeson (2012), p. 17 Archived 16 June 2021 at the Wayback Machine, cited to Eves.
59. Asensi, Victor; Asensi, Jose M. (March 2013). "Euler's right eye: the dark side of a bright scientist". Clinical Infectious Diseases. 57 (1): 158–159. doi:10.1093/cid/cit170. PMID 23487386.
60. Bullock, John D.; Warwar, Ronald E.; Hawley, H. Bradford (April 2022). "Why was Leonhard Euler blind?". British Journal for the History of Mathematics. 37: 24–42. doi:10.1080/26375451.2022.2052493. S2CID 247868159.
61. Gautschi 2008, pp. 9–10.
62. Marquis de Condorcet. "Eulogy of Euler – Condorcet". Archived from the original on 16 September 2006. Retrieved 30 August 2006.
63. Calinger 2016, pp. 530–536.
64. Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics. John Wiley & Sons. pp. 439–445. ISBN 978-0-471-54397-8.
65. Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. p. 166. ISBN 978-3-540-66572-4. Archived from the original on 17 June 2021. Retrieved 8 June 2021.
66. Wanner, Gerhard; Hairer, Ernst (2005). Analysis by its history (1st ed.). Springer Publishing. p. 63. ISBN 978-0-387-77036-9.
67. Ferraro 2008, p. 155.
68. Lagarias, Jeffrey C. (October 2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 556. arXiv:1303.1856. doi:10.1090/s0273-0979-2013-01423-x. MR 3090422. S2CID 119612431.
69. Feynman, Richard (1970). "Chapter 22: Algebra". The Feynman Lectures on Physics. Vol. I. p. 10.
70. Ferraro 2008, p. 159.
71. Davis, Philip J. (1959). "Leonhard Euler's integral: A historical profile of the gamma function". The American Mathematical Monthly. 66: 849–869. doi:10.2307/2309786. JSTOR 2309786. MR 0106810.
72. Nickalls, R. W. D. (March 2009). "The quartic equation: invariants and Euler's solution revealed". The Mathematical Gazette. 93 (526): 66–75. doi:10.1017/S0025557200184190. JSTOR 40378672. S2CID 16741834.
73. Dunham 1999, Ch. 3, Ch. 4.
74. Calinger 1996, p. 130.
75. Dunham 1999, p. 7.
76. Patterson, S. J. (1988). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics. Vol. 14. Cambridge: Cambridge University Press. p. 1. doi:10.1017/CBO9780511623707. ISBN 978-0-521-33535-5. MR 0933558. Archived from the original on 18 June 2021. Retrieved 6 June 2021.
77. Shiu, Peter (November 2007). "Euler's contribution to number theory". The Mathematical Gazette. 91 (522): 453–461. doi:10.1017/S0025557200182099. JSTOR 40378418. S2CID 125064003.
78. Stillwell, John (2010). Mathematics and Its History. Undergraduate Texts in Mathematics. Springer. p. 40. ISBN 978-1-4419-6052-8. Archived from the original on 27 July 2021. Retrieved 6 June 2021..
79. Dunham 1999, Ch. 1, Ch. 4.
80. Caldwell, Chris. "The largest known prime by year". PrimePages. University of Tennessee at Martin. Archived from the original on 8 August 2013. Retrieved 9 June 2021.
81. Hopkins, Brian; Wilson, Robin (2007). "Euler's science of combinations". Leonhard Euler: Life, Work and Legacy. Stud. Hist. Philos. Math. Vol. 5. Amsterdam: Elsevier. pp. 395–408. MR 3890500.
82. Alexanderson, Gerald (July 2006). "Euler and Königsberg's bridges: a historical view". Bulletin of the American Mathematical Society. 43 (4): 567. doi:10.1090/S0273-0979-06-01130-X.
83. Richeson 2012.
84. Gibbons, Alan (1985). Algorithmic Graph Theory. Cambridge University Press. p. 72. ISBN 978-0-521-28881-1. Archived from the original on 20 August 2021. Retrieved 12 November 2015.
85. Cauchy, A. L. (1813). "Recherche sur les polyèdres – premier mémoire". Journal de l'École polytechnique (in French). 9 (Cahier 16): 66–86. Archived from the original on 10 June 2021. Retrieved 10 June 2021.
86. L'Huillier, S.-A.-J. (1812–1813). "Mémoire sur la polyèdrométrie". Annales de mathématiques pures et appliquées. 3: 169–189. Archived from the original on 10 June 2021. Retrieved 10 June 2021.
87. Butcher, John C. (2003). Numerical Methods for Ordinary Differential Equations. New York: John Wiley & Sons. p. 45. ISBN 978-0-471-96758-3. Archived from the original on 19 June 2021. Retrieved 8 June 2021.
88. Calinger 2016, pp. 96, 137.
89. Ferraro 2008, pp. 171–180, Chapter 14: Euler's derivation of the Euler–Maclaurin summation formula.
90. Mills, Stella (1985). "The independent derivations by Leonhard Euler and Colin Maclaurin of the Euler–Maclaurin summation formula". Archive for History of Exact Sciences. 33 (1–3): 1–13. doi:10.1007/BF00328047. MR 0795457. S2CID 122119093.
91. Ojalvo, Morris (December 2007). "Three hundred years of bar theory". Journal of Structural Engineering. 133 (12): 1686–1689. doi:10.1061/(asce)0733-9445(2007)133:12(1686).
92. Youschkevitch, A. P. (1971). "Euler, Leonhard". In Gillispie, Charles Coulston (ed.). Dictionary of Scientific Biography. Vol. 4: Richard Dedekind – Firmicus Maternus. New York: Charles Scribner's Sons. pp. 467–484. ISBN 978-0-684-16964-4.
93. Davidson, Michael W. (February 2011). "Pioneers in Optics: Leonhard Euler and Étienne-Louis Malus". Microscopy Today. 19 (2): 52–54. doi:10.1017/s1551929511000046. S2CID 122853454.
94. Calinger 1996, pp. 152–153.
95. Home, R. W. (1988). "Leonhard Euler's 'anti-Newtonian' theory of light". Annals of Science. 45 (5): 521–533. doi:10.1080/00033798800200371. MR 0962700.
96. Li, Shengcai (October 2015). "Tiny bubbles challenge giant turbines: Three Gorges puzzle". Interface Focus. Royal Society. 5 (5): 20150020. doi:10.1098/rsfs.2015.0020. PMC 4549846. PMID 26442144.
97. Euler, Leonhard (1757). "Principes généraux de l'état d'équilibre d'un fluide" [General principles of the state of equilibrium of a fluid]. Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires (in French). 11: 217–273. Archived from the original on 6 May 2021. Retrieved 12 June 2021. Translated into English as Frisch, Uriel (2008). "Translation of Leonhard Euler's: General Principles of the Motion of Fluids". arXiv:0802.2383 [nlin.CD].
98. Gautschi 2008, p. 22.
99. Baron, Margaret E. (May 1969). "A note on the historical development of logic diagrams". The Mathematical Gazette. 53 (383): 113–125. doi:10.2307/3614533. JSTOR 3614533. S2CID 125364002.
100. Lemanski, Jens (2016). "Means or end? On the valuation of logic diagrams". Logic-Philosophical Studies. 14: 98–122.
101. Rodgers, Peter (June 2014). "A survey of Euler diagrams" (PDF). Journal of Visual Languages & Computing. 25 (3): 134–155. doi:10.1016/j.jvlc.2013.08.006. S2CID 2571971. Archived (PDF) from the original on 20 August 2021. Retrieved 23 July 2021.
102. Calinger 1996, pp. 144–145.
103. Pesic, Peter (2014). "Euler: the mathematics of musical sadness; Euler: from sound to light". Music and the Making of Modern Science. MIT Press. pp. 133–160. ISBN 978-0-262-02727-4. Archived from the original on 10 June 2021. Retrieved 10 June 2021.
104. Tegg, Thomas (1829). "Binary logarithms". London encyclopaedia; or, Universal dictionary of science, art, literature and practical mechanics: comprising a popular view of the present state of knowledge, Volume 4. pp. 142–143. Archived from the original on 23 May 2021. Retrieved 13 June 2021.
105. Euler 1739, p. 115.
106. Emery, Eric (2000). Temps et musique. Lausanne: L'Âge d'homme. pp. 344–345.
107. Mattheson, Johannes (1731). Grosse General-Baß-Schule. Vol. I. Hamburg. pp. 104–106. OCLC 30006387. Mentioned by Euler. Also: Mattheson, Johannes (1719). Exemplarische Organisten-Probe. Hamburg. pp. 57–59.
108. See:
• Perret, Wilfrid (1926). Some Questions of Musical Theory. Cambridge: W. Heffer & Sons. pp. 60–62. OCLC 3212114.
• "What is an Euler-Fokker genus?". Microtonality. Huygens-Fokker Foundation. Archived from the original on 21 May 2015. Retrieved 12 June 2015.
109. Euler 1739, p. 147.
110. Euler, Leonhard (1774). Eneström index 457. "De harmoniae veris principiis per speculum musicum repraesentatis". Novi Commentarii Academiae Scientiarum Petropolitanae. 18: 330–353. Retrieved 12 September 2022.
111. Gollin, Edward (2009). "Combinatorial and transformational aspects of Euler's Speculum Musicum". In Klouche, T.; Noll, Th. (eds.). Mathematics and Computation in Music: First International Conference, MCM 2007 Berlin, Germany, May 18–20, 2007, Revised Selected Papers. Communications in Computer and Information Science. Vol. 37. Springer. pp. 406–411. doi:10.1007/978-3-642-04579-0_40.
112. Lindley, Mark; Turner-Smith, Ronald (1993). Mathematical Models of Musical Scales : A New Approach. Bonn: Verlag für Systematische Musikwissenschaft. pp. 234–239. ISBN 9783922626664. OCLC 27789639. See also Nolan, Catherine (2002). "Music Theory and Mathematics". In Christensen, Th. (ed.). The Cambridge History of Western Music Theory. New York: Cambridge University Press. pp. 278–279. ISBN 9781139053471. OCLC 828741887.
113. Bailhache, Patrice (17 January 1997). "La Musique traduite en Mathématiques: Leonhard Euler". Communication au colloque du Centre François Viète, "Problèmes de traduction au XVIIIe siècle", Nantes (in French). Archived from the original on 28 November 2015. Retrieved 12 June 2015.
114. Calinger 1996, p. 123.
115. Calinger 1996, pp. 153–154
116. Euler, Leonhard (1747). Rettung der Göttlichen Offenbahrung gegen die Einwürfe der Freygeister [Defense of divine revelation against the objections of the freethinkers] (in German). Eneström index 92. Berlin: Ambrosius Haude and Johann Carl Spener. Archived from the original on 12 June 2021. Retrieved 12 June 2021 – via Euler Archive.
117. Marquis de Condorcet (1805). Comparison to the Last Edition of Euler's Letters Published by de Condorcet, with the Original Edition (PDF). Archived (PDF) from the original on 28 April 2015. Retrieved 26 July 2021. {{cite book}}: |work= ignored (help)
118. See:
• Brown, B. H. (May 1942). "The Euler–Diderot anecdote". The American Mathematical Monthly. 49 (5): 302–303. doi:10.2307/2303096. JSTOR 2303096.
• Gillings, R. J. (February 1954). "The so-called Euler–Diderot incident". The American Mathematical Monthly. 61 (2): 77–80. doi:10.2307/2307789. JSTOR 2307789.
• Struik, Dirk J. (1967). A Concise History of Mathematics (3rd revised ed.). Dover Books. p. 129. ISBN 978-0-486-60255-4.
119. Marty, Jacques (1988). "Quelques aspects des travaux de Diderot en " mathématiques mixtes "" [Some aspects of Diderot's work in general mathematics]. Recherches sur Diderot et sur l'Encyclopédie (in French). 4 (1): 145–147. Archived from the original on 24 September 2015. Retrieved 20 April 2012.
120. "Schweizerische Nationalbank (SNB) – Sechste Banknotenserie (1976)". Swiss National Bank. Archived from the original on 3 May 2021. Retrieved 15 June 2021.
121. "Schweizerische Nationalbank (SNB) – Siebte Banknotenserie (1984)". Swiss National Bank. Archived from the original on 23 April 2021. Retrieved 15 June 2021.
122. "E" (PDF). Members of the American Academy of Arts & Sciences, 1780–2017. American Academy of Arts and Sciences. pp. 164–179. Archived (PDF) from the original on 18 February 2019. Retrieved 17 February 2019. Entry for Euler is on p. 177.
123. Schmadel, Lutz D., ed. (2007). "(2002) Euler". Dictionary of Minor Planet Names. Berlin, Heidelberg: Springer Publishing. p. 162. doi:10.1007/978-3-540-29925-7_2003. ISBN 978-3-540-29925-7.
124. Fraser, Craig G. (11 February 2005). Leonhard Euler's 1744 book on the calculus of variations. Elsevier. ISBN 978-0-08-045744-4. In Grattan-Guinness 2005, pp. 168–180
125. Euler, Leonhard (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti [A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense] (in Latin). Bosquet. Archived from the original on 8 June 2021. Retrieved 8 June 2021 – via Euler archive.
126. Reich, Karin (11 February 2005). 'Introduction' to analysis. Elsevier. ISBN 978-0-08-045744-4. In Grattan-Guinness 2005, pp. 181–190
127. Ferraro, Giovanni (2007). "Euler's treatises on infinitesimal analysis: Introductio in analysin infinitorum, institutiones calculi differentialis, institutionum calculi integralis". In Baker, Roger (ed.). Euler Reconsidered: Tercentenary Essays (PDF). Heber City, UT: Kendrick Press. pp. 39–101. MR 2384378. Archived from the original (PDF) on 12 September 2022.
128. Reviews of Introduction to Analysis of the Infinite:
• Aiton, E. J. "Introduction to analysis of the infinite. Book I. Transl. by John D. Blanton. (English)". zbMATH. Zbl 0657.01013.
• Shiu, P. (December 1990). "Introduction to analysis of the infinite (Book II), by Leonard Euler (translated by John D. Blanton)". The Mathematical Gazette. 74 (470): 392–393. doi:10.2307/3618156. JSTOR 3618156.
• Ştefănescu, Doru. "Euler, Leonhard Introduction to analysis of the infinite. Book I. Translated from the Latin and with an introduction by John D. Blanton". Mathematical Reviews. MR 1025504.
129. Demidov, S. S. (2005). Treatise on the differential calculus. Elsevier. ISBN 978-0080457444. Archived from the original on 18 June 2021. Retrieved 12 November 2015. In Grattan-Guinness 2005, pp. 191–198.
130. Kleinert, Andreas (2015). "Leonhardi Euleri Opera omnia: Editing the works and correspondence of Leonhard Euler". Prace Komisji Historii Nauki PAU. Jagiellonian University. 14: 13–35. doi:10.4467/23921749pkhn_pau.16.002.5258.
131. Euler, Leonhard; Fuss, Nikola Ivanovich; Fuss, Paul (1862). Opera postuma mathematica et physica anno 1844 detecta quae Academiae scientiarum petropolitanae obtulerunt ejusque auspicus ediderunt auctoris pronepotes Paulus Henricus Fuss et Nicolaus Fuss. Imperatorskaia akademīia nauk (Russia). OCLC 9094558695.
132. Calinger 2016, pp. ix–x.
133. "The Eneström Index". Euler Archive. Archived from the original on 9 August 2021. Retrieved 27 May 2021.
134. Knapp, Susan (19 February 2007). "Dartmouth students build online archive of historic mathematician". Vox of Dartmouth. Dartmouth College. Archived from the original on 28 May 2010.
135. Klyve, Dominic (June–July 2011). "Euler Archive Moves To MAA Website". MAA FOCUS. Mathematical Association of America. Retrieved 9 January 2020.
136. "The Euler Archive". University of the Pacific. Archived from the original on 7 June 2021.
137. Plüss, Matthias. "Der Goethe der Mathematik". Swiss National Science Foundation. Archived from the original on 24 June 2021. Retrieved 16 June 2021.
138. Varadarajan, V. S. (2006). Euler through time : a new look at old themes. American Mathematical Society. ISBN 978-0-8218-3580-7. OCLC 803144928.
Sources
• Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica. 23 (2): 121–166. doi:10.1006/hmat.1996.0015.
• Calinger, Ronald (2016). Leonhard Euler: Mathematical Genius in the Enlightenment. Princeton University Press. ISBN 978-0-691-11927-4. Archived from the original on 13 July 2017. Retrieved 4 January 2017.
• Dunham, William (1999). Euler: The Master of Us All. Dolciani Mathematical Expositions. Vol. 22. Mathematical Association of America. ISBN 978-0-88385-328-3. Archived from the original on 13 June 2021. Retrieved 12 November 2015.
• Euler, Leonhard (1739). Tentamen novae theoriae musicae [An attempt at a new theory of music, exposed in all clearness, according to the most well-founded principles of harmony] (in Latin). St. Petersburg: Imperial Academy of Sciences. Archived from the original on 12 June 2021. Retrieved 12 June 2021 – via Euler archive.
• Ferraro, Giovanni (2008). The Rise and Development of the Theory of Series up to the Early 1820s. Springer Science+Business Media. ISBN 978-0-387-73467-5. Archived from the original on 29 May 2021. Retrieved 27 May 2021.
• Gekker, I. R.; Euler, A. A. (2007). "Leonhard Euler's family and descendants". In Bogolyubov, Nikolaĭ Nikolaevich; Mikhaĭlov, G. K.; Yushkevich, Adolph Pavlovich (eds.). Euler and Modern Science. Translated by Robert Burns. Mathematical Association of America. ISBN 978-0-88385-564-5. Archived from the original on 18 May 2016. Retrieved 12 November 2015.
• Gautschi, Walter (2008). "Leonhard Euler: His Life, the Man, and His Works". SIAM Review. 50 (1): 3–33. Bibcode:2008SIAMR..50....3G. CiteSeerX 10.1.1.177.8766. doi:10.1137/070702710. ISSN 0036-1445. JSTOR 20454060.
• Grattan-Guinness, Ivor, ed. (2005). Landmark Writings in Western Mathematics 1640–1940. Elsevier. ISBN 978-0-08-045744-4.
• Richeson, David S. (2012). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press. p. 17. ISBN 978-1-4008-3856-1.
Further reading
• Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward (2007). Euler at 300: An Appreciation. Mathematical Association of America. ISBN 978-0-88385-565-2.
• Bradley, Robert E.; Sandifer, Charles Edward, eds. (2007). Leonhard Euler: Life, Work and Legacy. Studies in the History and Philosophy of Mathematics. Vol. 5. Elsevier. ISBN 978-0-444-52728-8. Archived from the original on 19 June 2021. Retrieved 8 June 2021.
• Dunham, William (2007). The Genius of Euler: Reflections on his Life and Work. Mathematical Association of America. ISBN 978-0-88385-558-4.
• Hascher, Xavier; Papadopoulos, Athanase, eds. (2015). Leonhard Euler : Mathématicien, physicien et théoricien de la musique (in French). Paris: CNRS Editions. ISBN 978-2-271-08331-9. Archived from the original on 8 June 2021. Retrieved 8 June 2021.
• Sandifer, C. Edward (2007). The Early Mathematics of Leonhard Euler. Mathematical Association of America. ISBN 978-0-88385-559-1.
• Sandifer, C. Edward (2007). How Euler Did It. Mathematical Association of America. ISBN 978-0-88385-563-8.
• Sandifer, C. Edward (2015). How Euler Did Even More. Mathematical Association of America. ISBN 978-0-88385-584-3. Archived from the original on 16 June 2021. Retrieved 8 June 2021.
• Schattschneider, Doris, ed. (November 1983). "A Tribute to Leonhard Euler 1707–1783 (special issue)". Mathematics Magazine. 56 (5). JSTOR i326726.
External links
• Leonhard Euler at the Mathematics Genealogy Project
• The Euler Archive: Composition of Euler works with translations into English
• Opera-Bernoulli-Euler (compiled works of Euler, Bernoulli family, and contemporary peers)
• Euler Tercentenary 2007
• The Euler Society
• Euleriana at the Berlin-Brandenburg Academy of Sciences and Humanities
• Euler Family Tree
• Euler's Correspondence with Frederick the Great, King of Prussia
• Works by Leonhard Euler at LibriVox (public domain audiobooks)
• O'Connor, John J.; Robertson, Edmund F. "Leonhard Euler". MacTutor History of Mathematics Archive. University of St Andrews.
• Dunham, William (24 September 2009). "An Evening with Leonhard Euler". YouTube. Muhlenberg College: philoctetesctr (published 9 November 2009). (talk given by William Dunham at )
• Dunham, William (14 October 2008). "A Tribute to Euler - William Dunham". YouTube. Muhlenberg College: PoincareDuality (published 23 November 2011).
Leonhard Euler
• Euler–Lagrange equation
• Euler–Lotka equation
• Euler–Maclaurin formula
• Euler–Maruyama method
• Euler–Mascheroni constant
• Euler–Poisson–Darboux equation
• Euler–Rodrigues formula
• Euler–Tricomi equation
• Euler's continued fraction formula
• Euler's critical load
• Euler's formula
• Euler's four-square identity
• Euler's identity
• Euler's pump and turbine equation
• Euler's rotation theorem
• Euler's sum of powers conjecture
• Euler's theorem
• Euler equations (fluid dynamics)
• Euler function
• Euler method
• Euler numbers
• Euler number (physics)
• Euler–Bernoulli beam theory
• Namesakes
• Category
Links to related articles
Infinitesimals
History
• Adequality
• Leibniz's notation
• Integral symbol
• Criticism of nonstandard analysis
• The Analyst
• The Method of Mechanical Theorems
• Cavalieri's principle
Related branches
• Nonstandard analysis
• Nonstandard calculus
• Internal set theory
• Synthetic differential geometry
• Smooth infinitesimal analysis
• Constructive nonstandard analysis
• Infinitesimal strain theory (physics)
Formalizations
• Differentials
• Hyperreal numbers
• Dual numbers
• Surreal numbers
Individual concepts
• Standard part function
• Transfer principle
• Hyperinteger
• Increment theorem
• Monad
• Internal set
• Levi-Civita field
• Hyperfinite set
• Law of continuity
• Overspill
• Microcontinuity
• Transcendental law of homogeneity
Mathematicians
• Gottfried Wilhelm Leibniz
• Abraham Robinson
• Pierre de Fermat
• Augustin-Louis Cauchy
• Leonhard Euler
Textbooks
• Analyse des Infiniment Petits
• Elementary Calculus
• Cours d'Analyse
Differential equations
Classification
Operations
• Differential operator
• Notation for differentiation
• Ordinary
• Partial
• Differential-algebraic
• Integro-differential
• Fractional
• Linear
• Non-linear
• Holonomic
Attributes of variables
• Dependent and independent variables
• Homogeneous
• Nonhomogeneous
• Coupled
• Decoupled
• Order
• Degree
• Autonomous
• Exact differential equation
• On jet bundles
Relation to processes
• Difference (discrete analogue)
• Stochastic
• Stochastic partial
• Delay
Solutions
Existence/uniqueness
• Picard–Lindelöf theorem
• Peano existence theorem
• Carathéodory's existence theorem
• Cauchy–Kowalevski theorem
Solution topics
• Wronskian
• Phase portrait
• Phase space
• Lyapunov stability
• Asymptotic stability
• Exponential stability
• Rate of convergence
• Series solutions
• Integral solutions
• Numerical integration
• Dirac delta function
Solution methods
• Inspection
• Substitution
• Separation of variables
• Method of undetermined coefficients
• Variation of parameters
• Integrating factor
• Integral transforms
• Euler method
• Finite difference method
• Crank–Nicolson method
• Runge–Kutta methods
• Finite element method
• Finite volume method
• Galerkin method
• Perturbation theory
Applications
• List of named differential equations
Mathematicians
• Isaac Newton
• Gottfried Wilhelm Leibniz
• Leonhard Euler
• Jacob Bernoulli
• Émile Picard
• Józef Maria Hoene-Wroński
• Ernst Lindelöf
• Rudolf Lipschitz
• Joseph-Louis Lagrange
• Augustin-Louis Cauchy
• John Crank
• Phyllis Nicolson
• Carl David Tolmé Runge
• Martin Kutta
• Sofya Kovalevskaya
Calculus
Precalculus
• Binomial theorem
• Concave function
• Continuous function
• Factorial
• Finite difference
• Free variables and bound variables
• Graph of a function
• Linear function
• Radian
• Rolle's theorem
• Secant
• Slope
• Tangent
Limits
• Indeterminate form
• Limit of a function
• One-sided limit
• Limit of a sequence
• Order of approximation
• (ε, δ)-definition of limit
Differential calculus
• Derivative
• Second derivative
• Partial derivative
• Differential
• Differential operator
• Mean value theorem
• Notation
• Leibniz's notation
• Newton's notation
• Rules of differentiation
• linearity
• Power
• Sum
• Chain
• L'Hôpital's
• Product
• General Leibniz's rule
• Quotient
• Other techniques
• Implicit differentiation
• Inverse functions and differentiation
• Logarithmic derivative
• Related rates
• Stationary points
• First derivative test
• Second derivative test
• Extreme value theorem
• Maximum and minimum
• Further applications
• Newton's method
• Taylor's theorem
• Differential equation
• Ordinary differential equation
• Partial differential equation
• Stochastic differential equation
Integral calculus
• Antiderivative
• Arc length
• Riemann integral
• Basic properties
• Constant of integration
• Fundamental theorem of calculus
• Differentiating under the integral sign
• Integration by parts
• Integration by substitution
• trigonometric
• Euler
• Tangent half-angle substitution
• Partial fractions in integration
• Quadratic integral
• Trapezoidal rule
• Volumes
• Washer method
• Shell method
• Integral equation
• Integro-differential equation
Vector calculus
• Derivatives
• Curl
• Directional derivative
• Divergence
• Gradient
• Laplacian
• Basic theorems
• Line integrals
• Green's
• Stokes'
• Gauss'
Multivariable calculus
• Divergence theorem
• Geometric
• Hessian matrix
• Jacobian matrix and determinant
• Lagrange multiplier
• Line integral
• Matrix
• Multiple integral
• Partial derivative
• Surface integral
• Volume integral
• Advanced topics
• Differential forms
• Exterior derivative
• Generalized Stokes' theorem
• Tensor calculus
Sequences and series
• Arithmetico-geometric sequence
• Types of series
• Alternating
• Binomial
• Fourier
• Geometric
• Harmonic
• Infinite
• Power
• Maclaurin
• Taylor
• Telescoping
• Tests of convergence
• Abel's
• Alternating series
• Cauchy condensation
• Direct comparison
• Dirichlet's
• Integral
• Limit comparison
• Ratio
• Root
• Term
Special functions
and numbers
• Bernoulli numbers
• e (mathematical constant)
• Exponential function
• Natural logarithm
• Stirling's approximation
History of calculus
• Adequality
• Brook Taylor
• Colin Maclaurin
• Generality of algebra
• Gottfried Wilhelm Leibniz
• Infinitesimal
• Infinitesimal calculus
• Isaac Newton
• Fluxion
• Law of Continuity
• Leonhard Euler
• Method of Fluxions
• The Method of Mechanical Theorems
Lists
• Differentiation rules
• List of integrals of exponential functions
• List of integrals of hyperbolic functions
• List of integrals of inverse hyperbolic functions
• List of integrals of inverse trigonometric functions
• List of integrals of irrational functions
• List of integrals of logarithmic functions
• List of integrals of rational functions
• List of integrals of trigonometric functions
• Secant
• Secant cubed
• List of limits
• Lists of integrals
Miscellaneous topics
• Complex calculus
• Contour integral
• Differential geometry
• Manifold
• Curvature
• of curves
• of surfaces
• Tensor
• Euler–Maclaurin formula
• Gabriel's horn
• Integration Bee
• Proof that 22/7 exceeds π
• Regiomontanus' angle maximization problem
• Steinmetz solid
Topics in continuum mechanics
Divisions
• Solid mechanics
• Fluid mechanics
• Acoustics
• Vibrations
• Rigid body dynamics
Laws and Definitions
Laws
• Conservation of mass
• Conservation of momentum
• Navier-Stokes
• Bernoulli
• Poiseuille
• Archimedes
• Pascal
• Conservation of energy
• Entropy inequality
Definitions
• Stress
• Cauchy stress
• Stress measures
• Deformation
• Small strain
• Antiplane shear
• Large strain
• Compatibility
Solid and Structural mechanics
Solids
• Elasticity
• linear
• Hooke's law
• Transverse isotropy
• Orthotropy
• hyperelasticity
• Membrane elasticity
• Equation of state
• Hugoniot
• JWL
• hypoelasticity
• Cauchy elasticity
• Viscoelasticity
• Creep
• Concrete creep
• Plasticity
• Rock mass plasticity
• Viscoplasticity
• Yield criterion
• Bresler-Pister
• Contact mechanics
• Frictionless
• Frictional
Material failure theory
• Drucker stability
• Material failure theory
• Fatigue
• Fracture mechanics
• J-integral
• Compact tension specimen
• Damage mechanics
• Johnson-Holmquist
Structures
• Bending
• Bending moment
• Bending of plates
• Sandwich theory
Fluid mechanics
Fluids
• Fluid statics
• Fluid dynamics
• Navier–Stokes equations
• Bernoulli's principle
• Poiseuille equation
• Buoyancy
• Viscosity
• Newtonian
• Non-Newtonian
• Archimedes' principle
• Pascal's law
• Pressure
• Liquids
• Surface tension
• Capillary action
Gases
• Atmosphere
• Boyle's law
• Charles's law
• Gay-Lussac's law
• Combined gas law
Plasma
Acoustics
• Acoustic theory
• Aeroacoustics
Rheology
• Viscoelasticity
• Smart fluids
• Magnetorheological
• Electrorheological
• Ferrofluids
• Rheometry
• Rheometer
Scientists
• Bernoulli
• Boyle
• Cauchy
• Charles
• Euler
• Gay-Lussac
• Hooke
• Pascal
• Newton
• Navier
• Stokes
Awards
• Eringen Medal
• William Prager Medal
Authority control
International
• FAST
• ISNI
• VIAF
National
• Norway
• Chile
• Spain
• France
• BnF data
• Catalonia
• Germany
• Italy
• Israel
• Belgium
• 2
• United States
• Sweden
• Latvia
• Japan
• Czech Republic
• Australia
• Greece
• Korea
• 2
• Croatia
• Netherlands
• Poland
• Portugal
• Vatican
Academics
• CiNii
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
• Trove
Other
• Historical Dictionary of Switzerland
• RISM
• SNAC
• IdRef
| Wikipedia |
\begin{document}
\title{Stability for $ \Phi_{S, F,H}
\tableofcontents
\begin{abstract}
In this paper, we mainly consider the stability of $ \Phi_{S, F,H} $ harmonic map and $ \Phi_{T,F,H} $ harmonic map from or into $ \Phi $-SSU manifold. We mainly consider the stability of $ \Phi_{S, F,H} $ harmonic map and $ \Phi_{T,F,H} $ harmonic map from or into compact convex hypersurface. We also give some Theorems to know when a manifold is $ \Phi_{S, F,H} $ -stable or $ \Phi_{S, F,H} $ -unstable.
\end{abstract}
{\small
\noindent{\it Keywords and phrases}: Stability; $\Phi_{S, ,H} $ harmonic map; $\Phi_{T, F,H}$ harmonic map
\noindent {\it MSC 2010}: 58E15; 58E20 ; 53C27
}
\section{Introduction }
Harmonic map is an important topic in differential geomeotry.
For the research of stability on harmonic map, Xin\cite{xin1980some} proved the nonexistence of nonconstant stable harmonic map from sphere into compact manifold. Leung\cite{leung1982stability} proved the nonexistence of nonconstant stable harmonic map from compact manifold into sphere.
In \cite{Howard1985}, Howard proved the nonexistence of stable harmonic map from compact Riemannian manifold to $ \delta(n) $ pinched Riemannian manifold.
\begin{defn}Let $u:(M, g) \rightarrow(N, h)$ be a smooth map,
$$E_{sym}(u)=\int_{M}F\left(\frac{|u^*h|^{2}}{4}\right)+H(u)d\nu_{g},$$
whose critical point is called $ F $-symphonic map with potential which is symphonic map in \cite{nakauchi2011variational}.
\end{defn}
Han and Feng \cite{han2014monotonicity} has studied monotonicity formula and stability of $F $-symphonic map from or into $ \mathbb{S}^n. $ In \cite{caoxiangzhi202212}, we studied the stability of $ F $-symphonic map with potential from or into compact $ \Phi $-SSU manifold . In \cite{caoxiangzhi2022Liouville}, we studied Liouville theorem of $ F $-symphonic map with potential.
\begin{defn}Let $u:(M, g) \rightarrow(N, h)$ be a smooth map,
$$\Phi_{S,F,H}(u)=\int_{M}F\left(\frac{|S_u|^{2}}{4}\right)+H(u)d\nu_{g},$$
whose critical point is called $ \Phi_{S,F,H} $ harmonic map.
\end{defn}
When $ H=0 $, Han et al. \cite{han2022} studied the stability of. We also use the method of in \cite{torbaghan2022stability} to explore the unstability of map from $ S^n $ and into sphere.
\begin{defn}
Let RHS of \eqref{k2} be the index form $ I(V,W) $. For $ u:(M,g) \to (N,h), $ $ \Phi_{S,F, H} $-harmonic map $ u $ with potential is called stable if $ I(V,V)\geq 0 $ for any nonzero vector filed $V$. Otherwise, it is called unstable. A manifold $ (M,g) $ is called $ \Phi_{S,F, H} $-SU if it is neither the domain of $ \Phi_{S,F, H} $ stable harmonic map nor the target manifold of $ \Phi_{S,F, H} $ stable harmonic map. If the identiy map of $ (M,g) $ is $ \Phi_{S,F, H} $-stable, we call $ M $ $ \Phi_{S,F, H} $-stable, otherwise, it is called $ \Phi_{S,F, H} $-U.
\end{defn}
\begin{defn}[c.f. \cite{Han213213123123}\cite{Han2019HarmonicMA} ]
A Riemannian manifold $M^{m}$ is said to be $\Phi$-superstrongly unstable ( $\Phi$-SSU) if there exists an isometric immersion of $M^{m}$ in $R^{m+p}, p>0$ with its second fundamental form $B$ such that, for all unit tangent vectors $x$ to $M^{m}$ at very point , the following inequality holds:
\[
\langle Q_{x}(X),X\rangle=\sum_{i=1}^{m}\left(4\left\langle B\left(x, e_{i}\right), B\left(x, e_{i}\right)\right\rangle-\left\langle B(x, x), B\left(e_{i}, e_{i}\right)\right\rangle\right)<0,
\]
where $\left\{e_{i}\right\}_{i=1}^{m}$ is the local orthogonal frame field near $y$ of $M$.
\end{defn}
\begin{rem}
In \cite{ara2001instability}, Ara studied the relation between $ SSU $ manifold and Stability of $ F $ harmonic map. Obviously, $ \Phi$-$ SSU $ manifold is $ SSU $ manifold. $ \Phi$-$ SSU $ manifold was studied in (\cite{Han213213123123}\cite{Han2019HarmonicMA} ). This definition of $\Phi$-SSU was first introduced by Han and Wei in \cite{Han2019HarmonicMA}, and some interesting examples of $\Phi$-$S S U$ manifolds were studied.
\end{rem}
When $ F(x)=x,H=0 $ , Han \cite{Han213213123123} studied the stability of $ \Phi_{S} $ harmonic map from compact $ SSU $ manifold. However, there are rare studies of stability of $\Phi_{S,F}$ harmonic map from or into compact $\Phi$-SSU manifold.
Han et al. \cite{2021The} studied the stability of $ \Phi_{S} $ harmonic map from compact Convex Hypersurface or compact convex hypersurface. However, there are rare studies of stability of $\Phi_{S,F}$ harmonic map from compact Convex Hypersurface or compact convex hypersurface.
Ara \cite{ara2001stability} obtained Okayasu type theorem and Howard type theorem for $F$-harmonic map. Liu \cite{MR2259738} obtained Okayasu type theorem for $F$-harmonic map.
In \cite{torbaghan2022stability}, investigated the stability of $\alpha$-harmonic map and $\alpha$-stability of Riemannian manifold. We also use the method of in \cite{torbaghan2022stability} to explore the unstability of map. we will give sufficient conditions under which $ M $ is $\Phi_{S,F}$ unstable.
\begin{defn}Let $u:(M, g) \rightarrow(N, h)$ be a smooth map,
$$\Phi_{T,F,H}(u)=\int_{M}F\left(\frac{|T_u|^{2}}{4}\right)+H(u)d\nu_{g},$$
whose critical point is called $ \Phi_{T,F,H} $ harmonic map
\end{defn}
When $ H=0 $, In \cite{nakauchi2011}, defined C-stationary map. In \cite{Nakauchi2019a} gived the stress energy tensor of , In \cite{Nakauchi2022a}, Nakauchi studin the confomality of $ C $-stationary map, proved that any rotationally symmetric smooth map between 4-dimensional model spaces is a C-stationary map if and only if it is a conformal one. In \cite{Kawai2013}, Nakauchi proved that any stable C-stationary map from $ \mathbb{S}^{n}(n\geq 5) $ must be conformal.When the domain is $ S^n $ and $ F(x)=x,H=0 $, such kind of map has been studied in \cite{han2013stability}.
\begin{defn}
Let the index form $ I(V,W) $ be RHS of \eqref{gg1}. For $ u:(M,g) \to (N,h), $ $ \Phi_{T,F, H} $-harmonic map $ u $ with potential is called stable if $ I(V,V)\geq 0 $ for any nonzero vector filed $V$. Otherwise, it is called unstable. If the identiy map of $ M $ is $ \Phi_{T,F, H} $-stable, $ M $ is called $ \Phi_{T,F, H} $-stable, otherwise, it is called $ \Phi_{T,F, H} $-unstable. A manifold $ (M,g) $ is called $ \Phi_{S,F, H} $-SU if it is neither the domain of $ \Phi_{T,F, H} $ stable harmonic map nor the target manifold of $ \Phi_{T,F, H} $ stable harmonic map.
\end{defn}
The main motivation of this paper is to extend the Theorem in Han \cite{Han213213123123} to $ \Phi_{S, F,H} $ harmonic map and $ \Phi_{T,F,H} $ harmonic map.
We also use the method of in \cite{torbaghan2022stability} to explore the unstability of map. we will give sufficient conditions under which $ M $ is $\Phi_{T,F,H}$ unstable.
\begin{lem}[\cite{ara2001instability}]\label{dkl}
For any constant $a>0$, there is a strictly increasing and convex $C^{2}$ function $F:[0, \infty) \rightarrow[0, \infty)$ such that $t \cdot F^{\prime \prime}(t)<a \cdot F^{\prime}(t)$ for any $t>0$.
\end{lem}
\begin{rem}By \cite[Lemma 4.9]{ara2001instability}, the following functions $ F $ satisfies the conculsion of Lemma \ref{dkl}.
(1) $F_{1}(t)=t^{b+1}, 0<b<a$,
(2) $F_{2, n}(t)=\sum_{i=1}^{n} a_{i} t^{i}, n<a+1, a_{1}>0, a_{i} \geq 0(i=2, \cdots, n)$,
(3) $F_{3}(t)=\int_{0}^{t} e^{\int_{0}^{s} G(u) d u} d s$, where $G(u)$ is a continuous function and $u \cdot G(u)<a$
\end{rem}
\section{$ \Phi_{S,F,H} $ harmnonic maps}
\subsection{ $ \Phi_{S,F,H} $ harmnonic maps from $ \Phi$-SSU manifold }
\begin{lem}[The first variation, c.f. Theorem 2 in \cite{han2022}]
Let $u:(M, g) \rightarrow(N, h)$ be a smooth map, and let $u_{t}: M \rightarrow N,(-\varepsilon<t<\varepsilon)$ be a compactly supported variation such that $u_{0}=u$ and set $V=\left.\frac{\partial}{\partial t} u\right|_{t=0}$. Then we have
\[
\left.\frac{d}{d t} \Phi_{S,F,H}\left(u_{t}\right)\right|_{t=0}=-\int_{M} h\left(V,\tau_{F,H}\right) d v_{g},
\]
where $d v_{g}$ denotes the volume form on $M$, where $ \sigma_{u}(\cdot)=h(du(\cdot),du(e_j))du(e_j)+\frac{m-4}{4}|du |^2 du(\cdot), $
$ \tau_{F,H}(u)=\delta^\nabla\left(F^{\prime}\left(\frac{|S_u|^{2}}{2}\right)\sigma_{u}\right) -\nabla H(u)=0. $
\end{lem}
\begin{proof}
By Theorem 2 in \cite{han2022}, we have
\begin{equation*}
\begin{split}
&\frac{\partial}{\partial t} F\left(\frac{\left\|S_{u_{t}}\right\|^{2}}{4} \right) \\
&= \bigg[\operatorname{div}\left( F^\prime\left(\frac{\left\|S_{u_{t}}\right\|^{2}}{4} \right)X_{t}\right) -h\left(d \Psi\left(\frac{\partial}{\partial t}\right), F^\prime\left(\frac{\left\|S_{u_{t}}\right\|^{2}}{4} \right)\operatorname{div} \sigma_{u_{t}}+ \sigma_{u}\left( \nabla F^\prime\left(\frac{\left\|S_{u_{t}}\right\|^{2}}{4} \right)\right) \right) \bigg],
\end{split}
\end{equation*}
where$ X_t $ is defined by $ g(X_t,Y)=h(d\Psi(\frac{\partial}{\partial t},\sigma_{u}(Y))). $
Notice that
\begin{equation*}
\begin{split}
\frac{\partial}{\partial t} H(u)= h(d\Psi ( \frac{\partial}{\partial t} ),\nabla H(u)).
\end{split}
\end{equation*}
Combing the above two formulas, we finish the proof.
\end{proof}
\begin{lem}[The second variation, c.f. Theorem 3 in \cite{han2022}]
Let $u:(M, g) \rightarrow(N, h)$ be an F-stationary map. Let $u_{s, t}: M \rightarrow N(-\varepsilon<s, t<\varepsilon)$ be a compactly supported twoparameter variation such that $u_{0,0}=u$ and set $V=\left.\frac{\partial}{\partial t} u_{s, t}\right|_{s, t=0}, W=\left.\frac{\partial}{\partial s} u_{s, t}\right|_{s, t=0}$.
\begin{equation}\label{k2}
\begin{split}
&\left.\frac{\partial^{2}}{\partial s \partial t} \Phi_{S, F,H}\left(u_{s, t}\right)\right|_{s, t=0}\\
=&\int_{M} \operatorname{HessH}(V, W) d v_{g}+ \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left\langle\widetilde{\nabla} V, \sigma_{u}\right\rangle\left\langle\widetilde{\nabla} W, \sigma_{u}\right\rangle \mathrm{d} v_{g}\\
&+
\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} V, \widetilde{\nabla}_{e_j} W\right) \times\left.\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\right] \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), \tilde{\nabla}_{e j} W\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e j} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(R^{N}\left(V, \mathrm{~d} u\left(e_{i}\right)\right) W, \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}
\end{split}
\end{equation}
\end{lem}
\begin{proof}
The proof is similar to that \cite[theorem 5.1]{Han2013StabilityOF}. It follows from Theorem 3 in \cite{han2022} and
\begin{equation*}
\begin{split}
\frac{\partial^2}{\partial t\partial s} H(u)= \nabla^2 H(d\Psi ( \frac{\partial}{\partial t} ),d\Psi ( \frac{\partial}{\partial s} ))++h(\tilde{\nabla}_{\frac{\partial}{\partial s}}d\Psi ( \frac{\partial}{\partial t} ),\nabla H(u)).
\end{split}
\end{equation*}
\end{proof}
\begin{thm}\label{thm21a}
Let $ (M^m, g) $ be a compact $ \Phi
$-$ SSU $ manifold. For $x \in M$
$$ a=\min _{X \in U M, Y \in U M} \frac{-\left\langle Q_{x}^{M}(X), X\right\rangle_{M}}{8|B(X, X)|_{\mathbf{R}^{r}}|B(Y, Y)|_{\mathbf{R}^{r}}}>0,$$ Let $ F $ be the positive function determined by Lemma \ref{dkl}, $ \nabla^2 H \leq 0 $.
Let $ u $ be stable $ \Phi_{S,F, H} $-
harmonic map with potential from $ (M^m, g) $ into any Riemannian manifold $ N $, then $ u $ is constant.
\end{thm}
\begin{proof}
We use the same notations as in the proof of Theorem 6.1 in \cite{2021The} or \cite{caoxiangzhi202212}. We choose an orthogonal frame field $\left\{e_{1}, \cdots, e_{m+p}\right\}$ of $R^{m+p}$ such that $\left\{e_{i}\right\}_{i=1}^{m}$ are tangent to $M^{m},\left\{e_{\alpha}\right\}_{\alpha=m+1}^{m+p}$ are normal to $M^{m}$ and $\left.\nabla_{e_{i}} e_{j}\right|_{x}=0$, where $x$ is a fixed point of $M$. We take a fixed orthonormal basis of $R^{m+p}$ denoted by $E_{D}, D=$ $1, \cdots, m+p$ and set
\[
V_{D}=\sum_{i=1}^{m} v_{D}^{i} e_{i}, v_{D}^{i}=\left\langle E_{D}, e_{i}\right\rangle, v_{D}^{\alpha}=\left\langle E_{D}, e_{\alpha}\right\rangle, \alpha=m+1, \cdots, m+p,
\]
where $\langle\cdot, \cdot\rangle$ is the canonical Euclidean inner product.
Using averaging method, we have
\begin{equation*}
\begin{split}
&\sum_{D} \int_{M} \text{Hess} H(du(V_D),du(V_D)) +\sum_{D} \int_{M}F^{\prime}(\frac{|S_u |^2}{4})\left\langle(\Delta d u)\left(V_{D}\right), \sigma_{u}\left(V_{D}\right)\right\rangle d v_{g}\\
&=\int_{M} \sum_{i, j, D} v_{D}^{i} v_{D}^{j}\left\langle(\Delta d u)\left(e_{i}\right), \sigma_{u}\left(e_{j}\right)\right\rangle d v_{g} \\
&=\int_{M} \sum_{i}\left\langle(\Delta d u)\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right\rangle d v_{g}=\int_{M}\left\langle(\Delta d u), \sigma_{u}\right\rangle d v_{g}=\int_{M}\left\langle\delta d u, \delta \sigma_{u}\right\rangle d v_{g} \\
& =-\int_{M}\left\langle\delta d u, \operatorname{div}\left(\sigma_{u}\right)\right\rangle d v_{g}=0.
\end{split}
\end{equation*}
Then \eqref{k2} can be written as
\begin{equation}\label{k7a}
\begin{split}
&\sum_{D=1}^{m+p}I(du(V_D),du(V_D))=\\
& \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left\langle\widetilde{\nabla} V, \sigma_{u}\right\rangle\left\langle\widetilde{\nabla} W, \sigma_{u}\right\rangle \mathrm{d} v_{g}\\
&+
\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e i} d u\left(V_{D}\right), \widetilde{\nabla}_{e_j} d u\left(V_{D}\right)\right) \times\left.\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\right] \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), \tilde{\nabla}_{e_j} d u\left(V_{D}\right)\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(Ric^M(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}\\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left((\nabla^2 du)(e_i), \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}\\
&=\sum_{i=1}^{7} J_i.
\end{split}
\end{equation}
Now we estimate the seven terms respectively using (38), (39) in \cite{2021The},
\begin{equation}\label{k9a}
\begin{split}
J_1=&F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{A}\left\langle\tilde{\nabla} d u\left(V_{D}\right), \sigma_{u}\right\rangle^{2} \\
=& \sum_{A} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left[\sum_{i}\left\langle\tilde{\nabla}_{e_{i}} d u\left(V_{D}\right), \sigma_{u}\left(e_{i}\right)\right\rangle\right]\left[\sum_{j}\left\langle\tilde{\nabla}_{e_{j}} d u\left(V_{D}\right), \sigma_{u}\left(e_{j}\right)\right\rangle\right] \\
=& F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{A} \left[\sum_{i}\left(-v_{D}^{\alpha}B_{ik}^{\alpha} h\left(d u\left(e_{k}\right), \sigma_{u}\left(e_{i}\right)\right)+v_{D}^{k} h\left(\tilde{\nabla}_{e_{i}} d u\left(e_{k}\right), \sigma_{u}\left(e_{i}\right)\right)\right)\right] \\
&\times \sum_{A} \left[\sum_{i}\left(-v_{D}^{\alpha}B_{jl}^{\alpha} h\left(d u\left(e_{l}\right), \sigma_{u}\left(e_{j}\right)\right)+v_{D}^{k} h\left(\tilde{\nabla}_{e_{j}} d u\left(e_{k}\right), \sigma_{u}\left(e_{j}\right)\right)\right)\right] \\
=& F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{i} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] \\
&+F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{A}\left[\sum_{l} v_{A}^{l}\left[\sum_{i} h\left(\left(\nabla_{e_{i}} d u\right)\left(e_{l}\right), \sigma_{u}\left(e_{i}\right)\right)\right]\right]^{2} \\
=& F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{i} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] \\
&+ F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left[\sum_{l}\left[\sum_{i} h\left(\left(\nabla_{e_{l}} d u\right)\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right)\right]^{2}\right],
\end{split}
\end{equation}
and
\begin{equation}\label{k8a}
\begin{split}
J_2&=F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \widetilde{\nabla}_{e_j} d u\left(V_{D}\right)\right) \times\left.\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\right] \mathrm{d}\\
&=F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)
\bigg\{\sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(A^{\alpha}\left(e_{j}\right)\right)\right) \left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] \\
&+\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right),\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right)\right) \left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\bigg\},
\end{split}
\end{equation}
and
\begin{equation}\label{k10a}
\begin{split}
J_3=&F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right)\\
=&\sum h(du(A^\alpha(e_i)),du(e_j))h(du(e_i),du(A^\alpha(e_j)))\\
&+\sum h((\nabla_{e_k}du)((e_i)),du(e_j))h(du(e_i),(\nabla_{e_k}du)((e_j))),
\end{split}
\end{equation}
and
\begin{equation}\label{k11}
\begin{split}
J_4=&F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_j} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{i}\right)\right)\\
=&\sum h(du(A^\alpha A^\alpha(e_i)),du(e_j))h(du(e_i),du((e_j)))\\
&+\sum h((\nabla_{e_k}du)((e_i)),du(e_j))h((\nabla_{e_k}du)((e_i)),du(e_j)),
\end{split}
\end{equation}
and
\begin{equation}\label{k13a}
\begin{split}
J_5=&\frac{m-4}{2} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} d u\left(V_{D}\right), \mathrm{~d} u\left(e_{j}\right)\right)
\\
=&\frac{m-4}{2} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)\sum h(du(A^\alpha(e_i)),du(e_i))h(du(e_j),du(A^\alpha(e_j)))\\
&+\frac{m-4}{2} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)\sum h((\nabla_{e_k}du)((e_i)),du(e_i))h((\nabla_{e_k}du)((e_j)),du(e_j)),
\end{split}
\end{equation}
and
\begin{equation}\label{k11a}
\begin{split}
J_6=&F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(Ric^M(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) \left( h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\\
=& F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(\mathrm{trace}(A^\alpha)A^\alpha(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) \left( h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\\
&-F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(A^\alpha A^\alpha(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) \left( h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right).
\end{split}
\end{equation}
The last term $ J_7 $ can help us to cancel the terms above,
\begin{equation}\label{k11a}
\begin{split}
J_7=& F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j} h\left(\left(\nabla^{2} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\\
=&\sum_{i, j, k} e_{k}\left[ F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)h\left(\nabla_{e_{k}} d u\left(e_{i}\right), du\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\right]\\
&- F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right),\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\\
& -F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right),\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right)\right)\\
&- F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right) h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right) \\
&- F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right)\frac{m-4}{2} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{i}\right)\right) h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right), d u\left(e_{j}\right)\right)\\
&-F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left[\sum_{l}\left[\sum_{i} h\left(\left(\nabla_{e_{l}} d u\right)\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right)\right]^{2}\right],
\end{split}
\end{equation}
where we have used the formula
\begin{equation}\label{k14}
\begin{split}
e_kF^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)=F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) h\left(\left(\nabla_{e_{l}} d u\right)\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right).
\end{split}
\end{equation}
Plugging \eqref{k9a}\eqref{k8a}\eqref{k10a}\eqref{k12a}\eqref{k13a}\eqref{k11a} into \eqref{k7a}, we have
\begin{equation}\label{k15a}
\begin{split}
&\sum_{D} I\left(d u\left(V_{D}\right), d u\left(V_{D}\right)\right)\\
&=\int_{M} \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(A^{\alpha}\left(e_{j}\right)\right)\right) \times\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] d v_{g} \\
& +\int_{M} \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), d u\left(A^{\alpha}\left(e_{j}\right)\right)\right) d v_{g} \\
& +\int_{M} \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha} A^{\alpha}\left(e_{i}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right) d v_{g}\right. \\
&+\frac{m-4}{2} \int_{M} \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{i}\right)\right) h\left(d u\left(A^{\alpha}\left(e_{j}\right)\right), d u\left(e_{j}\right)\right) d v_{g} \\
&-\int_{M} \sum_{i, j, \alpha} h\left(d u\left(\operatorname{trace}\left(A^{\alpha}\right) A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{j}\right)\right) \\
& \times\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] d v_{g} \\
& +\int_{M} \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha} A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] d v_{g}.
\end{split}
\end{equation}
Choose normal coordinate such that $ h(du(e_i),du(e_j))=\lambda_i^2 \delta_{ij} $, then
\begin{equation*}
\begin{split}
h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)=\lambda_i^2(\lambda_i^2+\frac{m-4}{4}|du |^2)\delta_{ik}.
\end{split}
\end{equation*}
Hence , we get
\begin{equation*}
\begin{split}
& F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{i} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] \\
=&F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left \langle B(e_i,e_i) ,B(e_j,e_j) \right \rangle \lambda_i^2(\lambda_i^2+\frac{m-4}{4}|du |^2)\lambda_j^2(\lambda_j^2+\frac{m-4}{4}|du |^2).
\end{split}
\end{equation*}
By \eqref{k15a}, we get
\begin{equation*}
\begin{split}
&\sum_{D} I\left(d u\left(V_{D}\right), d u\left(V_{D}\right)\right) \\
\leq &\int_M F^{\prime\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \left \langle B(e_i,e_i) ,B(e_j,e_j) \right \rangle \lambda_i^2(\lambda_i^2+\frac{m-4}{4}|du |^2)\lambda_j^2(\lambda_j^2+\frac{m-4}{4}|du |^2)d v_{g}.\\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\sum_{i} \lambda_{i}^{2}\left(\lambda_{i}^{2}+\frac{m-4}{4}|d u|^{2}\right) \\
&\times \sum_{j}\left(4\left\langle B\left(e_{i}, e_{j}\right), B\left(e_{i}, e_{j}\right)\right\rangle-\left\langle B\left(e_{i}, e_{i}\right), B\left(e_{j}, e_{j}\right)\right\rangle\right)\\
\leq & \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\sum_{i} \lambda_{i}^{2}\left(\lambda_{i}^{2}+\frac{m-4}{4}|d u|^{2}\right) \\
& \frac{1}{2}\sum_{j}\left(4\left\langle B\left(e_{i}, e_{j}\right), B\left(e_{i}, e_{j}\right)\right\rangle-\left\langle B\left(e_{i}, e_{i}\right), B\left(e_{j}, e_{j}\right)\right\rangle\right).
\end{split}
\end{equation*}
where we used the assumption $t \cdot F^{\prime \prime}(t)<a \cdot F^{\prime}(t)$ and the fact that
\begin{equation*}
\begin{split}
\left\|S_{u}\right\|^{2}=\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right)=\frac{m-4}{4}|du|^4+|u^{*}h|^2=\lambda_i^2(\lambda_i^2+\frac{m-4}{4}|du |^2)
\end{split}
\end{equation*}
\end{proof}
From the above proof, it is obvious that
\begin{thm}
Let $ (M^n, g) $ be a compact $ \Phi
$-SSU manifold and $ F^{\prime\prime}(x)\leq 0, \nabla^2 H \leq 0 $. Let $ u $ be stable $ \Phi_{S,F, H} $-
harmonic map with potential from $ (M^m, g) $ into any Riemannian manifold $ (N,h) $, then $ u $ is constant.
\end{thm}
we get that
\begin{cor}
Let $ (M^m, g) $ be a compact $ \Phi
$-SSU manifold. For $a=\min _{X \in U M} \frac{-\left\langle Q_{x}^{M}(X), X\right\rangle_{M}}{2|B(X, X)|_{\mathbf{R}^{r}}^{2}}>0.$ Let $ F(x) $ is one of the following functions
(1) $F(t)=t^{b+1}, 0<b<a$,
(2) $F(t)=\sum_{i=1}^{n} a_{i} t^{i}, n<a+1, a_{1}>0, a_{i} \geq 0(i=2, \cdots, n)$,
(3) $F(t)=\int_{0}^{t} e^{\int_{0}^{s} G(u) d u} d s$, where $G(u)$ is a continuous function and $u \cdot G(u)<a$.
Let $ u $ be stable $ \Phi_{S,F, H} $-
harmonic map with potential from $ (M^m, g) $ into any Riemannian manifold $ N $, then $ u $ is constant.
\end{cor}
\subsection{ $ \Phi_{S,F,H} $ harmonic maps into $ \Phi$-SSU manifolds }
\begin{thm}\label{thm23a} Let $ (N^n, g) $ be a compact $ \Phi
$-$ SSU $ manifold. For $x \in M$
$$ a=\min _{X \in U N, Y \in U N} \frac{-\left\langle Q_{x}^{N}(X), X\right\rangle_{N}}{8|B(X, X)|_{\mathbf{R}^{r}}|B(Y, Y)|_{\mathbf{R}^{r}}}>0,$$ Let $ F $ be the positive function determined by Lemma \ref{dkl}, $ \nabla^2 H \leq 0 $.
Let $ u $ be stable $ \Phi_{S,F, H} $-
harmonic map with potential from $ (M^m, g) $ into any Riemannian manifold $ N $, then $ u $ is constant.
\end{thm}
\begin{proof}
We use the same notations as in the proof of Theorem 7.1 in \cite{2021The} or \cite{caoxiangzhi202212}.
Let $\left\{e_{1}, \cdots, e_{m}\right\}$ be a local orthonormal frame field of $M$ . Let $\left\{\epsilon_{1}, \cdots, \epsilon_{n}, \epsilon_{n+1}, \cdots, \epsilon_{n+p}\right\}$ be an orthonormal frame field of $R^{n+p}$, such that $\left\{\epsilon_{i}, \cdots, \epsilon_{n}\right\}$ are tangent to $N^{n}, \epsilon_{n+1}, \cdots, \epsilon_{n+p}$ are normal to $N^{n}$ and $\left.{}^{N}\nabla_{\epsilon_{b}} \epsilon_{c}\right|_{u(x)}=0$, where $x$ is a fixed point of $M$. As in \cite{2021The}, we fix an orthonormal basis $E_{D}$ of $R^{m+p}$, for $D=1, \cdots, m+p$ and set
\begin{equation*}
\begin{split}
V_{D}&=\sum_{b=1}^{n} v_{D}^{b} \epsilon_{a}, v_{D}^{b}=\left\langle E_{D}, \epsilon_{b}\right\rangle,\\
v_{D}^{\alpha}&=\left\langle E_{D}, \epsilon_{\alpha}\right\rangle, \quad for \quad \alpha=n+1, \cdots, n+p,\\
\nabla_{\epsilon_{b}} V_{D}&=\sum_{\alpha=n+1}^{n+p} \sum_{c=1}^{n} v_{D}^{\alpha} B_{b c}^{\alpha} \epsilon_{c}, 1 \leq b \leq n;
\end{split}
\end{equation*}
choose local frame such that
\begin{equation}\label{kkk}
\begin{split}
\sum_{i=1}^{m}u_{i}^{b}u_{i}^{c}=\lambda_b^2 \delta_{bc}.
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
&I(V_D,V_D)\\
=&\int_{M} \operatorname{HessH}(V_D, V_D) d v_{g}+ \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left\langle\widetilde{\nabla} V_D, \sigma_{u}\right\rangle\left\langle\widetilde{\nabla} V_D, \sigma_{u}\right\rangle \mathrm{d} v_{g}\\
&+
\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} V_D, \widetilde{\nabla}_{e_j} V_D\right) \times\left.\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\right] \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V_D, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} V_D, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V_D, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), \tilde{\nabla}_{e_j} V_D\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V_D, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e j}V_D, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(R^{N}\left(V, \mathrm{~d} u\left(e_{i}\right)\right) W, \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g}.} \\
\end{split}
\end{equation*}
Using (59) in \cite{2021The}, we have
\begin{equation}\label{4.2}
\begin{split}
&F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{A}\left\langle\tilde{\nabla} \left(V_{D}\right), \sigma_{u}\right\rangle^{2} \\
=& F^{\prime\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \left \langle B(\epsilon_b,\epsilon_b) ,B(\epsilon_c,\epsilon_c) \right \rangle \lambda_b^2(\lambda_b^2+\frac{m-4}{4}|du |^2)\lambda_c^2(\lambda_c^2+\frac{m-4}{4}|du |^2)\d v_{g}\\
\end{split}
\end{equation}
Notice that
\begin{equation}\label{}
\begin{split}
|S|^2= \left[ h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right]=\frac{m-4}{4}|du |^2\lambda_j^2 \delta_{jl}+\lambda_i^2 \delta_{il}\lambda_j^2 \delta_{ij}
\end{split}
\end{equation}
By the formula (62)-(66) in \cite{2021The}, we have
\begin{equation}\label{clp}
\begin{split}
&\sum_{D} I\left(V_{D}, V_{D}\right)\\
=&\int_M F^{\prime\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \left \langle B(e_i,e_i) ,B(e_j,e_j) \right \rangle \lambda_i^2(\lambda_i^2+\frac{m-4}{4}|du |^2)\lambda_j^2(\lambda_j^2+\frac{m-4}{4}|du |^2)d v_{g}\\
&+\int_{M}F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{b} \lambda_{b}^{2}\left(\lambda_{b}^{2}+\frac{m-4}{4}|d u|^{2}\right) \sum_{c}\left(4\left\langle B\left(\epsilon_{b}, \epsilon_{c}\right), B\left(\epsilon_{b}, \epsilon_{c}\right)\right\rangle-\left\langle B\left(\epsilon_{b}, \epsilon_{b}\right), B\left(\epsilon_{c}, \epsilon_{c}\right)\right\rangle\right) d v_{g}.
\end{split}
\end{equation}
Since $N$ is a $\Phi-\mathrm{SSU}$ manifold, if $u$ is not constant, we have
\[
\sum_{D} I\left(V_{D}, V_{D}\right)<0.
\]
\end{proof}
From \eqref{clp}, it is easy to get
\begin{thm}
Let $ (N^n, g) $ be a compact $ \Phi
$-SSU manifold and $ F^{\prime\prime}\leq 0, Hess H \leq 0 $. Then every stable $ \Phi_{S,F, H} $-
harmonic map u from $ (M^m, g) $ into any Riemannian manifold $ N $ is constant.
\end{thm}
\begin{cor}
Let $ (N^n, g) $ be a compact $ \Phi
$-SSU manifold and $ \alpha \leq 1$. Then every stable $ \Phi_{S,\alpha,H} $-
harmonic map u from $ (M^m, g) $ into any Riemannian manifold $ N $ is constant.
\end{cor}
\subsection{ $\Phi_{S,F,H}$ harmonic map when domain manifold is convex hypersurface}
\begin{thm}\label{9c}
Let $M \subset R^{m+1}$ be a compact convex hypersurface. Assume that the principal curvatures $\lambda_{i}$ of $M^{m}$ satisfy $0<\lambda_{1} \leq \cdots \leq \lambda_{m}$ and $3 \lambda_{m}<\sum_{i=1}^{m-1} \lambda_{i}$. Then every nonconstant $\Phi_{S,F,H}$-stationary map from $M$ to any compact Riemannian manifold $N$ is unstable if one of the two conditions hold,
(1) there exists a constant $c_{F}=\inf \left\{c \geq 0 | F^{\prime}(t) / t^{c}\right.$ is nonincreasing $\}$ such that
\begin{equation}\label{888}
\begin{split}
c_{F}<\frac{1}{4 \lambda_{m}^{2}} \min _{1 \leq i \leq m}\left\{\lambda_{i}\left(\sum_{k=1}^{m} \lambda_{k}-2 \lambda_{i}-2 \lambda_{m}\right)\right\},
\end{split}
\end{equation}
(2) when $F^{\prime \prime}(t)=F^{\prime}(t)$
\begin{equation}\label{999}
\begin{split}
\left\|u^{*} h\right\|^{2}<\frac{1}{\lambda_{m}^{2}} \min _{1 \leq i \leq m}\left\{\lambda_{i}\left(\sum_{k=1}^{m} \lambda_{k}-2 \lambda_{i}-2 \lambda_{m}\right)\right\} .
\end{split}
\end{equation}
\end{thm}
\begin{proof}
We modify the proof in \cite[Theorem 1]{MR2259738} and \cite[Theorem 3.1 ]{Li2017NonexistenceOS}. In the proof of this theorem, we also borrow the notations and settings from \cite[Theorem 1]{MR2259738} and \cite[Theorem 3.1 ]{Li2017NonexistenceOS}.
In order to prove the instability of $u: M^{n} \rightarrow N$, we need to consider some special variational vector fields along $u$. To do this, choosing an orthogonal frame field $\left\{e_{i}, e_{n+1}\right\}, i=1, \ldots, n$, of $\mathbf{R}^{n+1}$, such that $\left\{e_{i}\right\}$ are tangent to $M^{n} \subset \mathbf{R}^{n+1}, e_{n+1}$ is normal to $M^{n}$ and $\left.\nabla_{e_{i}} e_{j}\right|_{P}=0$. Meanwhile, taking a fixed orthonormal basis $E_{A}, A=1, \ldots, n+1$, of $\mathbf{R}^{n+1}$ and setting
(3.1) $ V_{A}=\sum_{i=1}^{n} v_{A}^{i} e_{i}, v_{A}^{i}=\left\langle E_{A}, e_{i}\right\rangle, v_{A}^{n+1}=\left\langle E_{A}, e_{n+1}\right\rangle$
where $\langle\cdot, \cdot\rangle$ denotes the canonical Euclidean inner product. Then $u_{*} V_{A} \in$ $\Gamma\left(u^{-1} T N\right)$ and
\begin{equation*}
\begin{split}
\tilde{\nabla}_{e_{i}}\left(\mathrm{d} u\left(\nabla_{e_{i}} V_{A}\right)\right)= & -v_{A}^{k} h_{i k} h_{i j}\left(\mathrm{d} u\left(e_{j}\right)\right)+v_{A}^{n+1}\left(\tilde{\nabla}_{e_{i}} h_{i j}\right)\left(\mathrm{d} u\left(e_{j}\right)\right) \\
& +v_{A}^{n+1} h_{i j} \tilde{\nabla}_{u_{+} e_{i}} u_{*} e_{j}
\end{split}
\end{equation*}
where, $h_{i j}$ denotes the components of the second fundamental form of $M^{n}$ in $\mathbf{R}^{n+1}$
\begin{equation}\label{9ccc}
\begin{split}
&\sum_{A} \int_{M^{n}}F{'}\left(\frac{\|S_u\|^{2}}{4}\right)\left\langle\Delta \sigma_{u}\left(V_{A}\right), \mathrm{d} u\left(V_{A}\right)\right\rangle \mathrm{d} v_g +\int_{M} \operatorname{HessH}(V, W) d v_{g}\\
=&\sum_{A} \int_{M^{n}} F{'}\left(\frac{\|S_u\|^{2}}{4}\right) v_{A}^{i} v_{A}^{j}\left\langle\Delta \mathrm{d} u\left(e_{i}\right),\sigma_{u}\left(e_{j}\right)\right\rangle \mathrm{d} v_g +\int_{M} \operatorname{HessH}(V, W) d v_{g}\\
=&\int_{M^{n}}f F{'}\left(\frac{\|S_u\|^{2}}{4}\right)\left\langle\Delta \mathrm{d} u\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right\rangle \mathrm{d} v_g+\int_{M} \operatorname{HessH}(V, W) d v_{g} \\
=&\int_{M^{n}}\left\langle\mathrm{d}^{*} \mathrm{d} u, \mathrm{d}^{*}\left(F{'}\left(\frac{\|S_u\|^{2}}{4}\right) \mathrm{d} u\right)\right\rangle \mathrm{d} v_g+\int_{M} \operatorname{HessH}(V, W) d v_{g}\\
=&-\int_{M^{n}}\left\langle\mathrm{d}^{*} \mathrm{d} u, \nabla H\right\rangle *1 +\int_{M} \operatorname{HessH}(V_A, V_A) d v_{g}=0,
\end{split}
\end{equation}
where we have used the fact that
\begin{equation}\label{bochner}
\begin{split}
-R^{N}\left(u_{*} V_{A}, \mathrm{d} u\left(e_{i}\right)\right) \mathrm{d} u\left(e_{i}\right)+u_{*} \operatorname{Ric}^{M^{n}}\left(V_{A}\right)=\Delta \mathrm{d} u\left(V_{A}\right)+\tilde{\nabla}^{2} \mathrm{d} u\left(V_{A}\right).
\end{split}
\end{equation}
so, by \eqref{k2}we have
\begin{equation}\label{k7}
\begin{split}
&\sum_{A=1}^{m+p}I(du(V_A),du(V_A))\\
&= \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left\langle\widetilde{\nabla} du(V_A), \sigma_{u}\right\rangle\left\langle\widetilde{\nabla} du(V_A), \sigma_{u}\right\rangle \mathrm{d} v_{g}\\
&+
\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} du(V_A), \widetilde{\nabla}_{e_j} du(V_A)\right) \times\left.\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\right] \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} du(V_A), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} du(V_A), \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} du(V_A), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), \tilde{\nabla}_{e_j} du(V_A)\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} du(V_A), \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} du(V_A), \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(Ric^M(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}\\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left((\nabla^2 du)(e_i), \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}\\
&:=\sum_{i=1}^{7} J_i.
\end{split}
\end{equation}
Since we have
\begin{equation}\label{gg}
\begin{split}
\tilde{\nabla}^{2} d u\left(V_{A}\right)=\widetilde{\nabla}_{e_{i}} \widetilde{\nabla}_{e_{i}}\left(d u\left(V_{A}\right)\right)-2 \widetilde{\nabla}_{e_{i}}\left(d u\left(\nabla_{e_{i}} V_{A}\right)\right)+d u\left(\nabla_{e_{i}} \nabla_{e_{i}} V_{A}\right).
\end{split}
\end{equation}
Thus, we get
\begin{equation}\label{}
\begin{split}
J_7= & \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left((\nabla^2 du)(e_i), \sigma_{u} \left(e_{j}\right)\right) { \mathrm{d} v_{g},}\\
=&\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} \widetilde{\nabla}_{e_{i}}\left(d u\left(V_{A}\right)\right)-2 \widetilde{\nabla}_{e_{i}}\left(d u\left(\nabla_{e_{i}} V_{A}\right)\right)+d u\left(\nabla_{e_{i}} \nabla_{e_{i}} V_{A}\right), \sigma_{u} \left(e_{j}\right)\right) { \mathrm{d} v_{g}}.
\end{split}
\end{equation}
However, we have
\begin{equation}\label{888a}
\begin{split}
&\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} \widetilde{\nabla}_{e_{i}}\left(d u\left(V_{A}\right)\right), \sigma_{u} \left(e_{j}\right)\right) { \mathrm{d} v_{g},}\\
&=-\int_{M} \sum_{A, i} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}}\left[F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sigma_{u}\left(V_{A}\right)\right]\right) d v_{g}\\
&=-\int_{M} \sum_{A, i} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}}\left[F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right)\right] \sigma_{u}\left(V_{A}\right)\right) d v_{g}\\
&-\int_{M} \sum_{A, i} h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \tilde{\nabla}_{e_{i}} \sigma_{u}\left(V_{A}\right)\right) d v_{g}\\
& =-\int_{M} \sum_{A, i} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}}\left[F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right)\right] \sigma_{u}\left(V_{A}\right)\right) d v_{g}\\
&-\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), d u\left(e_{j}\right)\right) d v_{g}\\
&-\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) d v_{g}\\
&-\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) d v_{g} .\\
&-\int_{M} \sum_{A, i} h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \tilde{\nabla}_{e_{i}} (\frac{m-4}{4}|du |^2du(V_A))\right) d v_{g}.
\end{split}
\end{equation}
After cancelling two terms from \eqref{k7} and \eqref{888a} and noticing that $$ \sigma_{u} (e_{j})=\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right]du(e_j), $$ we can get
\begin{equation}\label{11}
\begin{split}
&I(du(V_A),du(V_A))\\
&= \int_{M}\left\{F^{\prime \prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{A} \langle\widetilde{\nabla} d u\left(V_{A}\right), \sigma_{u}\rangle^{2} -h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}}\left[F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right)\right] \sigma_{u}\left(V_{A}\right)\right)\right\} d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) h\left(-2 \widetilde{\nabla}_{e_{i}}\left(d u\left(\nabla_{e_{i}} V_{A}\right)\right)\right.\left.+d u\left(\nabla_{e_{i}} \nabla_{e_{i}} V_{A}\right)-d u\left(\operatorname{Ric}^{M^{m}}\left(V_{A}\right)\right), \sigma_{u}\left(V_{A}\right)\right) d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) \big[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\big] d v_{g}\\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) d v_{g} \\
& + \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), d u\left(e_{j}\right)\right) d v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) d v_{g}\\
&-\int_{M} \sum_{A, i} h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \tilde{\nabla}_{e_{i}} (\frac{m-4}{4}|du |^2du(V_A))\right) d v_{g}.\\
\end{split}
\end{equation}
Notice that the last term of \eqref{11} can be cancelled by the term
$$ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} du(V_A), \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} du(V_A), \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g}, $$
and
$$\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) \bigg(\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\bigg) d v_{g}.\\$$
So \eqref{11} can be simplified as
\begin{equation}\label{12}
\begin{split}
&I(du(V_A),du(V_A))\\
&= \int_{M}\left\{F^{\prime \prime}\left(\frac{\left\|S_u \right\|^{2}}{4}\right) \sum_{A} \langle \widetilde{\nabla} d u\left(V_{A}\right), \sigma_{u}(V_A)\rangle^{2} -h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}}\left[F^{\prime}\left(\frac{\left\|S_u \right\|^{2}}{4}\right)\right] \sigma_{u}\left(V_{A}\right)\right)\right\} d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_u \right\|^{2}}{4}\right) h\left(-2 \widetilde{\nabla}_{e_{i}}\left(d u\left(\nabla_{e_{i}} V_{A}\right)\right)\right.\left.+d u\left(\nabla_{e_{i}} \nabla_{e_{i}} V_{A}\right)-d u\left(\operatorname{Ric}^{M^{m}}\left(V_{A}\right)\right), \sigma_{u}\left(V_{A}\right)\right) d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_u \right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) \big[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)\big] d v_{g}\\
& +\int_{M} F^{\prime}\left(\frac{\left\|S_u \right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) d v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|S_u \right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), d u\left(e_{j}\right)\right) d v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|S_u \right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) d v_{g}.
\end{split}
\end{equation}
This formula is similar to (18) in \cite{Li2017NonexistenceOS} after replacing $ u^*h $ by $ S_u $. Along the same line as in \cite{Li2017NonexistenceOS}, we can get the similar formula as (19)-(30) in \cite{Li2017NonexistenceOS} after replacing $ u^*h $ by $ S_u $.
If $ F^{\prime\prime} =F^\prime,$
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|S_u\right\|^{2}}{4}\right)\left\|S_u\right\|^{2}\left\{\lambda_{m}^{2}\left\|S_u\right\|^{2}\right.\\
&\left.+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\right\} d v_{g} .
\end{split}
\end{equation}
when $ F^{\prime\prime}(t)t\leq c_F F^{\prime}(t) , $
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|S_u\right\|^{2}}{4}\right)\left\|S_u\right\|^{2}\left\{\lambda_{m}^{2}c_F\right.\\
&\left.+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\right\} d v_{g} .
\end{split}
\end{equation}
\end{proof}
\subsection{ $\Phi_{S,F,H}$ harmonic map when target manifold is convex hypersurface }
\begin{thm}\label{9cc}
With the same assumptions on $ M^m $ as in Theorem \ref{9c}, every nonconstant $ \Phi_{S,F,H} $-
harmonic map from any compact Riemannian manifold $ N $ to $ M^m $ is unstable if \eqref{888} or \eqref{999} holds.
\end{thm}
\begin{proof}
We modify the proof in \cite[Theorem 3.1 ]{Li2017NonexistenceOS}. In the proof of this theorem, we also borrow the notations and settings from the proof of \cite[Theorem 3.1 ]{Li2017NonexistenceOS}.
In order to prove the instability of $u: N^{n} \rightarrow M^{m}$, we need to consider some special variational vector fields along $u$. To do this, we choose an orthonormal field $\left\{\epsilon_{\alpha}, \epsilon_{m+1}\right\}$, $\alpha=1, \ldots, m$, of $R^{m+1}$ such that $\left\{\epsilon_{\alpha}\right\}$ are tangent to $M^{m} \subset R^{m+1}, \epsilon_{m+1}$ is normal to $M^{m}$, $\left.M^{m} \nabla_{\epsilon_{\alpha}} \epsilon_{\beta}\right|_{P}=0$ and $B_{\alpha \beta}=\lambda_{\alpha} \delta_{\alpha \beta}$, where $B_{\alpha \beta}$ denotes the components of the second fundamental form of $M^{m}$ in $R^{m+1}$. Meanwhile, take a fixed orthonormal basis $E_{A}, A=1, \ldots, m+1$, of $R^{m+1}$ and set
\[
V_{A}=\sum_{\alpha=1}^{m} v_{A}^{\alpha} \epsilon_{\alpha}, v_{A}^{\alpha}=\left\langle E_{A}, \epsilon_{\alpha}\right\rangle, v_{A}^{m+1}=\left\langle E_{A}, \epsilon_{m+1}\right\rangle
\]
where $\langle\cdot, \cdot\rangle$ denotes the canonical Euclidean inner product.
By \eqref{k2}, we immediately get
\begin{equation*}
\begin{split}
& I(V_A,V_A)\\
=&\int_{M} \operatorname{HessH}(V_A, V_A) d v_{g}+ \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left\langle\widetilde{\nabla} V_A, \sigma_{u}\right\rangle\left\langle\widetilde{\nabla} V_A, \sigma_{u}\right\rangle \mathrm{d} v_{g}\\
&+
\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} V_A, \widetilde{\nabla}_{e_j} V_A\right) \times\left.\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\right] \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V_A, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} V_A, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V_A, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), \tilde{\nabla}_{e_j} V_A\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V_A, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} V_A, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(R^{N}\left(V, \mathrm{~d} u\left(e_{i}\right)\right) W, \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},} \\
&:= \sum_i J_i.
\end{split}
\end{equation*}
By (34) in \cite{Li2017NonexistenceOS}
\begin{equation*}
\begin{split}
J_1= \int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\alpha, \sigma_{u}(e_i))h(u_j^\alpha, \sigma_{u}(e_j)) dv_g.
\end{split}
\end{equation*}
By (35) in \cite{Li2017NonexistenceOS}
\begin{equation*}
\begin{split}
J_2&= \int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}^2h(u_i^\alpha \epsilon_\alpha, du(e_j))\bigg(h(du(e_i), du(e_j)) +\frac{m-4}{4}|du |^2g(e_i,e_j)\bigg) dv_g\\
&= \int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}^2h(u_i^\alpha \epsilon_\alpha, \sigma_{u}(e_i)) dv_g.
\end{split}
\end{equation*}
By (36) in \cite{Li2017NonexistenceOS}
\begin{equation*}
\begin{split}
J_3= \int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\alpha \epsilon_\alpha, du(e_j))h(u_i^\beta\epsilon_\beta, du(e_j)) dv_g
\end{split}
\end{equation*}
By (37) in \cite{Li2017NonexistenceOS}
\begin{equation*}
\begin{split}
J_4= \int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\alpha \epsilon_\alpha, du(e_j))h(du(e_i),u_j^\beta\epsilon_\beta) dv_g,
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
J_5&=\frac{m-4}{2} \int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \left( \sum_{i}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_i\right)\right)\right) ^2 d v_g \\
&=\frac{m-4}{2}\int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\beta\epsilon_{\beta}, du(e_i))h(u_j^\alpha\epsilon_{\alpha}, du(e_j)) dv_g.
\end{split}
\end{equation*}
By (38) in \cite{Li2017NonexistenceOS},
\begin{equation*}
\begin{split}
J_6&= \int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \left(\lambda_{\alpha}^2 -\sum_\beta \lambda_{\beta}\lambda_{\alpha}\right) h(u_i^\alpha \epsilon_\alpha, du(e_j))\bigg(h(du(e_i), du(e_j)) +\frac{m-4}{4}|du |^2g(e_i,e_j)\bigg) dv_g\\
&=\int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \left(\lambda_{\alpha}^2 -\sum_\beta \lambda_{\beta}\lambda_{\alpha}\right) h(u_i^\alpha \epsilon_\alpha, \sigma_{u}(e_i)) dv_g.
\end{split}
\end{equation*}
If $ F^{\prime\prime} =F^\prime$
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right)\left\|S_u\right\|^{2}\left\{\lambda_{m}^{2}\left\|S_u\right\|^{2}\right.\\
&\left.+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\right\} d v_{g}\\
+&\frac{m-4}{2}\int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\beta\epsilon_{\beta}, du(e_i))h(u_j^\alpha\epsilon_{\alpha}, du(e_j)) dv_g.
\end{split}
\end{equation}
when $ F^{\prime\prime}(t)t\leq c_F F^{\prime}(t) $
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|S_u\right\|^{2}}{4}\right)\left\|S_u\right\|^{2}\bigg(\lambda_{m}^{2}c_F+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\bigg) d v_{g} \\
+&\frac{m-4}{2}\int_M F^{\prime}\left(\frac{\left\|S_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\beta\epsilon_{\beta}, du(e_i))h(u_j^\alpha\epsilon_{\alpha}, du(e_j)) dv_g.
\end{split}
\end{equation}
\end{proof}
\subsection{Okayasu type result}
In this section, we give Okayasu type theorem(cf. Howard \cite{Howard1985,Howard1986}).
\begin{thm}
Let $ u : (M^m,g) \to (N^n,h) $ be stable conformal $ \Phi_{S, F,H} $-harmonic map with potential $ H $ with conformal factor $ \lambda $ from compact Riemannian manifold $ M $ into a compact simply connected $ \delta $-pinched n-dimensional
Riemannian manifold $ N$. If one of the following conditions holds,
(1) there exists a constant $ \frac{(m-2)^2}{m+1}c_F+<(\frac{m(m-4)}{4}+1)(n-2)-\frac{m^2}{2}$ and
\begin{equation*}
\begin{split}
&\left(\frac{(m-2)^2}{m+1}c_F+(\frac{m(m-4)}{4}+1)+\frac{m-4}{2}m+2m\right)\left\{\frac{n}{4} k_{3}^{2}(\delta)+k_{3}(\delta)+1\right\}\\
-&(\frac{m(m-4)}{4}+1)\frac{2 \delta}{1+\delta}(n-1)<0.
\end{split}
\end{equation*}
(2) $ F^{\prime\prime}=F^{\prime} $ and
\begin{equation*}
\begin{split}
\left((\frac{m(m-4)}{4}+1)m \lambda+(\frac{m(m-4)}{4}+1)+\frac{m-4}{2}m+2m\right)\left\{\frac{n}{4} k_{3}^{2}(\delta)+k_{3}(\delta)+1\right\}\\
-(\frac{m(m-4)}{4}+1)\frac{2 \delta}{1+\delta}(n-1) <0.
\end{split}
\end{equation*}
Then $ u $ must be a constant.
\end{thm}
\begin{proof}
As in the proof of Theorem \ref{thm2.3}.
As in \cite{MR2259738}, we assume the sectional curvature of N is equal to $ \frac{2\delta}{1+\delta} .$ Let the vector bundle $ E=TN\oplus \epsilon(N) ,$ here, $ \epsilon(N) $ is the trivial bundle. We use the notions in \cite{ara2001stability} \cite{MR2259738}, we can define a metric connection $ \nabla^{\prime\prime} $ as follows:
\begin{equation}\label{}
\begin{split}
&\nabla^{\prime\prime}_XY={}^{N}\nabla_XY-h(X,Y)e;\\
&\nabla^{\prime\prime}_Xe=X,
\end{split}
\end{equation}
We cite the functions in \cite{ara2001stability} or \cite{MR2259738},
\begin{equation*}
\begin{split}
k_{1}(\delta)=\frac{4(1-\delta)}{3 \delta}\left[1+\left(\sqrt{\delta} \sin \frac{1}{2} \pi \sqrt{\delta}\right)^{-1}\right],
k_{2}(\delta)=\left[\frac{1}{2}(1+\delta)\right]^{-1} k_{1}(\delta). \\
\end{split}
\end{equation*}
By \cite{ara2001stability} \cite{MR2259738}, we know that there exists a flat connection $ \nabla^{\prime} $ such that
\begin{equation*}
\begin{split}
\|\nabla^{\prime}-\nabla^{\prime\prime}\|\leq \frac{1}{2}k_3(\delta),
\end{split}
\end{equation*}
where $ k_3(\delta) $ is defined as
\begin{equation*}
\begin{split}
k_{3}(\delta)=k_{2}(\delta) \sqrt{1+\left(1-\frac{1}{24} \pi^{2}\left(k_{1}(\delta)\right)^{2}\right)^{-2}} .
\end{split}
\end{equation*}
Along the line in \cite{ara2001stability}, take a cross section $ W $ of $ E $ and let $ W^T $ denotes the $ TN $ component of $ W $ , by the second variation formula \eqref{second} for $ F $-sympohic map with potential, .
\begin{equation}\label{ckj}
\begin{split}
&I(W^T,W^T)\\
=&-\int_{M} \operatorname{HessH}(W^T, W^T) d v_{g} \\
&+\int_{M} F^{\prime \prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right)\left\langle \widetilde{\nabla} W^T, \sigma_{u}\right\rangle \left\langle \widetilde{\nabla} W^T, \sigma_{u}\right\rangle d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} W^T, \widetilde{\nabla}_{e_{j}} W^T\right) \left( \frac{m(m-4)}{4}+1\right) \lambda d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} W^T, d u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_{i}} W^T, d u\left(e_{j}\right)\right) d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} W^T, d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} W^T\right) d v_{g} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(R^{N}\left(W^T, d u\left(e_{i}\right)\right) W^T, d u\left(e_{i}\right)\right) (\frac{m(m-4)}{4}+1)\lambda d v_{g}.
\end{split}
\end{equation}
Let $\mathcal{W}:=\left\{W \in \Gamma(E) ; \nabla^{\prime} W=0\right\}$, then $\mathcal{W}$ with natural inner product is isomorphic to $\mathbf{R}^{n+1}$. Define a quadratic form $Q$ on $\mathcal{W}$ by
$$ Q(W):=.$$
Taking an orthonormal basis $\left\{W_{1}, W_{2}, \ldots, W_{n}, W_{n+1}\right\}$ of $\mathcal{W}$ with respect to its natural inner product such that $ W_1,W_2, \cdots, W_n $ is tangent to $N$, we obtain
Meanwhile, we observe that
\begin{equation}\label{}
\begin{split}
\tilde{\nabla}_{e_i} W^T &={ }^N \nabla_{u_* e_i} W^T \\
&=\nabla_{u_* e_i}^{\prime \prime} W^T+\left\langle W^T, u_* e_i\right\rangle e \\
&=\nabla_{u_* e_i}^{\prime \prime}(W-\langle W, e\rangle e)+\left\langle W^T, u_* e_i\right\rangle e \\
&=\left(\nabla_{u_* e_i}^{\prime \prime} W\right)^T-\langle W, e\rangle u_* e_i .
\end{split}
\end{equation}
notice that
\begin{equation}\label{586}
\begin{split}
|\left(\nabla_{u_{*} e_i}^{\prime \prime} W\right)^T | =|\sum_{j=1}^{n}\left \langle \nabla_{u_{*} e_i}^{\prime \prime}\left( W\right) , W_j\right \rangle W_j|\leq \frac{nk_3(\delta)}{2}|u_*(e_i)| |W|.
\end{split}
\end{equation}
Then, we have
\begin{equation}\label{}
\begin{split}
\sum_{i=1}^m\left|\tilde{\nabla}_{e_i} W^T\right|^2=& \sum_{i=1}^m\left|\left(\nabla_{u_*{e_i} }^{\prime \prime} W\right)^T\right|^2+\langle W, e\rangle^2|\mathrm{~d} u|^2 \\
&-2 \sum_{i=1}^m\langle W, e\rangle\left\langle\nabla_{u_* e_i}^{\prime \prime} W^T, u_* e_i\right\rangle .
\end{split}
\end{equation}
By \eqref{586}
\begin{equation*}
\begin{split}
\left \langle \tilde{\nabla}_{e_i} W^T,\sigma_{u}(e_i) \right \rangle =& \left \langle \tilde{\nabla}_{e_i} W^T,du(e_j) \right \rangle h(du(e_i),du(e_j))\\
&\leq (\frac{m(m-4)}{4}+1)\lambda |\tilde{\nabla}_{e_i} W^T| |du| .
\end{split}
\end{equation*}
Since \begin{equation*}
\begin{split}
\sigma_{u}(e_i)=&h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)du(e_j)+\frac{m-4}{4}|\mathrm{~d} u|^{2} du(e_i)\\
=&(\frac{m(m-4)}{4}+1)\lambda du(e_i)
\end{split}
\end{equation*}
So, \begin{equation*}
\begin{split}
\sum_{ij} \left \langle \tilde{\nabla}_{e_i} W^T,\sigma_{u}(e_i) \right \rangle\left \langle \tilde{\nabla}_{e_j} W^T,\sigma_{u}(e_j) \right \rangle \leq (\frac{m(m-4)}{4}+1)m \lambda^2 |\tilde{\nabla}_{e_i} W^T|^2 .
\end{split}
\end{equation*}
Since $ u $ is conformal map with conformal factor $ \lambda, $
\begin{equation*}
\begin{split}
\left \langle \tilde{\nabla}_{e_i} W^T ,\tilde{\nabla}_{e_j} W^T \right \rangle h(du(e_i),du(e_j)) \leq (\frac{m(m-4)}{4}+1) \lambda |\tilde{\nabla}_{e_i} W^T|^2 .
\end{split}
\end{equation*}
By \eqref{586},
\begin{equation*}
\begin{split}
&\sum_{ij}\left \langle \tilde{\nabla}_{e_i} W^T,du(e_j) \right \rangle \left \langle \tilde{\nabla}_{e_i} W^T,du(e_j) \right \rangle \leq |\tilde{\nabla}_{e_i} W^T|^2 |du|^2
\end{split}
\end{equation*}
Similarly, we also have
\begin{equation*}
\begin{split}
&\sum_{ij}\left \langle \tilde{\nabla}_{e_i} W^T,du(e_j) \right \rangle \left \langle \tilde{\nabla}_{e_j} W^T,du(e_i) \right \rangle \leq |\tilde{\nabla}_{e_i} W^T|^2 |du|^2
\end{split}
\end{equation*}
Since $N$ is $\delta$-pinched, similar to \cite[(5.4)]{MR2259738}, we have
\begin{equation*}
\begin{split}
&h\left(R^N\left(W^T, u_* e_i\right) u_* e_i, W^T\right)h(du(e_i),du(e_j))\\
\geq & (\frac{m(m-4)}{4}+1)\frac{2\lambda \delta}{1+\delta}\left\{\left|W^T\right|^2\left|u_* e_i\right|^2-\left\langle W^T, u_* e_i\right\rangle^2\right\}.
\end{split}
\end{equation*}
Substituting ()() into (\ref{ckj}), we obtain
\begin{equation*}
\begin{split}
I\left(W^T, W^T\right) \leq& \int_M F^{\prime}\left(\frac{|u^* h|^2}{2}\right) \lambda q(W) \mathrm{d}v_g,
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
q(W)=&\left(\frac{(m-2)^2}{m+1}c_F+(\frac{m(m-4)}{4}+1)+\frac{m-4}{2}m+2m\right)\left\{\sum_{i=1}^{m}\left|\left(\nabla_{u_{e} e_{i}}^{\prime \prime} W\right)^{T}\right|^{2}+\langle W, e\rangle^{2}|\mathrm{~d} u|^{2}\right.\\
&\left.\quad-2 \sum_{i=1}^{m}\langle W, e\rangle\left\langle\nabla_{u_{*} e_{i}}^{\prime \prime} W^{T}, u_{*} e_{i}\right\rangle\right\} \\
&-(\frac{m(m-4)}{4}+1)\frac{2 \delta}{1+\delta} \sum_{i=1}^{m}\left\{\left|W^{T}\right|^{2}\left|u_{*} e_{i}\right|^{2}-\left\langle W^{T}, u_{*} e_{i}\right\rangle^{2}\right\}.
\end{split}
\end{equation*}
By the similar argument in \cite{MR2259738}, we have
\begin{equation*}
\begin{split}
\operatorname{trace}(q)\leq \Phi_{n,F}(\delta)|du |^2.
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
\Phi_{n,F}(\delta)=\left(\frac{(m-2)^2}{m+1}c_F+(\frac{m(m-4)}{4}+1)+\frac{m-4}{2}m+2m\right)\left\{\frac{n}{4} k_{3}^{2}(\delta)+k_{3}(\delta)+1\right\}\\
-(\frac{m(m-4)}{4}+1)\frac{2 \delta}{1+\delta}(n-1).
\end{split}
\end{equation*}
Taking traces , we get
\begin{equation*}
\begin{split}
Tr_g I \leq& \int_{M} F^{\prime}\left(\frac{|u^*h|^2}{2}\right)\bigg[\lambda\Phi_{n,F}(\delta)|du |^2\bigg] \d v_g.
\end{split}
\end{equation*}
In the case that $ F^{\prime\prime}=F^{\prime} $
\begin{equation*}
\begin{split}
I\left(W^T, W^T\right) \leq& \int_M F^{\prime}\left(\frac{|u^* h|^2}{2}\right) \lambda q(W) \mathrm{d}v_g,
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
q(W)=&\left((\frac{m(m-4)}{4}+1)m \lambda+(\frac{m(m-4)}{4}+1)+\frac{m-4}{2}m+2m\right)\\
&\left\{\sum_{i=1}^{m}\left|\left(\nabla_{u_{e} e_{i}}^{\prime \prime} W\right)^{T}\right|^{2}+\langle W, e\rangle^{2}|\mathrm{~d} u|^{2}\right.\left.\quad-2 \sum_{i=1}^{m}\langle W, e\rangle\left\langle\nabla_{u_{*} e_{i}}^{\prime \prime} W^{T}, u_{*} e_{i}\right\rangle\right\} \\
&-(\frac{m(m-4)}{4}+1)\frac{2 \delta}{1+\delta} \sum_{i=1}^{m}\left\{\left|W^{T}\right|^{2}\left|u_{*} e_{i}\right|^{2}-\left\langle W^{T}, u_{*} e_{i}\right\rangle^{2}\right\}.
\end{split}
\end{equation*}
\end{proof}
\begin{cor}
Let $ (M,g) $ be a compact simply connected $ \delta $-pinched n-dimensional
Riemannian manifold , if there exists a constant $ c_F<\frac{n}{2}-1 $ and one of the following conditions holds,
(1) there exists a constant $ c_F<\frac{n}{2}-1 $ and
\begin{equation*}
\begin{split}
\left(\frac{4}{\lambda}c_{F}+2m+1\right)\left\{\frac{n}{4} k_{3}^{2}(\delta)+k_{3}(\delta)+1\right\}-\frac{2 \delta}{1+\delta}(n-1) <0.
\end{split}
\end{equation*}
(2) $ F^{\prime\prime}=F^{\prime} $ and
\begin{equation*}
\begin{split}
\left(\lambda^4 m+2m+1\right)\left\{\frac{n}{4} k_{3}^{2}(\delta)+k_{3}(\delta)+1\right\}-\frac{2 \delta}{1+\delta}(n-1) <0.
\end{split}
\end{equation*}
Then $M$ is $\Phi_{S,F,H}$-unstable.
\end{cor}
\subsection{Howard type result}
In this section, we show the Howard type theorem for $ F $-symphonic map. In \cite{takeuchi1991stability}, Takeuchi obtained Howard type theroem for $ p $ harmonic map. Later, in \cite{ara2001stability}, Ara generalized Takeuchi's result to $ F $ harmonic map.
\begin{thm}
Let $F:[0, \infty) \rightarrow[0, \infty)$ be a $C^{2}$ strictly increasing function, $\nabla^2 H$ is semipositive. is nonincreasing $\}$. Let $N$ be a compact simply-connected $\delta$-pinched $n$-dimensional Riemannian manifold. Assume that $n$ and $\delta$ satisfy $n>\frac{4}{m\lambda}c_F+\lambda+2m+1$ and
\begin{equation*}
\begin{split}
\Psi_{n, F}(\delta):=\int_{0}^{\pi}\bigg\{\bigg(\frac{(m-2)^2}{m+1}c_F+(\frac{m(m-4)}{4}+1)\lambda+3m\lambda\bigg) g_{2}(t, \delta)\left(\frac{\sin \sqrt{\delta} t}{\sqrt{\delta}}\right)^{n-1}\\
-(n-1) \delta \cos ^{2}(t) \sin ^{n-1}(t)\bigg\} \d t<0,
\end{split}
\end{equation*}
where $g_{2}(t, \delta)=\max \left\{\cos ^{2}(t), \delta \sin ^{2}(t) \cot ^{2}(\sqrt{\delta} t)\right\}$. Then for any compact Riemannian manifold $M$, every $\Phi_{S,F, H}$-stable conformal $ F $-symphonic map with potential $u: M \rightarrow N$ with conformal factor $ \lambda $ is constant.
\end{thm}
\begin{proof}
We follow \cite{ara2001stability}, let $ V^{y}=\nabla f\circ \rho_y,$ we can approximate $ f $ by $ f_k $, one can refer to \cite{ara2001stability} for the construction of $ f_k $. Let $ V_k^{y}=\nabla f_k\circ \rho_y $. Then $ V_k^y $ converges uniformly to $ V^y $ as $ k\to \infty $ . By the second variation formula for $ F $-symphonic map, we have
\begin{equation*}
\begin{split}
I(V^y,V^y)
\leq
&\int_{M} F^{\prime \prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \left\langle \widetilde{\nabla} V^y, \sigma_{u}\right\rangle \left\langle \widetilde{\nabla} V^y, \sigma_{u}\right\rangle d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} V^y, \widetilde{\nabla}_{e_{i}} V^y\right) (\frac{m(m-4)}{4}+1)\lambda d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} V^y, d u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_{i}} V^y, d u\left(e_{j}\right)\right) d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} V^y, d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} V^y\right) d v_{g} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, }^{m} h\left(R^{N}\left(V^y, d u\left(e_{i}\right)\right) V^y, d u\left(e_{i}\right)\right) (\frac{m(m-4)}{4}+1)\lambda d v_{g},\\
\end{split}
\end{equation*}
Notice that
\begin{equation*}
\begin{split}
\left\langle \widetilde{\nabla} V^y, \sigma_{u}\right\rangle \left\langle \widetilde{\nabla} V^y, \sigma_{u}\right\rangle=(\frac{m(m-4)}{4}+1)m\lambda^2 |\widetilde{\nabla}_{e_i} V^y |^2.
\end{split}
\end{equation*}
while
\begin{equation*}
\begin{split}
\|S_u\|^2= |u^{*}h |^2+|du|^4=m\lambda^2+m^2\lambda^2
\end{split}
\end{equation*}
Moreover, by \cite[(3.7)]{ara2001stability}, we have
\begin{equation*}
\begin{split}
\sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} V^y, d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} V^y\right)\leq g_2(\rho,\delta)|du|^4.
\end{split}
\end{equation*}
Thus, we have
\begin{equation*}
\begin{split}
I(V^y,V^y)\leq & \int_{M} F^{\prime\prime}\left(\frac{\left\|S_u\right\|^{2}}{4}\right)(\frac{m(m-4)}{4}+1)m\lambda^2 |\widetilde{\nabla}_{e_i} V^y |^2\\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_u\right\|^{2}}{4}\right) \sum_{i=1 }^{m} \bigg[(\frac{m(m-4)}{4}+1)\lambda|\tilde{\nabla}_{e_i} V^y|^2\\
&+3g_2(\rho,\delta)|du|^4+\lambda h\left( R^{N}\left(V^y, d u\left(e_{i}\right)\right) V^y, d u\left(e_{i}\right)\right)\bigg] d v_{g}.\\
\leq & \int_{M} F^{\prime}\left(\frac{\left\|S_u\right\|^{2}}{4}\right) \sum_{i=1 }^{m} \bigg[\frac{(m-2)^2}{m+1}c_F|\tilde{\nabla}_{e_i} V^y|^2+(\frac{m(m-4)}{4}+1)\lambda|\tilde{\nabla}_{e_i} V^y|^2\\
&+3g_2(\rho,\delta)|du|^4+\lambda h\left( R^{N}\left(V^y, d u\left(e_{i}\right)\right) V^y, d u\left(e_{i}\right)\right)\bigg] d v_{g}.
\end{split}
\end{equation*}
Taking limits, we have
\begin{equation*}
\begin{split}
&\lim\limits_{k\to \infty} \int_N I(V_{k}^{y},V_{k}^{y})dv_N(y)\\
\leq & \int_N \int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, }^{m} \bigg[\bigg(\frac{(m-2)^2}{m+1}c_F+(\frac{m(m-4)}{4}+1)\lambda\bigg)|\tilde{\nabla}_{e_i} V^y|^2\\
&+\lambda h\left( R^{N}\left(V^y, d u\left(e_{i}\right)\right) V^y, d u\left(e_{i}\right)\right)+2g_2(\rho,\delta)|du|^4\bigg] d v_{g} dv_N(y).
\end{split}
\end{equation*}
By \cite[(3.3),(3.4)]{ara2001stability}, we have
\begin{equation*}
\begin{split}
&\lim\limits_{k\to \infty} \int_N I(V_{k}^{y},V_{k}^{y})dv_N(y)\\
\leq & \int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right)\bigg( \bigg[ \bigg(\frac{(m-2)^2}{m+1}c_F+(\frac{m(m-4)}{4}+1)\lambda\bigg)g_2(\rho,\delta) |du |^2\Vol(S^{n-1})\left(\frac{\sin(\sqrt{\delta}t)}{\sqrt{\delta}} \right)^{n-1}\\
&\quad\quad\quad\quad\quad\quad\quad\quad -\lambda(n-1)|du |^2Vol(S^{n-1}) \int_{0}^{\pi}\delta\cos^2(\rho)\sin^{n-1}(\rho)d\rho\bigg]\\
&\quad\quad\quad\quad\quad\quad\quad\quad+2g_2(\rho,\delta)|du|^4 \Vol(S^{n-1}) \left(\frac{\sin(\sqrt{\delta}t)}{\sqrt{\delta}} \right)^{n-1}\bigg)\d v_{g} .
\end{split}
\end{equation*}
using the fact that $ |du |^2=\lambda m,$ by the same argument in \cite{{ara2001stability}}, we can get a contradiction.
\end{proof}
\subsection{Other result}
Motivated by the method in \cite{torbaghan2022stability}, we can establish
Han and Wei \cite{Han2019HarmonicMA} discussed the example of $\Phi$ SSU manifold and stability of $\Phi$ harmonic map which is the special case of $ F $ symphonic map in this paper.
\begin{thm}\label{thm2.1}
Let $u:\left(M^{m}, g\right) \longrightarrow\left(N^{n}, h\right)$ be a nonconstant conformal $\Phi_{S, F,H}$-harmonic map with conformal factor $ \lambda $ between Riemannian manifolds. Let $ \theta(v)=\sum_{a=1}^n |B(v,v_a) |^2 $, $f(x)=\max\{|B(v,v) |^2: v\in T_xN, |v|=1\}. $ Suppose that $ m\geq 6 $ and
\begin{equation*}
\begin{split}
\bigg( (4m-16)c_F|B|_\infty^2 +f(x)+2|B|_\infty^2\bigg)m
+(1+\frac{m^2-4m}{4}) \theta(v)\leq (1+\frac{m^2-4m}{4})Ric(v,v)
\end{split}
\end{equation*}
at each $x \in N$ and any unit vector $v \in T_{x} N$. Then, $u$ is $\Phi_{S, F,H}$-unstable.
\end{thm}
\begin{proof} We adpated the proof of Theorem 10 in \cite{torbaghan2022stability}.
\begin{equation*}
\begin{split}
&I_{u}(\omega^{\top},\omega^{\top})=\int_{M} \operatorname{Hess}H(\omega^{\top}, \omega^{\top}) d v_{g}+ \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left\langle\widetilde{\nabla} \omega^{\top}, \sigma_{u}\right\rangle\left\langle\widetilde{\nabla} \omega^{\top}, \sigma_{u}\right\rangle \mathrm{d} v_{g}\\
&+
\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} \omega^{\top}, \widetilde{\nabla}_{e_j} \omega^{\top}\right) \times g_{ij} (\lambda+\frac{m-4}{4}|du |^2)\mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} \omega^{\top}, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} \omega^{\top}, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), \tilde{\nabla}_{e j} \omega^{\top}\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} \omega^{\top}, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e j} \omega^{\top}, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(R^{N}\left(\omega^{\top}, \mathrm{~d} u\left(e_{i}\right)\right) \omega^{\top}, \mathrm{~d} u\left(e_{j}\right)\right) g_{ij} (\lambda+\frac{m-4}{4}|du |^2)\mathrm{d} v_{g},
\end{split}
\end{equation*}
By \cite[(30)(31)]{torbaghan2022stability}
\begin{equation*}
\begin{split}
&\left\langle\widetilde{\nabla} \omega^{\top}, \sigma_{u}\right\rangle\\
=&\left( h(du(e_i),du(e_j))\right) \left \langle B(du(e_i),du(e_j))) , \omega^{\perp}\right \rangle +\frac{m-4}{4}|du |^2\left \langle B(du(e_i),du(e_i))) , \omega^{\perp}\right \rangle
\end{split}
\end{equation*}
By \cite[(34)]{torbaghan2022stability}
\begin{equation*}
\begin{split}
\sum_{i, j} h\left(\widetilde{\nabla}_{e_i} \omega^{\top}, \widetilde{\nabla}_{e_j} \omega^{\top}\right) g_{ij} \leq |du |^2 \sum_i f(x) |du(e_i) |^2 ,
\end{split}
\end{equation*}
and So, by \cite[(30)(31)]{torbaghan2022stability}, we can get
\begin{equation*}
\begin{split}
\langle\tilde{\nabla}_{e_i} \omega^{\top}, \mathrm{~d} u\left(e_{j}\right)\rangle=\left \langle B(du(e_i),du(e_j))) , \omega^{\perp}\right \rangle
\end{split}
\end{equation*}
choose the orthonormal basis $ \{\theta_k,\theta_{\beta}\} $ on $ R^{n+p} $ such that $ \theta_k $ are tangent to $ N $ , $ \theta_{\beta} $ are normal to $ N $.
\begin{equation}\label{}
\begin{split}
&\sum_{k} I(\theta_k,\theta_k)\\
=& \sum_{k} \int_{M} \operatorname{Hess}H(\theta_k, \theta_k) d v_{g}+ \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\left\langle\widetilde{\nabla} \theta_k, \sigma_{u}\right\rangle\left\langle\widetilde{\nabla} \theta_k, \sigma_{u}\right\rangle \mathrm{d} v_{g}\\
&+ \sum_{k}
\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} \theta_k, \widetilde{\nabla}_{e_j} \theta_k\right) \times g_{ij} (\lambda+\frac{m-4}{4}|du |^2)\mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} \theta_k, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} \theta_k, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), \tilde{\nabla}_{e j} \theta_k\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} \theta_k, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} \theta_k, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(R^{N}\left(\theta_k, \mathrm{~d} u\left(e_{i}\right)\right) \theta_k, \mathrm{~d} u\left(e_{j}\right)\right) g_{ij} (\lambda+\frac{m-4}{4}|du |^2)\mathrm{d} v_{g}
\end{split}
\end{equation}
\begin{equation}\label{711}
\begin{split}
&\sum_{k} \left\langle\widetilde{\nabla} \theta_k, \sigma_{u}\right\rangle^2\\
=& \sum_{\beta}\left( \sum_{ij}\left( h(du(e_i),du(e_j))\right) \left \langle B(du(e_i),du(e_j))) ,\theta_{\beta}\right \rangle +\frac{m-4}{4}\sum_{i}|du |^2\left \langle B(du(e_i),du(e_i))) , \theta_{\beta}\right \rangle \right)^2 \\
\leq &2\sum_{\beta}\bigg(\sum_{ij}\left( h(du(e_i),du(e_j))\right) \left \langle B(du(e_i),du(e_j))) ,\theta_{\beta}\right \rangle\bigg)^2\\
&+2\sum_{\beta}\bigg(\frac{m-4}{4}\sum_{i}|du |^2\left \langle B(du(e_i),du(e_i))) , \theta_{\beta}\right \rangle\bigg)^2\\
\leq & 2\sum_{\beta}\sum_{ij}\bigg( h(du(e_i),du(e_j))\bigg)^2\sum_{ij}\bigg(\left( \left \langle B(du(e_i),du(e_j))) ,\theta_{\beta}\right \rangle\right) \bigg)^2\\
&+2\sum_{\beta}\bigg(\frac{m-4}{4}\sum_{i}|du |^2\left \langle B(du(e_i),du(e_i))) , \theta_{\beta}\right \rangle\bigg)^2\\
=& 2|u^*h|^2 |B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2+4\left( \frac{m-4}{4}\right)^2 |du |^4 |B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2\\
=&\left( 2|u^*h|^2+4\left( \frac{m-4}{4}\right)^2 |du |^4\right)|B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2 \\
\leq &(m-4)|S_u |^2|B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2,
\end{split}
\end{equation}
where in the last inequality, we have used the fact that $ m\geq 6. $
\begin{equation}\label{712}
\begin{split}
\sum_k (\lambda+\frac{m-4}{4}|du |^2) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} \theta_k, \widetilde{\nabla}_{e_j} \theta_k \right) g_{ij} \leq (\lambda+\frac{m-4}{4}|du |^2) \sum_i \theta(v_i) |du(e_i) |^2 ,
\end{split}
\end{equation}
and
\begin{equation}\label{713}
\begin{split}
&\sum_{k} \sum_{i=1}^{m}\sum_{j=1}^{m} h\left(\widetilde{\nabla}_{e_i} \theta_k, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\tilde{\nabla}_{e_j} \theta_k, \mathrm{~d} u\left(e_{i}\right)\right)\\
=&\sum_{\beta}\sum_{i=1}^{m}\sum_{j=1}^{m}\left \langle B(du(e_i),du(e_j))) , \theta_\beta \right \rangle^2\\
\leq& |B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2 ,
\end{split}
\end{equation}
and
\begin{equation}\label{714}
\begin{split}
&\sum_{k}\sum_{i=1}^{m}\sum_{j=1}^{m} h\left(\widetilde{\nabla}_{e_i} \theta_k, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\tilde{\nabla}_{e_i} \theta_k, \mathrm{~d} u\left(e_{j}\right)\right)\\
=&\sum_{\beta}\sum_{i=1}^{m}\sum_{j=1}^{m}\left \langle B(du(e_i),du(e_j))) , \theta_\beta \right \rangle^2 \leq |B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2,
\end{split}
\end{equation}
and
\begin{equation}\label{}
\begin{split}
&\sum_{k} \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} \theta_k, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} \theta_k, \mathrm{~d} u\left(e_{j}\right)\right)\\
=&\sum_{\beta}\sum_{i=1}^{m}\sum_{j=1}^{m}\left \langle B(du(e_i),du(e_i))) , \theta_\beta \right \rangle\left \langle B(du(e_j),du(e_j))) , \theta_\beta \right \rangle\\
\leq& f(x)|du |^2 \sum_i |du(e_i) |^2.
\end{split}
\end{equation}
Thus, set $ v_i=\frac{du(e_i)}{|du(e_i)|}, $ we get we have
\begin{equation}\label{}
\begin{split}
Tr_g I&= \int_{M} F^{\prime \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)\frac{m-4}{2}|S_u |^2|B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2 dv_g\\
&+\int_{M} F^{ \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right)(\lambda+\frac{m-4}{4}|du |^2) \sum_i \theta(v_i) |du(e_i) |^2 dv_g\\
&+2\int_{M} F^{ \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) |B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2 dv_g\\
&+\int_{M}F^{ \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) f(x)|du |^2 \sum_i |du(e_i) |^2 \\
&- (\lambda+\frac{m-4}{4}|du |^2)F^{ \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_i |du(e_i) |^2 Ric(v_i,v_i) dv_g\\
\leq& \int_M F^{ \prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_i |du(e_i) |^2 \bigg((4m-16)c_F|B|_\infty^2 |du |^2+2|B|_\infty^2 |du |^2+f(x)|du |^2\\
&+(\lambda+\frac{m-4}{4}|du |^2) \theta(v_i)-(\lambda+\frac{m-4}{4}|du |^2)Ric(v_i,v_i)\bigg) dv_g.
\end{split}
\end{equation}
\end{proof}
\begin{thm}\label{thm2.2}
Let $ u:(M^{n-1}, g) \to (N^{n}, h) $ be a
totally geodesic isometric immersion of a hypersurface
$ M^{n-1} $ to $ N^n $. Then $ u $ is $\Phi_{S, F,H}$-unstable if the Ricci curvature
of $ N^n $ is positive and $ F^\prime(x)>0. $
\end{thm}
\begin{proof}
Since $ u $ is totally geodesic isometric immersion, by \cite[(40)]{torbaghan2022stability}, we have
\begin{equation*}
\begin{split}
\nabla_{e_i}V =0.
\end{split}
\end{equation*}
The conclusion follows from
\begin{equation*}
\begin{split}
I(V,V)=&(1+\frac{m-4}{4}|du |^2)\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \mathrm{Ric}(V,V) \mathrm{d} v_{g}.
\end{split}
\end{equation*}
\end{proof}
\subsection{$\Phi_{S, F, H} $ -stable manifold}\label{cbm}
Motivated by the method in \cite{torbaghan2022stability}, in this subsection, we will give some condition under which M is $\Phi_{S, F} $ -stable or $\Phi_{S, F} $-U.
By Theorem \ref{thm21a} and Theorem \ref{thm23a}, we immediately get
\begin{thm}
Let $ (M^m, g) $ be a compact $ \Phi $-SSU Riemannian
manifold. Let $ a $ and $ F $ be defined in Theorem \ref{thm21a}, Then, $ M $ is $ \Phi_{S, F, H} $-SU and also $ \Phi_{S, F, H} $-U.
\end{thm}
\begin{thm}\label{stable1}
Let $ (M^m, g) $ be a compact Riemannian
manifold. Let $ l_F=\inf_{t\geq 0}\{ \frac{F^{\prime\prime}(t) }{F^{\prime}(t) } |F^{\prime\prime}(t) \geq a F^{\prime}(t) \} $. If
$1\leq m \leq 4 , F^{\prime} \geq 0, l_F \geq \frac{m(4-m)}{8}, \nabla^2 H \geq 0 $. Then, $ M $ is an $ \Phi_{S, F, H} $-stable manifold
\end{thm}
\begin{rem}
By \cite[Theorem 7.11]{Han2019HarmonicMA} , the dimension of compact $ \Phi $-SSU manifold must bigger than 4.
\end{rem}
\begin{proof}
Let $ \{ e_i\} $ be an orthonormal frame field on $ M $. We use the mthod in \cite[Theorem 13 ]{torbaghan2022stability} or \cite[section 3]{ara2001instability}. When $ u=id,\sigma_{u}(e_i)=(1+\frac{(m-4)m}{4})e_i $, we get the following formula,
\begin{equation}\label{}
\begin{split}
\left\langle\nabla V, \sigma_{u}\right\rangle\left\langle\nabla V, \sigma_{u}\right\rangle=(1+\frac{m^2-4m}{4})|div(V) |^2,
\end{split}
\end{equation}
and
\begin{equation}\label{k20}
\begin{split}
\left \langle \nabla_{e_i}v,\nabla_{e_j}v \right \rangle \left( h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right) =(1+\frac{m^2-4m}{4})|\nabla_{e_i} V |^2,
\end{split}
\end{equation}
and
\begin{equation*}
\begin{split}
&h\left(R^{N}\left(V, \mathrm{~d} u\left(e_{i}\right)\right) V, \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)+\frac{m-4}{4}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}\\
=&-(1+\frac{m^3-4m^2}{4}) \left \langle Ric(V), V \right \rangle.
\end{split}
\end{equation*}
Plugging the above three formulas into the index form , we get
\begin{equation}\label{369}
\begin{split}
I_{id}(V,V)=&\jifen{\nabla^2 H(V,V)}+\jifen{F^{\prime\prime}(\frac{m^2}{2}) \operatorname{div}(V)^2 (1+\frac{m^2-4m}{4}) }\\
&+\int_{M} F^{\prime}(\frac{m^{2}}{2}) \left\{\langle\nabla V, \nabla V\rangle-\sum_{t=1}^{m} h( R^N\left(V, e_{i}\right) e_{i}, V)\right\} (1+\frac{m^2-4m}{4})\d v_{g}\\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(\nabla_{e_i} V, e_{j}\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\nabla_{e_i} V, e_{i}\right) \sum_{j=1}^{m} h\left(\nabla_{e_j} V, e_{j}\right) \mathrm{d} v_{g} \\
\end{split}
\end{equation}
Next we deal with each terms of \eqref{369} respectively,
and
\begin{equation}\label{371}
\begin{split}
\sum_{ij} h\left(\nabla_{e_i} V, e_{j}\right)h\left(\nabla_{e_j} V, e_{i}\right)=& \frac{1}{2}|L_Vg |^2-\sum_i|\nabla_{e_i} V |^2 \\
\end{split}
\end{equation}
by the definition of divergence of vector field, we have
\begin{equation}\label{372}
\begin{split}
\sum_{ij} h\left(\nabla_{e_i} V, e_i\right)h\left(\nabla_{e_j} V, e_j\right)= \left( \sum_{i} \left \langle \nabla_{e_i} V, e_i\right \rangle\right) ^2\geq \div(V)^2,
\end{split}
\end{equation}
So we have (see \cite{Han2019HarmonicMA} )
\begin{equation}\label{370}
\begin{split}
&\int_{M} g\left(J_{2, i d}(V), V\right) \d V_{g}
=\int_{M}\left\{\frac{1}{2}|\mathcal{L}_V g|^{2}-(\operatorname{div} V)^{2}\right\} \d V_{g},
\end{split}
\end{equation}
by \eqref{369} \eqref{371}\eqref{372}\eqref{370}, we have
\begin{equation}\label{6.6}
\begin{split}
&I_{id}(V,V)\\
\geq &\int_M F^{\prime\prime}(\frac{(m-2)^2m}{16})(1+\frac{m^2-4m}{4})|div(V) |^2 dv_g\\
&+\int_M F^{\prime}(\frac{(m-2)^2m}{16}) (1+\frac{m^2-4m}{4}) (\frac{1}{2}|\mathcal{L}_V g|^{2}-(\operatorname{div} V)^{2}) dv_g \\
&+ \int_M F^{\prime}(\frac{(m-2)^2m}{16}) \sum \langle \nabla_{e_i} V,e_j\rangle^2 \d v_g\\
&+\int_M F^{\prime}(\frac{(m-2)^2m}{16}) \left( \frac{1}{2}|L_Vg |^2-\sum_i|\nabla_{e_i} V |^2\right) dv_g+ \frac{m-4}{2}F^{\prime}(\frac{(m-2)^2m}{16}) \int_M \div(V)^2 dv_g
\end{split}
\end{equation}
Noticing that
\begin{equation}\label{jfg}
\begin{split}
|L_Vg |^2 \geq \frac{4}{m} |\operatorname{div}(V)|^2
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\sum_{i,j} \langle \nabla_{e_i} V,e_j\rangle^2 \geq \sum_i|\nabla_{e_i} V |^2
\end{split}
\end{equation}
this inequality becomes an equality if $ V $ is a non-isometric conformal vector field.
Hence, we get
\begin{equation*}
\begin{split}
&I(V,V)\\
\geq& \int_M \bigg(F^\prime(\frac{(m-2)^2m}{16})(\frac{4}{m}l_F+\frac{m-4}{2})\bigg)|div(V) |^2 dv_g\\
&+\int_{M}F^{\prime}(\frac{(m-2)^2m}{16})\left( \frac{2}{m}+\frac{(m-2)^2}{4}(\frac{2}{m}-1)\right) |\div V |^2dv_g \\
\end{split}
\end{equation*}
By the assumption in the theorem, we can get
\begin{equation*}
\begin{split}
I(V,V)\geq 0.
\end{split}
\end{equation*}
\end{proof}
Take $ F(x)=x $ in Theorem \ref{stable1}
\begin{cor}\label{stable1}
Let $ (M, g) $ be a compact four dimensional Riemannian
manifold. Then, $ M $ is an $ \Phi_{S, H} $-stable manifold if
$ F^{\prime} \geq 0, \nabla^2 H \geq 0 $.
\end{cor}
\begin{thm}\label{thm2.4}
Let $ (M^m, g) $ be a compact Riemannian manifold. Assume that there exists a nonisometric conformal vector field $ V $ on $ M $. If $\nabla^2 H \leq 0 $ and
$$ F^{\prime\prime}(\frac{(m-2)^2m}{16}) + F^{\prime}(\frac{(m-2)^2m}{16})(\frac{4}{m}-1+\frac{m-4}{2}) <0, F^{\prime}(\frac{(m-2)^2m}{16})\leq 0. $$ Then, $ M $ is
$ \Phi_{S,F, H} $-U.
\end{thm}
\begin{proof}
For non-isometric conformal vector field $ V $ , we have
\begin{equation*}
\begin{split}
\frac{1}{2}|L_V g |^2=\frac{2}{m}(\operatorname{div} V)^2 \neq 0.
\end{split}
\end{equation*}
by the second variation formula and \eqref{371}, we get
\begin{equation}\label{}
\begin{split}
I_{id}(V,V)=& \jifen{\nabla^2 H(V,V)}+\jifen{fF^{\prime\prime}(\frac{m^2}{2}) div(V)^2}\\
&+\int_{M}F^{\prime}(\frac{m^{2}}{2}) \cdot\left\{\langle\nabla V, \nabla V\rangle-\sum_{t=1}^{m} h( R^N\left(V, e_{i}\right) e_{i}, V)\right\} dv_{g}\\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(\nabla_{e_i} V, e_{j}\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\nabla_{e_i} V, e_{i}\right) \sum_{j=1}^{m} h\left(\nabla_{e_j} V, e_{j}\right) \mathrm{d} v_{g} \\
\leq& \int_M F^{\prime\prime}(\frac{(m-2)^2m}{16}) div(V)^2 \d v_g+ \int_{M} F^{\prime}(\frac{(m-2)^2m}{16})(\frac{2}{m}-1+\frac{m-4}{2}) div(V)^2 \d v_g \\
&+\int_M F^{\prime}(\frac{(m-2)^2m}{16}) \left( \frac{1}{2}|L_Vg |^2-\sum_i|\nabla_{e_i} V |^2\right) \d v_g\\
&+\int_{M} F^{\prime}(\frac{(m-2)^2m}{16}) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right)^2\mathrm{d} v_{g}
\end{split}
\end{equation}
\end{proof}
\begin{thm}[The second variation formula]
Let $\psi$ : $(M, g) \longrightarrow(N, h)$ be an $\alpha-$ harmonic map and $\left\{\psi_{t, s}:\right.$ $M \longrightarrow N\}_{-\epsilon<s, t<\epsilon}$ be a $ 2 $-parameter smooth variation of $\psi$ such that $\psi_{0,0}=\psi$.$$
v=\left.\frac{\partial \psi_{t, s}}{\partial t}\right|_{s=t=0}, \quad \omega=\left.\frac{\partial \psi_{t, s}}{\partial s}\right|_{s=t=0},
$$ Then
\begin{equation*}
\begin{split}
&\left.\frac{\partial^2}{\partial t \partial s} E_\alpha(\psi)\right|_{t=s=0}=- \int_M h\left(B_\alpha(v), \omega\right)\\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{j}\right)\right) h\left(\mathrm{~d} u\left(e_{i}\right), _{e j} W\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e j} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
\end{split}
\end{equation*}
where
and the operator $J_\alpha(v) \in \Gamma\left(\psi^{-1} T N\right)$ is given by
$$
\begin{aligned}
B_\alpha(v) =&F^{\prime}(\frac{|d\psi |^2}{2}){} \operatorname{Tr}_g R^N(v, \sigma_{u}) \sigma_{u} + \operatorname{Tr}_g \nabla\bigg(\langle\nabla v,\sigma_{u}\rangle F^{\prime\prime}(\frac{|d\psi |^2}{2})\sigma_{u}\bigg) \\
&+ \operatorname{Tr}_g \nabla_{\cdot} \bigg(F^{\prime}(\frac{|d\psi |^2}{2}) h(du(e_i),du(\cdot)) \tilde{\nabla}_{e_i}V+F^{\prime}(\frac{|d\psi |^2}{2}) \frac{m-4}{4}\nabla_{\cdot} v \bigg)
\end{aligned}
$$
\end{thm}
\begin{thm}
Let $(M^m, g)$ be an Einstein manifold. Then, $M$ is an $\Phi_{S,F,H}$-stable manifold if and only if \begin{equation*}
\begin{split}
R_g \leq m\mu_1 (\frac{F^{\prime\prime}(\frac{(m-2)^2m}{16})}{F^{\prime}(\frac{(m-2)^2m}{16})}+1), m\geq 4, F^{\prime}(\frac{(m-2)^2m}{16}) > 0
\end{split}
\end{equation*}
where $\mu_{1}$ is the smallest positive eigenvalue of the Laplacian operating on functions.
\end{thm}
\begin{proof}
Firstly, we prove only if part modifying the proof of \cite[Theorem 17]{torbaghan2022stability}. Assume that $M$ is $\Phi_{S,F,H}$-stable and Let Ricci tensor equals $\lambda g$ for some constant $\lambda$.
When $ u=id $, by \eqref{370}\eqref{371}
\begin{equation}\label{kkp}
\begin{split}
I_{id}(V,V)=&(1+\frac{m^2-4m}{4})F^{\prime\prime}(\frac{(m-2)^2m}{16}) \jifen{ div(V)^2}+(1+\frac{m^2-4m}{4})F^{\prime}(\frac{(m-2)^2m}{16}) \int_M h\left(J_{2,id}(v), \omega\right)\\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(\nabla_{e_i} V, e_{j}\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\nabla_{e_i} V, e_{i}\right) \sum_{j=1}^{m} h\left(\nabla_{e_j} V, e_{j}\right) \mathrm{d} v_{g} \\
& \geq J_{F, i d}(V,V)
\end{split}
\end{equation}
Let \begin{equation*}
\begin{split}
J_{F, i d}(V,V)=(1+\frac{m^2-4m}{4})\jifen{F^{\prime\prime}(\frac{(m-2)^2m}{16}) div(V)^2}\\
+(1+\frac{m^2-4m}{4})\int_MF^{\prime}(\frac{(m-2)^2m}{16}) h\left(J_{2,id}(v), \omega\right)
\end{split}
\end{equation*}
We modify the formula in \cite[Theorem 17]{torbaghan2022stability}.
$$
\begin{aligned}
&g\left(J_{F, i d}(X+g r a d \kappa), X+\operatorname{grad} \kappa\right) \\
&=g\left(J_{F, i d}(X), X\right)+g\left(J_{F, i d}(\operatorname{grad\kappa }), \operatorname{grad} \kappa\right) .
\end{aligned}
$$
considering div $X=0$, by \eqref{370} we get
$$
\begin{aligned}
&\int_M g\left(J_{\alpha, i d}(X), X\right) \\
&=(1+\frac{m^2-4m}{4})F^{\prime}(\frac{(m-2)^2m}{16}) \int_M \frac{1}{2}\left|\mathcal{L}_X g\right|^2 d V_g \geq 0.
\end{aligned}
$$
By \eqref{kkp},we have
\begin{equation*}
\begin{split}
I_{id}(X,X)\geq 0.
\end{split}
\end{equation*}
Let $f$ is a smooth function on $M$ such that $\Delta f=\mu_1 f$.
Let $$D=(1+\frac{m^2-4m}{4})F^{\prime\prime}(\frac{(m-2)^2m}{16}),E=(1+\frac{m^2-4m}{4})F^{\prime}(\frac{(m-2)^2m}{16})$$
Then we have
$$
\begin{aligned}
&\int_M g\left(J_{F, i d}\left(\operatorname{grad} f\right), g r a d f\right) \\
&=D \int_M g\left(\operatorname{grad}\left(\Delta f\right), g r a d f\right) d V_g +E \int_M g\left(\operatorname{grad} \Delta f-\lambda g r a d f, g a d f\right) d V_g \\
&=\left( \mu_1 (D+E)-\lambda E\right) \int_M\left|\operatorname{grad} f\right|^2 d V_g
\end{aligned}
$$
We have
$$
\lambda \leq \mu_1 (\frac{D}{E}+1)=\mu_1 (\frac{F^{\prime\prime}(\frac{(m-2)^2m}{16})}{F^{\prime}(\frac{(m-2)^2m}{16})}+1) .
$$
By \eqref{kkp},we have
\begin{equation*}
\begin{split}
I_{id}(V,V)\geq 0.
\end{split}
\end{equation*}
Conversely, the proof of if part is almost the same as that in \cite[Theorem 17]{torbaghan2022stability}.
\end{proof}
\begin{thm}
Let $(M,g)$ be a compact manifold with $\left\langle\operatorname{Ric}^{M}(v), v\right\rangle_{M}=$ $\frac{1}{m} R_g$, for every unit vector $v$ at every point of $M$.If
$$ \lambda>(2+\frac{m^2-4m}{4})^{-1}(3+\frac{m^2-4m}{2})\frac{1}{m}R_g, F^{\prime} < 0, , F^{\prime\prime} \leq 0.$$
Then M is $\Phi_{S,F,H}$-U.
\end{thm}
\begin{proof}
Let $ J_{2,id}(V)= \nabla^{*}\nabla v -\Ric(V) $, Let $ V=(df)^{\sharp} $ such that $ -\bar{\Delta} V= \lambda V $
\begin{equation}\label{kkp}
\begin{split}
I_{id}(V,V)=&(1+\frac{m^2-4m}{4})F^{\prime\prime}(\frac{(m-2)^2m}{16}) \jifen{ div(V)^2}\\
&+(1+\frac{m^2-4m}{4})F^{\prime}(\frac{(m-2)^2m}{16}) \int_M h\left(J_{2,id}(V), V\right)\\
&+\int_{M} F^{\prime}\left(\frac{(m-2)^2m}{16}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(\nabla_{e_i} V, e_{j}\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{(m-2)^2m}{16}\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{k} \\
&+ \frac{m-4}{2} \int_{M} F^{\prime}\left(\frac{(m-2)^2m}{16}\right) \sum_{i=1}^{m} h\left(\nabla_{e_i} V, e_{i}\right) \sum_{j=1}^{m} h\left(\nabla_{e_j} V, e_{j}\right) \mathrm{d} v_{g} \\
\leq &F^{\prime}(\frac{(m-2)^2m}{16})\bigg[ (1+\frac{m^2-4m}{4})\int_M h\left(J_{2,id}(V), V\right) \d v_g\\
&+\int_{M} \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{g}\bigg] \\
\end{split}
\end{equation}
where $ J_{2,id}(V)=-\nabla^{*}\nabla -Ric(V) $
By \cite[(8.4)]{Han2019HarmonicMA} ,
\begin{equation*}
\begin{split}
\int_{M} \sum_{i, j=1}^{m}\left\langle\nabla_{e_{i}} v, e_{j}\right\rangle\left\langle e_{i}, \nabla_{e_{j}} v\right\rangle d x=\int_{M}-V(\operatorname{div} V)-\left\langle\operatorname{Ric}^{M}(V), V\right\rangle d x\\
= \frac{1}{2}|L_Vg |^2-\sum_i|\nabla_{e_i} V |^2
\end{split}
\end{equation*} and also \begin{equation*}
\begin{split}
\int_M h\left(J_{2,id}(v), V\right)=& \int_M \frac{1}{2}|L_Vg |^2-\sum_i|\nabla_{e_i} V |^2\\
=&\int_M \langle-\nabla^{*}\nabla V -\Ric(V), V \rangle \d v_g\\
=& \int_M \langle -\bar{\Delta} V -2\Ric(V), V \rangle \d v_g
\end{split} \end{equation*} Thus, \begin{equation*}
\begin{split}
&(1+\frac{m^2-4m}{4})\int_M h\left(J_{2,id}(V), V\right) \d v_g
+\int_{M} \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{g}\\
=& \int_M ((2+\frac{m^2-4m}{4})\lambda-(3+\frac{m^2-4m}{2})\frac{1}{m}R_g) |df |^2\d v_g
\end{split} \end{equation*}
\end{proof}
\section{$ \Phi_{T,F,H} $ harmonic map }
\subsection{$ \Phi_{T,F,H} $ harmonic map from $ \Phi$-SSU manifold }
\begin{thm}[The first variation, c.f. \cite{Han2013StabilityOF}]
Let $u:(M, g) \rightarrow(N, h)$ be a smooth map, and let $u_{t}: M \rightarrow N,(-\varepsilon<t<\varepsilon)$ be a compactly supported variation such that $u_{0}=u$ and set $V=\left.\frac{\partial}{\partial t} u\right|_{t=0}$. Then we have
\[
\left.\frac{d}{d t} \Phi_{T,F,H}\left(u_{t}\right)\right|_{t=0}=-\int_{M} h\left(V,\tau_{T,F,H}(u)\right) d v_{g},
\]
where $ \sigma_{u}(\cdot)=h(du(\cdot),du(e_j))du(e_j)-\frac{1}{m}|du |^2 du(\cdot), $
$ \tau_{T,F,H}(u)=\delta^\nabla\left(F^{\prime}\left(\frac{|T_u|^{2}}{2}\right)\sigma_{u}\right) -\nabla H(u)=0. $
\end{thm}
\begin{proof}
\begin{equation*}
\begin{split}
&\frac{\partial}{\partial t} F\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right)=F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) \sum_{i=1}^{m} e_{i} g\left(X_{t}, e_{i}\right) -F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) \sum_{i=1}^{m}\left[h\left(d \Psi\left(\frac{\partial}{\partial t}\right), \tilde{\nabla}_{e_{i}} \sigma_{u_{i}}\left(e_{i}\right)\right)\right]\\
&=F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) \sum_{i=1}^{m}\left[g\left(\nabla_{e_{i}} X_{t}, e_{i}\right)+g\left(X_{t}, \nabla_{e_{i}} e_{i}\right)\right]-F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) \sum_{i=1}^{m} h\left(d \Psi\left(\frac{\partial}{\partial t}\right), \tilde{\nabla}_{e_{i}} \sigma_{u_{t}}\left(e_{i}\right)\right)\\
&=F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) \operatorname{div}_{g}\left(X_{t}\right) -F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) \sum_{i=1}^{m} h\left(d \Psi\left(\frac{\partial}{\partial t}\right), \tilde{\nabla}_{e_{i}} \sigma_{u_{t}}\left(e_{i}\right)-\sigma_{u_{t}}\left(\nabla_{e_{i}} e_{i}\right)\right)\\
&=\operatorname{div}\left(F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) X_{t}\right)-g\left(X_{t}, \operatorname{grad}\left(F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right)\right)\right)\\
&-F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) \sum_{i=1}^{m} h\left(d \Psi\left(\frac{\partial}{\partial t}\right), \tilde{\nabla}_{e_{i}} \sigma_{u_{t}}\left(e_{i}\right)-\sigma_{u_{i}}\left(\nabla_{e_{i}} e_{i}\right)\right)\\
&=\operatorname{div}\left(F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) X_{t}\right)-h\left(d \Psi\left(\frac{\partial}{\partial t}\right), F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right) d i v_{g} \sigma_{u_{t}}\right)+\sigma_{u_{t}}\left(\operatorname{grad}\left(F^{\prime}\left(\frac{\left\|T_{u_{t}}\right\|^{2}}{4}\right)\right)\right) \text {. }
\end{split}
\end{equation*}
\end{proof}
\begin{thm}[ The second variation formula, see \cite{Han2013StabilityOF} theorem 5.1 ]
Let $u:(M, g) \rightarrow(N, h)$ be an F-stationary map. Let $u_{s, t}: M \rightarrow N(-\varepsilon<s, t<\varepsilon)$ be a compactly supported twoparameter variation such that $u_{0,0}=u$ and set $V=\left.\frac{\partial}{\partial t} u_{s, t}\right|_{s, t=0}, W=\left.\frac{\partial}{\partial s} u_{s, t}\right|_{s, t=0}$. Then
\begin{equation}\label{gg1}
\begin{split}
&\left.\frac{\partial^2}{\partial s \partial t} \Phi_F\left(u_{s, t}\right)\right|_{s, t=0}\\
=&\int_M \nabla^2H(V,W)dv_g+ \int_M F^{\prime \prime}\left(\frac{\left\|T_u\right\|^2}{4}\right)\left\langle\tilde{\nabla} V, \sigma_u\right\rangle\left\langle\tilde{\nabla} W, \sigma_u\right\rangle d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, \tilde{\nabla}_{e_j} W\right) T_u\left(e_i, e_j\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_j\right)\right) h\left(\tilde{\nabla}_{e_i} W, d u\left(e_j\right)\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_j\right)\right) h\left(d u\left(e_i\right), \tilde{\nabla}_{e_j} W\right) d v_g \\
&-\frac{2}{m} \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_i\right)\right) h\left(d u\left(e_j\right), \tilde{\nabla}_{e_j} W\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(R^N\left(V, d u\left(e_i\right)\right) W, d u\left(e_j\right)\right) T_u\left(e_i, e_j\right) d v_g .
\end{split}
\end{equation}
where $\langle,\rangle$ is the inner product on $ T^* M \otimes u^{-1} T N$ and $R^N$ is the curvature tensor of $N$, where $ T_u=u^*h-\frac{1}{m}\|du\|^2 g, \sigma_{u}(\cdot)=h(du(\cdot),du(e_j))du(e_j)-\frac{1}{m}|du |^2 du(\cdot) $
\end{thm}
\begin{thm}\label{thm3.3} Let $ (M^m, g) $ be a compact $ \Phi
$-$ SSU $ manifold. For $x \in M$
$$ a=\min _{X \in U M, Y \in U M} \frac{-\left\langle Q_{x}^{M}(X), X\right\rangle_{M}}{8|B(X, X)|_{\mathbf{R}^{r}}|B(Y, Y)|_{\mathbf{R}^{r}}}>0,$$ Let $ F $ be the positive function determined by Lemma \ref{dkl}, $ \nabla^2 H \leq 0 $.
Let $ u $ be stable $ \Phi_{T,F, H} $-
harmonic map with potential from $ (M^m, g) $ into any Riemannian manifold $ N $, then $ u $ is constant.
\end{thm}
\begin{proof}
We use the same notations as in the proof of Theorem 6.1 in \cite{2021The}. we modify the proof in \cite{caoxiangzhi202212}. We choose an orthogonal frame field $\left\{e_{1}, \cdots, e_{m+p}\right\}$ of $R^{m+p}$ such that $\left\{e_{i}\right\}_{i=1}^{m}$ are tangent to $M^{m},\left\{e_{\alpha}\right\}_{\alpha=m+1}^{m+p}$ are normal to $M^{m}$ and $\left.\nabla_{e_{i}} e_{j}\right|_{x}=0$, where $x$ is a fixed point of $M$. We take a fixed orthonormal basis of $R^{m+p}$ denoted by $E_{D}, D=$ $1, \cdots, m+p$ and set
\[
V_{D}=\sum_{i=1}^{m} v_{D}^{i} e_{i}, v_{D}^{i}=\left\langle E_{D}, e_{i}\right\rangle, v_{D}^{\alpha}=\left\langle E_{D}, e_{\alpha}\right\rangle, \alpha=m+1, \cdots, m+p,
\]
where $\langle\cdot, \cdot\rangle$ is the canonical Euclidean inner product.
Let RHS of \eqref{gg1} be the index form $ I(V,W) $.
\begin{equation}\label{kkj}
\begin{split}
I(du(V_D),du(V_D))=\sum_{i=0}^{6}J_i
\end{split}
\end{equation}
similar to \eqref{k2},
\begin{equation}\label{k91}
\begin{split}
J_1=&F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A}\left\langle\tilde{\nabla} d u\left(V_{D}\right), \sigma_{u}\right\rangle^{2} \\
=& \sum_{A} F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right)\left[\sum_{i}\left\langle\tilde{\nabla}_{e_{i}} d u\left(V_{D}\right), \sigma_{u}\left(e_{i}\right)\right\rangle\right]\left[\sum_{j}\left\langle\tilde{\nabla}_{e_{j}} d u\left(V_{D}\right), \sigma_{u}\left(e_{j}\right)\right\rangle\right] \\
=& F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A} \left[\sum_{i}\left(-v_{D}^{\alpha}B_{ik}^{\alpha} h\left(d u\left(e_{k}\right), \sigma_{u}\left(e_{i}\right)\right)+v_{D}^{k} h\left(\tilde{\nabla}_{e_{i}} d u\left(e_{k}\right), \sigma_{u}\left(e_{i}\right)\right)\right)\right] \\
&\times \sum_{A} \left[\sum_{i}\left(-v_{D}^{\alpha}B_{jl}^{\alpha} h\left(d u\left(e_{l}\right), \sigma_{u}\left(e_{j}\right)\right)+v_{D}^{k} h\left(\tilde{\nabla}_{e_{j}} d u\left(e_{k}\right), \sigma_{u}\left(e_{j}\right)\right)\right)\right] \\
=& F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{i} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] \\
&+F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A}\left[\sum_{l} v_{A}^{l}\left[\sum_{i} h\left(\left(\nabla_{e_{i}} d u\right)\left(e_{l}\right), \sigma_{u}\left(e_{i}\right)\right)\right]\right]^{2} \\
=& F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{i} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] \\
&+ F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right)\left[\sum_{l}\left[\sum_{i} h\left(\left(\nabla_{e_{l}} d u\right)\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right)\right]^{2}\right],
\end{split}
\end{equation}
and
\begin{equation}\label{k81}
\begin{split}
J_2&=F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j} h\left(\widetilde{\nabla}_{e_i} du(V_D), \widetilde{\nabla}_{e_j} du(V_D)\right) \times\left.\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\right] \mathrm{d}\\
&=F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right)
\bigg\{\sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(A^{\alpha}\left(e_{j}\right)\right)\right) \left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)-\frac{1}{m}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] \\
&+\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right),\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right)\right) \left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)-\frac{1}{m}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\bigg\},
\end{split}
\end{equation}
and
\begin{equation}\label{k101}
\begin{split}
J_3=&F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} du(V_D), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_i} du(V_D), \mathrm{~d} u\left(e_{j}\right)\right)\\
=&\sum h(du(A^\alpha(e_i)),du(e_j))h(du(e_i),du(A^\alpha(e_j)))\\
&+\sum h((\nabla_{e_k}du)((e_i)),du(e_j))h(du(e_i),(\nabla_{e_k}du)((e_j)))
\end{split}
\end{equation}
and
\begin{equation}\label{k111}
\begin{split}
J_4=&F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_i} du(V_D), \mathrm{~d} u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_j} du(V_D), \mathrm{~d} u\left(e_{i}\right)\right)\\
=&\sum h(du(A^\alpha A^\alpha(e_i)),du(e_j))h(du(e_i),du((e_j)))\\
&+\sum h((\nabla_{e_k}du)((e_i)),du(e_j))h((\nabla_{e_k}du)((e_i)),du(e_j)),
\end{split}
\end{equation}
and
\begin{equation}\label{k121}
\begin{split}
J_5=&\frac{2}{m} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} du(V_D), \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} du(V_D), \mathrm{~d} u\left(e_{j}\right)\right)
\\
=&\frac{2}{m} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum h(du(A^\alpha(e_i)),du(e_i))h(du(e_j),du(A^\alpha(e_j)))\\
&+\frac{2}{m} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum h((\nabla_{e_k}du)((e_i)),du(e_i))h((\nabla_{e_k}du)((e_j)),du(e_j)),
\end{split}
\end{equation}
and
\begin{equation}\label{k131}
\begin{split}
J_6=&F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(Ric^M(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) \left( h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\\
=& F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(\mathrm{trace}(A^\alpha)A^\alpha(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) \left( h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right)\\
&-F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(A^\alpha A^\alpha(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) \left( h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right).
\end{split}
\end{equation}
The last term $ J_7 $ can help us to cancel the manny terms including $ \nabla du $ above
\begin{equation}\label{k141}
\begin{split}
J_7=& F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j} h\left(\left(\nabla^{2} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)-\frac{1}{m}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\\
=&\sum_{i, j, k} e_{k}\left[ F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)h\left(\nabla_{e_{k}} d u\left(e_{i}\right), du\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)-\frac{1}{m}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\right]\\
&- F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right),\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)-\frac{1}{m}|d u|^{2} g\left(e_{i}, e_{j}\right)\right]\\
&- F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right),\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right)\right)\\
&- F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, k} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right) h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{j}\right)\right) \\
&- F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\frac{2}{m} h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{i}\right), d u\left(e_{i}\right)\right) h\left(\left(\nabla_{e_{k}} d u\right)\left(e_{j}\right), d u\left(e_{j}\right)\right)\\
&-F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right)\left[\sum_{l}\left[\sum_{i} h\left(\left(\nabla_{e_{l}} d u\right)\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right)\right]^{2}\right],
\end{split}
\end{equation}
where we have used the formula
\begin{equation}\label{k151}
\begin{split}
e_kF^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right)=F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) h\left(\left(\nabla_{e_{l}} d u\right)\left(e_{i}\right), \sigma_{u}\left(e_{i}\right)\right)
\end{split}
\end{equation}
Plugging \eqref{k81}\eqref{k91}\eqref{k101}\eqref{k111}\eqref{k121}\eqref{k131} and \eqref{k141} into \eqref{kkj},
\begin{equation}\label{k161}
\begin{split}
&\sum_{D} I\left(d u\left(V_{D}\right), d u\left(V_{D}\right)\right)\\
=& F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{i} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] \\
+&\int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(A^{\alpha}\left(e_{j}\right)\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)-\frac{1}{m}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), d u\left(A^{\alpha}\left(e_{j}\right)\right)\right) d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i, j, \alpha} h\left(d u\left(A^{\alpha} A^{\alpha}\left(e_{i}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right) d v_{g}\right. \\
&+\frac{2}{m}\int_{M}F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{i}\right)\right) h\left(d u\left(A^{\alpha}\left(e_{j}\right)\right), d u\left(e_{j}\right)\right) d v_{g} \\
&-\int_{M}F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \sum_{i, j, \alpha} h\left(d u\left(\operatorname{trace}\left(A^{\alpha}\right) A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{j}\right)\right) \\
&\times\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)+\frac{m-4}{4}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] d v_{g} \\
&+\int_{M}F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \sum_{i, j, \alpha} h\left(d u\left(A^{\alpha} A^{\alpha}\left(e_{i}\right)\right), d u\left(e_{j}\right)\right)\left[h\left(d u\left(e_{i}\right), d u\left(e_{j}\right)\right)-\frac{1}{m}|d u|^{2} g\left(e_{i}, e_{j}\right)\right] d v_{g} .
\end{split}
\end{equation}
Choose normal coordinate such that $ h(du(e_i),du(e_j))=\lambda_i^2 \delta_{ij} $
\begin{equation*}
\begin{split}
& F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{i} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] \\
=&F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right)\left \langle B(e_i,e_i) ,B(e_j,e_j) \right \rangle \lambda_i^2(\lambda_i^2-\frac{1}{m}|du |^2)\lambda_j^2(\lambda_j^2-\frac{1}{m}|du |^2)F^{\prime\prime}.
\end{split}
\end{equation*}
Hence, by \cite{Han2013StabilityOF}, we get
\begin{equation*}
\begin{split}
&\sum_{D} I\left(d u\left(V_{D}\right), d u\left(V_{D}\right)\right) \\
&\leq \int_M F^{\prime\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \left \langle B(e_i,e_i) ,B(e_j,e_j) \right \rangle \lambda_i^2(\lambda_i^2-\frac{1}{m}|du |^2)\lambda_j^2(\lambda_j^2-\frac{1}{m}|du |^2) d v_{g}\\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i} \lambda_{i}^{2}\left(\lambda_{i}^{2}-\frac{1}{m}|d u|^{2}\right) \sum_{j}\left(4\left\langle B\left(e_{i}, e_{j}\right), B\left(e_{i}, e_{j}\right)\right\rangle-\left\langle B\left(e_{i}, e_{i}\right), B\left(e_{j}, e_{j}\right)\right\rangle\right)\\
&\leq \int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{i} \lambda_{i}^{2}\left(\lambda_{i}^{2}-\frac{1}{m}|d u|^{2}\right) \frac{1}{2}\sum_{j}\left(4\left\langle B\left(e_{i}, e_{j}\right), B\left(e_{i}, e_{j}\right)\right\rangle-\left\langle B\left(e_{i}, e_{i}\right), B\left(e_{j}, e_{j}\right)\right\rangle\right).
\end{split}
\end{equation*}
\end{proof}
\subsection{ $ \Phi_{T,F,H} $ harmonic map into $\Phi$-SSU manifold }
\begin{thm} \label{thm34}
Let $ (N^n, g) $ be a compact $ \Phi
$-$ SSU $ manifold. For $x \in M$
$$ a=\min _{X \in U N, Y \in U N} \frac{-\left\langle Q_{x}^{N}(X), X\right\rangle_{N}}{8|B(X, X)|_{\mathbf{R}^{r}}|B(Y, Y)|_{\mathbf{R}^{r}}}>0,$$ Let $ F $ be the positive function determined by Lemma \ref{dkl}, $ \nabla^2 H \leq 0 $.
Let $ u $ be stable $ \Phi_{S,F, H} $-
harmonic map with potential from $ (M^m, g) $ into any Riemannian manifold $ N $, then $ u $ is constant.
\end{thm}
\begin{proof}
We use the same notations as in the proof of Theorem 7.1 in \cite{2021The}. We modify the proof in \cite{caoxiangzhi202212}.
Let $\left\{e_{1}, \cdots, e_{m}\right\}$ be a local orthonormal frame field of $M$ . Let $\left\{\epsilon_{1}, \cdots, \epsilon_{n}, \epsilon_{n+1}, \cdots, \epsilon_{n+p}\right\}$ be an orthonormal frame field of $R^{n+p}$, such that $\left\{\epsilon_{i}, \cdots, \epsilon_{n}\right\}$ are tangent to $N^{n}, \epsilon_{n+1}, \cdots, \epsilon_{n+p}$ are normal to $N^{n}$ and $\left.{}^{N}\nabla_{\epsilon_{b}} \epsilon_{c}\right|_{u(x)}=0$, where $x$ is a fixed point of $M$. As in \cite{2021The}, we fix an orthonormal basis $E_{D}$ of $R^{m+p}$, for $D=1, \cdots, m+p$ and set
\begin{equation*}
\begin{split}
V_{D}&=\sum_{b=1}^{n} v_{D}^{b} \epsilon_{a}, v_{D}^{b}=\left\langle E_{D}, \epsilon_{b}\right\rangle,\\
v_{D}^{\alpha}&=\left\langle E_{D}, \epsilon_{\alpha}\right\rangle, \quad for \quad \alpha=n+1, \cdots, n+p,\\
\nabla_{\epsilon_{b}} V_{D}&=\sum_{\alpha=n+1}^{n+p} \sum_{c=1}^{n} v_{D}^{\alpha} B_{b c}^{\alpha} \epsilon_{c}, 1 \leq b \leq n;
\end{split}
\end{equation*}
choose local frame such that
\begin{equation}\label{kkk}
\begin{split}
\sum_{i=1}^{m}u_{i}^{b}u_{i}^{c}=\lambda_b^2 \delta_{bc}.
\end{split}
\end{equation}
Inspired by the formula (62)-(66) in \cite{2021The}, similar to \eqref{k9}, using (59) in \cite{2021The}, the we have
\begin{equation}
\begin{split}
&F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A}\left\langle\tilde{\nabla} \left(V_{D}\right), \sigma_{u}\right\rangle^{2} \\
=& \sum_{A} F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right)\left[\sum_{i}\left\langle\tilde{\nabla}_{e_{i}} \left(V_{D}\right), \sigma_{u}\left(e_{i}\right)\right\rangle\right]\left[\sum_{j}\left\langle\tilde{\nabla}_{e_{j}} \left(V_{D}\right), \sigma_{u}\left(e_{j}\right)\right\rangle\right] \\
=& F^{\prime \prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{A} B_{ik}^{\alpha} B_{jl}^{\alpha}\left[\sum_{i} h\left(d u\left(e_{i}\right), \sigma_{u}\left(e_{k}\right)\right)\right] \left[\sum_{j} h\left(d u\left(e_{j}\right), \sigma_{u}\left(e_{l}\right)\right)\right] .
\end{split}
\end{equation}
By the formula (62)-(66) in \cite{2021The}, we have
\begin{equation}\label{po}
\begin{split}
\sum_{D} I\left(V_{D}, V_{D}\right)\leq& \int_M F^{\prime\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \left \langle B(\epsilon_{b},\epsilon_{b}) ,B(\epsilon_{c},\epsilon_{c}) \right \rangle \lambda_b^2(\lambda_b^2-\frac{1}{m}|du |^2)\lambda_c^2(\lambda_c^2-\frac{1}{m}|du |^2) d v_{g}\\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{b} \lambda_{b}^{2}\left(\lambda_{b}^{2}+\frac{m-4}{4}|d u|^{2}\right)\\
&\times \sum_{c}\left(4\left\langle B\left(\epsilon_{b}, \epsilon_{c}\right), B\left(\epsilon_{b}, \epsilon_{c}\right)\right\rangle-\left\langle B\left(\epsilon_{b}, \epsilon_{b}\right), B\left(\epsilon_{c}, \epsilon_{c}\right)\right\rangle\right) \d v_{g}\\
\leq &\int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right)\sum_{b} \lambda_{b}^{2}\left(\lambda_{b}^{2}+\frac{m-4}{4}|d u|^{2}\right)\\
&\times \frac{1}{2} \sum_{c}\left(4\left\langle B\left(\epsilon_{b}, \epsilon_{c}\right), B\left(\epsilon_{b}, \epsilon_{c}\right)\right\rangle-\left\langle B\left(\epsilon_{b}, \epsilon_{b}\right), B\left(\epsilon_{c}, \epsilon_{c}\right)\right\rangle\right) d v_{g}.
\end{split}
\end{equation}
Since $N$ is a $\Phi-\mathrm{SSU}$ manifold, if $u$ is not constant, we have
\[
\sum_{D} I\left(V_{D}, V_{D}\right)<0.
\]
\end{proof}
From the above proof and \eqref{po}, it is immediately to get
\begin{thm}
Let $ (N^n, g) $ be a compact $ \Phi
$-SSU manifold and $ F^{\prime\prime}\leq 0, Hess H \leq 0 $. Then every stable $ \Phi_{S,F, H} $-
harmonic map u from $ (M^m, g) $ into any Riemannian manifold $ N $ is constant.
\end{thm}
\subsection{ $\Phi_{T,F,H}$ harmonic map when domain manifold is convex hypersurface}
\begin{thm}[c.f. \cite{Li2017NonexistenceOS}]\label{thm1}
Let $M \subset R^{m+1}$ be a compact convex hypersurface. Assume that the principal curvatures $\lambda_{i}$ of $M^{m}$ satisfy $0<\lambda_{1} \leq \cdots \leq \lambda_{m}$ and $3 \lambda_{m}<\sum_{i=1}^{m-1} \lambda_{i}$. Then every nonconstant $\Phi_{T,F,H}$-stationary map with potential from $M$ to any compact Riemannian manifold $N$ is $ \Phi_{T,F,H} $-unstable if one of the following two conditions hold.
(1) there exists a constant $c_{F}=\inf \left\{c \geq 0 | F^{\prime}(t) / t^{c}\right.$ is nonincreasing $\}$ such that
\begin{equation}\label{6a}
\begin{split}
c_{F}<\frac{1}{4 \lambda_{m}^{2}} \min _{1 \leq i \leq m}\left\{\lambda_{i}\left(\sum_{k=1}^{m} \lambda_{k}-2 \lambda_{i}-2 \lambda_{m}\right)\right\},
\end{split}
\end{equation}
(2) $F^{\prime \prime}(t)=F^{\prime}(t).$
\begin{equation}\label{7a}
\begin{split}
\left\|u^{*} h\right\|^{2}<\frac{1}{\lambda_{m}^{2}} \min _{1 \leq i \leq m}\left\{\lambda_{i}\left(\sum_{k=1}^{m} \lambda_{k}-2 \lambda_{i}-2 \lambda_{m}\right)\right\} .
\end{split}
\end{equation}
\end{thm}
\begin{proof}
We modify the proof in \cite[Theorem 1]{MR2259738} and \cite[Theorem 3.1 ]{Li2017NonexistenceOS}. In the proof of this theorem, we also borrow the notations and settings from \cite[Theorem 1]{MR2259738} and \cite[Theorem 3.1 ]{Li2017NonexistenceOS}.
We modify the proof of Theorem \ref{9c}.
In order to prove the instability of $u: M^{n} \rightarrow N$, we need to consider some special variational vector fields along $u$. To do this, choosing an orthogonal frame field $\left\{e_{i}, e_{n+1}\right\}, i=1, \ldots, n$, of $\mathbf{R}^{n+1}$, such that $\left\{e_{i}\right\}$ are tangent to $M^{n} \subset \mathbf{R}^{n+1}, e_{n+1}$ is normal to $M^{n}$ and $\left.\nabla_{e_{i}} e_{j}\right|_{P}=0$. Meanwhile, taking a fixed orthonormal basis $E_{A}, A=1, \ldots, n+1$, of $\mathbf{R}^{n+1}$ and setting
(3.1) $ V_{A}=\sum_{i=1}^{n} v_{A}^{i} e_{i}, v_{A}^{i}=\left\langle E_{A}, e_{i}\right\rangle, v_{A}^{n+1}=\left\langle E_{A}, e_{n+1}\right\rangle$
where $\langle\cdot, \cdot\rangle$ denotes the canonical Euclidean inner product. Then $u_{*} V_{A} \in$ $\Gamma\left(u^{-1} T N\right)$ and
\begin{equation*}
\begin{split}
\tilde{\nabla}_{e_{i}}\left(\mathrm{d} u\left(\nabla_{e_{i}} V_{A}\right)\right)= & -v_{A}^{k} h_{i k} h_{i j}\left(\mathrm{d} u\left(e_{j}\right)\right)+v_{A}^{n+1}\left(\tilde{\nabla}_{e_{i}} h_{i j}\right)\left(\mathrm{d} u\left(e_{j}\right)\right) \\
& +v_{A}^{n+1} h_{i j} \tilde{\nabla}_{u_{+} e_{i}} u_{*} e_{j}
\end{split}
\end{equation*}
where, $h_{i j}$ denotes the components of the second fundamental form of $M^{n}$ in $\mathbf{R}^{n+1}$
Similar to \eqref{9ccc} , we can also infer that
\begin{equation*}
\begin{split}
\int_{M}F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \left \langle (\Delta du)(V_A) ,\sigma_{u}(V_A) \right \rangle dv_g+\int_{M} \operatorname{HessH}(V_A, V_A)=0.
\end{split}
\end{equation*}
By \eqref{bochner} we have
\begin{equation}\label{}
\begin{split}
&I(du(V_A),du(V_A)) \\
=& \int_M F^{\prime \prime}\left(\frac{\left\|T_u\right\|^2}{4}\right)\left\langle\tilde{\nabla} du(V_A), \sigma_u\right\rangle\left\langle\tilde{\nabla} du(V_A), \sigma_u\right\rangle d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} du(V_A), \tilde{\nabla}_{e_j} du(V_A)\right) T_u\left(e_i, e_j\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} du(V_A), d u\left(e_j\right)\right) h\left(\tilde{\nabla}_{e_i} du(V_A), d u\left(e_j\right)\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} du(V_A), d u\left(e_j\right)\right) h\left(d u\left(e_i\right), \tilde{\nabla}_{e_j} du(V_A)\right) d v_g \\
&-\frac{2}{m} \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} du(V_A), d u\left(e_i\right)\right) h\left(d u\left(e_j\right), \tilde{\nabla}_{e_j} du(V_A)\right) d v_g \\
&+ \int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(du(Ric^M(e_i)), \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}\\
&+ \int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left((\nabla^2 du)(e_i), \mathrm{~d} u\left(e_{j}\right)\right) {\left[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\right] \mathrm{d} v_{g},}
\end{split}
\end{equation}
by\eqref{gg}
\begin{equation}
\begin{split}
&I(du(V_A),du(V_A))\\
&= \int_{M}\left\{F^{\prime \prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{A} (\widetilde{\nabla} d u\left(V_{A}\right), \sigma_{u})^{2} -h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}}\left[F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right)\right] \sigma_{u}\left(V_{A}\right)\right)\right\} d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) h\left(-2 \widetilde{\nabla}_{e_{i}}\left(d u\left(\nabla_{e_{i}} V_{A}\right)\right)\right.\left.+d u\left(\nabla_{e_{i}} \nabla_{e_{i}} V_{A}\right)-d u\left(\operatorname{Ric}^{M^{m}}\left(V_{A}\right)\right), \sigma_{u}\left(V_{A}\right)\right) d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) \big[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\big] d v_{g}\\
& +\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) d v_{g} \\
& - \frac{2}{m} \int_{M} F^{\prime}\left(\frac{\left\|S_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), d u\left(e_{j}\right)\right) d v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) d v_{g}\\
&+\int_{M} \sum_{A, i} h\left(\tilde{\nabla}_{e_{i}} d u\left(V_{A}\right), F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \tilde{\nabla}_{e_{i}} (\frac{1}{m}|du |^2du(V_A))\right) d v_{g}. \end{split}
\end{equation}
the last term is cancelled by
\begin{equation*}
\begin{split}
- \frac{2}{m} \int_{M} F^{\prime}\left(\frac{\left\|T_{u}\right\|{ }^{2}}{4}\right) \sum_{i=1}^{m} h\left(\widetilde{\nabla}_{e_i} V, \mathrm{~d} u\left(e_{i}\right)\right) \sum_{j=1}^{m} h\left(\tilde{\nabla}_{e_j} W, \mathrm{~d} u\left(e_{j}\right)\right) \mathrm{d} v_{g} ,
\end{split}
\end{equation*}
and
\begin{equation}\label{}
\begin{split}
\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) \big[-\frac{1}{m}|\mathrm{~d} u|^{2} g\left(e_{i}, e_{j}\right)\big] d v_{g}.
\end{split}
\end{equation}
Thus, \eqref{11} can be simplified as
\begin{equation}
\begin{split}
&I(du(V_A),du(V_A))\\
&= \int_{M}\left\{F^{\prime \prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{A} ( \widetilde{\nabla} d u\left(V_{A}\right), \sigma_{u})^{2} -h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}}\left[F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right)\right] T_{u}\left(V_{A}\right)\right)\right\} d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) h\left(-2 \widetilde{\nabla}_{e_{i}}\left(d u\left(\nabla_{e_{i}} V_{A}\right)\right)\right.\left.+d u\left(\nabla_{e_{i}} \nabla_{e_{i}} V_{A}\right)-d u\left(\operatorname{Ric}^{M^{m}}\left(V_{A}\right)\right), \sigma_{u}\left(V_{A}\right)\right) d v_{g} \\
&+\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) \big[h\left(\mathrm{~d} u\left(e_{i}\right), \mathrm{d} u\left(e_{j}\right)\right)\big] d v_{g}\\
& +\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{i, j, A} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} d u\left(V_{A}\right)\right) d v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), d u\left(e_{j}\right)\right) d v_{g} \\
&-\int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \sum_{A, i, j} h\left(\widetilde{\nabla}_{e_{i}} d u\left(V_{A}\right), d u\left(e_{j}\right)\right) h\left(d u\left(V_{A}\right), \widetilde{\nabla}_{e_{i}} d u\left(e_{j}\right)\right) d v_{g}.
\end{split}
\end{equation}
This formula is similar to (18) in \cite{Li2017NonexistenceOS} after replacing $ u^*h $ by $ T_u $. by the similar compuation as in \cite{Li2017NonexistenceOS}, we can get the similar formula as (19)-(30) in \cite{Li2017NonexistenceOS} after after replacing $ u^*h $ by $ T_u $.
If $ F^{\prime\prime} =F^\prime$
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right)\left\|T_u\right\|^{2}\left\{\lambda_{m}^{2}\left\|T_u\right\|^{2}\right.\\
&\left.+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\right\} d v_{g} .
\end{split}
\end{equation}
when $ F^{\prime\prime}(t)t\leq c_F F^{\prime}(t) $
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right)\left\|T_u\right\|^{2}\left\{\lambda_{m}^{2}c_F\right.\\
&\left.+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\right\} d v_{g} .
\end{split}
\end{equation}
\end{proof}
\subsection{ $\Phi_{T,F,H}$ harmonic map when target manifold is convex hypersurface}
\begin{thm}
With the same assumptions on $ M^m $as in Theorem \ref{thm1}, every nonconstant $ \Phi_{T,F,H} $-
stationary map with potential from any compact Riemannian manifold $ N $ to $ M^m $ is $ \Phi_{T,F,H} $-unstable if \eqref{6a} or \eqref{7a} holds.
\end{thm}
\begin{proof}
We modify the proof in \cite[Theorem 3.1 ]{Li2017NonexistenceOS}. In the proof of this theorem, we also borrow the notations and settings from the proof of \cite[Theorem 3.1 ]{Li2017NonexistenceOS}. The proof is the same as that in Theorem \ref{9cc}.
In order to prove the instability of $u: N^{n} \rightarrow M^{m}$, we need to consider some special variational vector fields along $u$. To do this, we choose an orthonormal field $\left\{\epsilon_{\alpha}, \epsilon_{m+1}\right\}$, $\alpha=1, \ldots, m$, of $R^{m+1}$ such that $\left\{\epsilon_{\alpha}\right\}$ are tangent to $M^{m} \subset R^{m+1}, \epsilon_{m+1}$ is normal to $M^{m}$, $\left.M^{m} \nabla_{\epsilon_{\alpha}} \epsilon_{\beta}\right|_{P}=0$ and $B_{\alpha \beta}=\lambda_{\alpha} \delta_{\alpha \beta}$, where $B_{\alpha \beta}$ denotes the components of the second fundamental form of $M^{m}$ in $R^{m+1}$. Meanwhile, take a fixed orthonormal basis $E_{A}, A=1, \ldots, m+1$, of $R^{m+1}$ and set
\[
V_{A}=\sum_{\alpha=1}^{m} v_{A}^{\alpha} \epsilon_{\alpha}, v_{A}^{\alpha}=\left\langle E_{A}, \epsilon_{\alpha}\right\rangle, v_{A}^{m+1}=\left\langle E_{A}, \epsilon_{m+1}\right\rangle,
\]
where $\langle\cdot, \cdot\rangle$ denotes the canonical Euclidean inner product.
Recall \eqref{gg1}
\begin{equation*}
\begin{split}
I(V_{A},V_{A}) &:=\sum J_i,
\end{split}
\end{equation*}
By (34) in \cite{Li2017NonexistenceOS},
\begin{equation*}
\begin{split}
J_1= \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\alpha, \sigma_{u}(e_i))h(u_j^\alpha, \sigma_{u}(e_j)) dv_g,
\end{split}
\end{equation*}
by (35) in \cite{Li2017NonexistenceOS},
\begin{equation*}
\begin{split}
J_2= \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \lambda_{\alpha}^2h(u_i^\alpha \epsilon_\alpha, du(e_j))\bigg(h(du(e_i), du(e_j)) -\frac{1}{m}|du |^2g(e_i,e_j)\bigg) dv_g.
\end{split}
\end{equation*}
By (36) in \cite{Li2017NonexistenceOS},
\begin{equation*}
\begin{split}
J_3= \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\alpha \epsilon_\alpha, du(e_i))h(u_j^\beta\epsilon_\beta, du(e_j)) dv_g.
\end{split}
\end{equation*}
By (37) in \cite{Li2017NonexistenceOS},
\begin{equation*}
\begin{split}
J_4= \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\alpha \epsilon_\alpha, \sigma_{u}(e_i))h(u_j^\beta\epsilon_\beta,du(e_j)) dv_g
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
J_5=&-\frac{2}{m} \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \left( \sum_{i}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_i\right)\right)\right) ^2 d v_g \\
=&-\frac{2}{m}\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \lambda_{\alpha}\lambda_{\beta}h(u_i^\beta\epsilon_{\beta}, du(e_i))h(u_j^\alpha\epsilon_{\alpha}, du(e_j)) dv_g\leq 0.
\end{split}
\end{equation*}
By (38) in \cite{Li2017NonexistenceOS},
\begin{equation*}
\begin{split}
J_6= \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \left(\lambda_{\alpha}^2 -\sum_\beta \lambda_{\beta}\lambda_{\alpha}\right) h(u_i^\alpha \epsilon_\alpha, du(e_j))\bigg(h(du(e_i), du(e_j)) -\frac{1}{m}|du |^2g(e_i,e_j)\bigg) dv_g.
\end{split}
\end{equation*}
If $ F^{\prime\prime} =F^\prime$
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right)\left\|T_u\right\|^{2}\left\{\lambda_{m}^{2}\left\|T_u\right\|^{2}\right.\\
&\left.+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\right\} d v_{g},
\end{split}
\end{equation}
when $ F^{\prime\prime}(t)t\leq c_F F^{\prime}(t) $
\begin{equation}\label{}
\begin{split}
I(du,du) \leq & \int_{M} F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right)\left\|T_u\right\|^{2}\left\{\lambda_{m}^{2}c_F\right.\\
&\left.+\max _{1 \leq i \leq m}\left\{\left[2 \lambda_{i}+2 \lambda_{m}-\left(\sum_{k} \lambda_{k}\right)\right] \lambda_{i}\right\}\right\} d v_{g} .
\end{split}
\end{equation}
\end{proof}
\
\subsection{Other result}
Motivated by the method in \cite{torbaghan2022stability}, we can establish
\begin{thm}\label{thm2.1}
Let $u:\left(M^{m}, g\right) \longrightarrow\left(N^{n}, h\right)$ be a nonconstant $\Phi_{T,F, H}$-harmonic map with potential. Suppose that $ \nabla^2H $ is semi-negative, $ F^{\prime} \geq 0. $Then, $u$ is $\Phi_{T,F, H}$-stable.
\end{thm}
\begin{proof} Let $ \{e_i\}_{i=1}^{m} $ be a local orthonormal frame field
on $ M $ and $ \omega $ be a parallel vector field in $ R^{n+p} $. Recall the index form of $ F $-symphonic map(c.f. \cite{ara2001stability}), We take $ V=\omega^\top $
\begin{equation}\label{pok1}
\begin{split}
I(\omega^\top,\omega^\top)=&-\int_{M} \operatorname{Hess}H(\omega^\top, \omega^\top) d v_{g} \\
&+\int_{M} F^{\prime \prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right)\left\langle \widetilde{\nabla} \omega^\top, \sigma_{u}\right\rangle \left\langle \widetilde{\nabla} \omega^\top, \sigma_{u}\right\rangle d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} \omega^\top, \widetilde{\nabla}_{e_{j}} \omega^\top\right) T_u\left(e_i, e_j\right) d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} \omega^\top, d u\left(e_{j}\right)\right) h\left(\widetilde{\nabla}_{e_{i}} \omega^\top, d u\left(e_{j}\right)\right) d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(\widetilde{\nabla}_{e_{i}} \omega^\top, d u\left(e_{j}\right)\right) h\left(d u\left(e_{i}\right), \widetilde{\nabla}_{e_{j}} \omega^\top\right) d v_{g} \\
& +\int_{M} F^{\prime}\left(\frac{\left\|u^{*} h\right\|^{2}}{4}\right) \sum_{i, j=1}^{m} h\left(R^{N}\left(\omega^\top, d u\left(e_{i}\right)\right) \omega^\top, d u\left(e_{j}\right)\right) T_u\left(e_i, e_j\right) d v_{g},
\end{split}
\end{equation}
Since $ u $ is conformal, $\sigma_{u}=0,T_u=0 $. The proof is simpler than that of Theorem. So, by \eqref{pok1}, we can get
\begin{equation}\label{}
\begin{split}
I(\omega^\top,\omega^\top)=
& \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^{2}}{4}\right) \left \langle B(du(e_i),du(e_j)), \omega^\perp \right \rangle^2 \d v_g
\end{split}
\end{equation}
\end{proof}
Similar to Theorem \ref{thm2.2}, we have
\begin{thm}\label{thm2.2} Let $u:\left(M^{n-1}, g\right) \longrightarrow\left(N^{n}, h\right)$ be a nonconstant $\Phi_{T,F, H}$-harmonic map with potential.
Let $ u:(M^{n-1}, g) \to (N^n, h) $ be a
totally geodesic isometric immersion of a hypersurface
$ M^{n-1} $ to $ N^n $. Then $u$ is FBH-unstable if the Ricci curvature
of $ N^n $ is positive.
\end{thm}
\begin{proof}
Since $ u $ is the totally geodesic, then
\begin{equation*}
\begin{split}
\tilde{\nabla} V=0.
\end{split}
\end{equation*}
So, by the index form of $\Phi_{T,F,H}$-harmonic map with potential $H$, we have
\begin{equation*}
\begin{split}
I(V,V)=-\int_{M} \F Ric(V,V)\d v_g.
\end{split}
\end{equation*}
\end{proof}
\begin{thm}\label{thm2.1} Let $u:\left(M^{m}, g\right) \longrightarrow\left(N^{n}, h\right)$ be a nonconstant conformal $\Phi_{T,F, H}$-harmonic map with potential with conformal factor $\lambda$ between Riemannian manifolds. Suppose that $F^\prime(0)\geq 0,\nabla^2 H <0$ and $N$ is the totally geodesic submanifold of $R^{n+p}$. Then, $u$ is $\Phi_{T,F, H}$-unstable.
\end{thm}
\begin{proof} Recall the second variation formula of $\Phi_{T,F, H}$-harmonic map with potential,
\begin{equation}
\begin{split}
I(V,W)<& \int_M F^{\prime \prime}\left(\frac{\left\|T_u\right\|^2}{4}\right)\left\langle\tilde{\nabla} V, \sigma_u\right\rangle\left\langle\tilde{\nabla} W, \sigma_u\right\rangle d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, \tilde{\nabla}_{e_j} W\right) T_u\left(e_i, e_j\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_j\right)\right) h\left(\tilde{\nabla}_{e_i} W, d u\left(e_j\right)\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_j\right)\right) h\left(d u\left(e_i\right), \tilde{\nabla}_{e_j} W\right) d v_g \\
&-\frac{2}{m} \int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(\tilde{\nabla}_{e_i} V, d u\left(e_i\right)\right) h\left(d u\left(e_j\right), \tilde{\nabla}_{e_j} W\right) d v_g \\
&+\int_M F^{\prime}\left(\frac{\left\|T_u\right\|^2}{4}\right) \sum_{i, j=1}^m h\left(R^N\left(V, d u\left(e_i\right)\right) W, d u\left(e_j\right)\right) T_u\left(e_i, e_j\right) d v_g .
\end{split}
\end{equation}
where we have used the fact that $\|T_u\|^2=|u^*h |^2-\frac{1}{m}|du |^4, T_u=0, \sigma_u=0$.
By \eqref{711}, \eqref{712},\eqref{713},\eqref{714} and the argument in \cite[Theorem 10]{torbaghan2022stability} we have
.
\begin{equation*}
\begin{split}
Tr_g I <
&2\int_{M} F^{ \prime}\left(0\right) |B|_\infty^2 |du |^2 \sum_i |du(e_i) |^2 \d v_g=0. \\
\end{split}
\end{equation*}
\end{proof}
\subsection{$\Phi_{T, F}$-stable manifold}
We can obtain similr theorem as in section \ref{cbm}, but the proof is simpler since $ T_u=0 $.
\begin{defn}
For $ id:(M,g) \to (M,g) $ , if $ id $ is $\Phi_{T, F,H} $ stable , then $ (M,g) $ is called $\Phi_{T, F,H} $ -stable manifold.
\end{defn}
By Theorem \ref{thm3.3} and Theorem \ref{thm34}, we immediately get
\begin{thm}
Let $ (M^m, g) $ be a compact $ \Phi $-SSU Riemannian
manifold. Let $ a $ and $ F $ be defined in Theorem \ref{thm3.3}, Then, $ M $ is $ \Phi_{T, F, H} $-SU and also $ \Phi_{T, F, H} $-U.
\end{thm}
\begin{thm}\label{stable1}
Let $ (M^m, g) $ be a compact Riemannian
manifold. If
$ F^{\prime} \geq 0, \nabla^2 H \geq 0 $. Then, $ M $ is an $ \Phi_{T, F, H} $-stable.
\end{thm}
\begin{rem}
By \cite[Theorem 7.11]{Han2019HarmonicMA} , the dimension of compact $ \Phi $-SSU manifold must bigger than 4.
\end{rem}
\begin{proof}
Let $ \{ e_i\} $ be an orthonormal frame field on $ M $. we use the mthod in \cite[Theorem 13 ]{torbaghan2022stability} or \cite[section 3]{ara2001instability}. When $ u=id,\sigma_{u}(e_i)=0,T_u=0$, we get the following formula,
Plugging the above three formulas into the index form , we get
\begin{equation}\label{369}
\begin{split}
I_{id}(V,V)=&\jifen{\nabla^2 H(V,V)}\\
&+\int_{M} F^{\prime}\left(0\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(\nabla_{e_i} V, e_{j}\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(0\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{k} \\
&-\frac{2}{m}\int_{M} F^{\prime}\left(0\right) \sum_{i=1}^{m} h\left(\nabla_{e_i} V, e_{i}\right) \sum_{j=1}^{m} h\left(\nabla_{e_j} V, e_{j}\right) \mathrm{d} v_{g} \\
\end{split}
\end{equation}
Next we deal with each terms of \eqref{369} respectively,
By \eqref{371} \eqref{370} \eqref{372}, we have
\begin{equation}\label{6.6}
\begin{split}
&I_{id}(V,V)\\
\geq & \int_M F^{\prime}(0) \sum \langle \nabla_{e_i} V,e_j\rangle^2 \d v_g\\
&+\int_M F^{\prime}(0) \left( \frac{1}{2}|L_Vg |^2-\sum_i|\nabla_{e_i} V |^2\right) dv_g-\frac{2}{m}F^{\prime}(0) \int_M \div(V)^2 dv_g
\end{split}
\end{equation}
Hence, by \eqref{jfg}, we get
\begin{equation*}
\begin{split}
&I(V,V)\\
\geq& \int_M F^\prime(0)(-\frac{2}{m})|div(V) |^2 dv_g+\int_{M}F^{\prime}(0)\frac{2}{m} |\div V |^2dv_g \\
\end{split}
\end{equation*}
By the assumption in the theorem, we can get
\begin{equation*}
\begin{split}
I(V,V)\geq 0.
\end{split}
\end{equation*}
\end{proof}
\begin{thm}\label{thm2.4}
Let $ (M^m, g) $ be a compact Riemannian manifold. Assume that there exists a nonisometric conformal vector field $ V $ on $ M $. If $\nabla^2 H< 0 $ and
$ F^{\prime}(0)\leq 0. $ Then, $ M $ is
$ \Phi_{T,F, H} $-U.
\end{thm}
\begin{proof}
For non-isometric conformal vector field $ V $ , we have
\begin{equation}\label{cmk}
\begin{split}
\frac{1}{2}|L_V g |^2=\frac{2}{m}(\operatorname{div} V)^2 \neq 0.
\end{split}
\end{equation}
by the second variation formula and \eqref{371}, we get
\begin{equation}\label{}
\begin{split}
I_{id}(V,V)=& \jifen{\nabla^2 H(V,V)}\\
&+\int_{M} F^{\prime}\left(0\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(\nabla_{e_i} V, e_{j}\right) \mathrm{d} v_{g} \\
&+\int_{M} F^{\prime}\left(0\right) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right) h\left(e_{i}, \nabla_{e_j} V\right) \mathrm{d} v_{k} \\
&- \frac{2}{m} \int_{M} F^{\prime}\left(0\right) \sum_{i=1}^{m} h\left(\nabla_{e_i} V, e_{i}\right) \sum_{j=1}^{m} h\left(\nabla_{e_j} V, e_{j}\right) \mathrm{d} v_{g} \\
<& \int_{M} F^{\prime}(0)(-\frac{2}{m}) div(V)^2 \d v_g \\
&+\int_M F^{\prime}(0) \left( \frac{1}{2}|L_Vg |^2-\sum_i|\nabla_{e_i} V |^2\right) \d v_g\\
&+\int_{M} F^{\prime}(0) \sum_{i, j=1}^{m} h\left(\nabla_{e_i} V, e_{j}\right)^2\mathrm{d} v_{g}
\end{split}
\end{equation}
\end{proof}
By \eqref{cmk}, The proof is done.
\end{document} | arXiv |
\begin{document}
\title[]{Positive self-similar Markov processes obtained by resurrection} \author{ Panki Kim \quad Renming Song \quad and \quad Zoran Vondra\v{c}ek} \thanks{P. Kim: This research is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. NRF-2021R1A4A1027378).} \thanks{R. Song: Research supported in part by a grant from the Simons Foundation (\#960480, Renming Song)} \thanks{Z. Vondra\v{c}ek: Research supported in part by the Croatian Science Foundation under the project 4197.}
\date{}
\begin{abstract} In this paper we study positive self-similar Markov processes obtained by
(partially) resurrecting a strictly $\alpha$-stable process at its first exit time from $(0,\infty)$. We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes.
\end{abstract} \maketitle
\noindent {\bf AMS 2020 Mathematics Subject Classification}: Primary 60G18; Secondary 60G51, 60G52, 60J76.
\noindent {\bf Keywords and phrases}: Positive self-similar Markov process, Lamperti transform, L\'evy process, jump kernel, resurrection
\section{Introduction}\label{s:intro} A $[0,\infty)$-valued standard Markov process (see \cite{BG68}) $X=(X_t, {\mathbb P}_x)$, $t\ge 0$, $x\ge0$, is called a \emph{positive self-similar Markov process} (pssMp) if there exists $\alpha>0$ such that for any $x>0$ and $c>0$, the law of $(cX_{c^{-\alpha}t}:\, t\ge 0)$ under ${\mathbb P}_x$ is equal to the law of $(X_{t}:\, t\ge 0)$ under ${\mathbb P}_{cx}$. One refers to $\alpha$ as the self-similarity index. We will say that $X$ is a pssMp with the origin as a trap (or that $X$ is absorbed at the origin) if once $X$ hits the origin it stays there forever. Self-similar processes were introduced by Lamperti in \cite{Lam72} where he established a one-to-one correspondence between pssMps up to the first exit time from $(0, \infty)$ and possibly killed L\'evy processes. A detailed description of this correspondence, usually called the Lamperti transform, is given in Section \ref{s:prelim}.
A canonical example of a pssMp with origin as a trap is an $\alpha$-stable process in ${\mathbb R}$ absorbed at the origin upon exiting $(0,\infty)$. To be more precise, let $\eta=(\eta_t)_{t\ge 0}$ be a strictly $\alpha$-stable process in ${\mathbb R}$, $\alpha\in(0,2)$. Its L\'evy measure has a density \begin{equation}\label{e:stable-density}
\nu(x)=c_+\, x^{-1-\alpha}{\bf 1}_{(x>0)}+c_-\, |x|^{-1-\alpha}{\bf 1}_{(x<0)}, \quad x\in {\mathbb R}, \end{equation} where $c_+, c_-\ge 0$ and $c_+=c_-$ if $\alpha=1$. Let $\rho:={\mathbb P}(\eta_1\ge 0)={\mathbb P}(\eta_1>0)$
be the positivity parameter, and set $\widehat{\rho}:=1-\rho$. The process $\eta$ will be parameterized so that \begin{equation}\label{e:c+c-} c_+=\frac{\Gamma(\alpha+1)}{\Gamma(\alpha\rho)\Gamma(1-\alpha\rho)}\, ,\quad c_-=\frac{\Gamma(\alpha+1)}{\Gamma(\alpha\widehat{\rho})\Gamma(1- \alpha\widehat{\rho})}. \end{equation} Throughout the paper, we will exclude the cases of only one-sided jumps. More precisely, the set of permissible parameters $(\alpha, \rho)$ is given by $$ \{(\alpha,\rho): \alpha\in (0,1), \rho\in (0,1)\}\cup \{(\alpha, \rho): \alpha\in (1,2), \rho\in (1-1/\alpha, 1/\alpha)\} \cup \{(\alpha, \rho)=(1,1/2)\}, $$ cf.~\cite[p.399]{KPW14}. We denote by ${\mathbb P}_x$, $x>0$, the law of $\eta$ starting at $x$.
Let $\tau=\tau_{(0,\infty)}:=\inf\{t>0: \eta_t\in (-\infty, 0]\}$ be the first exit time of $\eta$ from $(0,\infty)$. At time $\tau$, we send the process to $0$ where it stays forever, and thus arriving at the process $X^{\ast}_t:=\eta_t{\bf 1}_{(t<\tau)}$, $t\ge 0$. The process $X^{\ast}=(X^{\ast}_t, {\mathbb P}_x)$ is a pssMp of index $\alpha$, cf.~\cite[Section 3.1]{CC06}. If $T_0:=\inf\{t>0: X^{\ast}_t=0\}$, then $T_0=\tau_{(0,\infty)}<\infty$ and $X^{\ast}_{T_0-}>0$ a.s. We denote by $\xi^{\ast}$ the L\'evy process associated with $X^{\ast}$ through the Lamperti transform.
In this paper, we introduce a large class of positive self-similar processes that can be obtained by
(partially) resurrecting the strictly $\alpha$-stable process $\eta$ at the first exit time $\tau$. If $z=\eta_{\tau}<0$ is the position where $\eta$ lands at the exit from $(0,\infty)$, we return the process into
$[0,\infty)$ according to a probability distribution $p(z, \cdot)$. If the process is returned to 0, it stays there forever. More precisely, to ensure self-similarity, we consider probability kernels $p:(-\infty, 0)\times {\mathcal B}([ 0, \infty))\to [0, 1]$ satisfying the scaling condition \begin{equation}\label{e:p-scaling-measure} p(\lambda z, \lambda A)= p(z,A)\, \quad \text{for all }z<0, A\in {\mathcal B}( [0, \infty)) \text{ and }\lambda >0. \end{equation} All such kernels arise in the following way: Let $\phi$ be a probability measure on ${\mathcal B}([ 0, \infty))$. Then \begin{equation}\label{e:p-phi-measure} p(z,A):=
\phi(|z|^{-1}{A}),\quad \text{for all } z<0 \text{ and } A\in {\mathcal B}( [ 0, \infty)) \end{equation} satisfies \eqref{e:p-scaling-measure}. Conversely, if $p(\cdot, \cdot)$ satisfies \eqref{e:p-scaling-measure} and if we set $\phi(A)=p( -1, A)$, then $p(\cdot, \cdot)$ is of the form \eqref{e:p-phi-measure}. We call $p(\cdot, \cdot)$ the \emph{return kernel}. Note that it follows from \eqref{e:p-scaling-measure} that $$ \mathfrak{p}:=1-p(z, \{0\}) $$ is independent of $z<0$.
Let $j(x,z):=\nu(z-x)$ be the jump kernel of $\eta$. Set \begin{equation}\label{e:int-kernel-0-measure} q_0(x,A) :=\int_{(-\infty,0)}j(x,z)p(z,A)\, dz, \quad x>0, A\in {\mathcal B}( [0, \infty)), \end{equation} and note that $$ q_0(x, \{0\})=\int_{(-\infty,0)}j(x,z)p(z,\{0\})\, dz=(1-\mathfrak{p})\int_{-\infty}^0 c_- (x-z)^{-1-\alpha}\, dz =(1-\mathfrak{p}) \frac{c_-}{\alpha}x^{-\alpha}. $$ We define a \emph{resurrection kernel} $q$ as the restriction of $q_0(x, \cdot)$ to $(0,\infty)$: \begin{equation}\label{e:int-kernel-measure} q(x,A) :=\int_{(-\infty,0)}j(x,z)p(z,A)\, dz, \quad x>0, A\in {\mathcal B}((0, \infty)). \end{equation} The idea behind the kernel $q_0(x, \cdot)$ is that if $x=\eta_{\tau_-}$, then instead of sending $\eta$ to the origin at time $\tau$ (thus obtaining the pssMp $X^{\ast}$), with probability $1-\mathfrak{p}$ we send $\eta$ to the origin, and with probability $\mathfrak{p}=p(1, (0,\infty))$ we restart (or resurrect) it according to the normalized kernel $q(x, \cdot)$. If $\mathfrak{p}\in (0,1)$ we call the process \emph{partially resurrected}, and when $\mathfrak{p}=1$ we say that it is fully resurrected (or just resurrected). By Lemma \ref{l:q-always-density}, $q(x,\cdot)$ is absolutely continuous with respect to the Lebesgue measure, and its density will be denoted by $q(x,y)$, $x,y>0$. By resurrecting according to the normalized density we arrive at a pssMp $\overline{X}$ absorbed at the origin with jump kernel $J(x,y):=j(x,y)+q(x,y)$. The precise construction will be carried out in Section \ref{s:res-proc} by means of the Lamperti transform.
Examples of pssMp that can be obtained by such resurrection include the path censored process from \cite{KPW14} (also called the trace process on $(0, \infty)$ of the stable process) in which case the return kernel is equal to the Poisson kernel of $\eta$ with respect to $(-\infty,0)$; the process from \cite{DROV17, Von21}
related to nonlocal problems with Neumann boundary condition in which case the return kernel is equal to $j(z, y)dy/\int^\infty_0j(z, u)du$;
and the absolute value process $|\eta |=(|\eta_t|)_{t\ge 0}$ in which case $p(z,A)=\delta_{-z}(A)$, see Subsection \ref{ss:examples-return}. In all three examples above $\mathfrak{p}=1$ so we have full resurrection of $\eta$. If $p(z,A)=(1- \mathfrak{p})\delta_{0}(A)+\mathfrak{p} \delta_{-z}(A)$, we recover the ricocheted process from \cite{Bud18, KPV21}.
One of the key questions that we will address in this paper is the behavior of $\overline{X}$ at its absorption time at 0. In case $\mathfrak{p}<1$ it is clear from the Lamperti trichotomy (see Section \ref{s:prelim}) that the partially resurrected process $\overline{X}$ will be absorbed at 0 in finite time by a jump. In case $\mathfrak{p}=1$, the absorption time may be infinite or finite, and in the latter case it turns out that $\overline{X}$ is continuously absorbed at 0. We answer the question of finite or infinite lifetime by studying the behavior of the L\'evy process $\overline{\xi}=(\overline{\xi}_t, {\bf P}_x)$, $t\ge 0, x\in {\mathbb R}$, associated to $\overline{X}$ through the Lamperti transform. Let $\overline{\Psi}$ denote the characteristic exponent of $\overline{\xi}$. We will write ${\bf P}_0$ as ${\bf P}$ and denote expectation with respect to ${\bf P}$ by ${\bf E}$. When $\mathfrak{p}<1$, it is obvious that
${\bf E}\overline{\xi}_1=-\infty$. Our first main result is about the finiteness of ${\bf E}|\overline{\xi}_1|$ and provides an explicit expression of ${\bf E}\overline{\xi}_1$ when $\mathfrak{p}=1$.
\begin{thm}\label{t:derivative-at-zero} Suppose $\mathfrak{p}=1$.
It holds that ${\bf E}|\overline{\xi}_1|<\infty$ if and only if \begin{equation}\label{e:phi-int-log}
\int_{(0,{\infty})}|\log y|\phi(dy)<\infty. \end{equation} In this case \begin{equation}\label{e:derivative-at-zero}
{\bf E}[\overline{\xi}_1]=i\overline{\Psi}'(0) =\Gamma(\alpha)\frac{\sin(\pi \alpha\widehat\rho)}{\pi} \left(\pi\cot(\pi\alpha\widehat\rho)+ \int_{(0,{\infty})}(\log y)\phi(dy)\right). \end{equation} \end{thm}
It follows from \cite[Theorem 7.2]{Kyp14} and the Lamperti trichotomy (see Section \ref{s:prelim}) that (i) If ${\bf E}[\overline{\xi}_1] \ge 0$, then $\limsup_{t\to \infty}\overline{\xi}_t=+\infty$, hence the absorption time of $\overline{X}$ is infinite, and
(ii) If ${\bf E}[\overline{\xi}_1] < 0$, then $\lim_{t\to \infty}\overline{\xi}_t=-\infty$, hence the absorption time of $\overline{X}$ is finite ${\mathbb P}_x$-a.s.~and $\overline{X}$ is continuously absorbed at 0. Therefore, since $\alpha\widehat{\rho}\in (0,1)$, to deduce the long term behavior of $\overline{X}$ it suffices to determine the sign of the expression in the parenthesis in \eqref{e:derivative-at-zero}, see Corollaries \ref{c:s}--\ref{c:s2}.
In case when $\overline{X}$ is absorbed at zero in finite time,
we give a definitive answer on the existence of its recurrent positive self-similar extensions. \begin{thm}\label{t:R_e} Suppose that $\overline{X}$ is absorbed at zero in finite time. If \begin{equation}\label{e:kappa_0} \kappa_0:=\sup\{\kappa\in (0,{\infty}):\, \int_{(0,{\infty})}u^{\kappa}\phi(du)<\infty\} \in (0, \infty], \end{equation} then (1) $\overline{X}$ has a positive self-similar recurrent extension which leaves 0 continuously; and (2) there exists $\kappa^\ast\in (0, \alpha)$ such that, for any $\beta \in (0, \kappa^\ast)$, $\overline{X}$ has a positive self-similar recurrent extension which leaves 0 by a jump associated with an excursion measure of the form $c\beta x^{-1-\beta}dx, x>0$.
Conversely, if \eqref{e:kappa_0} does not hold, then $\overline{X}$ has no positive self-similar recurrent extension. \end{thm}
Positive self-similar Markov processes and their associated L\'evy processes have been extensively studied in the last 15 years. We mention here the Lamperti stable processes \cite{CC06, CPP10}, hypergeometric processes \cite{KPR10, KP13}, double hypergeometric L\'evy processes \cite{KPV21}, and $\beta$-processes \cite{Ku10}. The interest in those families of processes was mostly motivated by the Wiener-Hopf factorization.
Our motivation for studying pssMps comes from our research program on the potential theory of Markov processes with jump kernels degenerate at the boundary. In \cite{KSV21, KSV22}, we introduced a large class of symmetric Markov processes in ${\mathbb R}^d_+$ with jump kernels decaying at the boundary and systematically studied their potential theory. The trace process of a symmetric $\alpha$-stable process on ${\mathbb R}^d_+$ is degenerate in the sense that its jump kernel blows up at the boundary, see \cite{BGPR21}. The same feature is true also for the process studied in \cite{DROV17, Von21}. In \cite{KSV22b} we studied the potential theory of a large class of symmetric Markov processes in ${\mathbb R}^d_+$ with jump kernels blowing up at the boundary. The main examples of such processes are rotationally symmetric $\alpha$-stable processes resurrected upon exiting the upper-half-space. The current paper concentrates on the one dimensional case, but we do not assume that the processes are symmetric. The jump kernel $J(x,y)=j(x,y)+q(x,y)$ of our resurrected process exhibits unusual and interesting behavior when $y\to 0$. Depending on the return kernel $p(z,\cdot)$, $J(x, y)$ may tend to $\infty$ at various rates as $y\to 0$. Since $j(x,y)$ is bounded away from the diagonal, this explosion is due to the resurrection kernel.
In order to state the precise result about behavior of the resurrection kernel,
we first need a definition. \begin{defn}
{\rm Let $g:(0,\infty) \to (0,\infty)$ and $\beta_1, \beta_2 \in {\mathbb R}$.
\begin{enumerate}
\item[(i)]
We say that $g$ satisfies the lower weak scaling condition at zero $L_1(\beta_1)$ (resp. at infinity $L^1(\beta_1)$) if there exists $c\in(0,1]$ such that
$$
\frac{g(R)}{g(r)} \geq c \left(\frac{R}{r}\right)^{\beta_1} \quad \text{for all} \quad r\leq R<
1\;(\text{resp.}\;
1\le r\leq R).
$$
\item[(ii)] We say that $g$ satisfies the upper weak scaling condition at zero $U_1(\beta_2)$ (resp. at infinity $U^1(\beta_2)$) if there exists $C\in [1, \infty)$ such that $$ \frac{g(R)}{g(r)} \leq C \left(\frac{R}{r}\right)^{\beta_2} \quad \text{for all} \quad r\leq R< 1\;(\text{resp.}\; 1\le r\leq R). $$
\end{enumerate} } \end{defn} Here is our third main result in which we assume that the restriction of the measure $\phi$ to $(0, \infty)$ is absolutely continuous with respect to the Lebesgue measure and, with slight abuse of notation, denote its density also by $\phi$. The notation $a\asymp b$ means that $c\le b/a \le c^{-1}$ for some $c\in (0,1)$.
\begin{thm}\label{t:estimates-of-q} Suppose that the density $\phi$ is strictly positive. (1) If $x \le y\le 5x$, then $$ q(x,y)\asymp q(y,x)\asymp x^{-1-\alpha}\asymp y^{-1-\alpha}. $$
\noindent (2) Suppose $\phi$ satisfies the lower weak scaling condition $L_1(\beta_1)$ at zero with $\beta_1>-1-\alpha $. Then for $5x\le y$, $$ q(y,x)\asymp (y-x)^{-1-\alpha}\int_{\frac{x}{y-x}}^{1} \phi(t)\frac{dt}{t} \asymp y^{-1-\alpha} \int_{\frac{x}{y-x}}^{1} \phi(t)\frac{dt}{t}. $$ Further, if $\phi$ also satisfies the upper weak scaling condition $U_1(\beta_2)$ at zero with $\beta_2<0$, for $5x\le y$, $$ q(y,x)\asymp (y-x)^{-1-\alpha} \phi\big( \frac{x}{y-x}\big)\asymp y^{-1-\alpha} \phi\big( \frac{x}{y}\big).$$ (3) Suppose $\phi$ satisfies the upper weak scaling condition $U^1(\gamma_2)$ at infinity with $\gamma_2<0 $. Then for $5x\le y$, \begin{equation}\label{e:estimate-q-21} q(x,y) \asymp (y-x)^{-1-\alpha}\int_0^{ \frac{y-x}{x}} t^{\alpha} \phi(t){dt} \asymp y^{-1-\alpha}\int_1^{\frac{y}{x}-1}t^{\alpha} \phi(t){dt}. \end{equation} Further, if $\phi$ also satisfies the lower weak scaling condition $L^1(\gamma_1)$ at infinity with $\gamma_1<-1-\alpha$, for $5x\le y$, $$ q(x,y) \asymp (y-x)^{-1-\alpha} \big(\frac{x}{y-x}\big)^{-1-\alpha} \phi\big(\frac{y-x}{x}\big) \asymp x^{-1-\alpha}\phi\big(\frac{y}{x}\big). $$ \end{thm}
As a consequence of this theorem we can deduce, see Corollary \ref{c:estimates-of-q}, that the jump kernel $J(x,y)$ of the path censored process goes to $\infty$ at rate $y^{-\alpha\rho}$ as $y\to 0$, and that the jump kernel of the process with the resurrection kernel $j(z, y)dy/\int^\infty_0j(z, u)du$ goes to infinity at rate $\log(1/y)$ as $y\to 0$.
Stable process conditioned to stay positive and censored stable process can be regarded as resurrected stable processes, see \cite{Ber93} and \cite[Remark 3.3]{KPW14}. However, they do not fall into the framework of resurrected stable processes of this paper. To cover these processes, we introduce a larger class of pssMps in Section \ref{s:modified-jump} of this paper. This larger class incudes stable processes conditioned to stay positive, stable processes conditioned to hit 0 continuously, and censored stable processes as examples. The jump kernel $J(x,y)=j(x,y)+q(x,y)$ of our resurrected stable process can be regarded as a modification of the original kernel $j(x,y)$. The jump kernels of the class of pssMps in Section \ref{s:modified-jump} are of the more general form $j(x,y)\sB(x,y)$ where $\sB:(0,\infty)\times (0,\infty)\to (0,\infty)$. In case of multidimensional isotropic stable process, the analogous procedure is quite standard when $\sB$ is bounded from below and above by two positive constants, leading to the so-called stable-like processes. The situation when the function $\sB(x,y)$ decays and vanishes at the boundary of the state space was recently studied in \cite{KSV21, KSV22} in the multidimensional case of
the upper-half-space in ${\mathbb R}^d$ and the symmetric jump kernel $j(x,y)=|x-y|^{-d-\alpha}$.
The paper \cite{KSV22b} deals with the multidimensional case in the upper-half-space, the same jump kernel $|x-y|^{-d-\alpha}$, with $\sB(x,y)$ exploding at the boundary. The symmetric 1-dimensional case is also covered in that paper, but differently to the current paper which is mostly focused on self-similarity, the main concern of \cite{KSV22b} is on potential-theoretic questions.
{\bf Organization of the paper}: In the next section we recall some preliminary results related to pssMps and their connection to L\'evy processes through the Lamperti transform, in particular the Lamperti trichotomy and the relationship between infinitesimal generators of those processes. We also introduce two families of return kernels and show how some of examples for pssMp from literature fit into these families.
Section \ref{s:res-proc} is central to the paper. Starting from the general return kernel described by the measure $\phi$, we first look at the regular step process with the resurrection kernel $q(x,y)$ as its jump kernel, and its counterpart through the Lamperti transform -- a compound Poisson process $\chi$. We compute the L\'evy measure of $\chi$ and its characteristic function. For particular examples given by \eqref{e:phi-special} and \eqref{e:phi-special-exp} we obtain more precise expressions. Then we construct the resurrected process $\overline{X}$, and compute the characteristic exponent $\overline{\Psi}$ of the corresponding L\'evy process $\overline{\xi}$. In Subsection \ref{ss:X-lifetime} we study the behavior of $\overline{X}$ by analyzing the derivative of the characteristic exponent $\overline{\Psi}$ of $\overline{\xi}$ at zero and give a proof of Theorem \ref{t:derivative-at-zero}. In Subsection \ref{ss:res-proc} we prove Theorem \ref{t:R_e}. Finally, we provide several concrete examples illustrating the behavior of $\overline{X}$ at its absorption time.
In Section \ref{s:sym-int-kernel} we give a necessary and sufficient condition on $\phi$ making the resurrection kernel symmetric, i.e., $q(x,y)=q(y,x)$ for all $x,y>0$, cf.~Theorem \ref{t:q-symmetric}. If, in addition, the underlying $\alpha$-stable process $\eta$ is symmetric, the resulting resurrected process $\overline{X}$ will be also symmetric.
In Section \ref{s:estimates-q} we provide a proof of Theorem \ref{t:estimates-of-q}. The key estimates are given in Lemma \ref{l:estimates-of-q} where it is assumed that $|x-y|=1$. By using scaling of the resurrection kernel, this suffices to prove the theorem. A particular example of a slightly modified density from the family \eqref{e:phi-special} gives additional insight of possible behaviors at zero and at infinity, see Corollary \ref{c:estimates-of-q}.
In Section \ref{s:modified-jump} we put the resurrected process in a more general context of processes with modified jump kernel. Given a process with jump kernel $j(x,y)$, we multiply it by a function $\sB(x,y)$ which changes the behavior of the original kernel. We show that the jump kernel of the resurrected process can be thought of as being modified by the function which explodes at the boundary, namely at zero. We end the paper by looking at the symmetric $\alpha$-stable case modified by a function $\sB(x,y)$, and establish the behavior of the modified process at its lifetime.
For the reader's convenience, we summarize the known examples of resurrected stable processes that can be included in our framework. Path-censored stable processes, the processes studied in \cite{DROV17, Von21}, the absolute value of a stable process and the ricocheted stable processes are examples of our resurrected stable processes. Censored stable processes, stable processes conditioned to stay positive, and stable processes conditioned to hit 0 continuously are included in the more general framework of Section \ref{s:modified-jump}.
{\bf Notation}: We use ``:='' to indicate definitions. Define $a\land b := \min\{a, b\}$ and $a\vee b := \max\{a, b\}$. We write $f\asymp g$ if $f,g$ are nonnegative functions, $c^{-1}g\le f\le cg$ with some constant $c \in (0,\infty)$. We call $c$ the comparability constant. Lower case letters $c_i, i=1,2, \dots$ are used to denote the constants in the proofs and the labeling of these constants starts anew in each proof.
$\Gamma$ denotes the Gamma function defined as $\Gamma(x)=\int_0^\infty y^{x-1}e^{-y}d y$, $\psi$ denotes the digamma function defined as $\psi(x)=\frac{d}{dx}\log \Gamma(x)$, and $B$ denotes the beta function defined by $B(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$.
\section{Preliminaries}\label{s:prelim} \subsection{Lamperti correspondence}\label{ss:lam-corr} We start this preliminary section by briefly describing the correspondence between positive self-similar Markov processes and 1-dimensional L\'evy processes, usually called the Lamperti transform. Let $\xi=(\xi_t, {\bf P}_x)$, $t\ge 0, x\in {\mathbb R},$ be a possibly killed L\'evy process sent to $-\infty$ at death. Define the integrated exponential process $I=(I_t)_{t\ge0}$ by $$ I_t:=\int_0^t e^{\alpha \xi_s} ds, \quad t\ge 0, $$ and let $\varphi$ be its inverse: $$ \varphi(t):=\inf\{s>0:\, I_s >t\}, \quad t\ge 0. $$ For each $x>0$, define ${\mathbb P}_x:={\bf P}_{\log x}$ and $$ X_t:=\exp\{\xi_{\varphi(t)}\}{\bf 1}_{(t < I_{\infty})}, \quad t\ge 0. $$ Then $X=(X_t, {\mathbb P}_x)$, $t\ge 0, x>0$, is a pssMp of index $\alpha$ with absorption time $\zeta=I_{\infty}$. (See \cite{PS18} for an in-depth analysis of the absorption time.) Conversely, for a pssMp $X=(X_t, {\mathbb P}_x)$, $t\ge 0, x>0$, of index $\alpha$, let $$ S_t:=\int^t_0X^{-\alpha}_udu $$ and let $T(\cdot)$ be its inverse $$ T(t):=\inf\{u>0:\, S_u >t\}, \quad t\ge 0. $$ For any $x\in {\mathbb R}$, define ${\bf P}_x:={\mathbb P}_{e^x}$ and $\xi_s:=\log X_{T_s}$. Then $\xi=(\xi_t, {\bf P}_x)$, $t\ge 0, x\in {\mathbb R}$, is a possibly killed L\'evy process. Moreover, we have the following three exhausting scenarios (see \cite[Theorem 13.1]{Kyp14}) that we refer to as the {\it Lamperti trichotomy}: \begin{itemize}
\item[(1)]
${\mathbb P}_{x}(\zeta=\infty)=1$
for all $x>0$ in which case $\limsup_{t\to \infty}\xi_t=\infty$;
\item[(2)]
${\mathbb P}_{x}(\zeta<\infty, X_{\zeta-}=0)=1$
for all $x>0$ in which case $\lim_{t\to \infty}\xi_t=-\infty$;
\item[(3)]
${\mathbb P}_{x}(\zeta<\infty, X_{\zeta-}>0)=1$
for all $x>0$ in which case $\xi$ is killed at an independent exponentially distributed random time. \end{itemize} In case (2), we will say that $X$ is continuously absorbed at $0$.
Now we recall a few facts about 1-dimensional L\'evy processes. Let $\xi=(\xi_t, {\bf P}_x)$, $t\ge0, x\in {\mathbb R}$, be a 1-dimensional L\'evy process with characteristic triple $(d,\sigma, \nu)$. We will write ${\bf P}_0$ as ${\bf P}$ and denote expectation with respect to ${\bf P}$ by ${\bf E}$. Then \begin{equation}\label{e:char-fn} {\bf E} \left[e^{i\theta \xi_t}\right]=e^{-t\Psi(\theta)}, \quad \theta\in {\mathbb R}, \end{equation} where the characteristic exponent $\Psi$ is given by \begin{equation}\label{e:char-exp}
\Psi(\theta)=d i\theta +\frac{1}{2}\sigma^2 \theta^2 +\int_{{\mathbb R}}\left(1-e^{i\theta x}+i\theta x{\bf 1}_{(|x|\le 1)}\right)\nu(dx), \quad \theta \in {\mathbb R}. \end{equation} Recall that $\xi_1$ has finite expectation if and only if
$\int_{|y|\ge 1}|y|\nu(dy)<\infty$, cf.~\cite[Theorem 25.3, Example 25.12]{Sat14}. In this case, by differentiating \eqref{e:char-fn} we get that ${\bf E}[\xi_1]=i\Psi'(0)$.
If $\xi$ is killed at an independent exponential time of parameter $q\ge 0$ (when $\xi$ is sent either to a cemetery $\partial$ or to $-\infty$), the characteristic exponent becomes $\widetilde{\Psi}(\theta)=\Psi(\theta)+q$. Thus for the killed L\'evy process $\widetilde{\xi}$, the killing rate is equal to $\widetilde{\Psi}(0)$.
Let ${\mathcal A}$ be the infinitesimal generator of the semigroup of $\xi$ (possibly killed at rate $q\ge 0$) acting on $C_0({\mathbb R})$ (continuous functions vanishing at infinity). Then, cf.~\cite[Theorem 31.5]{Sat14}, $C_0^2({\mathbb R})\subset {\mathcal D}({\mathcal A})$, and for $f\in C_0^2({\mathbb R})$, $$ {\mathcal A} f(x)=-qf(x) -
df'(x)+\frac{1}{2}\sigma^2 f''(x)+\int_{{\mathbb R}}\left(f(x+y)-f(x)-f'(x)y {\bf 1}_{(|y|\le 1)}\right) \nu(dy). $$
Let $X$ be the pssMp of index $\alpha$ corresponding to the L\'evy process $\xi$. Its infinitesimal generator ${\mathcal L}$ can be described as follows (cf.~\cite[Theorem 1]{CC06}, where we take the usual cutoff function $\ell(y)=y{\bf 1}_{[-1,1]}(y)$): If $f:[0,\infty]\to {\mathbb R}$ is such that $f$, $xf'$ and $x^2 f''$ are continuous on $[0,\infty]$ then it belongs to the domain of ${\mathcal L}$ and \begin{eqnarray}\label{e:pssMp-inf-gen} {\mathcal L} f(x)&=&-qx^{-\alpha}f(x) +x^{1-\alpha}\left(- d +\frac{1}{2}\sigma^2\right) f'(x)+\frac{1}{2}\sigma^2 x^{2-\alpha}f''(x) \nonumber \\ & & +x^{-\alpha} \int_{(0, \infty)} \left(f(ux)-f(x)- x f'(x)(\log u){\bf 1}_{[-1,1]}(\log u)\right) \mu(du), \end{eqnarray} where $\mu(du)=\nu(du)\circ \log u$. By the change of variables $y=\log u$, we get \begin{eqnarray}\label{e:pssMp-inf-gen-2} {\mathcal L} f(x)&=&-qx^{-\alpha}f(x) +x^{1-\alpha}\left(- d+\frac{1}{2}\sigma^2\right)f'(x)+\frac{1}{2}\sigma^2 x^{2-\alpha}f''(x) \nonumber \\ & & +x^{-\alpha}\int_{{\mathbb R}}\left(f(xe^y)-f(x)-xf'(x)y{\bf 1}_{[-1,1]}(y)\right)\nu(dy), \end{eqnarray} which corresponds to the formula in \cite[p.~4]{PR13}. In case $\nu$ has a density (which we also denote by $\nu$), the integral in \eqref{e:pssMp-inf-gen-2} can be (after a change of variables) written in the form $$ \int_0^{\infty}\left(f(z)-f(x)-xf'(x)(\log z/x){\bf 1}_{[-1,1]}(\log z/x )\right)\nu(\log z/x)\frac{dz}{z} $$ showing that the intensity of jumps from $x$ to $z$ (i.e.~the jump kernel of $X$) is given by $z^{-1}\nu(\log z/x )$.
\subsection{Strictly stable process absorbed at 0 and censored process}\label{ss:sspcp}
Recall that $\eta=(\eta_t, {\mathbb P}_x)$ denotes a strictly $\alpha$-stable process in ${\mathbb R}$, $\alpha\in(0,2)$. Thus $\eta$ is a L\'evy process with characteristic exponent given by \eqref{e:char-exp}, where $\sigma=0$ and the L\'evy measure $\nu$ has density given by \eqref{e:stable-density}. Moreover, it holds that $d=a:=(c_+-c_-)/(\alpha-1)$ when $\alpha\neq 1$, and we specify $d=a=0$ when $\alpha=1$, cf.~\cite[p.~398]{KPW14}.
Recall also that $\tau=\tau_{(0,\infty)}:=\inf\{t>0: \eta_t\in (-\infty, 0]\}$, $X^{\ast}_t:=\eta_t{\bf 1}_{(t<\tau)}$, $t\ge 0$, and $\xi^{\ast}$ is the L\'evy process associated to $X^{\ast}$ through the Lamperti transform. Then $\xi^{\ast}$ is a killed L\'evy process. Its L\'evy measure $\mu$ was computed in \cite[Section 3.1]{CC06}, see also \cite[(6)]{KPW14}. It holds that $\mu$ has a density \begin{equation}\label{e:levy-xi*} \mu(x) =c_+\frac{e^x}{(e^x-1)^{1+\alpha}}{\bf 1}_{(x>0)}+ c_-\frac{e^x}{(1-e^x)^{1+\alpha}}{\bf 1}_{(x<0)}, \quad x\in {\mathbb R}, \end{equation} and $\xi^{\ast}$ is killed at rate \begin{equation}\label{e:killing-rate} \frac{c_-}{\alpha}=\frac{\Gamma(\alpha)}{\Gamma(\alpha\widehat{\rho})\Gamma(1- \alpha\widehat{\rho})}. \end{equation} The characteristic exponent $\Psi^{\ast}$ of $\xi^{\ast}$ can be found in \cite[(13.46)]{Kyp14}: \begin{equation}\label{e:char-exp-xi*} \Psi^{\ast}(\theta)=\frac{\Gamma(\alpha-i\theta)}{\Gamma(\alpha\widehat{\rho}-i\theta)}\, \frac{\Gamma(i\theta+1)}{\Gamma(i\theta+1-\alpha\widehat{\rho})}, \quad \theta \in {\mathbb R}. \end{equation} The infinitesimal generator of $\xi^*$ is given by \begin{equation}\label{e:inf-ge-xi-star} {\mathcal A}^* f(x)=-\frac{c_-}{\alpha}f(x)-bf'(x)+\int_{{\mathbb R}}\left(f(x+y)-f(x)-f'(x)y {\bf 1}_{[-1,1]}(y)\right) \mu(dy), \end{equation} where $b\in {\mathbb R}$ is a linear term which we will identify shortly. By \eqref{e:pssMp-inf-gen-2}, the infinitesimal generator of $X^*$ is equal to \begin{eqnarray*} {\mathcal L}^* f(x)&=& -\frac{c_-}{\alpha}x^{-\alpha}f(x)
-b x^{1-\alpha}f'(x)+x^{-\alpha}\int_{{\mathbb R}}\left(f(xe^y)-f(x)-xf'(x)y{\bf 1}_{[-1,1]}(y)\right)\mu(y)dy\\ &=& -\frac{c_-}{\alpha}x^{-\alpha}f(x) -b x^{1-\alpha}f'(x)\\ & & +x^{-\alpha} \int^\infty_0 \left(f(z)-f(x)-xf'(x)(\log z/x){\bf 1}_{[-1,1]}(\log(z/x))\right)\mu(\log(z/x))z^{-1}dz. \end{eqnarray*} We have that \begin{eqnarray*} \mu\left(\log \frac{z}{x}\right)&=&c_+\frac{\frac{z}{x}}{\left(\frac{z}{x}-1\right)^{1+\alpha}}{\bf 1}_{(z>x)}+c_-\frac{\frac{z}{x}}{\left(1-\frac{z}{x}\right)^{1+\alpha}}{\bf 1}_{(z<x)}\\
&=& x^{\alpha} z \left(c_+ |z-x|^{-1-\alpha}{\bf 1}_{(z>x)}+c_- |z-x|^{-1-\alpha}{\bf 1}_{(z<x)}\right)\\ &=& x^{\alpha} z\, \nu(z-x). \end{eqnarray*} Therefore, with $j(x,z):=\nu(z-x)$, \begin{eqnarray}\label{e:gen-killed} {\mathcal L}^* f(x)&=& -\frac{c_-}{\alpha}x^{-\alpha}f(x) -b x^{1-\alpha}f'(x)\nonumber \\
& &+
\int^\infty_0
\left(f(z)-f(x)-xf'(x)(\log z/x){\bf 1}_{[-1,1]}(\log(z/x))\right)j(x,z)dz. \end{eqnarray} By comparing this expression with the form of the infinitesimal generator of $X^*$ given in \cite[Theorem 2]{CC06}, we see that \begin{equation}\label{e:linear-term} b= - a- \int^\infty_0 \left((\log u){\bf 1}_{[-1,1]}(\log u)-(u-1){\bf 1}_{[-1,1]}(u-1)\right)\nu(u-1)\, du. \end{equation}
Let $\Psi(\theta):=\Psi^{\ast}(\theta)-\Psi^{\ast}(0)=\Psi^{\ast}(\theta)-c_- /\alpha$. Then $\Psi$ is the characteristic exponent of an unkilled L\'evy processes $\xi$. More precisely, $\xi^{\ast}$ is equal in distribution to $\xi$ killed at an independent exponential time with parameter $c_-/\alpha$. The infinitesimal generator of $\xi$ is \begin{equation}\label{e:inf-gen-xi} {\mathcal A} f(x)= -bf'(x)+\int_{{\mathbb R}}\left(f(x+y)-f(x)-f'(x)y {\bf 1}_{[-1,1]}(y)\right) \mu(dy). \end{equation} It is immediate from \eqref{e:levy-xi*} that
$\int_{|y|\ge 1}|y|\mu(dy)<\infty$, hence ${\mathbb E}|\xi_1|<\infty$. Let $X=(X_t, {\mathbb P}_x)$ be the pssMp of index $\alpha$ corresponding to $\xi$ through the Lamperti transform. The effect of removing the killing term from the generator of $\xi^{\ast}$ is to remove the killing term from the generator of $X^{\ast}$. Hence, the infinitesimal generator ${\mathcal L}$ of $X$ is given by the right-hand side of \eqref{e:gen-killed} with $-(c_-/\alpha)x^{-\alpha}f(x)$ removed: \begin{equation}\label{e:generator-LL-X} {\mathcal L} f(x)=-b x^{1-\alpha}f'(x)+\int^\infty_0\left(f(y)-f(x)-xf'(x)(\log y/x){\bf 1}_{[-1,1]}(\log(y/x))\right)j(x,y)dy. \end{equation} Considered on $(0,\infty)$, $X^{\ast}$ is a stable process killed upon exiting $(0,\infty)$. By removing the killing term in the infinitesimal generator, we end up with the process $X$ -- the (not necessarily symmetric) \emph{censored $\alpha$-stable process} on $(0,\infty)$. The censored process $X$ can be also regarded as a resurrected process with the resurrection kernel $q(x,A)={\mathbb P}_x(\eta_{\tau_-}\in A)$ -- it is continued exactly at the position from which $\eta$ has jumped out from $(0,\infty)$ (thus effectively suppressing this jump). Note that this type of resurrection does not fall into our setting. Censored processes were introduced in \cite{BBC} in a more general multi-dimensional context for rotationally symmetric stable processes.
\subsection{Examples of return kernels}\label{ss:examples-return}
An example of pssMp of index $\alpha$ related to $\eta$ is its absolute value process $|\eta |=(|\eta_t|)_{t\ge 0}$. One can view
$|\eta|$ also as a resurrected process: at time $\tau$, if $z=\eta_{ \tau}$, we resurrect at $-z>0$ according to the resurrection kernel $q$ with $p(z,A)=\delta_{-z}(A)$.
We discuss now two families of return kernels with $\mathfrak{p}=1$. First note that if the measure $\phi$ is absolutely continuous
with respect to the Lebesgue measure on $[0, \infty)$ with a density (which we denote by the same letter), then the return kernel $p(z, \cdot)$ has a density $p(z,y)$, $y>0$, and \eqref{e:p-scaling-measure}--\eqref{e:p-phi-measure} imply that $$
p(z,y)=\phi\left(\frac{y}{|z|}\right)\frac{1}{|z|}. $$
In the first family of return kernels the probability measure $\phi$ has a density which decays polynomially at infinity: For $\beta>0$ and $\gamma>\beta$, let \begin{equation}\label{e:phi-special} \phi(t)=\phi_{\beta,\gamma}(t)= \frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}t^{\beta-1}(1+t)^{-\gamma}. \end{equation} Motivation for this family comes from two particular examples. The first one is the path-censored process introduced in \cite{KPW14}. This process is obtained from $\eta$ by removing parts of the path in $(-\infty,0]$. More formally, define $A_t:=\int_0^t {\bf 1}_{(\eta_s>0)}ds$ and let $\tau_t:=\inf\{s>0:\, A_s>t\}$ be its right-continuous inverse. The process $\theta=(\theta_t)_{t\ge 0}$, defined by $\theta_t=\eta_{\tau_t}$, is a strong Markov process on $(0,\infty)$, called the \emph{path-censored} process of $\eta$ on $(0,\infty)$. The part of the process $\theta$ until its first hitting time of $0$ can be described in the following way: Let $x=\eta_{\tau-}\in (0,\infty)$ be the position from which $\eta$ jumps out of $(0,\infty)$, and $z=\eta_{\tau}<0$ be the position where $\eta$ lands at the exit from $(0,\infty)$. The distribution of the returning position of $\eta$ to $(0,\infty)$ has the density $P_{(-\infty, 0)}(z,y)$ called the Poisson kernel: If $\sigma:=\inf\{t>0:\, \eta_t\in [0,\infty)\}$, then ${\mathbb P}_z(\eta_{\sigma}\in A)=\int_A P_{(-\infty,0)}(z,y)\, dy$, $A\in {\mathcal B}((0, \infty))$. The exact formula for this Poisson kernel is given by (e.g. \cite[Lemma 1.1]{Kyp18} which contains a minor typo: the $\alpha$ there should be $\alpha\rho$), $$
P_{(-\infty,0)}(z,y)=\frac{1}{\Gamma(1-\alpha\rho)\Gamma(\alpha\rho)}\left(\frac{y}{|z|}\right)^{-\alpha\rho}(|z|+y)^{-1}=\phi_{1-\alpha\rho, 1}\left(\frac{y}{|z|}\right)\frac{1}{|z|}. $$ In the second example the return kernel is equal to the normalized jump kernel, see \cite{DROV17, Von21}, \begin{align} \label{e:njk}
p(z,y)=\frac{j(z,y)}{\int_0^{\infty}j(z,u)du}=\alpha |z|^{\alpha}(|z|+y)^{-1-\alpha}=\phi_{1,1+\alpha}\left(\frac{y}{|z|}\right)\frac{1}{|z|}. \end{align} Here we used that $j(z,y)=c_+(y-z)^{-1-\alpha}$ for $z<0$, $y>0$.
In the second family, the probability measure $\phi$ has a density which decays exponentially at infinity: For $a,\beta, \gamma>0$, let \begin{equation}\label{e:phi-special-exp} \phi(t)=\phi_{a,\beta,\gamma}(t)=\frac{\gamma a^{\frac{\beta}{\gamma}}}{\Gamma(\frac{\beta}{\gamma})}t^{\beta-1} e^{-at^{\gamma}}, \quad t>0. \end{equation} Then $$
p(z,y)=\frac{\gamma a^{\frac{\beta}{\gamma}}}{\Gamma(\frac{\beta}{\gamma})}\frac{y^{\beta-1}}{|z|^{\beta}} e^{-a\left({y}/{|z|}\right)^{\gamma}}. $$
\section{Resurrected process}\label{s:res-proc}
Let us go back to the strictly $\alpha$-stable process $\eta$. Instead of killing $\eta$ upon exiting $(0,\infty)$ to get $X^{\ast}$, or restarting at $\eta_{\tau-}$ to get the censored process $X$, we look at the exit point $\eta_{\tau}=z<0$ and (partially) resurrect according to some probability kernel $p(z,A)$, $A\in {\mathcal B}( [ 0, \infty))$. We assume that $p(z, \cdot)$ satisfies \eqref{e:p-scaling-measure}. Recall from \eqref{e:p-phi-measure} that such kernels are in one-to-one correspondence with probability measures $\phi$ on ${\mathcal B}([ 0, \infty))$ through the relation $$
p(z,A)=\phi(|z|^{-1}{A}), \quad z<0. $$ If the measure $p(z,\cdot)$ has a density with respect to the Lebesgue measure, we will denote it by $p(z,y)$, $y>0$. It is immediate that \begin{equation}\label{e:p-scaling} p(\lambda z, \lambda y)=\lambda^{-1} p(z,y)\, \quad \text{for all }z<0, \text{ a.e. }y>0 \text{ and all }\lambda >0. \end{equation} Recall that $\nu(x)$ is the L\'evy density given in \eqref{e:stable-density}. For $x,z\in {\mathbb R}$, let $j(x,z):=\nu(z-x)$. If $x>0$ and $z<0$, we have that \begin{equation}\label{e:j-nonsym} j(x,z)=c_-(x-z)^{-1-\alpha}. \end{equation} Further, $j$ enjoys the following scaling property: \begin{align} \label{e:spj} j(\lambda x, \lambda z)=\lambda^{-1-\alpha}j(x,z), \quad x>0, z<0, \lambda >0. \end{align} Recall that the resurrection kernel $q(x, A)$ was defined in \eqref{e:int-kernel-measure} as \begin{align} \label{e:qpph} q(x,A):= \int_{ (-\infty, 0) }j(x,z)p(z,A)\, dz
=\int_{ (-\infty, 0)}\frac{c_-}{(x-z)^{1+\alpha}}\phi\left(\frac{A}{|z|}\right) dz \quad x>0, A\in {\mathcal B}((0, \infty)). \end{align}
\subsection{Compound Poisson processes corresponding to resurrection kernels }\label{ss:cpp}
In this subsection we study the pssMp defined through the resurrection kernel $q$ and the corresponding L\'evy process. From the scaling properties of $p$ in \eqref{e:p-scaling-measure}
and $j$ in \eqref{e:spj} we get that the resurrection kernel $q$ satisfies $$ q(\lambda x, \lambda A)=\lambda^{-\alpha}q(x,A), \quad x>0, A\in {\mathcal B}((0, \infty)),\lambda >0.
$$ In particular, $q (1,A)=x^{\alpha }q(x,xA)$, implying that for any $g:(0, \infty) \to {\mathbb R}$, \begin{equation}\label{e:integral-1-x} \int g(y)q(1,dy)= x^{ \alpha}\int g(y/x)q(x,dy). \end{equation}
\begin{lemma}\label{l:q-always-density} For all $x>0$, the measure $q(x, \cdot)$ has a density $q(x,y)$ given by \begin{equation}\label{e:q-always-density} q(x,y):=c_-\int_{(0,\infty)} \left(x+\frac{y}{t}\right)^{-1-\alpha}t^{-1}\phi(dt). \end{equation} \end{lemma} \noindent{\bf Proof.} By \eqref{e:qpph}, for any $A\in {\mathcal B}((0, \infty))$ we have \begin{eqnarray*} q(x,A)&=&c_- \int_0^\infty (x+z)^{-1-\alpha}\left(\int_{(0, \infty)} {\bf 1}_{A/z}(t)\phi(dt)\right) dz\\ &=&c_- \int_{(0, \infty)} \int^\infty_0(x+z)^{-1-\alpha}{\bf 1}_A(zt)\, dz\, \phi(dt)\\ &=& c_- \int_{(0, \infty)} \int^\infty_0\left(x+\frac{y}{t}\right)^{-1-\alpha}t^{-1} {\bf 1}_A(y)\, dy\, \phi(dt)\\ &=&\int_A \left(c_- \int_{(0, \infty)}\left(x+\frac{y}{t}\right)^{-1-\alpha}t^{-1}\, \phi(dt) \right) dy, \end{eqnarray*} where in the second and last equalities we used Tonelli's theorem and in the penultimate equality the change of variables $y=tz$. {
$\Box$
}
We record here a simple consequence of \eqref{e:q-always-density}: For all $y>0$, \begin{equation}\label{e:simple-bound-on-q} q(1,y)\le c_- (y^{-1-\alpha}+1). \end{equation} Indeed, \begin{eqnarray*} q(1,y)&=&c_-\left(\int_{(0,1)}\left(1+\frac{y}{t}\right)^{-1-\alpha}\frac{\phi(dt)}{t}+\int_{[1,\infty)}\left(1+\frac{y}{t}\right)^{-1-\alpha}\frac{\phi(dt)}{t}\right)\\ &\le & c_- \left( y^{-1-\alpha}\int_{(0,1)}t^{\alpha}\phi(dt)+\int_{[1,\infty)}t^{-1}\phi(dt)\right)\le c_- (y^{-1-\alpha}+1). \end{eqnarray*}
Note that $$ q(x):= q(x,(0, \infty))=\int_{-\infty}^0j(x,z)p(z,(0, \infty))dz =\mathfrak{p} c_-\int_{-\infty}^0(x-z)^{-1-\alpha}dz= \mathfrak{p} \frac{c_-}{\alpha}x^{-\alpha}, $$ so that $$ Q(x,A):=\frac{q(x,A)}{q(x)}, \quad x>0, A\in {\mathcal B}((0, \infty)), $$ is a well-defined probability kernel satisfying $$ Q(\lambda x, \lambda A)=Q(x,A), \quad x>0, A\in {\mathcal B}((0, \infty)), \lambda >0. $$ It follows from Lemma \ref{l:q-always-density} that both $q(x, \cdot)$ and $Q(x,\cdot)$ have densities $q(x,y)$, resp.~$Q(x,y)$, satisfying $$ q(\lambda x, \lambda y)=\lambda^{-1-\alpha}q(x,y), \quad Q(\lambda x, \lambda y)=\lambda^{-1}Q(x,y), \quad x,y>0, \lambda >0. $$
We define now $\Pi(x,\cdot)$ to be the image measure of $Q(e^x,\cdot)$ under the mapping $y\mapsto e^y$: $\Pi(x,A):=Q(e^x, e^A)$. In particular, for every $g:{\mathbb R}\to {\mathbb R}$, \begin{equation}\label{e:pi-q-change} \int_{{\mathbb R}}g(y)\Pi(x,dy)= \int_{(0,{\infty})}g(\log y)Q(e^x, dy). \end{equation} Note that $\Pi(\cdot, \cdot)$ is translation invariant, that is, for all $u\in {\mathbb R}$, $$ \Pi(x+u, A+u)=Q(e^x e^u, e^u e^A)=Q(e^x, e^A)=\Pi(x,A). $$ Let $$ \Pi(A):= \Pi(0,A) =Q(1,e^A), \quad A\in {\mathcal B}({\mathbb R}), $$ and $$ \pi(A):=q(1,e^A)=q(1)Q(1,e^A)= q(1)\Pi(A), \quad A\in {\mathcal B}({\mathbb R}). $$ Clearly, $\Pi$ is a probability measure on ${\mathcal B}({\mathbb R})$ and $\pi$ a finite measure, hence a L\'evy measure. Both $\Pi$ and $\pi$ have densities (which again by an abuse of notation we denote by the same letters) satisfying \begin{equation}\label{e:densities-Q-q} \Pi(y)=Q(1,e^y)e^y \quad \text{and } \quad \pi(y)=q(1,e^y)e^y=e^{-\alpha y}q(e^{-y},1). \end{equation} It follows from \eqref{e:simple-bound-on-q} that \begin{equation}\label{e:simple-bound-on-pi} \pi(y)\le c_- (e^{-(1+\alpha)y}+1)e^y=c_- (e^{-\alpha y}+e^y). \end{equation} In particular, $\pi(y)$ is bounded in every neighborhood of 0.
Let $\chi$ be a compound Poisson process with characteristic exponent $\Psi^\chi(\theta)=\int_{{\mathbb R}}(1-e^{i\theta y})\pi(dy)$. The jump distribution of $\chi$ is given by the measure $\Pi$ and the jump rate is $q(1)$. The infinitesimal generator of $\chi$ is given by $$ {\mathcal A} f(x)=\int_{{\mathbb R}}(f(x+y)-f(x))\pi(dy). $$
Let $Y=(Y_t, {\mathbb P}_x)$ be the pssMp of index $\alpha$ related to $\chi$ through the Lamperti transform. The infinitesimal generator of $Y$ is, according to \eqref{e:pssMp-inf-gen-2}, equal to \begin{eqnarray}\label{e:new-number-for-LL} {\mathcal L}^Yf(x)&=&x^{-\alpha} \int_{{\mathbb R}}(f(xe^u)-f(x))\pi(du)=x^{-\alpha} \int_{(0,{\infty})} (f(xu)-f(x))q(1,du) \nonumber \\ &=& \int_{(0,{\infty})}(f(y)-f(x))q(x,dy), \end{eqnarray} where we used \eqref{e:pi-q-change} in the second equality, and \eqref{e:integral-1-x} in the third equality. This shows that $Y=(Y_t, {\mathbb P}_x)$ is a regular step process defined by the Markov kernel $Q(x,A)$, $A\in {\mathcal B}((0, \infty))$, and the holding function $q(x)$, cf.~\cite[I.12]{BG68}.
\begin{thm}\label{t:ft-pi} Suppose that $p(z,\cdot)$ is given by \eqref{e:p-phi-measure}. Then $$ \hat{\pi}(\theta):=\int_{{\mathbb R}}e^{i\theta y}\pi(dy)=\frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)} \int_{(0, \infty)}u^{i\theta}\phi(du). $$ \end{thm} \noindent{\bf Proof.} We have for $A\in {\mathcal B}({\mathbb R})$, \begin{eqnarray} \pi(A)&=&q(1,e^A)=c_- \int_0^\infty (1+z)^{-1-\alpha}\phi\left(\frac{e^A}{z}\right)dz \nonumber \\ &=&c_- \int_0^\infty(1+z)^{-1-\alpha}\left(\int_{(0, \infty)} {\bf 1}_{e^A/z}(y)\phi(dy)\right)dz \nonumber \\ &=&c_- \int_0^\infty(1+z)^{-1-\alpha}\left(\int_{(0, \infty)} {\bf 1}_A(\log(yz))\phi(dy)\right)dz \nonumber \\ &=&c_- \int_{(0, \infty)}\left( \int_0^\infty {\bf 1}_A(\log(yz))(1+z)^{-1-\alpha}dz\right)\phi(dy). \label{e:pi} \end{eqnarray} Therefore, by using \cite[8.380.1-3]{GR07}, \begin{eqnarray*} \hat{\pi}(\theta) &=&\int_{{\mathbb R}}e^{i\theta y}\pi(dy) = c_- \int_{(0, \infty)}\left( \int_0^\infty e^{i\theta\log(yz)}(1+z)^{-1-\alpha}dz \right)\phi(dy)\\ &=&c_- \int_{(0, \infty)}\left( \int_0^\infty (yz)^{i\theta}(1+z)^{-1-\alpha}dz\right)\phi(dy)\\ &=& c_- \left( \int_0^\infty z^{i\theta}(1+z)^{-1-\alpha}dz\right) \left(\int_{(0, \infty)}y^{i\theta}\phi(dy)\right)\\ &=& \frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)}\, \int_{(0, \infty)}y^{i\theta}\phi(dy). \end{eqnarray*} {
$\Box$
}
This theorem allows us to rewrite the density of the jump distribution of $\chi$ as a convolution of a subprobability and a probability distribution on ${\mathbb R}$, cf.~\cite[p.411, 2nd paragraph]{KPW14}. Define $$ \tau(A):=\phi(e^A), \quad A\in {\mathcal B}({\mathbb R}), \quad\text{and} \quad f(y):=\frac{\alpha e^y}{(1+e^y)^{1+\alpha}}, \quad y\in {\mathbb R}. $$ Then $f$ is a probability density on ${\mathbb R}$ and $$ \hat{f}(\theta):=\int_{{\mathbb R}}e^{i\theta y}f(y)dy =\frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)}, \quad \hat{\tau} (\theta):=\int_{{\mathbb R}}e^{i\theta y}\tau(dy)= \int_{(0, \infty)}y^{i\theta}\phi(dy). $$
\begin{corollary}\label{c:density-Pi} It holds that $$ \Pi(y)=(f\ast \tau)(y)=\alpha e^{ y} \int_{(0, \infty)} \frac{t^\alpha} {(t+e^{y})^{1+\alpha}}\phi(dt), \quad y\in {\mathbb R}. $$ \end{corollary} \noindent{\bf Proof.} The first equality is an immediate consequence of the equality $\hat{\Pi}=\hat{f}\hat{\tau}$. For the second equality, we rewrite \begin{align*}\label{e:demsity-Pi-general} \Pi(y)&= \int_{{\mathbb R}} \frac{\alpha e^{y-u}}{(1+e^{y-u})^{1+\alpha}}\tau(du)=\alpha e^y \int_{(0, \infty)} \frac{ t^{-1}}{(1+t^{-1}e^{y})^{1+\alpha}} \phi(dt) =\alpha e^{ y} \int_{(0, \infty)} \frac{t^\alpha} {(t+e^{y})^{1+\alpha}}\phi(dt). \nonumber \end{align*} {
$\Box$
}
\begin{corollary}\label{c:ft-pi} (a) Assume that $\phi$ is given by \eqref{e:phi-special}. Then \begin{equation}\label{e:hat-pi} \hat{\pi}(\theta)= \frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)} \frac{\Gamma(\beta+i\theta)\Gamma(\gamma-\beta-i\theta)}{\Gamma(\beta)\Gamma(\gamma-\beta)} . \end{equation}
\noindent (b) Assume that $\phi$ is given by \eqref{e:phi-special-exp}. Then \begin{equation}\label{e:hat-pi-phi2} \hat{\pi}(\theta)= \frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)} \frac{a^{-\frac{i\theta}{\gamma}}\Gamma\left(\frac{\beta+i\theta}{\gamma}\right)
}{\Gamma\left(\frac{\beta}{\gamma}\right)}. \end{equation}
\noindent (c) Assume that $\phi=\delta_a$, $a\in (0,{\infty})$. Then $$ \hat{\pi}(\theta)= \frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)} \, a^{i\theta}. $$ \end{corollary} \noindent{\bf Proof.} (a) It follows from Theorem \ref{t:ft-pi} and \cite[8.380.1-3]{GR07} that \begin{eqnarray*} \hat{\pi} (\theta)&=&\frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)} \frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}\int_0^{\infty} \frac{u^{i\theta +\beta-1}}{(1+u)^{\gamma}}\, du\\ &=&\frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta)\Gamma(1+i \theta)}{\Gamma(\alpha)} \frac{\Gamma(\beta+i\theta)\Gamma(\gamma-\beta-i\theta)}{\Gamma(\beta)\Gamma(\gamma-\beta)} . \end{eqnarray*}
\noindent (b) This follows from Theorem \ref{t:ft-pi} by noting (after the change of variables $v=au^{\gamma}$) that $$ \int_0^{\infty}u^{i\theta+\beta-1}e^{-au^{\gamma}} \, du= \gamma^{-1}a^{-\frac{\beta+i\theta}{\gamma}}\Gamma\left(\frac{\beta+i\theta}{\gamma}\right). $$
\noindent (c) This is clear. {
$\Box$
}
In case $\phi$ is given by \eqref{e:phi-special}, we can also compute the density $\Pi(y)$ of the jump distribution of $\chi$. By using Corollary \ref{c:density-Pi} and the change of variables $u=t^{-1}e^y$
we have \begin{eqnarray}\label{e:tilde-pi-exact} \lefteqn{\Pi(y) = \frac{\alpha\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)} e^y \int_0^{\infty} \left(t+e^y\right)^{-1-\alpha}t^{\beta+\alpha-1}(1+t)^{-\gamma}dt }\nonumber \\ &=& \frac{\alpha\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)} e^{-(\gamma-\beta) y}\int_0^{\infty} u^{\gamma-\beta}(1+ue^{-y})^{-\gamma}(1+u)^{-1-\alpha}du \nonumber \\ &=&\frac{\alpha\Gamma(\gamma)B(1+\gamma-\beta, \beta-\alpha)}{\Gamma(\beta)\Gamma(\gamma-\beta)} e^{-(\gamma-\beta) y}\, {_2}{\mathcal F}_1(\gamma, 1+\gamma-\beta; 1+\alpha+\gamma; 1-e^{-y}), \end{eqnarray} where the last line follows from \cite[3.197.5]{GR07}. Here ${_2}{\mathcal F}_1$ is the hypergeometric function.
\begin{example}\label{ex:trace}{\rm (a) Recall that $\alpha \rho<1$. Take $\beta=1-\alpha \rho$ and $\gamma=1$ in \eqref{e:phi-special}, we get \begin{equation}\label{e:res-trace}
p(z,y)=\frac{1}{\Gamma(1-\alpha \rho)\Gamma(\alpha \rho)}\frac{|z|^{\alpha \rho}}{y^{\alpha\rho}}(|z|+y)^{-1}. \end{equation} Since $\Pi(y)=(\alpha/c_-)\pi(y)$, it follows from Corollary \ref{c:ft-pi} that the characteristic function of the jump distribution $\Pi$ of $\chi$ is equal to $$ \frac{\Gamma(\alpha-i\theta)\Gamma(1+i\theta)}{\Gamma(\alpha)} \frac{\Gamma(1-\alpha\rho+i\theta)\Gamma(\alpha \rho-i\theta))}{\Gamma(1-\alpha \rho)\Gamma(\alpha \rho)} . $$ Since $\Gamma(1-\alpha \rho)\Gamma(\alpha \rho)=\frac{\pi}{\sin(\pi\alpha\rho)}$, cf.~\cite[8.334.3]{GR07}, we see that the characteristic function of the jump distribution of $\chi$ coincides with the one in \cite[Proposition 4.2, (9)]{KPW14}. This means that the return kernel given in \eqref{e:res-trace} corresponds to the path censored (or trace) process studied in \cite{KPW14}. This can be also seen by recognizing $p(z,y)$ from \eqref{e:res-trace} as the Poisson kernel $P_{(-\infty,0)}(z,y)$ of the stable process $\eta$ (see Subsection \ref{ss:examples-return} above). By using \eqref{e:tilde-pi-exact} with $\beta=1-\alpha\rho$ and $\gamma=1$, we find $$ \Pi(y)=\frac{\alpha\Gamma(\alpha \rho+1)\Gamma(\alpha\widehat{\rho}+1)}{\Gamma(1-\alpha\rho)\Gamma(\alpha\rho)\Gamma(\alpha+2)}
\, e^{-\alpha\rho y}\, _2{\mathcal F}_1(1, \alpha\rho+1; \alpha+2; 1-e^{-y}),\quad y\in {\mathbb R}, $$ cf.~\cite[(13)]{KPW14}.
\noindent (b) Take $\beta=1$ and $\gamma=1+\alpha$ in \eqref{e:phi-special}. Then \begin{equation}\label{e:rez-Z}
p(z,y)=\frac{\Gamma(1+\alpha)}{\Gamma(\alpha)}|z|^{\alpha}(|z|+y)^{-1-\alpha}=\alpha|z|^{\alpha}(|z|+y)^{-1-\alpha}, \end{equation} which is the normalized jump kernel in \eqref{e:njk}. It follows from Corollary \ref{c:ft-pi} that the characteristic function of the jump distribution of $\chi$ is equal to $$ \hat{\Pi} (\theta)=\frac{\Gamma(\alpha-i\theta)^2 \Gamma(1+i\theta)^2}{\Gamma(\alpha)^2}, $$ and the density is $$ \Pi(y)=\alpha^2 B(1+\alpha, 1-\alpha)e^{-\alpha y} \, _2{\mathcal F}_1(1+\alpha, 1+\alpha; 2+2\alpha; 1-e^{-y}),\quad y\in {\mathbb R}. $$
\noindent (c) Suppose that $\phi=\delta_a$, $a>0$. Then $$ \Pi(y)= \alpha e^{ y} \frac{a^\alpha} {(a+e^{y})^{1+\alpha}}, \quad y\in {\mathbb R}. $$ } \end{example}
We end this subsection with a necessary and sufficient condition for $\chi_1$ to have finite expectation.
\begin{prop}\label{p:chi-exp}
${\bf E}|\chi_1|<\infty$ if and only if \eqref{e:phi-int-log} holds true. \end{prop} \noindent{\bf Proof.} It follows from \cite[Theorem 25.3, Example 25.12]{Sat14} that
${\bf E}|\chi_1|<\infty$ if and only if
$\int_{|y|\ge 1}|y|\pi(dy)<\infty$.
We first assume \eqref{e:phi-int-log}. We will show that $\int_{{\mathbb R}}|y|\pi(dy)<\infty$. By using \eqref{e:pi} we have \begin{eqnarray*}
\int_{{\mathbb R}}|y|\pi(dy)&=&c_- \int_{(0, \infty)}\left(\int_0^\infty
|\log(yz)|(1+z)^{-1-\alpha}dz \right)\phi(dy) \\ &\le & c_- \int_{(0, \infty)}\left(\int_0^\infty
(|\log y| +|\log z|)|(1+z)^{-1-\alpha}dz \right)\phi(dy) \\ &=& c_-
\int_{(0, \infty)}|\log y| \left(\int_0^\infty (1+z)^{-1-\alpha}dz\right)\phi(dy)\\
& & + c_-
\int_{(0, \infty)}\left(\int_0^\infty
|\log z|(1+z)^{-1-\alpha}dz\right)\phi(dy) <\infty. \end{eqnarray*} Finiteness follows from the assumption and the fact that both integrals with respect to $dz$ are finite.
We now assume that
$\int_{|y|\ge 1}|y|\pi(dy)<\infty$. Then by \eqref{e:pi} \begin{align*} \infty&>\int_{y \le -1}(-y)\pi(dy) \ge c_- \int_{(0, 1)}\left(\int_0^\infty (-\log(yz)) (1+z)^{-1-\alpha} {\bf 1}_{\log(yz)\le -1} dz \right)\phi(dy) \\ & \ge c_- \int_{(0, 1)}\left(\int_0^\infty
(-\log y)(1+z)^{-1-\alpha}{\bf 1}_{\log z\le -1} dz \right)\phi(dy)\\ &= c_-\left(\int_0^{1/e} (1+z)^{-1-\alpha}dz\right) \int_{(0, 1)}(-\log y) \phi(dy). \end{align*} Thus, $\int_{(0, 1)}(-\log y) \phi(dy)< \infty.$ Similarly, \begin{align*} \infty&>\int_{y \ge 1}y\pi(dy) \ge c_- \int_{(1, \infty)}\left(\int_0^\infty \log(yz)(1+z)^{-1-\alpha} {\bf 1}_{\log(yz)\ge 1} \, dz \right)\phi(dy) \\ & \ge c_- \int_{(1, \infty)}\left(\int_0^\infty \log y(1+z)^{-1-\alpha} {\bf 1}_{\log z\ge 1} \, dz \right) \phi(dy) \\ & = c_-\left(\int_{e}^\infty (1+z)^{-1-\alpha}dz\right) \int_{(1, \infty)}(\log y) \phi(dy). \end{align*} Thus, $\int_{(1, \infty)} ( \log y ) \phi(dy)<\infty$. We have shown that \eqref{e:phi-int-log} holds.
{
$\Box$
}
\subsection{Resurrected process}\label{ss:res-proc} Let $\Psi(\theta)=\Psi^{\ast}(\theta)-\Psi^{\ast}(0)=\Psi^{\ast}(\theta)-c_-/\alpha$ be the characteristic exponent of the L\'evy process $\xi$ corresponding to the censored $\alpha$-stable process $X$ through the Lamperti transform. We define $$ \Psi^{\sharp}(\theta):=\Psi(\theta)+(1-\mathfrak{p})\frac{c_-}{\alpha}=\Psi^{\ast}(\theta)-\mathfrak{p}\frac{c_-}{\alpha}, $$ and let $\xi^{\sharp}$ be the (killed) L\'evy process with the characteristic exponent $\Psi^{\sharp}$.
We will add to $\xi^{\sharp}$ an independent compound Poisson process, denoted by $\chi$, with characteristic exponent $\Psi^{\chi}$ given by \begin{equation}\label{e:ch-exp-CPP} \Psi^{\chi}(\theta)=\int_{{\mathbb R}}(1-e^{i\theta y})\pi(dy) = \mathfrak{p} \frac{c_-}{\alpha}-\hat{\pi}(\theta), \quad \theta\in {\mathbb R}. \end{equation} The effect of this procedure is that, instead of completely removing the killing from $\xi^{\ast}$, we remove part of the killing (i.e., $\mathfrak{p}c_-/\alpha$) and add jumps according to $\Pi$ at the exponential rate $\widehat{\pi}(0)=\mathfrak{p}c_-/\alpha$. Let $\overline{\xi}:= \xi^{\sharp} +\chi$ be this new L\'evy process. By \eqref{e:char-exp-xi*}, its characteristic exponent $\overline{\Psi}$ is given by \begin{align}\label{e:char-exp-final} \overline{\Psi}(\theta) = \Psi^{\sharp}(\theta)+\Psi^{\chi}(\theta) =\frac{\Gamma(\alpha-i\theta)}{\Gamma(\alpha\widehat{\rho}-i\theta)}\, \frac{\Gamma(i\theta+1)}{\Gamma(i\theta+1-\alpha\widehat{\rho})}- \hat{\pi} (\theta). \end{align} Note that $$ \overline{\Psi}(0)=\frac{\Gamma(\alpha)}{\Gamma(\alpha\widehat{\rho})\Gamma(1-\alpha\widehat{\rho})}-\widehat{\pi}(0)=\frac{c_-}{\alpha}(1-\mathfrak{p}). $$ If $\mathfrak{p}=1$, then $\overline{\Psi}(0)=0$, implying that in the Lamperti trichotomy the case (3) does not occur. If $\mathfrak{p}<1$, $\overline{\xi}$ is a killed L\'evy process with rate $\overline{\Psi}(0)$ and thus ${\bf E}{\overline \xi}_1=-\infty$.
Let $\overline{X}=(\overline{X}_t, {\mathbb P}_x)$ be the pssMp of index $\alpha$ with origin as a trap corresponding to $\overline{\xi}$ through the Lamperti transform. This process can be described as follows: Consider the strictly $\alpha$-stable process $\eta=(\eta_t, {\mathbb P}_x)$, and set as before $\tau=\inf\{t>0: \eta_t\le 0\}$. Then $\tau<\infty$ a.s. At time $\tau$, with probability $\mathfrak{p}$ we resurrect according to the return kernel $p(\eta_{\tau}, y)$, $y>0$, and with probability $1-\mathfrak{p}$ kill the process and send it to the origin. This amounts to adding the resurrection kernel $q(x,y)$ to the jump kernel $j(x,y)$ of $\eta$. The process $\overline{X}$ is a pssMp of index $\alpha$ with origin as a trap and jump kernel $J(x,y):=j(x,y)+q(x,y)$. Indeed, let ${\mathcal A}^{\xi}$ be the infinitesimal generator of $\xi$ and ${\mathcal L}^X$ the infinitesimal generator of $X$ given by \eqref{e:generator-LL-X}. The infinitesimal generator of $\overline{\xi}$ is obtained by adding $\pi$ to the L\'evy measure $\nu$ of $\xi$, and taking into account the killing term. Hence for $g\in C_0^2({\mathbb R})$, $$ {\mathcal A}^{\overline{\xi}}g=-\frac{c_-}{\alpha}(1-\mathfrak{p})g+{\mathcal A}^{\xi}g+{\mathcal A}^{\chi}g. $$ By using the relation between generators of the L\'evy process and the corresponding the Lamperti transformed pssMp of index $\alpha$, together with \eqref{e:new-number-for-LL}, we see that the infinitesimal generator of $\overline{X}$ is equal to $$ {\mathcal L}^{\overline{X}}f(x)= - \frac{c_-}{\alpha}(1-\mathfrak{p})x^{-\alpha}+ {\mathcal L}^X f(x)+\int_0^{\infty}(f(y)-f(x))q(x,y)dy. $$
Assume that $p(z,y)$ is given by \eqref{e:p-phi-measure}. Then by \eqref{e:c+c-}, \eqref{e:char-exp-final} and Theorem \ref{t:ft-pi}, \begin{align}\label{e:overline-Psi-general} &\overline{\Psi}(\theta)=\frac{\Gamma(\alpha-i\theta)\Gamma(i\theta+1)}{\Gamma(\alpha\widehat{\rho}-i\theta)\Gamma(i\theta+1-\alpha\widehat{\rho})}\, -\frac{c_-}{\alpha}\frac{\Gamma(\alpha-i\theta) \Gamma(i\theta+1)}{\Gamma(\alpha)} \int_{(0,{\infty})}u^{i\theta}\phi(du)\nonumber \\ &=\Gamma(\alpha-i\theta)\Gamma(i\theta+1)\left(\frac{1}{\Gamma(\alpha\widehat{\rho}-i\theta)\Gamma(i\theta+1-\alpha\widehat{\rho})}-\frac{1}{\Gamma(\alpha\widehat{\rho})\Gamma(1-\alpha\widehat{\rho})} \int_{(0,{\infty})}u^{i\theta}\phi(du)\right)\nonumber\\ &=\frac{\Gamma(\alpha-i\theta) \Gamma(i\theta+1)}{\pi} \left(\sin(\pi(\alpha\widehat{\rho}-i\theta)) -\sin(\pi\alpha\widehat{\rho}) \int_{(0,{\infty})}u^{i\theta}\phi(du)\right). \end{align} In the third line we used the identity $\Gamma(z)\Gamma(1-z)=\pi/\sin (\pi z)$ twice. \begin{remark}\label{r:ricochet} {\rm
For the ricocheted stable process from \cite{KPV21}, the measure $\phi$ determining the return kernel is equal to $(1-\mathfrak{p})\delta_0+\mathfrak{p}\delta_1$. In this case $$ \overline{\Psi}(\theta)=\frac{\Gamma(\alpha-i\theta) \Gamma(i\theta+1)}{\pi} \left(\sin(\pi(\alpha\widehat{\rho}-i\theta)) -\mathfrak{p}\sin(\pi\alpha\widehat{\rho})\right) $$ which recovers \cite[(4.2)]{KPV21}, } \end{remark}
\subsection{Behavior of $\overline{X}$ at absorption time }\label{ss:X-lifetime} If $\mathfrak{p}<1$, it follows from the Lamperti trichotomy that case (3) occurs, hence $\overline{X}$ is absorbed at 0 by a jump. In the remaining part of this subsection we therefore assume that $\mathfrak{p}=1$.
Recall that ${\bf E}|\xi_1|<\infty$ and, under assumption \eqref{e:phi-int-log}, also ${\bf E}|\chi_1|<\infty$, cf.~Proposition \ref{p:chi-exp}. Therefore, under assumption \eqref{e:phi-int-log},
${\bf E}|\overline{\xi}_1|<\infty$, $\overline{\Psi}'(0)$ exists, and ${\bf E}[\overline{\xi}_1]=i\overline{\Psi}'(0)$. Thus, combining \cite[Theorem 7.2]{Kyp14} with the Lamperti trichotomy, we get that if $i\overline{\Psi}'(0) \ge 0$, then $\limsup_{t\to \infty}\overline{\xi}_t=+\infty$, hence the absorption time of $\overline{X}$ is infinite; and if $i\overline{\Psi}'(0)< 0$, then $\lim_{t\to \infty}\overline{\xi}_t=-\infty$, hence the absorption time of $\overline{X}$ is finite ${\mathbb P}_x$-a.s.~and $\overline{X}$ is continuously absorbed at 0.
\noindent \textbf{Proof of Theorem \ref{t:derivative-at-zero}:}
The equivalence of $\mathbf{E}|\overline{\xi}_1|<\infty$ and \eqref{e:phi-int-log} follows from Proposition \ref{p:chi-exp}. Put $$ f_1(\theta):=B(\alpha-i\theta, 1+i\theta),\quad f_2(\theta):= \int_{(0,{\infty})}u^{i\theta}\phi(du) $$ $$ f_3(\theta) :=\frac{\sin(\pi(\alpha\widehat\rho-i\theta))}{\pi} -\frac{\sin(\pi \alpha\widehat\rho \, )}{\pi}f_2(\theta), $$ where $B$ denotes the beta function. Then by \eqref{e:overline-Psi-general}, $ \overline\Psi(\theta)=\Gamma(1+\alpha)f_1(\theta)f_3(\theta). $ Since $f_2(0)=1$ and $f_3(0)=0$, we have \begin{align} \label{oP0'} \overline\Psi'(0)=\Gamma(1+\alpha)(f'_1(0)f_3(0)+f_1(0)f'_3(0))= \Gamma(1+\alpha)B(\alpha, 1)f'_3(0)= \Gamma(\alpha)f'_3(0). \end{align} Using the assumption \eqref{e:phi-int-log}, $$
f'_2(0)=i \int_{(0,{\infty})}(\log u)u^{i\theta}\phi(du)|_{\theta=0}=i\int_{(0,{\infty})}(\log u)\phi(du), $$ and so $$ f'_3(0)=-i\cos(\pi\alpha\widehat\rho)-\frac{\sin(\pi \alpha\widehat\rho)}{\pi}f'_2(0)= -i\cos(\pi\alpha\widehat\rho)-i\frac{\sin(\pi \alpha\widehat\rho)}{\pi} \int_{(0,{\infty})}(\log u)\phi(du). $$ Therefore by \eqref{oP0'} \begin{align}\label{e:i-Psi-prime} i\overline\Psi'(0) &=\Gamma(\alpha)\left(\cos(\pi\alpha\widehat\rho)+\frac{\sin(\pi \alpha\widehat\rho)}{\pi} \int_{(0,{\infty})}(\log u)\phi(du)\right)\nonumber\\ &=\Gamma(\alpha)\frac{\sin(\pi \alpha\widehat\rho)}{\pi} \left(\pi\cot(\pi\alpha\widehat\rho)+ \int_{(0,{\infty})}(\log u)\phi(du)\right). \nonumber \end{align} {
$\Box$
}
Since $(\alpha-1)_+ < \alpha\widehat{\rho}<1$, we see from the display above that the sign of $i\overline{\Psi}'(0)$ depends on the sign of $$ \pi\cot(\pi\alpha\widehat\rho)+ \int_{(0,{\infty})}(\log u)\phi(du). $$ Note that $\int_{(0,{\infty})}(\log u)\phi(du)$ may depend on $\alpha$. For example, when $\phi$ is given by \eqref{e:phi-special} with $\beta=1$ and $\gamma=1+\alpha$, it holds that $\int_{(0,{\infty})}(\log u)\phi(du)=\psi(1)-\psi(\alpha)$, where $\psi$ is the digamma function.
Let $L_\phi:=- \int_{(0,{\infty})}(\log u)\phi(du)$. Define $\mathrm{arccot}: {\mathbb R}\to (0,\pi)$ as a strictly decreasing and continuous function. Set $$ a_{\phi}: =\frac{1}{ \pi}\mathrm{arccot}\left( \frac{L_\phi}{\pi}\right) $$ and note that $a_{\phi}\in (0,1)$. \begin{corollary}\label{c:s} Suppose $\mathfrak{p}=1$. (a) If $\alpha \le 1+a_{\phi}$,
then $\overline{X}$ is (continuously) absorbed at $0$ at an a.s.-finite time if and only if $\alpha \widehat{\rho}>a_{\phi}$. (b) If $\alpha>1+a_{\phi}$, then the absorption time of $\overline{X}$ is always finite ${\mathbb P}_x$-a.s.. \end{corollary} \noindent{\bf Proof.} (a) If $\alpha \le 1$, since $0< \alpha\widehat{\rho}<1$, we see that $\pi\cot(\pi\alpha\widehat\rho)-L_\phi<0$ if and only if $$ \widehat\rho> \frac{1}{ \alpha\pi}\mathrm{arccot} \left(\frac{L_\phi}{\pi}\right)=\frac{a_{\phi}}{ \alpha}. $$ If $1<\alpha \le 1+a_{\phi}$, then $$ \cot(\pi(\alpha-1)) \ge \cot(\pi a_{\phi})= \frac{L_\phi}{\pi} $$ and so we also have that $\pi\cot(\pi\alpha\widehat\rho)- L_\phi<0$ if and only if $\widehat\rho>\frac{a_{\phi}}{ \alpha}$.
\noindent (b) If $\alpha > 1+a_{\phi}$ then $\cot(\pi(\alpha-1)))< \frac{L_\phi}{\pi}$ and we always have $$ \pi\cot(\pi\alpha\widehat\rho)- L_\phi < \pi\cot(\pi(\alpha-1))- L_\phi \le 0. $$ {
$\Box$
}
In case $L_\phi=-\int_{(0,{\infty})}(\log u)\phi(du)$ is independent of $\alpha$, we can be slightly more precise. Let $$ \rho(\alpha):=1-\frac{1}{\alpha \pi}\mathrm{arccot}\left(\frac{L_\phi}{\pi}\right), $$ so that $\pi \cot\big(\pi \alpha(1-\rho(\alpha))\big)- L_\phi=0$. Notice that $\rho(a_{\phi})=0$ (this need not be true in case $L_\phi$ depends on $\alpha$). Since $L_\phi$ does not depend on $\alpha$, the function $\alpha\mapsto \rho(\alpha)$ is strictly increasing, hence $a_{\phi}$ is the only zero of $\rho(\alpha)$.
\begin{corollary}\label{c:s2} Suppose that $\mathfrak{p}=1$ and $L_\phi$ does not depend on $\alpha$. \begin{itemize} \item[(i)] If $\alpha\in (0, a_{\phi})$, then the absorption time of $\overline{X}$ is infinite; \item[(ii)] If $\alpha\in [a_{\phi}, 1+a_{\phi}]$, then the absorption time of $\overline{X}$ is finite if and only if $\rho>\rho(\alpha)$; \item[(iii)] If $\alpha\in (1+a_{\phi}, 2)$, then the absorption time of $\overline{X}$ is finite. \end{itemize} \end{corollary} \noindent{\bf Proof.} This is a direct consequence of Corollary \ref{c:s} and the discussion above. {
$\Box$
}
\subsection{Recurrent extension}\label{ss:rec-ext} Recall that the origin is a trap for $\overline{X}$. If $\overline{X}$ is absorbed in 0 at finite time, one can ask if there exists a positive self-similar recurrent extension of $\overline{X}$. The general result is given in \cite[Theorem 1]{Fit06} and \cite[Theorems 1 and 2]{Riv07}: (i) There exists a unique positive self-similar recurrent extension of $\overline{X}$ which leaves 0 continuously if and only if there exists $\kappa \in(0,\alpha)$ such that \begin{equation}\label{e:cont-rec-ext} {\bf E}\left[e^{\kappa \overline{\xi}_1}\right]=1, \end{equation} and (ii) For $\beta\in(0,\alpha)$ there exists a positive self-similar recurrent extension of $\overline{X}$ which leaves 0 by a jump associated with an excursion measure of the form $c\beta x^{-1-\beta} dx$ if and only if \begin{equation}\label{e:jump-rec-ext} {\bf E}\left[e^{\beta \overline{\xi}_1}\right]<1. \end{equation}
Note that ${\bf E}[e^{\kappa \overline{\xi}_1}]={\bf E}[e^{i(-i\kappa)\overline{\xi}_1}]=e^{-\overline{\Psi}(-i\kappa)}$ for all $\kappa\ge 0$ for which the expectation is finite. Let $\varphi:{\mathbb R}\to (-\infty, +\infty]$ be defined by $\varphi(\kappa):=-\overline{\Psi}(-i\kappa)$, so that $$ {\bf E}\left[e^{\kappa \overline{\xi}_t}\right]=e^{t\varphi(\kappa)}. $$ Hence, \eqref{e:cont-rec-ext} is equivalent to the existence of $\kappa\in (0,\alpha)$ such that $\varphi(\kappa)=0$, and \eqref{e:jump-rec-ext} is equivalent to $\varphi(\beta)<0$. Note that, by H\"older's inequality, $\varphi$ is convex.
\noindent \textbf{Proof of Theorem \ref{t:R_e}:} Let $$ h(\kappa):=\sin(\pi(\alpha\widehat{\rho}-\kappa))-\sin(\pi\alpha\widehat{\rho}) \int_{(0,{\infty})}u^{\kappa}\phi(du). $$ Clearly, $h(0)=(1-\mathfrak{p})\sin(\pi\alpha\widehat{\rho})\ge 0$ since $\alpha\widehat{\rho}\in (0,1)$, and note that from \eqref{e:overline-Psi-general} \begin{align}\label{e:Psik} -\varphi(\kappa)=\overline{\Psi}(-i\kappa) &=\frac{\Gamma(\alpha-\kappa)\Gamma(\kappa+1)}{\pi} h(\kappa). \end{align}
If $\kappa_0<\alpha$, then $\kappa_0+\epsilon <\alpha$ for all small $\epsilon >0$. Since $\int_{(0,\infty)}u^{\kappa_0+\epsilon}\phi(du)=+\infty$ by definition of $\kappa_0$, we get that $h(\kappa_0+\epsilon)=-\infty$.
Assume that \eqref{e:kappa_0} holds true, i.e., $\kappa_0>0$. If $\kappa_0 \ge\alpha$, then \begin{eqnarray*} h(\alpha-)&=&\sin(\pi(\alpha\widehat{\rho}-\alpha))-\sin(\pi\alpha\widehat{\rho})\lim_{\kappa\uparrow \alpha} \int_{(0,{\infty})}u^{\kappa}\phi(du) \\ &=&-\sin(\pi\alpha \rho)-\sin(\pi\alpha\widehat{\rho})\lim_{\kappa\uparrow \alpha} \int_{(0,{\infty})}u^{\kappa}\phi(du) <0, \end{eqnarray*} where in the last inequality we used the assumptions $\alpha\rho\in (0, 1)$ and $\alpha\widehat{\rho}\in (0, 1)$. Therefore, $h((\alpha\wedge (\kappa_0+\epsilon)) -)<0$ for all small $\epsilon >0$. If $\kappa\in ( 0,\kappa_0)$, \begin{align} \label{e:hprime} h'(\kappa)=-\pi\cos(\pi(\alpha\widehat{\rho}-\kappa))-\sin(\pi\alpha\widehat{\rho})\left( \int_{(0,{\infty})}(\log u)u^{\kappa}\phi(du) \right), \end{align} which is justified by \eqref{e:kappa_0}.
Assume that $\mathfrak{p}=1$. Since by the assumption that $\overline{X}$ is absorbed at 0 in finite time, this happens continuously, and therefore ${\bf E}[\overline{\xi}_1] \in [-\infty, 0)$. If \eqref{e:phi-int-log} holds true, since $i\overline{\Psi}'(0)<0$, by Theorem \ref{t:derivative-at-zero} and \eqref{e:hprime}, $$ h'(0+)=-\pi\cos(\pi\alpha\widehat{\rho})-\sin(\pi\alpha\widehat{\rho}) \int_{(0,{\infty})}(\log u)\phi(du)>0, $$
implying that $h$ is strictly positive in some neighborhood of zero.
Note that by \eqref{e:kappa_0}, we have $\int_{(1,{\infty})}(\log u)u^{\kappa}\phi(du)<\infty$ for $\kappa\in ( 0,\kappa_0)$ and thus $\int_{(1,{\infty})}(\log u)\phi(du)\le \int_{(1,{\infty})}(\log u)u^{\kappa}\phi(du)<\infty$. Consequently, if \eqref{e:phi-int-log} does not holds, then we have $\int_{(0,1)}(\log u)\phi(du)=-\infty$.
By the monotone convergence theorem, $$ \lim_{\kappa \downarrow 0}
\int_{(0,1)}(\log u)u^{\kappa}\phi(du)=
-\lim_{\kappa \downarrow 0}\int_{(0,1)}(-\log u)u^{\kappa}\phi(du)=-\infty. $$ We now see from \eqref{e:hprime} that $$ \liminf_{\kappa \downarrow 0} h'(\kappa)=-\pi\cos(\pi\alpha\widehat{\rho})-\sin(\pi\alpha\widehat{\rho}) \left(\lim_{\kappa \downarrow 0} \int_{(0,{\infty})}(\log u)u^{\kappa}\phi(du) \right) =\infty, $$ implying again that $h$ is strictly positive in some neighborhood of zero.
If $\mathfrak{p}<1$, then $h(0)>0$, so again we see that $h$ s strictly positive in some neighborhood of zero.
Together with $h((\alpha\wedge (\kappa_0+\epsilon)) -)<0$ for all small $\epsilon >0$, this implies the existence of $\kappa^{\ast}\in (0, \alpha)$ such that $h(\kappa^{\ast})=0$, hence also $\varphi(\kappa^{\ast})=0$. Thus $\overline{X}$ has a positive self-similar recurrent extension. Furthermore, by the convexity of $\varphi$, for every $\beta\in (0,\kappa^{\ast})$ we have $\varphi(\beta)<0$. This means that ${\bf E}[e^{\beta \overline{\xi}_1}]<1$. By \cite[Theorem 1]{Riv07}, there exists a positive self-similar recurrent extension of $\overline{X}$ which leaves 0 by a jump associated with the excursion measure $c\beta x^{-1-\beta}dx, x>0$.
Assume that \eqref{e:kappa_0} is false, that is $\kappa_0 = 0$. Then $h(\epsilon)=-\infty$ for all $\epsilon>0$, and we see from \eqref{e:Psik} that $\varphi(\beta)=+\infty$ for all $\beta\in (0,\alpha)$ and consequently ${\bf E}[e^{\beta \overline{\xi}_1}]=+\infty$. Hence $\overline{X}$ does not have a recurrent extension. {
$\Box$
}
\begin{remark}\label{r:rec-ext-jump} {\rm It is easy to find examples of probability measures $\phi$ on $(0,\infty)$ satisfying \eqref{e:phi-int-log} but not \eqref{e:kappa_0} giving rise to pssMp that are (continuously) absorbed at zero in finite time, but not having a positive self-similar recurrent extension. One such example is the measure with density ${\bf 1}_{(2,\infty)}(t)\frac{2(\log 2)^2}{t(\log t)^3}$. } \end{remark}
\subsection{Examples}\label{ss:examples} In this subsection we analyze a list of examples. Recall that $\psi$ is the digamma function.
\begin{example}\label{ex:beta-gamma} {\rm We look at our main example in which $\phi$ is given by \eqref{e:phi-special}.
Let $f(t)=(1+t)^{-\gamma}$ with $\gamma>0$.
The Mellin transform of $f$ is, by \cite[p.1131, 17.43.7]{GR07}, equal to $$ M_f(s):=\int_0^{\infty}f(t)t^{s-1}dt=\frac{\Gamma(s)\Gamma(\gamma-s)}{\Gamma(\gamma)}. $$ Since $$ \int_0^{\infty}f(t)(\log t)t^{s-1} dt=M_f'(s)= \frac{1}{\Gamma(\gamma)} \Big(\Gamma'(s)\Gamma(\gamma-s)-\Gamma(s)\Gamma'(\gamma-s)\Big), $$ we get that $$ \int_0^{\infty}(\log t)\phi(t)dt =\frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}M_f'(\beta)=\psi(\beta)-\psi(\gamma-\beta). $$ Hence by \eqref{e:derivative-at-zero}, \begin{equation}\label{e:derivative-at-zero-special} i\overline{\Psi}'(0)=\Gamma(\alpha)\frac{\sin(\pi \alpha\widehat\rho)}{\pi} \left(\pi\cot(\pi\alpha\widehat\rho)+ \psi(\beta)-\psi(\gamma-\beta)\right). \end{equation}
\noindent {\bf (a)} Suppose that $\gamma=1$ in \eqref{e:phi-special}. Then $\beta\in (0,1)$, and by the reflection formula for the digamma function, $\psi(1-\beta)-\psi(\beta)=\pi\cot(\pi \beta)$, see \cite[8.365.8]{GR07}. Elementary calculation gives that \begin{equation} i\overline{\Psi}'(0)=-\Gamma(\alpha)\frac{\sin(\pi(\alpha\widehat{\rho}-\beta))}{\sin(\pi \beta)}, \nonumber \end{equation} and the sign of $i\overline{\Psi}'(0)$ depends on the sign of $\sin(\pi(\alpha\widehat{\rho}-\beta))$.
\noindent {\it Case 1}: $\alpha=1$. Then $\widehat{\rho}=1/2$, and $\sin(\pi(\alpha\widehat{\rho}-\beta))=\sin(\pi/2-\pi\beta)=\cos(\pi \beta)$. Therefore, $i\overline{\Psi}'(0)>0$ if $\beta\in (1/2,1)$, $i\overline{\Psi}'(0)=0$ if $\beta=1/2$, and $i\overline{\Psi}'(0)<0$ if $\beta\in (0,1/2)$.
\noindent {\it Case 2}: $\alpha\in (0,1)\cup (1,2)$. Since $\alpha\widehat{\rho}\in (0,1)$, we have that $\alpha\widehat{\rho}-\beta\in (-1,1)$. Thus, $\sin(\pi(\alpha\widehat{\rho}-\beta))=0$ if and only if $\rho=1-\frac{\beta}{\alpha}$. Since we must have that $\rho>0$ and $\rho< 1/\alpha$, we get two critical values: $\alpha_*=\beta$ and $\alpha^*:=1+\alpha_*=1+\beta$, see Corollary \ref{c:s2}. If $\alpha\in (0, \alpha_*)$, then $i\overline{\Psi}'(0)>0$; If $\alpha\in [\alpha_*, \alpha^*]$, then $i\overline{\Psi}'(0)>0$ if $\rho>1-\frac{\beta}{\alpha}$, $i\overline{\Psi}'(0)=0$ if $\rho=1-\frac{\beta}{\alpha}$, and $i\overline{\Psi}'(0)<0$ if $\rho<1-\frac{\beta}{\alpha}$; If $\alpha\in ( \alpha^*,2)$, then $i\overline{\Psi}'(0)<0$.
\begin{figure}
\caption{Example \ref{ex:beta-gamma} (a): Left: $\beta=1/3$, $\gamma=1$, $\alpha_*=1/3$, $\alpha^*=4/3$; Right: $\beta=2/3$, $\gamma=1$, $\alpha_*=2/3$, $\alpha^*=5/3$; $i\overline{\Psi}'(0)>0$ in the shaded region, $i\overline{\Psi}'(0)=0$ on the red line. }
\end{figure}
We further assume that $\beta=1-\alpha\rho$. The pssMp $\overline{X}$ is then the part (until the first hitting of zero) of the trace process of the $\alpha$-stable process $\eta$ in $(0, \infty)$ (or the path-censored $\alpha$-stable process). We have that $\alpha \widehat{\rho}-\beta =\alpha \widehat{\rho}+\alpha \rho-1 =\alpha-1$. Thus, if $\alpha<1$, then $\alpha \widehat{\rho}-\beta\in (-1,0)$, and therefore $i\overline{\Psi}'(0)>0$. If $\alpha=1$, then $\alpha \widehat{\rho}-\beta=0$, and therefore $i\overline{\Psi}'(0)=0$. If $\alpha>1$, then $\alpha \widehat{\rho}-\beta\in (0,1)$, and therefore $i\overline{\Psi}'(0)<0$. This shows that $\overline{X}$ has infinite absorption time in case $\alpha\in(0,1]$ and hits zero in finite time when $\alpha\in (1,2)$. Of course, since $\overline{X}$ is the trace process, this fact is well known.
\noindent {\bf (b)} Let us now consider the case $\beta=1$ and $\gamma=\alpha+1$. In this case \begin{eqnarray}\label{e:Psi'-for-Z} i\overline{\Psi}'(0) &=&\frac{\Gamma(\alpha)\sin(\pi \alpha\widehat\rho)}{\pi}\left(\pi\cot(\pi \alpha\widehat\rho))+(\psi(1)-\psi(\alpha))\right). \end{eqnarray} The equation $\pi\cot(\pi(\alpha-1))=\psi(\alpha)-\psi(1)$ (obtained by formally taking $\rho=1/\alpha$), has a unique solution $\alpha^{\ast}$ in $(1,2)$ which can be numerically computed. It turns out that $\alpha^{\ast}\approx 1.44386$ with corresponding $\rho^{\ast}=1/\alpha^{\ast}\approx 0.692588$. Further, solving $\pi\cot(\pi \alpha (1-\rho))+(\psi(1)-\psi(\alpha))=0$ for $\rho$, we get a unique solution $$ \rho(\alpha)=1-\frac{1}{\alpha \pi}\mathrm{arccot} \left(\frac{\psi(1)-\psi(\alpha)}{\pi}\right). $$ It is easy to see that $\lim_{\alpha \downarrow 0}\rho(\alpha)=-\infty$, $\lim_{\alpha\uparrow 2}\rho(\alpha)=1$, hence by continuity there exists $\alpha_*$ such that $\rho(\alpha_*)=0$, and consequently, $\rho(\alpha)>0$ for $\alpha\in (0,\alpha_*)$. Numerically we obtain $\alpha_*\approx 0.596051$.
Therefore, we conclude that (i) If $\alpha\in (0,\alpha_*]$, then $i\overline{\Psi}(0)>0$ for all $\rho\in (0,1)$; (ii) if $\alpha\in (\alpha_*, \alpha^*)$, then $i\overline{\Psi}(0)>0$ for $\rho>\rho(\alpha)$, $i\overline{\Psi}(0)=0$ for $\rho=\rho(\alpha)$, $i\overline{\Psi}(0)<0$ for $\rho<\rho(\alpha)$; (iii) if $\alpha\in [\alpha^*, 2)$, then $i\overline{\Psi}(0)<0$ for all admissible $\rho$. \begin{figure}
\caption{Example
\ref{ex:beta-gamma} (b): $\alpha_*\approx 0.596501$, $\alpha^*\approx 1.44386$; $i\overline{\Psi}'(0)>0$ in the shaded region, $i\overline{\Psi}'(0)=0$ on the red line.}
\label{cap:2}
\end{figure} } \end{example}
\begin{example}\label{ex:reflected-process} {\rm In case $\phi=\delta_a$, \begin{equation}\label{e:derivative-at-zero-case2} i\overline{\Psi}'(0)=\Gamma(\alpha)\frac{\sin(\pi \alpha\widehat\rho)}{\pi} \left(\pi\cot(\pi\alpha\widehat\rho)-\log a\right). \end{equation} If $a=1$, the corresponding pssMp $\overline{X}$ of index $\alpha$ is the absolute value of the strictly $\alpha$-stable L\'evy process. The sign of $i\overline{\Psi}'(0)$ depends on whether $\alpha\widehat{\rho}$ is less than, equal, or larger than 1/2. Similar analysis can be done for any $a>0$. \begin{figure}
\caption{Example \ref{ex:reflected-process}: $a=1$; $i\overline{\Psi}'(0)>0$ in the shaded region, $i\overline{\Psi}'(0)=0$ on the red line.}
\end{figure} } \end{example}
\begin{example}\label{ex:phi-exp-decay}{\rm We now look at the example in which $\phi$ is given by \eqref{e:phi-special-exp}. Then $\phi$ is a probability density on $(0,{\infty})$. Its Mellin transform is given by \begin{eqnarray*} M_{\phi}(s)&:=&\int_0^{\infty}\phi(t) t^{s-1}dt=\frac{\gamma a^{\frac{\beta}{\gamma}}}{\Gamma(\frac{\beta}{\gamma})}\int_0^{\infty}t^{s+\beta-2} e^{-at^{\gamma}}dt\\ &=&\frac{\gamma a^{\frac{\beta}{\gamma}}}{\Gamma(\frac{\beta}{\gamma})}\, \gamma^{-1} a^{-\frac{s+\beta-1}{\gamma}} \Gamma\left(\frac{s+\beta-1}{\gamma}\right) =\frac{ \Gamma\left(\frac{s+\beta-1}{\gamma}\right)}{\Gamma\left(\frac{\beta}{\gamma}\right)}\, a^{\frac{1-s}{\gamma}}. \end{eqnarray*} Thus we have $$ \int_0^{\infty}\phi(t)(\log t)t^{s-1}dt=M_{\phi}'(1)=\frac{ \Gamma\left(\frac{s+\beta-1}{\gamma}\right)}{\gamma\Gamma\left(\frac{\beta}{\gamma}\right)}\left(\psi\left(\frac{s+\beta-1}{\gamma}\right)- \log a\right)a^{\frac{1-s}{\gamma}}, $$ and finally, $$ \int_0^{\infty}\phi(t)(\log t)dt=M_{\phi}'(1)=\frac{1}{\gamma}\left(\psi\left(\frac{\beta}{\gamma}\right)- \log a\right). $$ Therefore, $$ i\overline{\Psi}'(0)=\Gamma(\alpha)\frac{\sin(\pi \alpha\widehat\rho)}{\pi} \left(\pi\cot(\pi\alpha\widehat\rho)-\frac{1}{\gamma}\left(\psi\left(\frac{\beta}{\gamma}\right)- \log a\right)\right). $$ \begin{figure}
\caption{Example \ref{ex:phi-exp-decay}: Left: $a=1$, $\beta=0.1$, $\gamma=1$, $\alpha_*\approx 0.906821$, $\alpha^*\approx 1.906821$; Right: $a=1$, $\beta=100000$, $\gamma=1$, $\alpha_*\approx 0.0847945$, $\alpha^*\approx 1.0847945$; $i\overline{\Psi}'(0)>0$ in the shaded region, $i\overline{\Psi}'(0)=0$ on the red line. }
\end{figure} } \end{example}
We end this subsection with the analysis of the behavior at lifetime (absorption time) of the censored process. To the best of our knowledge, this has not been completely done before, but see \cite[p.976]{CC06}.
\begin{example}\label{ex:censored} {\rm In this example we consider the L\'evy process with characteristic exponent $$ \Psi(\theta)=\frac{\Gamma(\alpha-i\theta)}{\Gamma(\alpha\widehat{\rho}-i\theta)}\, \frac{\Gamma(i\theta+1)}{\Gamma(i\theta+1-\alpha\widehat{\rho})}-\frac{c_-}{\alpha}. $$ The corresponding pssMp $X$ is the (not necessarily symmetric) censored $\alpha$-stable process. It is straightforward to calculate that \begin{eqnarray}\label{e:Psi'-for-censored} i\Psi'(0) =\frac{\Gamma(\alpha)\sin(\pi \alpha\widehat\rho)}{\pi}\left(\pi\cot(\pi \alpha\widehat\rho)-(\psi(1)-\psi(\alpha))\right). \end{eqnarray} Notice the similarity with the expression in Example
\ref{ex:beta-gamma} (b) and the difference being the sign in front of $(\psi(1)-\psi(\alpha))$. The equation $\pi\cot(\pi(\alpha-1))=\psi(1)-\psi(\alpha)$ (obtained by formally taking $\rho=1/\alpha$), has a unique solution $\alpha^{\ast}$ in $(1,2)$ which can be numerically computed. It turns out that $\alpha^{\ast}\approx 1.56735$ with corresponding $\rho^{\ast}=1/\alpha^{\ast}\approx 0.63802$. For every $\alpha\in (0,\alpha^{\ast})$ equation $\pi \cot(\pi\alpha(1-\rho))=\psi(1)-\psi(\alpha)$ has a unique solution given by $$ \rho(\alpha)=1-\frac{1}{\alpha \pi}\mathrm{arccot} \left(\frac{\psi(1)-\psi(\alpha)}{\pi}\right) $$ which is strictly increasing in $\alpha$. Moreover, it can be shown that $\rho(\alpha)>0$ for every $\alpha\in (0,2)$, and $\lim_{\alpha\downarrow 0}\rho(\alpha)=0$ (so formally we can take $\alpha_*=0$). When $\rho>\rho(\alpha)$ we have $i\Psi'(0)>0$, for $\rho=\rho(\alpha)$ it holds that $i\Psi'(0)=0$, while for $\rho<\rho(\alpha)$, $i\Psi'(0)<0$. When $\alpha\in [\alpha^{\ast}, 2)$, for every admissible $\rho$ we have that $i\Psi'(0)<0$. Note also that for every $\rho\in [\rho^{\ast},1)$ it holds that $i\Psi'(0)>0$. \begin{figure}
\caption{Example \ref{ex:censored}: $\alpha^*\approx 1.56735$; $i\overline{\Psi}'(0)>0$ in the shaded region, $i\overline{\Psi}'(0)=0$ on the red line.}
\end{figure} } \end{example}
\section{Symmetric resurrection kernels}\label{s:sym-int-kernel} The goal of this section is to find a sufficient and necessary condition for the resurrection kernel to be symmetric, namely that it holds $q(x,y)=q(y,x)$. When the resurrection kernel is symmetric and the strictly $\alpha$-stable process $\eta$ is also symmetric (that is $c_+=c_-$, or, equivalently, $\rho=1/2$), the jump kernel $J(x,y)=j(x,y)+q(x,y)$ of the resurrected process $\overline{X}$ is also symmetric. In particular, the process $\overline{X}$ is symmetric with respect to the Lebesgue measure in $(0,{\infty})$.
We first give a necessary and sufficient condition for symmetry in terms of the L\'evy measure $\pi$ of the compound Poisson process $\chi$.
\begin{prop}\label{p:q-symmetric-levy} Let $\pi$ be the L\'evy measure of the compound Poisson process $\chi$. Then $q(x,y)=q(y,x)$ for all $x,y>0$, if and only if, \begin{equation}\label{e:q-symmetric-levy} \pi(-y)=e^{(\alpha-1)y}\pi(y). \end{equation} \end{prop} \noindent{\bf Proof.} Recall that $\pi(y)=e^{-\alpha y}q(e^{-y},1)$. Suppose that $q$ is symmetric. Then by symmetry and scaling \begin{eqnarray*} \pi(-y)&=&e^{\alpha y}q(e^y,1)=e^{\alpha y}q(1,e^y)=e^{\alpha y} (e^y)^{-1-\alpha}q(e^{-y},1)\\ &=&e^{-y}q(e^{-y},1)=e^{-y} e^{\alpha y}\pi(y)=e^{(\alpha-1)y}\pi(y). \end{eqnarray*} The converse is similar. {
$\Box$
}
Recall from Lemma \ref{l:q-always-density} that $$ q(x,y)=c_- \int_{(0,{\infty})} \left(x+\frac{y}{v}\right)^{-1-\alpha}\frac{\phi(dv)}{v}. $$
The proof of the next technical lemma is given in the appendix. \begin{lemma}\label{l:q-symmetric} Suppose that $m$ is a signed Borel measure on $(0, \infty)$ such that $$
\int_{(0, \infty)}(1+xu)^{-1-\alpha}|m|(du)<\infty, \quad \text{for all } x>0 $$ and \begin{equation}\label{e:unique1} \int_{(0, \infty)}(1+xu)^{-1-\alpha}m(du)=0, \quad \text{for all } x>0. \end{equation} Then $m$ is the zero measure on $(0, \infty)$. \end{lemma}
\begin{thm}\label{t:q-symmetric} It holds that $q(x,y)=q(y,x)$ for all $x,y>0$, if and only if \begin{equation}\label{e:q-symmetric} \phi_*(dt)=t^{\alpha-1}\phi(dt), \quad \mbox{ on } (0, \infty), \end{equation} where $\phi_*$ is the pushforward of the restriction of the measure $\phi$ to $(0, \infty)$ under the map $x\to 1/x$. In case when the restriction of the measure $\phi(dt)$ to $(0, \infty)$ has a density $\phi(t)$ with respect to the Lebesgue measure, the measure equality above reduces to $\phi(t^{-1})=t^{1+\alpha}\phi(t)$ for almost every $t>0$. \end{thm} \noindent{\bf Proof.} We have that \begin{eqnarray*} q(y,x)&=&c_- \int_{(0, \infty)} \left(y+\frac{x}{v}\right)^{-1-\alpha}\frac{\phi(dv)}{v}= c_- \int_{(0, \infty)} (yv+x)^{-1-\alpha} v^{\alpha}\phi(dv) \\ &=&c_-\int_{(0, \infty)} \left(\frac{y}{u}+x\right)^{-1-\alpha}u^{-\alpha}\phi_*(du). \end{eqnarray*} If \eqref{e:q-symmetric} holds, then the last integral in the display above is equal to $$ c_- \int_{(0, \infty)} \left(\frac{y}{u}+x\right)^{-1-\alpha}u^{-\alpha}u^{\alpha-1}\phi(du) =c_- \int_{(0, \infty)} \left(x+\frac{y}{u}\right)^{-1-\alpha}\frac{\phi(du)}{u} =q(x,y). $$
Conversely, assume that $q$ is symmetric. Then we must have that $$ \int_{(0, \infty)} \left(\frac{y}{u}+x\right)^{-1-\alpha} u^{-\alpha}\phi_*(du)= \int_{(0, \infty)} \left(\frac{y}{u}+x\right)^{-1-\alpha}\frac{\phi(du)}{u}, $$ for all $x,y>0$. By taking $y=1$ and rewriting, we get that $$ \int_{(0, \infty)} (1+xu)^{-1-\alpha}u\phi_*(du)= \int_{(0, \infty)} (1+xu)^{-1-\alpha} u^{\alpha}\phi(du),\quad \text{for all }x>0. $$ The claim now follows from Lemma \ref{l:q-symmetric}. {
$\Box$
}
\begin{corollary}\label{c:phi-sym-general} Let $\phi:(0, \infty)\to [0,\infty)$ be such that $\int_0^{\infty}\phi(t)dt=1$. Then $\phi$ satisfies \eqref{e:q-symmetric} if and only if \begin{equation}\label{e:phi-sym-general} \phi(t)=\frac{f(t+t^{-1})}{(1+t)^{1+\alpha}}, \end{equation} for $f:[2,\infty)\to[0,\infty)$. \end{corollary} \noindent{\bf Proof.} Assume that $\phi$ is given by \eqref{e:phi-sym-general}. Let $g(t):=f(t+t^{-1})$. Then $g(t^{-1})=g(t)$ implying that $\phi$ satisfies \eqref{e:q-symmetric}.
Conversely, if $\phi$ satisfies \eqref{e:q-symmetric}, define $g(t):=\phi(t)(1+t)^{1+\alpha}$. Then $g(t^{-1})=g(t)$. For $s\ge 2$, solving $t+t^{-1}=s$, we
get two solutions: $t=(s+\sqrt{s^2-4})/2 \ge1$ and $t^{-1}=(s-\sqrt{s^2-4})/2\le 1$. Define $f(s):=g(t)=g(t^{-1})$. Since $s=t+t^{-1}$, we see that $g(t)=f(t+t^{-1})$. {
$\Box$
}
\begin{prop}\label{p:region-symmetry} Suppose that $\mathfrak{p}=1$ and $q$ is symmetric. Then ${\bf E}[\overline{\xi}_1]< 0$ if $\alpha > 1$ and $\widehat\rho \in [1/(2\alpha), 1/\alpha)$, ${\bf E}[\overline{\xi}_1]=0$ if $\alpha =1$ and $\widehat\rho = 1/2$, and ${\bf E}[\overline{\xi}_1]>0$ if $\widehat\rho \in (0, 1/(2\alpha)]$.
In particular, if $\alpha \ge 3/2$, then the absorption time of $\overline{X}$ is finite ${\mathbb P}_x$-a.s. and $X$ is continuously absorbed at 0. Also, if $\alpha\le 1/2$, then the absorption time of $\overline{X}$ is infinite. \end{prop} \noindent{\bf Proof.} By Theorem \ref{t:q-symmetric}, \begin{align*} \int_{(0, \infty)} (\log u)\phi(du)&= \int_{(0, 1)} (\log u)\phi(du)+ \int_{(1, \infty)} (\log u)\phi(du)\\ &=\int_{(1, \infty)}(\log 1/u)\phi_*(du)+\int_{(1, \infty)}(\log u)\phi(du) \\ &=\int_{(1, \infty)} (\log u)(-u^{\alpha-1}+1)\phi(du), \end{align*} which is negative for $\alpha > 1$, zero for $\alpha =1$, and positive for $\alpha < 1$. It follows from \eqref{e:derivative-at-zero} that the sign of $i\overline{\Psi}'(0) $ depends on the sign of $$ \pi\cot(\pi\alpha \widehat \rho)- \int_{(1, \infty)} (\log u)(u^{\alpha-1}-1)\phi(du). $$ This expression is negative for $\alpha > 1$ and $\widehat\rho \in [1/(2\alpha), 1/\alpha)$, zero for $\alpha =1$ and $\widehat\rho = 1/2$, and positive for $\alpha <1$ and $\widehat\rho \in (0, 1/(2\alpha)]$.
The absorption claim for $\alpha\ge 3/2$ (which is equivalent to $1-1/\alpha\ge 1/(2\alpha)$) follows from the assumption $\widehat{\rho}>1-1/\alpha$. {
$\Box$
}
\begin{figure}
\caption{Illustration of Proposition \ref{p:region-symmetry}: In the shaded region $i\overline{\Psi}'(0)>0$, in the white region
$i\overline{\Psi}'(0) < 0$, in the yellow region the sign of $i\overline{\Psi}'(0)$ is undetermined.}
\end{figure}
\begin{example}\label{ex:symmetric-phi} {\rm (a) Let $\phi$ be as in \eqref{e:phi-special}: $$ \phi(t)=\frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma(\gamma-\beta)}t^{\beta-1}(1+t)^{-\gamma}. $$ Imposing condition \eqref{e:q-symmetric} on $\phi$ implies that $\gamma=\alpha+2\beta-1$. Since $\gamma>\beta$, we get that $\beta >1-\alpha$. Thus $$ \phi(t)=\frac{\Gamma(\alpha+2\beta-1)}{\Gamma(\beta)\Gamma(\alpha+\beta-1)} t^{\beta-1}(1+t)^{1-\alpha-2\beta}. $$
Note that if $\beta=1$ then symmetry requires that $\gamma=\alpha+1$, so we get Example
\ref{ex:beta-gamma} (b). In this case, the regions for the sign of $i\overline{\Psi}'(0)$ are completely determined, see Figure \ref{cap:2}.
\noindent (b) If $\phi$ is as in \eqref{e:phi-special-exp}, then it cannot lead to a symmetric resurrection kernel.
\noindent (c) Let $\phi=\delta_a$, $a>0$. Then the measure $\phi$ satisfies \eqref{e:q-symmetric} if and only if $a=1$. } \end{example}
\section{Sharp two-sided estimates of the resurrection kernel }\label{s:estimates-q} In this section we establish sharp two-sided estimates of $q(x,y)$ under minimal assumptions. First notice that it follows from \eqref{e:q-always-density} that $q$ enjoys the following scaling $$ q(\lambda x, \lambda y)=\lambda^{-1-\alpha}q(x,y), \quad x,y >0, \ \lambda>0. $$ This implies that for all $0<x<y$ we have \begin{align} q(x,y)=&(y-x)^{-1-\alpha}q\left(\frac{x}{y-x}, \frac{x}{y-x}+1\right) \label{e:scalq1}\\ q(y,x)=&(y-x)^{-1-\alpha}q\left(\frac{x}{y-x}+1, \frac{x}{y-x}\right). \label{e:scalq2} \end{align} Thus it suffices to get the estimates of $q(x,x+1)$ and $q(x+1,x)$, $x>0$.
We first look at the simple case when the measure $\phi$ has compact support in $(0,{\infty})$. Then it is easy to see that $$ q(x,x+1)=c_- \int_{(0, \infty)} \left(x+1+\frac{x}{t}\right)^{-1-\alpha} \frac{\phi(dt)}{t}\asymp 1 $$ and $$ q(x+1,x)=c_-\int_{(0, \infty)} \left(x+\frac{x+1}{t}\right)^{-1-\alpha} \frac{\phi(dt)}{t}\asymp 1. $$ Thus in this case by \eqref{e:scalq1}-\eqref{e:scalq2}, we have $$
q(x, y)\asymp |x-y|^{-1-\alpha}, \quad x, y\in (0, \infty). $$
In the remainder of this section, we assume that $\phi$ is absolutely continuous and has a strictly positive density.
Assume that $|y-x|=1$ so that either $y=x+1$ or $y=x-1$. Then by Lemma \ref{l:q-always-density}, $$ q(x+1,x) = c_- \int_0^{\infty}\left(x+1+\frac{x}{t}\right)^{-1-\alpha} \phi(t)\frac{dt}{t} $$ and $$ q(x,x+1) = c_-\int_0^{\infty}\left(x+\frac{x+1}{t}\right)^{-1-\alpha} \phi(t)\frac{dt}{t}. $$
\begin{lemma}\label{l:estimates-of-q} (1) If $x\ge 1/4$, then $ q(x+1,x)\asymp q(x,x+1)\asymp x^{-1-\alpha}. $
\noindent (2) Suppose $\phi$ satisfies the lower weak scaling condition $L_1(\beta_1)$ at zero with $\beta_1>-1-\alpha $. Then for $x\le 1/4$, \begin{equation}\label{e:estimate-q-1} q(x+1,x)\asymp \int_{x}^{1} \phi(t)\frac{dt}{t}.\end{equation} Further, if $\phi$ also satisfies the upper weak scaling condition $U_1(\beta_2)$ at zero with $\beta_2<0$, then \begin{equation}\label{e:estimate-q-10} q(x+1,x)\asymp \phi(x).\end{equation} (3) Suppose $\phi$ satisfies the upper weak scaling condition $U^1(\gamma_2)$ at infinity with $\gamma_2<0$. Then \begin{equation}\label{e:estimate-q-2} q(x,x+1) \asymp \int_0^{1/x} t^{\alpha} \phi(t){dt} \asymp \int_1^{1/x} t^{\alpha} \phi(t){dt}.\end{equation} Further, if $\phi$ also satisfies the lower weak scaling condition $L^1(\gamma_1)$ at infinity with $\gamma_1<-1-\alpha$, \begin{equation}\label{e:estimate-q-20} q(x,x+1) \asymp x^{-1-\alpha} \phi(1/x).\end{equation} \end{lemma} \noindent{\bf Proof.} Without loss of generality, we neglect the constant $c_-$ in the proof. First note that \begin{align}\label{e:est1} 0<\int_0^{\infty}\frac{t^\alpha\phi(t)}{(1+t)^{\alpha+1}}dt \le \int_0^{\infty}\phi(t)dt =1. \end{align}
\noindent (1) {\bf Case 1:} $1/4 \le x \le 4$. Then $$ x+1+\frac{x}{t}\ge 1+\frac{1}{4t}\ge \frac14\left(1+\frac{1}{t}\right) \quad \text{ and} \quad x+1+\frac{x}{t} \le 4+1 +\frac{1}{t}=5\left(1+\frac{1}{t}\right). $$
Also, $$ x+\frac{x+1}{t}\ge \frac14 +\frac{1}{t} \ge \frac14 \left(1+\frac{1}{t}\right) \quad \text{ and} \quad x+\frac{x+1}{t} \le 4 +\frac{4+1}{t}\le 5\left(1+\frac{1}{t}\right). $$ Thus, by \eqref{e:est1} we get $$ q(x+1,x)\asymp q(x,x+1) \asymp \int_0^{\infty}\left(1+\frac{1}{t}\right)^{-1-\alpha}\phi(t)\frac{dt}{t} = \int_0^{\infty}\frac{t^\alpha\phi(t)}{(1+t)^{\alpha+1}}dt \asymp 1. $$
\noindent {\bf Case 2:} $ x \ge 4$. Then we have that $$ x+\frac{x}{t}\le x+1+\frac{x}{t}\le 2x+2\frac{x}{t}, $$ and hence $$ x+1+\frac{x}{t} \asymp x\left(1+\frac{1}{t}\right). $$ Also, $$ x+\frac{x+1}{t}\le (x+1)\left(1+\frac{1}{t}\right) \quad \text{ and } \quad x+\frac{x+1}{t}\ge \frac{x+1}{2}\left(1+\frac{1}{t}\right) $$ imply $$ x+\frac{x+1}{t}\asymp (x+1)\left(1+\frac{1}{t}\right)\asymp x\left(1+\frac{1}{t}\right). $$ Thus, by \eqref{e:est1} $$ q(x+1,x)\asymp q(x,x+1) \asymp x^{-1-\alpha} \int_0^{\infty}\left(1+\frac{1}{t}\right)^{-1-\alpha}\phi(t)\frac{dt}{t}\asymp x^{-1-\alpha}. $$
\noindent (2) Assume $x \le 1/4$ and let $$ q(x+1,x)=\int_0^{x}+ \int_{x}^{1}+\int_{1}^{\infty}=:I+II+III. $$
For $0<t<x$, we have that $$ x+1=\frac{t(x+1)}{x}\frac{x}{t} \le (x+1)\frac{x}{t}\le 2\frac{x}{t}, $$
hence $x+1+\frac{x}{t}\asymp \frac{x}{t}$. Since $\phi$ satisfies the lower weak scaling condition at zero $L_1(\beta_1)$ with $\beta_1>-1-\alpha $, we have $$\int_0^{x} t^{\alpha} \phi(t){dt} =\phi(x)\int_0^{x} t^{\alpha} (\phi(t)/\phi(x)){dt} \le c^{-1} \phi(x)\int_0^{x} t^{\alpha} (t/x)^{\beta_1}{dt} =c_1 x^{\alpha+1} \phi(x).$$ Thus, \begin{align*} I \asymp \int_0^{x} \left(\frac{x}{t}\right)^{-1-\alpha} \phi(t)\frac{dt}{t} = x^{-1-\alpha}\int_0^{x} t^{\alpha} \phi(t){dt} \le c_1 \phi(x). \end{align*}
For $x\le t <\infty$, we have $$ 1\le x+1+\frac{x}{t}\le \frac14+1+1, $$ hence $x+1+x/t\asymp 1$. Therefore, \begin{eqnarray*} II &\asymp & \int_{x}^{1} \phi(t)\frac{dt}{t}. \end{eqnarray*} Finally, $$ III \asymp \int_1^{\infty}\phi(t)\frac{dt}{t} \le \int_1^{\infty}\phi(t){dt} \le 1. $$ Since $$ \int_{x}^{1} \phi(t)\frac{dt}{t} \ge \phi(x) \int_{x}^{2x} \frac{\phi(t)dt}{ \phi(x)t}+ 2 \int_{1/2}^{1} \phi(t){dt} \ge c \phi(x) \int_{x}^{2x} \frac{t^{\beta_1-1}}{x^{\beta_1}}dt+c_2
\asymp \phi(x)+1,
$$ we get \eqref{e:estimate-q-1}. Moreover, if $\phi$ also satisfies $U_1(\beta_2)$ with $\beta_2<0$, then $$ c_3 \phi(x) = c \phi(x)\int_{x}^{1} (t/x)^{\beta_1}\frac{dt}{t} \le \int_{x}^{1} \phi(t)\frac{dt}{t} \le C \phi(x)\int_{x}^{1} (t/x)^{\beta_2}\frac{dt}{t} =c_4 \phi(x), $$ so we get \eqref{e:estimate-q-10}.
(3) Assume $x \le 1/4$. $$ q(x,x+1)= \int_0^{1/x}+\int_{1/x}^{\infty}=:I+II. $$
For $0<t<2$, $$ x+\frac{x+1}{t}\le x+1 +\frac{x+1}{t} \le 2\frac{x+1}{t}+(x+1)t, $$ hence $$ x+\frac{x+1}{t}\asymp \frac{x+1}{t}\asymp \frac{1}{t}. $$ For $2<t <1/x$, $$ x+\frac{x+1}{t} \le \frac{1}{t}+\frac{2}{t}=\frac{3}{t}, $$
hence $x+(x+1)/t\asymp 1/t$. Therefore, $$ I \asymp \int_0^{1/x} \left(\frac{1}{t}\right)^{-1-\alpha} \phi(t)\frac{dt}{t} = \int_0^{1/x} t^{\alpha} \phi(t){dt}. $$
When $t>1/x$, $$ \frac{x+1}{t}=\frac{x+1}{x}\, \frac{x}{t}\le \frac{2}{x}\, \frac{x}{t}\le (2t)\frac{x}{t}=2x. $$ Thus, $x+(x+1)/t\le 3x$, hence $x+(x+1)/t \asymp x$. Moreover, using that fact that $\phi$ satisfies the upper weak scaling condition at infinity $U^1(\gamma_2)$ with $\gamma_2<0 $, we have $$ \int_{1/x}^{\infty} \phi(t)\frac{dt}{t} =\phi(1/x)\int_{1/x}^{\infty} \frac{\phi(t)}{\phi(1/x)t} dt
\le C \phi(1/x)\int_{1/x}^{\infty} (tx)^{\gamma_2}\frac{dt}{t} =c_5 \phi(1/x).$$ Therefore, $$ II\asymp x^{-1-\alpha}\int_{1/x}^{\infty} \phi(t)\frac{dt}{t} \le c_5 x^{-\alpha-1}\phi(1/x). $$ Now, using \begin{align*} \int_0^{1/x} t^{\alpha} \phi(t){dt} &\ge \phi(1/x)\int_{1/(2x)}^{1/x}\frac{\phi(t)}{\phi(1/x)} t^{\alpha} {dt}+\int_{0}^{1} t^{\alpha} \phi(t){dt}\\ &\ge C^{-1} \phi(1/x)\int_{1/(2x)}^{1/x}(xt)^{\gamma_2} t^{\alpha} {dt}+c_6\asymp x^{-\alpha-1}\phi(1/x)+1,\end{align*} we get \eqref{e:estimate-q-2}. Finally, if $\phi$ also satisfies $L^1(\gamma_1)$ with $\gamma_1<-1-\alpha$, then \begin{align*} &c_7 x^{-\alpha-1}\phi(1/x) \le c \phi(1/x)\int_1^{1/x} t^{\alpha} (tx)^{\gamma_1} {dt} \le \int_1^{1/x} t^{\alpha} \phi(t){dt} \\ & \le C \phi(1/x)\int_1^{1/x} t^{\alpha} (tx)^{\gamma_2} {dt} \le c_8 x^{-\alpha-1}\phi(1/x), \end{align*} and so we get \eqref{e:estimate-q-20}. {
$\Box$
}
\noindent \textbf{Proof of Theorem \ref{t:estimates-of-q}:} The result follows immediately from Lemma \ref{l:estimates-of-q} and the scaling relations \eqref{e:scalq1}--\eqref{e:scalq2}. {
$\Box$
}
As a consequence of Theorem \ref{t:estimates-of-q} we can get estimates of the L\'evy density $\pi(u)=q(1,e^u)e^u$. We state the next corollary in its simple form. \begin{corollary}\label{c:estimates-of-pi} Suppose that $\phi$ satisfies both the lower and upper scaling conditions at zero and infinity as in Theorem \ref{t:estimates-of-q}. Then $$ \pi(u)\asymp \left\{ \begin{array}{ll}
1,& |u| \le \log 5, \\
e^u\phi(e^u), & |u|>\log 5. \end{array} \right. $$ \end{corollary}
We first apply Theorem \ref{t:estimates-of-q} to a generalization of the function $\phi$ given in \ref{e:phi-special}. For $\beta \ge 0$, $\gamma \ge \beta$ and $\delta_+, \delta_- \in {\mathbb R}$, let \begin{align} \label{e:phasy} \phi(t)\asymp \left(\log(e+t)\right)^{\delta_+}\left(\log(e+t^{-1})\right)^{\delta_-}{t^{\beta-1}(1+t)^{-\gamma}}, \end{align} and $\int_0^{\infty}\phi(t)dt=1$, which implies that $\delta_-<-1$ if
$\beta = 0$ and $\delta_+<-1$ if $\gamma = \beta$. Then $$ q(x,y)\asymp\int_0^{\infty}\left(x+\frac{y}{t}\right)^{-1-\alpha} \frac{\left(\log(e+t)\right)^{\delta_+}\left(\log(e+t^{-1})\right)^{\delta_-}}{t^{2-\beta}(1+t)^{\gamma}}\, dt. $$ It is straightforward to check that the function $\phi$ satisfies the scaling conditions assumed in Theorem \ref{t:estimates-of-q}. Hence we have the following result.
\begin{corollary}\label{c:estimates-of-q} Suppose $\phi$ is a probability density on $(0, \infty)$ satisfying \eqref{e:phasy}. If $x \le y\le 5x$, then $$ q(x,y)\asymp q(y,x)\asymp x^{-1-\alpha}\asymp y^{-1-\alpha}. $$ If $5x\le y$, then \begin{equation}\label{e:estimate-q-3} q(y,x)\asymp
y^{-1-\alpha} \left\{ \begin{array}{ll} (y/x)^{1-\beta}(\log (y/x))^{\delta_-}, & \beta<1,\\ \left\{\begin{array}{ll} (\log (y/x))^{1+\delta_-}, & \delta_- > -1, \\ \log(\log( y/x)), & \delta_- =-1, \\ 1, &\delta_-<-1, \end{array}\right. & \beta=1,\\ 1, & \beta>1, \end{array} \right. \end{equation} and \begin{equation} q(x,y) \asymp y^{-1-\alpha} \left\{ \begin{array}{ll} (y/x)^{\alpha+\beta-\gamma}(\log (y/x))^{\delta_+}, &\gamma-\beta <\alpha ,\\ \left\{\begin{array}{ll} (\log (y/x))^{1+\delta_+}, & \delta_+ > -1, \\ \log(\log(y/x)) , & \delta_+ =-1, \\ 1, &\delta_+ <-1, \end{array}\right. & \gamma-\beta =\alpha ,\\ 1, & \gamma-\beta >\alpha . \end{array} \right. \end{equation}\label{e:estimate-q-4} \end{corollary} Note that in the case $\gamma-\alpha <\beta<1$, or in the case $\gamma-\alpha =\beta=1$ and $\delta_+ \wedge \delta_-\ge -1$, both $q(x,y)$ and $q(y,x)$ explode when $x\to 0$. This leads to the following path interpretation: The intensity of jumps to and away from points near 0 is much higher than in case of the stable process. Thus, on average, large jump to and away from points near 0 are more probable.
In the symmetric case, we have $\gamma=\alpha+2\beta-1$ and $\delta_+=\delta_-$, see Example \ref{ex:symmetric-phi} (a). Hence $-\alpha-\beta+\gamma=-(1-\beta)$, and the estimates for $q(x,y)$ and $q(y,x)$ coincide.
Now we apply Theorem \ref{t:estimates-of-q} to the function $\phi$ given in \eqref{e:phi-special-exp}.
\begin{corollary}\label{c:estimates-of-qq} Suppose $\phi$ is a probability density on $(0, \infty)$ satisfying \begin{align} \label{e:phi_ex} \phi(t) \asymp t^{\beta-1} e^{-at^{\gamma}}, \quad t>0 \end{align} where $a,\gamma>0$ and $\beta>0$. If $x \le y\le 5x$, then $$ q(x,y)\asymp q(y,x)\asymp x^{-1-\alpha}\asymp y^{-1-\alpha}. $$ If $5x\le y$, then \begin{equation}\label{e:estimate-q-14a} q(y,x)\asymp y^{-1-\alpha} \left\{ \begin{array}{ll} (y/x)^{-\beta+1}, & 0<\beta<1,\\ \log (y/x)& \beta=1,\\ 1, & \beta>1, \end{array} \right. \end{equation} and \begin{equation}\label{e:estimate-q-14} q(x,y) \asymp y^{-1-\alpha}\int_1^{\frac{y}{x}} t^{\alpha+\beta-1} e^{-at^{\gamma}}{dt} \asymp y^{-1-\alpha}. \end{equation} \end{corollary} \noindent{\bf Proof.} For the second comparison in \eqref{e:estimate-q-14}, see \eqref{e:estimate-q-21} and $$ 0< \int_1^{4} t^{\alpha+\beta-1} e^{-at^{\gamma}}{dt} \le \int_1^{\frac{y}{x}-1} t^{\alpha+\beta-1} e^{-at^{\gamma}}{dt} \le\int_1^{\infty} t^{\alpha+\beta-1} e^{-at^{\gamma}}{dt} <\infty, \quad 5x\le y. $$ {
$\Box$
}
In case $\beta \le 1$, we see that $q(y,x)$ explodes as $x\to 0$, but $q(x,y)$ stays bounded. This means that the process will have tendency for big jumps to points close to the origin.
As a consequence of Corollary \ref{c:estimates-of-qq}
we can derive that, when a probability density $\phi$ on $(0, \infty)$ satisfies \eqref{e:phi_ex}, $\pi(u)\asymp 1$ for $|u|\le \log 5$, $\pi(u)\asymp e^{-u\alpha}$ for $u>\log 5$, and $$ \pi(u)\asymp \left\{\begin{array}{ll} e^{u\beta}, & 0<\beta <1, \\
|u|e^u, & \beta=1, \\ e^u, & \beta >1, \end{array}\right. $$ when $u<-\log 5$.
\section{Modified jump kernel}\label{s:modified-jump} \subsection{General case}\label{ss:2-general} Stable process conditioned to stay positive is a pssMp that can be regarded as a resurrected stable process. However, it does not fall into the framework of resurrected stable processes of this paper. In this section we introduce a larger class of pssMps by modifying the jump kernel of the pssMp $X=(X_t,{\mathbb P}_x)$ of index $\alpha$ defined in Subsection \ref{ss:sspcp}. Thus, $X$ is a not necessarily symmetric censored process. Let $j(x,y)=\nu(y-x)$, where $\nu$ is defined in \eqref{e:stable-density}. We define a new jump kernel by $$ J(x,y):=\sB(x,y)j(x,y),\quad x,y>0, $$ where $\sB:(0,\infty)\times (0,\infty)\to (0,\infty)$ is a function satisfying the following properties:
\noindent \textbf{(B1)} Homogeneity: $\sB(\lambda x, \lambda y)=\sB(x,y)$ for all $x,y>0$ and all $\lambda >0$.
\noindent \textbf{(B2)} Integrability: (a) $y\mapsto e^{-\alpha y} \sB(1,e^y)$ is integrable at $\infty$ and $y\mapsto e^y \sB(1,e^y)$ is integrable at $-\infty$; (b)
$y\mapsto \sB(1,e^y)|y|^{1-\alpha}$ is integrable at 0; (c) For all $x \in (0, \infty)$, $y\mapsto \sB(x,y)$ is locally integrable in $(0, \infty)\setminus \{x\}$.
\noindent \textbf{(B3)} Regularity: If $\alpha\in [1,2)$, there exist $\theta>\alpha-1$ and $C>0$ such that $$
|\sB(x,x)-\sB(x,y)|\le
C\left(\frac{|x-y|}{x\wedge y}\right)^{\theta} $$
for $|x-y| \le (x\wedge y)/4$. If $\alpha<1$, there exists $C>0$ such that $ \sB(x,y)\le C$
for $|x-y| \le (x\wedge y)/4$.
Without loss of generality, from now on, we assume that $\sB(1,1)=1$.
For any $\sB(\cdot, \cdot)$ satisfying \textbf{(B1)}--\textbf{(B3)}, we will construct a pssMp with the jump kernel $J$ above via the Lamperti transform of a certain L\'evy process.
We first show that the jump kernel $J(x,y)=j(x,y)+q(x,y)$ of the resurrected process $\overline{X}$ is of the form introduced above. Indeed, $J(x,y)$ can be rewritten as $$ J(x,y)=j(x,y)+q(x,y)=j(x,y)\left(1+\frac{q(x,y)}{j(x,y)}\right)=\sB(x,y)j(x,y), $$ where we define $\sB(x,y):=1+q(x,y)/j(x,y)$ for $y\neq x$, and $\sB(x,x)=1$. Clearly, $\sB(x,y)$ satisfies \textbf{(B1)}. Next, by \eqref{e:levy-xi*} and \eqref{e:densities-Q-q}, $$ \sB(1, e^y)=1+\frac{q(1,e^y)}{j(1,e^y)}=1+\frac{\pi(y)}{e^y \nu(e^y-1)}=1+\frac{\pi(y)}{\mu(y)}, $$ so that $\sB(1,e^y)\mu(y)=\mu(y)+\pi(y)$ is a L\'evy density, i.e., $\int_{{\mathbb R}}(1\wedge y^2)\sB(1,e^y)\mu(y)dy<\infty$.
Indeed, since $\mu(y)\asymp e^{-\alpha y}$ at $+\infty$, $\mu(y)\asymp e^{-y}$ at $-\infty$, and $\mu(y)\asymp |y|^{-1-\alpha}$ near zero, we know that \textbf{(B2)} holds. Finally, it follows from Theorem \ref{t:estimates-of-q} that for $x<y<(5/4)x$ or $y<x<(5/4)y$ it holds that $q(x,y)\asymp q(y,x)\asymp x^{-1-\alpha}\asymp y^{-1-\alpha}$. Hence, if $|x-y|<(x\wedge y)/4$, \begin{eqnarray*}
|\sB(x,y)-\sB(x,x)|&=&\frac{q(x,y)}{j(x,y)}\le (c_+\vee c_-)|x-y|^{1+\alpha}q(x,y)\\
&\le & C|x-y|^{1+\alpha}(x^{-1-\alpha}\vee y^{-1-\alpha})=C\left(\frac{|y-x|}{x\wedge y}\right)^{1+\alpha}. \end{eqnarray*} Thus \textbf{(B3)} holds with $\theta=1+\alpha$.
As examples of this general setting we also mention the $\alpha$-stable process conditioned to stay positive and the $\alpha$-stable process conditioned to hit 0 continuously, see \cite{Cha96, CD05}. The jump kernel of the former is $$ J(x, y)=\frac{y^{\alpha\widehat\rho}}{x^{\alpha\widehat\rho}}\, j(x, y), \quad x, y>0 $$ and the latter $$ J(x, y)=\frac{y^{\alpha\widehat\rho-1}}{x^{\alpha\widehat\rho-1}}\, j(x, y), \quad x, y>0. $$ It is straightforward to show that for every $\gamma\in (-1,\alpha)$, the function $\sB(x,y)=(y/x)^{\gamma}$ satisfies conditions \textbf{(B1)}-\textbf{(B3)}. In fact, \textbf{(B1)} and \textbf{(B2)}(b)--(c) clearly hold. For \textbf{(B2)}(a), $\sB(1, e^y)=e^{\gamma y}$, so $e^{-\alpha y}\sB(1,e^y)=e^{-(\alpha-\gamma)y}$ and is integrable at $\infty$ if and only if $\gamma <\alpha$. Also, $e^y \sB(1,e^y)=e^{(1+\gamma)y}$ and is integrable at $–\infty$ if and only if $\gamma >-1$. For \textbf{(B3)}, without loss of generality assume that $\gamma\neq 0$
and consider $x, y\in (0, \infty)$ with $|y-x|<(x\wedge y)/4$. Then \begin{equation}\label{e:B3-6.2}
|\sB(x,y)-\sB(x,x)|=\left| \left(\frac{y}{x}\right)^{\gamma}-1\right|=
\frac{|\gamma| u^{\gamma-1}|y-x|}{x^{\gamma}}, \end{equation} where $u$ is between $x$ and $y$. If $x<y$, then $x=x\wedge y$ and $y\le (5/4)x$. If $\gamma\ge 1$, then the right-hand side above is less than $(5/4)^{\gamma-1}\gamma (y-x)/x$. If $\gamma <1$, then we estimate the right-hand side with
$|\gamma| (y-x)/x$. Thus in both cases \textbf{(B3)} holds with $\theta=1$. If $y<x$, then we replace $x^{\gamma}$ with $y^{\gamma}$ in \eqref{e:B3-6.2} and argue as before.
See Proposition \ref{p:expectation-non-symmetry} for more on this example.
Given an arbitrary $\sB(x,y)$ satisfying \textbf{(B1)}-\textbf{(B3)}, and the jump kernel $j(x,y)$ of the censored process $X$, we now construct a a pssMp $\overline{X}=(\overline{X}_t, {\mathbb P}_x)$ of index $\alpha$ corresponding to the jump kernel $J(x,y)=\sB(x,y)j(x,y)$ via the Lamperti transform of a certain L\'evy process. Define $$ \mu^{\sB}(y):=\sB(1,e^y)\mu(y)=\sB(1,e^y)\left(c_+\frac{e^y}{(e^y-1)^{1+\alpha}}{\bf 1}_{(y>0)}+c_-\frac{e^y}{(1-e^y)^{1+\alpha}}{\bf 1}_{(y<0)}\right). $$ By the assumptions \textbf{(B2)}(a) and (c) we have that
$\int_{|y|>1}\mu^{\sB}(y)dy <\infty$, while by \textbf{(B2)}(b) and (c) we get that
$\int_{|y|\le 1} y^2\mu^{\sB}(y)dy <\infty$. Thus $\mu^{\sB}$ is a L\'evy measure. Further, let $\overline{\xi}$ denote the L\'evy process with infinitesimal generator \begin{equation}\label{e:inf-gen-overline-xi} \overline{{\mathcal A}} f(x)= -\overline{b}f'(x)+\int_{{\mathbb R}}\left(f(x+y)-f(x)-f'(x)y {\bf 1}_{[-1,1]}(y)\right) \mu^{\sB}(y)dy, \end{equation} where $\overline{b}\in {\mathbb R}$. Let $\overline{X}=(\overline{X}_t, {\mathbb P}_x)$ be the pssMp of index $\alpha$ obtained from $\overline{\xi}$ through the Lamperti transform. By using a calculation similar to the one we used to obtain ${\mathcal L}^*$ in Section \ref{s:prelim}, together with the homogeneity of $\sB$ and \textbf{(B1)}, we see that the infinitesimal generator of $\overline{X}$ is \begin{align}\label{e:inf-gen-overline-X} \overline{{\mathcal L}} f(x)&= -\overline{b} x^{1-\alpha}f'(x)+x^{-\alpha}\int_{{\mathbb R}}\left(f(xe^y)-f(x)-xf'(x)y{\bf 1}_{[-1,1]}(y)\right) \mu^{\sB}(y)dy\\ &=-\overline{b} x^{1-\alpha}f'(x)+ \int_0^\infty \left(f(z)-f(x)-xf'(x)(\log z/x){\bf 1}_{[-1,1]}(\log(z/x))\right)\sB(x,z)j(x,z)dz.\nonumber \end{align} This shows that the jump kernel of $\overline{X}$ is precisely $J(x,y)=\sB(x,y)j(x,y)$, see the last sentence of Subsection \ref{ss:sspcp}.
\subsection{Symmetric case}\label{ss:2-symmetric} In this subsection we assume that $\eta$ is a \emph{symmetric} $\alpha$-stable process. Then $\rho=1/2$, $c_+=c_-=:c$, and $a=0$. For simplicity, we will assume that $c=1$. We first consider the case that $\sB(x,y)$ is identically 1. Recall that $X^*$ is the process $\eta$ killed upon exiting $(0,{\infty})$, and the constant $b$ in the linear term of its infinitesimal generator in \eqref{e:linear-term} equal to $$ b= - \int_0^\infty
\left((\log u){\bf 1}_{[-1,1]}(\log u)-(u-1){\bf 1}_{[-1,1]}(u-1)\right)|u-1|^{-1-\alpha}\, du. $$
\begin{lemma}\label{l:lin-term-alt} It holds that $$
-b=\lim_{\epsilon\to 0} \int_{{\mathbb R}, |e^y-1|>\epsilon} y {\bf 1}_{[-1,1]}(y)\mu(y) dy =\mathrm{p.v.} \int_{-1}^1 y \mu(y) dy. $$ \end{lemma} \noindent{\bf Proof.} We first note that by using symmetry, for $\epsilon\in (0,1)$ we have $$
\int_{(0,{\infty}), |u-1|>\epsilon} (u-1) {\bf 1}_{[-1,1]}(u-1)|u-1|^{-1-\alpha} du=\int_{{\mathbb R}}v {\bf 1}_{(\epsilon<|v|\le 1)}|v|^{-1-\alpha}dv=0. $$ Therefore \begin{align*}
I(\epsilon):= & \int_{{\mathbb R}, |e^y-1|>\epsilon} y {\bf 1}_{[-1,1]}\mu(y) dy=
\int_{{\mathbb R}, |e^y-1|>\epsilon}
y {\bf 1}_{[-1,1]}(y)\frac{e^y}{|e^y-1|^{1+\alpha}} dy\\
=&
\int_{(0, \infty), |u-1|>\epsilon}
(\log u) {\bf 1}_{[-1,1]}(\log u)|u-1|^{-1-\alpha}du\\ = &
\int_{(0, \infty), |u-1|>\epsilon}
\left( (\log u) {\bf 1}_{[-1,1]}(\log u)-(u-1){\bf 1}_{[-1,1]}(u-1)\right)|u-1|^{-1-\alpha}du. \end{align*} By letting $\epsilon \to 0$ we obtain that $\lim_{\epsilon\to 0}I(\epsilon)=-b$ which is the first equality in the statement. For the second, \begin{eqnarray*} I(\epsilon)&=& \int_{-1}^{\log(1-\epsilon)} y\mu(y)\, dy+\int_{\log(1+\epsilon)}^1 y\mu(y)\, dy\\ &=&\left(\int_{-1}^{\log(1-\epsilon)} y\mu(y)\, dy+\int_{-\log(1-\epsilon)}^1 y\mu(y)\, dy\right)+ \int_{\log(1+\epsilon)}^{-\log(1-\epsilon)}y\mu(y)\, dy\\ &=:&I_1(\epsilon)+I_2(\epsilon). \end{eqnarray*} Suppose $\alpha\in [1,2)$ (for $\alpha\in (0,1)$ the integral $I_2(\epsilon)$ is convergent). For $y\in (0,1/2)$ it holds that $y\mu(y)\le c_1 y^{-\alpha}$ for some $c_1>0$, hence \begin{align*} I_2(\epsilon) &\le c_1\int_{\log(1+\epsilon)}^{-\log(1-\epsilon)} y^{-\alpha}dy\\ &\le c_2 \begin{cases} \left(\log(1+\epsilon)^{1-\alpha}-(-\log(1-\epsilon))^{1-\alpha}\right)\le c_3 \epsilon^{2-\alpha}& \text{for } \alpha\in (1,2)\\ \log\left(\frac{-\log(1-\epsilon)} {\log(1+\epsilon)} \right) &\text{for } \alpha=1 \end{cases} \to 0 \end{align*} as $\epsilon \to 0$. Since we have already proved that $\lim_{\epsilon\to 0}I(\epsilon)$ exists we can conclude that $$ \lim_{\epsilon\to 0}I_1(\epsilon)=\mathrm{p.v.} \int_{-1}^1 y \mu(y) dy. $$ {
$\Box$
}
\begin{remark}\label{r:lin-term-alt}{\rm The existence of the principal value integral $\mathrm{p.v.} \int_{-1}^1 y \mu(y) dy$ can be alternatively proved in the following way. First note that $$
\int_{{\mathbb R},\epsilon < |y|\le 1}y \mu(y) dy=\int_{\epsilon}^1 y(\mu(y)-\mu(-y))dy. $$ Secondly, $\mu(y)-\mu(-y)=y^{-\alpha}((1-\alpha)+O(y^2))$ as $y\downarrow 0$, showing that the right-hand side above is convergent. } \end{remark}
Let $\overline{\xi}$ be a L\'evy process with L\'evy density $\mu^{\sB}(y)=\mu(y)\sB(1,e^y)$ and linear term \begin{equation}\label{e:linear-term-sym} \overline{b}=b-\int_{-1}^1y(\sB(1,e^y)-1)\mu(y)dy, \end{equation} cf.~\eqref{e:inf-gen-overline-xi}. Note that the integral is convergent because of \textbf{(B3)}.
Let $\overline{X}$ be the corresponding pssMp of index $\alpha$. The jump kernel of $\overline{X}$ is $J(x,y)=\sB(x,y)|x-y|^{-1-\alpha}$ and the infinitesimal generator of $\overline{X}$ is given in \eqref{e:inf-gen-overline-X}. The following is an analog of Lemma \ref{l:lin-term-alt}.
\begin{lemma}\label{l:lin-term-alt-2} It holds that $$
-\overline{b}=\lim_{\epsilon\to 0} \int_{{\mathbb R}, |e^y-1|>\epsilon} y {\bf 1}_{[-1,1]}(y)\mu^{\sB}(y) dy =\mathrm{p.v.} \int_{-1}^1 y \mu^{\sB}(y) dy. $$ \end{lemma} \noindent{\bf Proof.} We have \begin{align*}
& \int_{{\mathbb R}, |e^y-1|>\epsilon} y {\bf 1}_{[-1,1]}\mu^{\sB}(y) dy=
\int_{{\mathbb R}, |e^y-1|>\epsilon}
y {\bf 1}_{[-1,1]}(y)\sB(1,e^y)\frac{e^y}{|e^y-1|^{1+\alpha}} dy\\ & =
\int_{{\mathbb R}, |e^y-1|>\epsilon} y {\bf 1}_{[-1,1]}(y)\sB(1,1)\frac{e^y}{|e^y-1|^{1+\alpha}} dy\\
& \quad +\int_{{\mathbb R}, |e^y-1|>\epsilon} y {\bf 1}_{[-1,1]}(y)(\sB(1,e^y)-\sB(1,1))\frac{e^y}{|e^y-1|^{1+\alpha}} dy\\ & = :J_1(\epsilon)+J_2(\epsilon). \end{align*} By Lemma \ref{l:lin-term-alt}, and since $\sB(1,1)=1$, $\lim_{\epsilon\to 0} J_1 (\epsilon)= -b$. On the other hand, by using \textbf{(B3)} if $\alpha \ge 1$, we conclude that $$ \lim_{\epsilon\to 0}J_2(\epsilon)
=\int_{{\mathbb R}} y{\bf 1}_{[-1,1]}(y)(\sB(1,e^y)-1)\frac{e^y}{|e^y-1|^{1+\alpha}}dy = \int_{-1}^1 y(\sB(1,e^y)-1)\mu(y)dy. $$ This proves the first equality in the statement. For the second statement, note that for
$u\mapsto \sB(1,u)$ is by \textbf{(B3)} bounded in a neighborhood of 1. Hence, $y\mu^{\sB}(y)=y\mu(y)\sB(1,e^y)\le c_1 y^{-\alpha}$, and we obtain the conclusion in the same way as in Lemma \ref{l:lin-term-alt}. In the case $\alpha\in (0,1)$, since the integral is absolutely convergent, we use the dominated convergence theorem. {
$\Box$
}
In the context of pssMps it is natural to write the generator in the form \eqref{e:inf-gen-overline-X} which involves a cutoff function. On the other hand, in the multidimensional setting of regional non-local operators, such as the infinitesimal generator of a censored $\alpha$-stable process, generators are usually written as principal value integrals. In the context of jump kernels decaying at the boundary, such operators were studied in \cite[Section 3.2]{KSV21} when $\sB$ is symmetric (see \textbf{(B4)} below). In the next result we reconcile these two approaches in the current setting. Let $$ \tilde{{\mathcal L}}f(x):=\textrm{ p.v. } \int^\infty_0 (f(z)-f(x))J(x,z)dz=\lim_{\epsilon\to 0}
\int_{(0, \infty), |z-x|>\epsilon}
(f(z)-f(x))\sB(x,z)|x-z|^{-1-\alpha} dz. $$
\begin{lemma}\label{l:LLL} If $f \in C_c^2((0,{\infty}))$, then $\tilde{{\mathcal L}}f(x)$ is well defined and $\tilde{{\mathcal L}}f=\overline{{\mathcal L}}f$. \end{lemma} \noindent{\bf Proof.} By \textbf{(B2)}(c), for any compact set $K\subset (0, \infty)$ and $\epsilon>0$,
$$\int_{z \in K, |z-x|\ge \epsilon}\sB(x,z)dz \le c(x, K, \epsilon)<\infty.$$ Using this and \textbf{(B3)}, one can follow the proofs of \cite[Lemma 3.3 and Proposition 3.4]{KSV21} and show that $\tilde{{\mathcal L}}f$ is well defined for $f \in C_c^2((0,{\infty}))$.
By the change of variables $z=xe^y$ we have: \begin{align*} &
\int_{(0, \infty), |z-x|>\epsilon}
(f(z)-f(x))\sB(x,z)|x-z|^{-1-\alpha} dz\\
&=\int_{{\mathbb R}, |xe^y-x|>\epsilon}\left(f(xe^y)-f(x)\right)\sB(x, xe^y)|x-xe^y|^{-1-\alpha}xe^y\, dy\\
&=x^{-\alpha} \int_{{\mathbb R}, |e^y-1|>\epsilon/x}\left(f(xe^y)-f(x)\right)\sB(1, e^y)|1-e^y|^{-1-\alpha}e^y\, dy\\
&=x^{-\alpha} \int_{{\mathbb R}, |e^y-1|>\epsilon/x}\left(f(xe^y)-f(x)\right)\mu^{\sB}(y)\, dy\\
&=x^{-\alpha}\left(\int_{{\mathbb R}, |e^y-1|>\epsilon/x}\Big(f(xe^y)-f(x)-xf'(x)y{\bf 1}_{[-1,1]}(y)\Big)\mu^{\sB}(y)\, dy \right.\\
& \qquad \qquad \qquad \left. +xf'(x)\int_{{\mathbb R}, |e^y-1|>\epsilon/x}y{\bf 1}_{[-1,1]}(y)\mu^{\sB}(y)\, dy\right)\\ &=: x^{-\alpha}(J_1(\epsilon)+xf'(x) J_2(\epsilon)). \end{align*} By the dominated convergence theorem, $$ \lim_{\epsilon \to 0}J_1(\epsilon) =\int_{{\mathbb R}}\Big(f(xe^y)-f(x)-xf'(x)y{\bf 1}_{[-1,1]}(y)\Big)\mu^{\sB}(y)\, dy. $$ Since $\tilde{{\mathcal L}} f(x)$ is well defined, we see that there also exists $$
\lim_{\epsilon\to 0}J_2(\epsilon)=\lim_{\epsilon\to 0}\int_{{\mathbb R}, |e^y-1|>\epsilon/x}y{\bf 1}_{[-1,1]}(y)\mu^{\sB}(y)\, dy=: - \tilde{b}. $$ Thus $$ \tilde{{\mathcal L}} f(x)=- \tilde{b}
x^{1-\alpha} f'(x)+ x^{-\alpha}\int_{{\mathbb R}}\Big(f(xe^y)-f(x)-xf'(x)y{\bf 1}_{[-1,1]}(y)\Big)\mu^{\sB}(y)\, dy. $$ By Lemma \ref{l:lin-term-alt-2} we see that $ \tilde{b}=\overline{b}$ and thus $\tilde{{\mathcal L}}=\overline{{\mathcal L}}$. {
$\Box$
}
Now we turn to the question of the behavior of the pssMp $\overline{X}$ at its absorption time. We assume that \begin{equation}\label{e:B-cond-exp}
\int_{-\infty}^{-1}|y|e^y \sB(1,e^y)dy + \int_1^{\infty}ye^{-\alpha y}\sB(1,e^y)dy <\infty. \end{equation}
Then $\int_{{\mathbb R}, |y|\ge 1}|y|\mu^{\sB}(y)dy<\infty$, hence $\overline{\xi}_1$ has finite expectation given by \begin{equation}\label{e:B-exp}
{\mathbb E} \overline{\xi}_1=-\overline{b}+\int_{{\mathbb R}, |y|\ge 1}y\mu^{\sB}(y)dy, \end{equation} cf.~\cite[Theorem 25.3, Example 25.12]{Sat14}.
For $\gamma\in {\mathbb R}$ let $$ \sigma_{\gamma}(x):=\frac{e^{(1+\gamma)x}}{(e^x-1)^{1+\alpha}}-\frac{e^{-(1+\gamma)x}}{(1-e^{-x})^{1+\alpha}} =\frac{e^{-x} (e^{(\gamma-\alpha+1)x}-1)}{(1-e^{-x})^{1+\alpha}} , \quad x>0. $$ The next lemma follows immediately from the second expression of $\sigma_{\gamma}$ above. \begin{lemma}\label{l:sign-of-sigma} For every $x>0$ it holds that $\sigma_{\gamma}(x)>0$ for $\alpha < 1+\gamma$, $\sigma_{\gamma}(x)=0$ for $\alpha=1+\gamma$, and $\sigma_{\gamma}(x)>0$ for $\alpha >1+\gamma$. \end{lemma}
In the next result, we will also assume that, in addition to \textbf{(B1)}-\textbf{(B3)}, $\sB$ satisfies
\noindent \textbf{(B4)} Symmetry: $\sB(x,y)=\sB(y,x)$ for all $x,y>0$.
\begin{prop}\label{p:expectation-symmetry}
Let $\overline{X}$ be a pssMp with the infinitesimal generator $\overline{{\mathcal L}}$ given in \eqref{e:inf-gen-overline-X} where the jump kernel is $\sB(x,y)|x-y|^{-1-\alpha}$ and the linear term given in \eqref{e:linear-term-sym}. Assume that $\sB$ satisfies \textbf{(B1)}-\textbf{(B4)} and \eqref{e:B-cond-exp}. Let $\overline\xi$ be the corresponding L\'evy process through the Lamperti transform. Then ${\bf E} \overline\xi_1>0$ if $\alpha\in (0,1)$, ${\bf E} \overline\xi_1=0$ if $\alpha=1$, and ${\bf E} \overline\xi_1<0$ if $\alpha\in (1,2)$. \end{prop} \noindent{\bf Proof.} Note that by Lemma \ref{l:lin-term-alt-2} and the fact that $\sB(1,e^y)=\sB(e^y,1)=\sB(1, e^{-y})$, it holds that $$ -\overline{b}=\mathrm{p.v.}\int_{-1}^1 y\mu^{\sB}(y)dy=\lim_{\epsilon\to 0}\int_{\epsilon}^1y(\mu^{\sB}(y)-\mu^{\sB}(-y))dy= \lim_{\epsilon\to 0}\int_{\epsilon}^1 y\sigma_0(y)\sB(1,e^y)dy. $$ Similarly, $$
\int_{{\mathbb R}, |y|\ge 1}y \mu^{\sB}(y)dy =\int_1^{\infty}y (\mu^{\sB}(y)-\mu^{\sB}(-y))dy =\int_1^{\infty}y \sigma_0(y)\sB(1,e^y)dy. $$ The claim now follows from \eqref{e:B-exp} and Lemma \ref{l:sign-of-sigma}. {
$\Box$
}
We can also cover some cases with non-symmetric $\sB(x,y)$.
\begin{prop}\label{p:expectation-non-symmetry}
Let $\overline{X}$ be a pssMp with the infinitesimal generator $\overline{{\mathcal L}}$ given in \eqref{e:inf-gen-overline-X} where the jump kernel $J(x,y) =(y/x)^{\gamma}|x-y|^{-1-\alpha}$ with $\gamma\in (-1,\alpha)$ and the linear term is given in \eqref{e:linear-term-sym}. Let $\overline\xi$ be the corresponding L\'evy process through the Lamperti transform. Then $ {\bf E} \overline\xi_1>0$ if $\alpha\in (0,1+\gamma)$, ${\bf E} \overline\xi_1=0$ if $\alpha=1+\gamma$, and ${\bf E} \overline\xi_1<0$ if $\alpha\in (1+\gamma,2)$.
In particular, if $\gamma\in [\alpha/2,\alpha)$ then $ {\bf E} \overline\xi_1>0$. \end{prop} \noindent{\bf Proof.} Since $\sB(x,y)=(y/x)^{\gamma}$ with $\gamma\in (-1, \alpha)$, we have that \eqref{e:B-cond-exp} holds and that $\mu^{\sB}(y)-\mu^{\sB}(-y)=e^{\gamma y}\mu(y)-e^{-\gamma y}\mu(-y)=\sigma_\gamma(y)$ for $y >0$. By Lemma \ref{l:lin-term-alt-2}, it holds that $$ -\overline{b}=\mathrm{p.v.}\int_{-1}^1 y\mu^{\sB}(y)dy=\lim_{\epsilon\to 0}\int_{\epsilon}^1y(\mu^{\sB}(y)-\mu^{\sB}(-y))dy= \lim_{\epsilon\to 0}\int_{\epsilon}^1 y\sigma_\gamma (y)dy. $$ Similarly, $$ \int_{
{\mathbb R}, |y|\ge 1}y \mu^{\sB}(y)dy =\int_1^{\infty}y (\mu^{\sB}(y)-\mu^{\sB}(-y))dy =\int_1^{\infty}y \sigma_\gamma(y)dy. $$ The claim now follows from Lemma \ref{l:sign-of-sigma}. {
$\Box$
}
We end this subsection with a class of examples of modifying functions,
satisfying \textbf{(B1)}-\textbf{(B4)}, which appeared in our papers \cite{KSV21, KSV22} on the potential theory of Dirichlet forms with jump kernels decaying at the boundary. For $\beta\ge 0$ and $\gamma\ge 0$ with $\gamma=0$ if $\beta=0$, we define \begin{align}\label{wt_B} \widetilde{B} (x,y)=\left(\frac{x\wedge y}{x\vee y}\right)^{\beta}\left(\log\left(1+\frac{x\vee y}{x\wedge y}\right)\right)^{\gamma}. \end{align} It is easy to check that $\widetilde{B} (x,y)$ satisfies \textbf{(B1)}-\textbf{(B4)}. Since
$ \frac{x\wedge y}{|x-y|}\wedge 1 \asymp \frac{x\wedge y}{x\vee y}$ and $\frac{x\vee y}{|x-y|}\wedge 1\asymp 1$, $\widetilde{B}(x,y)$ is comparable to the $\sB(x,y)$
in \cite[(1.8)]{KSV21} with $\beta=\beta_1$, $\gamma=\beta_3$ and $\beta_2=\beta_4=0$
When $\sB(x,y)$ is equal to $c\widetilde{B}(x,y)$, we have that the L\'evy measure $\mu^{\sB}(y)$ of $\xi$ is equal to $c$ times \begin{eqnarray*}
&&\frac{e^y}{|e^y-1|^{1+\alpha}}\left(\frac{1\wedge e^y}{1\vee e^y}\right)^{\beta}\left(\log\left(1+\frac{1\vee e^y}{1\wedge e^y}\right)\right)^{\gamma}\\
&=&\frac{e^y}{|e^y-1|^{1+\alpha}}\left({\bf 1}_{(y<0)} e^{y\beta}\left(\log(1+e^{-y})\right)^{\gamma}+{\bf 1}_{(y>0)}e^{-y\beta}\left(\log(1+e^{y})\right)^{\gamma}\right)\\
&=& \frac{e^y}{|e^y-1|^{1+\alpha}}e^{-|y|\beta}\left(\log(1+e^{|y|})\right)^{\gamma}. \end{eqnarray*} If $\gamma=0$ (so there is no logarithmic term), we see that $\mu^{\sB}$ is the L\'evy measure of a Lamperti stable process in the sense of \cite{CPP10}.
Clearly, if $\sB(x,y)$ is comparable to $\widetilde{B}(x,y)$, then the corresponding L\'evy measure $\mu_{\ast}^{\sB}$ satisfies $$ \mu_{\ast}^{\sB}(y)\asymp \mu^{\sB}(y), \quad y\in {\mathbb R}\setminus \{0\}. $$
\section{Appendix}\label{s:appendix}
\noindent \textbf{Proof of Lemma \ref{l:q-symmetric}:} We claim that for any $0\le j\le k$, \begin{equation}\label{e:unique2} \int_{(0, \infty)} u^j(1+xu)^{-1-\alpha-k} m(du)=0, \quad x>0. \end{equation} \eqref{e:unique2} is valid for $k=0$ by assumption. Note that, for $k=1, 2, \dots$, $$
|\frac{\partial^k}{\partial x^k}(1+xu)^{-1-\alpha}|\le (1+xu)^{-1-\alpha}, \quad x>0, u>0. $$ Combining this with the integrability assumption of the lemma, we can exchange the order of the differentiation and integration when we take the derivative of the left hand side of \eqref{e:unique1}. Taking derivative with respect to $x$ in \eqref{e:unique1} we get \begin{equation}\label{e:unique-add} \int_{(0, \infty)} u(1+xu)^{-1-\alpha-1} m(du)=0, \quad x>0, \end{equation} and so \eqref{e:unique2}
is valid for $k=j=1$. Since \begin{align*} & \int_{(0, \infty)} (1+xu)^{-1-\alpha-1} m(du)\\ &= \int_{(0, \infty)} (1+xu)^{-1-\alpha}m(du)- \int_{(0, \infty)} xu(1+xu)^{-1-\alpha-1}m(du)=0, \quad x>0, \end{align*} (where the last equality follows from the assumption and \eqref{e:unique-add}), we get that \eqref{e:unique2} is valid for $k=1$ and $j=0$. Now suppose that \eqref{e:unique2} is valid for $0\le j\le k$. Taking derivative with respect to $x$ in \eqref{e:unique2}, we get $$ \int_{(0, \infty)} u^{j+1}(1+xu)^{-1-\alpha-k-1} m(du)=0, \quad x>0. $$ Thus \eqref{e:unique2} is valid for $1\le j\le k+1$. Noting that \begin{align*} & \int_{(0, \infty)} (1+xu)^{-1-\alpha-k-1}m(du)\\ &= \int_{(0, \infty)} (1+xu)^{-1-\alpha-k} m(du)- \int_{(0, \infty)} xu(1+xu)^{-1-\alpha-k-1}m(du)=0, \quad x>0, \end{align*} we get that \eqref{e:unique2} is valid for $0\le j\le k+1$.
Taking $x=1$, we get that for any $0\le j\le k$, $$ \int_{(0, \infty)} \frac{u^j}{(1+u)^k}(1+u)^{-1-\alpha} m(du)=0. $$ Since the linear span of the set $\{\frac{u^j}{(1+u)^k}: 0< j< k\}$ is an algebra of real-valued continuous functions on $(0, \infty)$ which separates points of $(0, \infty)$ and vanishes at infinity, by the Stone-Weierstrass Theorem, the linear span of the set $\{\frac{u^j}{(1+u)^k}: 0< j< k\}$
is dense in $C_\infty(0, \infty)$ with respect to the uniform topology. Thus for all $g\in C_\infty(0, \infty)$, $$ \int_{(0, \infty)} g(u)(1+u)^{-1-\alpha}m(du)=0, $$ which implies $(1+u)^{-1-\alpha}m(du)$ is the zero measure on $(0, \infty)$. Therefore $m$ is the zero measure on $(0, \infty)$. {
$\Box$
}
\textbf{Acknowledgment}: We thank the referees for insightful comments and suggestions that led to improvements of the paper. We also thank Pierre Patie for helpful comments on a preliminary version of this paper.
\small
\vskip 0.1truein
\parindent=0em
{\bf Panki Kim}
Department of Mathematical Sciences and Research Institute of Mathematics,
Seoul National University, Seoul 08826, Republic of Korea
E-mail: \texttt{[email protected]}
{\bf Renming Song}
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
E-mail: \texttt{[email protected]}
{\bf Zoran Vondra\v{c}ek}
Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia,
Email: \texttt{[email protected]}
\end{document} | arXiv |
Vector fields on spheres
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in $n$-dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known,[1] by direct construction using Clifford algebras, that there were at least $\rho (n)-1$ such fields (see definition below). Adams applied homotopy theory and topological K-theory[2] to prove that no more independent vector fields could be found. Hence $\rho (n)-1$ is the exact number of pointwise linearly independent vector fields that exist on an ($n-1$)-dimensional sphere.
Technical details
In detail, the question applies to the 'round spheres' and to their tangent bundles: in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers $\rho (n)$ determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of $n$ odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem), so the case $n$ even is an extension of that. Adams showed that the maximum number of continuous (smooth would be no different here) pointwise linearly-independent vector fields on the ($n-1$)-sphere is exactly $\rho (n)-1$.
The construction of the fields is related to the real Clifford algebras, which is a theory with a periodicity modulo 8 that also shows up here. By the Gram–Schmidt process, it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis at each point.
Radon–Hurwitz numbers
The Radon–Hurwitz numbers $\rho (n)$ occur in earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) on the Hurwitz problem on quadratic forms.[3] For $n$ written as the product of an odd number $A$ and a power of two $2^{B}$, write
$B=c+4d,0\leq c<4$.
Then[3]
$\rho (n)=2^{c}+8d$.
The first few values of $\rho (2n)$ are (from (sequence A053381 in the OEIS)):
2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ...
For odd $n$, the value of the function $\rho (n)$ is one.
These numbers occur also in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real $n\times n$ matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by Beno Eckmann. They are now applied in areas including coding theory and theoretical physics.
References
1. James, I. M. (1957). "Whitehead products and vector-fields on spheres". Proceedings of the Cambridge Philosophical Society. 53 (4): 817–820. doi:10.1017/S0305004100032928. S2CID 119646042.
2. Adams, J. F. (1962). "Vector Fields on Spheres". Annals of Mathematics. 75 (3): 603–632. doi:10.2307/1970213. JSTOR 1970213. Zbl 0112.38102.
3. Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. p. 127. ISBN 0-521-42668-5. Zbl 0785.11022.
• Porteous, I.R. (1969). Topological Geometry. Van Nostrand Reinhold. pp. 336–352. ISBN 0-442-06606-6. Zbl 0186.06304.
• Miller, H.R. "Vector fields on spheres, etc. (course notes)" (PDF). Retrieved 10 November 2018.
| Wikipedia |
\begin{document}
\begin{abstract} In this paper we analyse the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter. \vskip5pt \noindent \textsc{Keywords}: Geometric evolutions; rigidity results; minimizing movements; variational methods. \vskip5pt \noindent \textsc{AMS subject classifications:} 53C44; 53C24; 49M25; 49Q20. \end{abstract}
\title{Long time behaviour of discrete volume preserving\ mean curvature flows}
\tableofcontents
\section*{Introduction} We study the asymptotic behaviour of a discrete-in-time approximation of the mean curvature flow with constrained volume. This geometric evolution consists in a family of evolving sets $ [0,+\infty)\ni
t\mapsto E_t\subset\mathbb{R}^N$, whose normal velocities $V(x,t)$ at any point $x\in \partial E_t$ are proportional to the scalar mean curvature $H_{\partial E_t}(x)$ of $\partial E_t$ at $x$ corrected by a constant forcing term which guarantees that the measure of $|E_t|$ is preserved during the flow: \begin{equation}\label{e:equazione modello} V(x,t) = \overline H_{\partial E_t} - H_{\partial E_t}(x), \qquad \overline H_{\partial E_t} := \frac{1}{\mathcal{H}^{N-1}(\partial E_t)}\int_{\partial E_t} H_{\partial E_t}(x) \, d\mathcal{H}^{N-1}(x) . \end{equation} Under suitable assumptions, this geometric flow is a model for coarsening phenomena in physical systems. For example, one can consider mixtures that, after a first relaxation time, can be described by two subdomains of nearly pure phases far from equilibrium, evolving in a way to minimize the total interfacial area between the phases while keeping their volume constant (further details on the physical background can be found in \cite{CRCT95, MuSe13,TW72,W61}).
It is well-known that a typical evolution of \eqref{e:equazione modello} develops singularities of different kinds in finite time: components shrinking to points and disappearance, collisions and merging of domains, pinch-offs etc{\ldots} Compared with the more familiar unconstrained mean curvature flow, the possible changes in the topology of the evolving sets $E_t$ of \eqref{e:equazione modello} are even wider, because of the nonlocal character of the flow and the subsequent lack of comparison principles. There exist singular solutions also in the two dimensional case (see \cite{M, MS}).
It is then clear that the analysis of the long time behaviour of systems exhibiting coarsening requires the introduction of suitable notions of weak solutions which allow the formation of singularities and extend the flow past the singular times. This is a well-established feature of curvature flows, and many definitions of weak solutions have been introduced in the literature. Here, we follow the method proposed independently by Almgren, Taylor and Wang \cite{ATW} and by Luckhaus and Sturzenhecker \cite{LS}, based on De Giorgi's minimizing movement approach. The authors consider an implicit time-discretization of the flow, which is regarded as a gradient flow of the perimeter functional with respect to a metric resembling the $L^2$-distance. The limiting time-continuous flow constructed with this method is usually referred to as flat flow. This approach has the advantage to be easily adapted to volume constrained mean curvature flow as shown in \cite{MSS}, producing global-in-time solutions which then permit to analyse the equilibrium configurations reached in the long time asymptotics.
Previous results on the long time behaviour of the volume constrained mean curvature flow are mostly confined to the case of smooth solutions, starting from specific classes of initial regular sets ensuring the existence of global regular solutions; for example uniformly convex and nearly spherical initial sets (see \cite{H, ES}), or nearly strictly stable initial sets in the three and four dimensional flat torus (see \cite{Nii}). For more general initial data, the long time behaviour in the context of flat flows of convex and star-shaped sets (see \cite{BCCN, KK}) has been characterized only up to (possibly diverging in the case of \cite{BCCN}) translations.
In this paper we characterize the long-time limits of the discrite-in-time approximate flows constructed by the Euler implicit scheme introduced in \cite{ATW, LS} plus the volume constraint. Our main result in Theorem \ref{mainthm} establishes that for every initial bounded set $E_0$ with finite perimeter, any discrete-in-time mean curvature flow with volume constraint converges exponentially fast to a finite union of disjoint balls with equal radii.
\subsection{Exponential convergence of dissipations} The main issue here is to prove that the evolving sets remain uniformly bounded for all times and that the long-time limit is truly unique (and not just up to translations).
This is a non-trivial fact, since the discrete scheme (as well as its continuous counterpart) does not satisfies comparison principles. To the best of our knowledge, the boundedness of the flow was not known previously for any weak notion of the nonlocal mean curvature flow. The problem is that, summing the dissipations of the discrete schemes formally provides a $L^2$ (with respect to time) bound on the velocity of the evolving set, rather than the necessary $L^1$ bound: roughly speaking, one needs to sum the square roots of the dissipations rather than the dissipations themself.
By a compactness argument and a characterization of the stationary configurations of the discrete flow (based on a recent theorem by Delgadino and Maggi \cite{DM} on Caccioppoli sets with constant weak mean curvature), the connected components of the evolving sets converge up to translations to balls.
Therefore, in order to improve the estimate on the dissipations, we compare at each time step the energy of the actual minimizers with that of a properly placed union of balls (one nearby each connected component of the evolving sets). In this way, it turns out to be necessary to estimate the dissipation of a nearly spherical set with respect to a close-by ball (this condition is definitively reached by each connected component of the flow thanks to the compactness argument above) with the actual dissipation of the flow. Indeed, if this happens, one can control the sum of the dissipations for all times larger than any time $t$ from above by the energy dissipated at time $t$, thus clearly leading to an exponential decay of the dissipations and in turn of the velocity of the evolvings sets.
The comparison of the dissipation of the flow and the one with respect to balls is equivalent to prove a control of the $L^2$-norm of the function parametrizing a nearly spherical set with the $L^2$-norm of the oscillation of its mean curvature. In the case of zero oscillation, this is nothing but Alexandrov's theorem on the characterization of the spheres as the unique embedded hypersurfaces with constant mean curvature and thus the aforementioned estimate may be regarded as a quantitative version of it. Its proof in turn relies on a result proven by Krummel and Maggi \cite[Theorem~1.10]{KM}, which can be seen as a sort of higher order Fuglede-type estimate involving the first variation of the perimeter rather than the perimeter itself. For different quatitative versions of Alexandrov's theorem see also \cite{CV, CM17, D+18}.
\subsection{Comments and open questions} It remains an open question to extend the results of the present paper to the flat flows, that is to obtain uniform estimates as the time step converges to zero. We conjecture the result to hold true, but the compactness arguments used in this paper in order to gain the closeness to sphere is not quantitative and does not allow to pass to the limit. Moreover, it is worth mentioning that, both nonlocal flat flows and discrete-in-time nonlocal flows are not uniquely defined. There are different ways to impose the volume constraint (by constraining the volume at each time-step, or by penalization, or by tuning suitably a forcing term). The flows so defined can be different (also because of the congenital non-uniqueness of mean curvature flows), but we expect that the asymptotics are always finite unions of equal volume balls.
Finally, we stress that the main Theorem \ref{mainthm} is sharp for what concerns the limiting configurations. Indeed, we can show that the union of suitably distant balls with equal radii is a stationary solution for the discrete flow and, hence, a possible asymptotics. However, such solutions are clearly unstable, because whatever small increase in one of the radii (and consequent decrease in the others) will drive the evolution towards a different asymptotic limit. This is easily seen for two disjoint balls with equal radii: any configuration of concentric balls with the same total volume but slightly perturbed radii will have strictly lower perimeter and thus will converge to a single ball (see Corollary~\ref{cor:menodidue}). In this regard, we conjecture that, generically, the evolution of an initial set $E_0$ will converge to a {\it unique} limiting ball with the same volume.
\subsection{Structure of the paper} In the first section we prove the Fuglede-type inequality for nearly spherical sets with almost constant mean curvature. Section 2 is then devoted to the introduction of the incremental problem at the basis of the discrete-in-time mean curvature flow. Finally, in the last section of the paper we prove the exponential convergence of the discrete flow to a union of well-separated balls with the equal volume.
\section{A quantitative Alexandrov Theorem for nearly spherical sets}
In this section we prove a variant of a stability inequality by Krummel and Maggi \cite[Theorem 1.10]{KM} related to Alexandrov's theorem, whose statement we recall for the reader's convenience. To this aim, and for later use, we fix the following notation: $B$ denotes the unit ball of $\mathbb{R}^N$. Given ${m}>0$, we denote by $B^{(m)}$ the ball centered at the origin with volume $m$ and we denote by $r(m)$ its radius $\Big(\frac{m}{\omega_N}\Big)^{\frac1N}$. For any measurable function $\varphi:\partial B^{(m)} \to (-1, +\infty)$, we denote by $E_{\varphi,m}$ the set in $\mathbb{R}^N$ whose boundary is described through the radial graph of $\varphi$, namely \begin{equation}\label{schifo}
E_{\varphi, m}:= \big\{t x \in \mathbb{R}^N : \, x\in\partial B^{(m)},\; 0\le t \le 1+ \varphi(x)\big\}. \end{equation} In the case $m=\omega_N$ we will write $B$ instead of $B^{(\omega_N)}$ (to denote the unit ball) and $ E_{\varphi}$ instead of $ E_{\varphi, \omega_N}$. If the parametrizing function $\varphi$ has small $L^\infty$-norm, then the set $E_{\varphi, m}$ can be regarded as a graphical radial pertubation of the ball $B^{(m)}$. We finally recall that given a sufficiently regular set $E$, $H_{\partial E}$ stands for the sum of the principal curvatures of $\partial E$ (with respect to the orientation given by the outward normal).
\begin{theorem}[\cite{KM}]\label{th:KM}
There exist $\delta\in (0,\frac12)$ and $C>0$ with the following property: For any $f \in C^1(\partial B) \cap H^2(\partial B)$ such that $\|f\|_{C^1}\le \delta$ and $\mathrm{bar}(E_f)=0$,
we have $$
\|f\|_{H^1(\partial B)}\le C \| H_{\partial E_f} - (N-1)) \|_{L^2(\partial B)} \,.
$$ \end{theorem} \begin{remark} In fact, in \cite{KM} the theorem is stated under the assumption $f\in C^{1,1}(\partial B)$. However, it is immediate to see by a standard approximation argument that the statement holds true under the weaker hypothesis $f \in C^1(\partial B) \cap H^2(\partial B)$. \end{remark}
The stability inequality for nearly spherical sets with almost constant mean curvature that we are going to use in the next sections is the following.
\begin{theorem}\label{Aleq}
There exist $\delta\in (0,\frac12)$ and $C>0$ with the following property: For any $f \in C^1(\partial B) \cap H^2(\partial B)$ such that $\|f\|_{C^1}\le \delta$, $|E_f|=\omega_N$ and $\mathrm{bar}(E_f)=0$,
we have
\begin{equation}\label{e:aleq}
\|f\|_{H^1(\partial B)}\le C \| H_{\partial E_f} - \overline H_{\partial E_f} \|_{L^2(\partial B)} \,, \end{equation} where we have set $$ \overline H_{\partial E_f} : = -\kern -,375cm\int_{\partial B}H_{\partial E_f}(x+f(x)x)\, d\mathcal{H}^{N-1}\,. $$ \end{theorem}
\begin{proof} First of all we notice that, if we take the constant $C$ in \eqref{e:aleq} to be bigger than $\sqrt{\frac{N}2\omega_{N}}$, then it is enough to consider only the case
$\|H_{\partial E_f} - \overline H_{\partial E_f}\|_2\leq 1$.
We write explicit formulas for the perimeter and its first variations for the sets $E_f$: by using the area formula we get \begin{equation}\label{eq:per}
P(E_f) = \int_{\partial B} (1+f)^{N-1} (1 + (1+f)^{-2} |\nabla f|^2)^{\frac 12} \, d\mathcal{H}^{N-1}\,. \end{equation} Recall that $H_{\partial E_f} $ is the first variation of the perimeter at $E_f$, that is, \begin{equation} \label{e:var1-1} \begin{split} \delta P(E_f) [\psi] := &\frac{d}{dt}P(E_{f+t\psi}) = \int_{\partial E} (H_{\partial E_f} \nu_{\partial E_f}\big) \cdot x\, \psi(x) \, d\mathcal{H}^{N-1}(x)\\ &= \int_{\partial B} (H_{\partial E_f} \nu_{\partial E_f}\big)(p) \cdot x
\;\psi (1+f)^{N-1} (1 + (1+f)^{-2} |\nabla f|^2)^{\frac 12}\, d\mathcal{H}^{N-1}\,, \end{split} \end{equation} where for simplicity of notation we have set $p=(1+f(x))x$. On the other hand, by differentiating under the integral sign, from \eqref{eq:per} we get \begin{equation} \label{e:var1-2} \begin{split} \delta P(E_f) [\psi]
& =\int_{\partial B} (N-1) (1+f)^{N-2} (1 + (1+f)^{-2} |\nabla f|^2)^{\frac 12} \psi \, d\mathcal{H}^{N-1} \\
&\qquad-\int_{\partial B} \frac{(1+f)^{N-4}}{(1 + (1+f)^{-2} |\nabla f|^2)^{\frac 12}} |\nabla f|^2 \psi\, d\mathcal{H}^{N-1}\\
&\qquad+ \int_{\partial B} \frac{(1+f)^{N-3}}{(1 + (1+f)^{-2} |\nabla f|^2)^{\frac 12}} \nabla f \cdot \nabla \psi\, d\mathcal{H}^{N-1}, \end{split} \end{equation} for all $\psi \in C^1(\partial B)$, where $\nabla f$ and $\nabla \psi$ are the tangent gradient of $f$ and $\psi$ on $\partial B$, respectively.
In the following, with a slight abuse of notation, by the symbol $O(g)$ we mean any function $h$ of the form $h(x)=r(x)g(x)$, where $|r(x)|\leq C$ for all $x\in \partial B$, with $C>0$ being a constant depending only on the apriori $C^1$-bound $\|f\|_{C^1}\leq\frac12$.
Since the normal to $E_f$ at a point $p=(1+f(x))x$ with $x\in \partial B$ is given by \[
\nu_{{\partial E_f}} (p) = \frac{1}{\sqrt{1+(1+f)^{-2}|\nabla f|^2}}\left(-\frac{\nabla f}{1+f} + x\right), \] one gets
\begin{equation}\label{onegets}
\nu_{{\partial E_f}} (p) \cdot x = \frac{1}{\sqrt{1+(1+f)^{-2}|\nabla f|^2}}.
\end{equation} Therefore, by \eqref{e:var1-1}, \eqref{onegets} and a simple Taylor expansion, we have that \begin{align}\label{e:lin1} \delta P(E_f) [\psi] &= \int_{\partial B } (1+R_1) \,H_{\partial E_f}(p) \Big (1+ (N-1) f + R_2 \Big ) \psi \, d\mathcal{H}^{N-1}, \end{align}
with $|R_1|= O(|\nabla f|^2)$ and $|R_2|= O(|f|^2)$. Similarly, using \eqref{e:var1-2} and a Taylor expansion, we get \begin{align}\label{e:lin2} \delta P(E_f) [\psi] &= \int_{\partial B} \big(
(N-1) + (N-1)(N-2) f\,+ O(|f|^2) + O(|\nabla f|^2)\big)\psi\, d\mathcal{H}^{N-1}\notag\\ &\quad + \int_{\partial B} \left(\nabla f + h\right)\cdot \nabla \psi \, d\mathcal{H}^{N-1}, \end{align} where $h$ is a vector field satisfying \begin{equation}\label{bacca}
|h|\leq C (|f|+|\nabla f|^2)|\nabla f|\,,
\end{equation} with $C>0$ depending only on the apriori $C^1$-bound $\|f\|_{C^1}\leq \frac12$. By comparing \eqref{e:lin1} and \eqref{e:lin2}, and recalling that
$|R_2|= O(|f|^2)$, we infer that \begin{align}\label{e:lin3} &\int_{\partial B} \left(-(N-1)f\psi + \nabla f\cdot \nabla \psi\right)\, d\mathcal{H}^{N-1}\notag\\ &= \int_{\partial B} (1+R_1)\big( H_{\partial E_f}(p) - (N-1)\big)(1+(N-1)f+R_2)\psi\, d\mathcal{H}^{N-1}\notag\\
&+\int_{\partial B} \big(-h\cdot \nabla \psi + \big(O(|f|^2) + O(|\nabla f|^2)\big)\psi\big) \, d\mathcal{H}^{N-1}. \end{align} Observe that testing \eqref{e:lin3} with $\psi = 1$ and setting
$R_3= (1+R_1)(1+(N-1)f+R_2)-1$, we get $R_3 = O(|f|) + O(|\nabla f|^2)$ and \begin{equation}\label{e:lin4} \int_{\partial B} (1+R_3)\big( H_{\partial E_f}(p) - (N-1)\big)\, d\mathcal{H}^{N-1} =
\int_{\partial B} \left(O(|f|) + O(|\nabla f|^2)\right)\, d\mathcal{H}^{N-1}\,. \end{equation} In particular, for $F :=(1+R_3) H_{\partial E_f}$ and recalling that the overline denotes the average on $\partial B$, we infer that for every $\eta\in (0,\frac12)$, if $\delta$ is sufficiently small, we have that \[ \left\vert\overline F-(N-1)-\kern -,375cm\int_{\partial B} (R_3+1)\right\vert \leq \eta \] and, always assuming $\delta$ small enough, \begin{align}
|\overline H_{\partial E_f}-(N-1)|&\leq |\overline H_{\partial E_f}-\overline F| + \left\vert\overline F-(N-1)-\kern -,375cm\int_{\partial B} (R_3+1)\right\vert+(N-1)\left\vert-\kern -,375cm\int_{\partial B} R_3\right\vert\notag\\ &\leq\left\vert -\kern -,375cm\int_{\partial B} R_3 H_{\partial E_f} \right\vert + 2\eta\leq\notag\\
&\leq \omega_N^{-1}\|R_3\|_2 \|H_{\partial E_f} - \overline H_{\partial E_f}\|_2
+ |\overline H_{\partial E_f}-(N-1)| \int_{\partial B}\left\vert R_3 \right\vert\, d\mathcal{H}^{N-1}+\notag\\ &\qquad + (N-1)\int_{\partial B}\left\vert R_3 \right\vert\, d\mathcal{H}^{N-1}+2\eta\notag\\
&\leq \eta \|H_{\partial E_f} - \overline H_{\partial E_f}\|_2 + \eta |\overline H_{\partial E_f}-(N-1)| + 3\eta\notag\\
&\leq \eta |\overline H_{\partial E_f}-(N-1)| + 4\eta,\notag \end{align}
where in the last inequality we have used $\|H_{\partial E_f} - \overline H_{\partial E_f}\|_2\leq 1$. In particuar, there exists a constant $\lambda$ such that \begin{equation}\label{e:vicinanza media} \overline H_{\partial E_f}=N-1 + \lambda,\qquad
|\lambda|\leq \frac{4\eta}{1-\eta}. \end{equation} Note that $\lambda$ is arbitralily small, if $\delta$ (and hence $\eta$) is chosen accordingly.
Now the proof can be concluded as follows. Consider $\kappa = 1+\frac{\lambda}{N-1}$. If $\delta$ is small enough, then $\kappa\in (\frac12, 2)$. Consider also the set $\kappa E_{f}= E_{u}$ with $u:=\kappa-1+\kappa f$. Then, \[
\overline H_{E_u} = N-1, \qquad \|u\|_{H^1}\leq \kappa \|f\|_{H^1}+ \sqrt{N\omega_N}\frac{\lambda}{N-1}, \text{ and } \|u\|_{C^1}\leq \kappa \|f\|_{C^1}+ \frac{\lambda}{N-1}\,. \]
In particular, if $\delta$ and $\eta$ are sufficiently small (and, therefore, such is $\lambda$), we are in position to apply Theorem~\ref{th:KM} (because $\|u\|_{C^1}$ becomes arbitrarily small) and infer that \[
\|u\|_{H^1}\leq C \|H_{E_u} - N+1\|_{L^2} = \kappa^{-1} \,C\|H_{E_f} - \overline H_{E_f}\|_{L^2}
\leq 2C\|H_{E_f} - \overline H_{E_f}\|_{L^2}. \] In particular, we can estimate the $L^2$ norm of the gradient of $f$: \begin{equation}\label{eq:gradiente}
\|\nabla f\|_{L^2}\leq \kappa^{-1}\|u\|_{H^1}\leq 4C\|H_{E_f} - \overline H_{E_f}\|_{L^2}.
\end{equation} Finally, since $|E_f|=|B|$, i.e., \[ \frac{1}{N}\int_{\partial B}(1+f)^N\,d\mathcal{H}^{N-1} = \omega_N, \] we get by Taylor expansion \begin{equation}\label{e:lin6}
\big\vert\int_{\partial B} f\,d\mathcal{H}^{N-1}\big\vert = \int_{\partial B} O(|f|^2) \,d\mathcal{H}^{N-1}. \end{equation} This implies that \begin{equation}\label{e:media}
|\overline f| \leq C\, \|f\|_{L^2}^2\leq C\delta \, \|f\|_{L^2}. \end{equation} By Poincar\'e inequality we have that \begin{align}\label{e:norma 2}
\|f\|_{L^2} & \leq 2\|f - \overline f\|_{L^2} + 2|\overline f| \big(N\omega_N\big)^{\frac12}\leq C\|\nabla f\|_{L^2}+ 2|\overline f| \big(N\omega_N\big)^{\frac12}. \end{align} Inserting \eqref{e:media} in \eqref{e:norma 2} and combining with \eqref{eq:gradiente} we deduce \eqref{e:aleq} if $\delta$ is sufficiently small.
\end{proof}
\begin{remark}\label{rm:acdc} By a simple rescaling argument it is clear that Theorem~\ref{Aleq} holds also for the sets $E_{\varphi,m}$ parametrized over a ball $B^{(m)}$ with volume $m$, with constants $\delta$ and $C$ depending on $m$. Clearly, the dependence of such constants on $m$ can be made uniform when $m$ varies in compact subsets of $(0,+\infty)$. \end{remark}
\begin{remark} The estimate \eqref{e:aleq} is optimal for what concerns the power of the norms. To see this, it is enough to consider sets $E_{\varepsilon f}$ with $f$ a functions in the second eigenspace of the Laplace-Beltrami operator on the sphere (with corresponding eigenvalue $2N$) and $\varepsilon>0$ sufficiently small: then, computing the $L^2$-norm of the average of the mean curvature yields \[
\| H_{\partial E_{\varepsilon f}} - \overline H_{\partial E_{\varepsilon f}} \|_{L^2(\partial B)}\leq
C \|\varepsilon f\|_{L^2(\partial B)}, \]
for a dimensional constant $C<0$. \end{remark}
\section{The incremental problem}
We start by introducing the incremental minimum problem which defines the discrite-in-time volume preserving mean curvature flow. To this purpose, let $E\neq \emptyset$ be a bounded measurable set. Notice that the topological boundary of $E$ depends on its representative; from now on, we will assume that $E$ coincides with its Lebesgue representative, i.e., with its points of density equal to one. We let \begin{equation}\label{defsigndist} d_E(x)\ =\ \textup{dist}(x,E)-\textup{dist}(x,\mathbb{R}^N\setminus E) \end{equation} be the signed distance function to $\partial E$. Let ${m}>0$ be fixed, representing the volume of the evolving set.
Fix a time step $h>0$ and consider the problem \begin{equation}\label{varprob}
\min \left\{P(F)\ +\ \frac{1}{h}\int_F d_E(x)\,dx:\, | F|={m} \right\}. \end{equation}
Note that $$ \int_Fd_E(x)\, dx-\int_E d_E(x)\, dx=\int_{E\Delta F}\mathrm{dist}(x, \partial E)\, dx $$ so that \eqref{varprob} is equivalent to $$ \min \left\{P(F)\ +\ \frac{1}{h}\int_{E\Delta F}\mathrm{dist}(x, \partial E)\, dx
:\, |F|={m} \right\}. $$ Given two sets $E,\, F$, we let $$ \mathcal{D} (F,E):= \int_{E\Delta F}\mathrm{dist}(x, \partial E)\, dx\,. $$ In order to to prove the existence of a solution to \eqref{varprob}, we need some preliminary results. We start with the following non-vanishing estimate for sets of finite perimeter and finite measure.
\begin{lemma}\label{lm:nonvanishing} There exists a constant $C=C(N)\in (0, \frac12)$ such that if $E\subset\mathbb{R}^N$ is a set of finite perimeter and finite measure, then, setting $Q:=(0,1)^N$, we have $$
\sup_{z\in \mathbb{Z}^N}|E\cap (z+Q)|\geq c(N)\min\Bigl\{\Big(\frac{|E|}{P(E)}\Big)^N, 1\Bigr\}\,. $$ \end{lemma} \begin{proof} Settting $$
\beta:=\sup_{z\in \mathbb{Z}^N}|E\cap (z+Q)|\,, $$ assume that $\beta\leq \frac12$. Then,
by the Relative Isoperimetric Inequality we have \begin{align*}
P(E)& \geq \sum_{z\in \mathbb{Z}^N}P(E, z+Q)\geq c(N)\sum_{z\in \mathbb{Z}^N}|E\cap (z+Q)|^{\frac{N-1}{N}}\\
&\geq \frac{c(N)}{\beta^{\frac1N}}\sum_{z\in \mathbb{Z}^N}|E\cap (z+Q)|=
\frac{c(N)}{\beta^{\frac1N}} |E| \end{align*} and the conclusion easily follows. \end{proof}
We are now in a position to prove the following proposition, which in particular establishes the existence of a solution to \eqref{varprob}. The crucial point in the following statement is that the choice of the penalization parameter $\Lambda$ depends only on the bounds on the perimeter and on the prescribed measure $m$, and thus can be made uniform along the minimizing movements scheme. \begin{proposition}\label{prop:penal}
Given $m$, $M>0$, there exists $\Lambda_0=\Lambda_0(m, M, h, N)>0$ such that for any bounded set $E$ of finite perimeter, with $P(E)\leq M$ and $ |E|=m$, and for any $\Lambda\geq\Lambda_0$, any solution solution $\overline F$ of \begin{equation}\label{varprob-pen}
\min \left\{P(F)\ +\frac1h\mathcal{D}(F, E)+\Lambda\big||F|-m\big|:\, F\subset\mathbb{R}^N\textrm{ \em measurable}\right\} \end{equation}
satisfies the volume constraint $|\overline F|=m$. In particular, \eqref{varprob} and \eqref{varprob-pen} are equivalent. \end{proposition} \begin{proof}
First of all, note that the existence of a solution to \eqref{varprob-pen} follows by standard arguments, see for instance \cite{MSS}. We argue by contradiction by assuming for every $n$ the existence of a set $E_n$, with $|E_n|=m$ and $P(E_n)\leq M$, and a set $$
\overline F_n\in \operatorname*{argmin}\left\{P(F)\ +\frac1h\mathcal{D}(F, E_n)+n\big||F|-m\big|:\, F\subset\mathbb{R}^N\text{ measurable} \right\} $$
such that $|\overline F_n|\neq m$. In the sequel, we may assume $|\overline F_n|<m$, as the other case can be treated analogously. Testing with $E_n$, by minimality we have \begin{equation}\label{trbounds}
P(\overline F_n)\ +\frac1h\mathcal{D}(\overline F_n, E_n)+n\big||\overline F_n|-m\big|\leq P(E_n)\leq M\,. \end{equation} In particular, \begin{equation}\label{plus}
|\overline F_n|\to m \end{equation} as $n\to\infty$. In turn, by Lemma~\ref{lm:nonvanishing} there exist a constant $c_0>0$, depending only on $m$, $M$ and $N$, and $z_n\in \mathbb{Z}^N$ such that $$
|\overline F_n\cap (z_n+Q)|\geq c_0 $$
for all $n$ sufficiently large. Thus, by replacing $\overline F_n$, $E_n$ by $\overline F_n-z_n$, $E_n-z_n$, respectively, and appealing to the well-known campactness properties of sets of finite perimeter, we may assume that up to a (not relabelled) subsequence we have $\overline F_n\to F_\infty$ in $L^1_{loc}$, with $|F_\infty|\geq|F_\infty\cap Q|\geq c_0>0$.
The idea now is to modify the sets $\overline F_n$ by applying a local dilation in order to impose the volume constraint. In this construction, we follow closely \cite[Section\til2]{EF}. We only outline the main steps.
First of all, arguing as in Step 1 of \cite[Section\til2]{EF}, for any fixed $y_0\in \partial^*F_\infty$ and for any given small $\varepsilon>0$ we may find a radius $r>0$ and a point $x_0$ in a small neighborhood of $y_0$ such that
$$
| \overline F_n\cap B_{r/2}(x_0)|<\varepsilon r^N\,, \quad | \overline F_n\cap B_{r}(x_0)|>\frac{\omega_Nr^N}{2^{N+2}}
$$
for all $n$ sufficiently large. In the following, to simplify the notation we assume that $x_0=0$ and we write $B_r$ instead of $B_r(0)$. For a sequence $0<\sigma_n<1/2^N$ to be chosen, we introduce the following bilipschitz maps:
$$
\Phi_n(x):=
\begin{cases}
(1-\sigma_n(2^N-1))x & \text{if $|x|\leq \frac r2$,}\\
x+\sigma_n\Bigl(1-\frac{r^N}{|x|^N}\Bigr)x & \text{$\frac r2\leq |x|<r$,}\\
x & \text{$|x|\geq r$.}
\end{cases}
$$
Setting $\widetilde F_n:=\Phi_n(\overline F_n)$, we have as in Step 3 of \cite[Section\til2]{EF}
\begin{equation}\label{EFper}
P(\overline F_n, {B_r})-P(\widetilde F_n, {B_r})\geq -2^NNP (\overline F_n, {B_r})\sigma_n\geq
-2^NN M\sigma_n\,.
\end{equation}
Moreover, as in Step 4 of \cite[Section\til2]{EF} we have $$
|\widetilde F_n|-|\overline F_n|\geq \sigma_nr^N\Bigl[c\frac{\omega_N}{2^{N+2}}-\varepsilon(c+(2^N-1)N)\Bigr] $$
for a suitable constant $c$ depending only on the dimension $N$. If we fix $\varepsilon$ so that the negative term in the square bracket does not exceed half the positive one, then we have
\begin{equation}\label{EFvol}
|\widetilde F_n|-|\overline F_n|\geq \sigma_nr^NC_1\,,
\end{equation}
with $C_1>0$ depending on $N$.
In particular, from this inequality it is clear that we can choose $\sigma_n$ so that $|\widetilde F_n|=m$ for $n$ large; this implies $\sigma_n\to 0$, thanks to \eqref{plus}.
Now, arguing as in \cite[Equations (2.12) and (2.13)]{AFM}, we obtain \begin{equation}\label{difsim}
|\widetilde F_n\Delta \overline F_n|\leq C_3\sigma_nP(\overline F_n, B_r)\leq C_3\sigma_n M\,. \end{equation}
Set now $$ i_n:=\min_{\overline B_r}\textup{dist}(\cdot, \partial E_n) $$ and note that if $i_n>0$, then either $B_r\subset E_n$ or $B_r\subset E_n^c$. In the first case, we have $(E_n\Delta \overline F_n)\cap B_r=B_r\setminus \overline F_n$ and, in turn, $$
|B_r\setminus \overline F_n|\geq |B_{r/2}\setminus \overline F_n|=|B_{r/2}|-| \overline F_n\cap B_{r/2}|>\Big(\frac{\omega_N}{2^N}-\varepsilon\Big) r^N\geq \frac{\omega_N}{4^N}r^N\,, $$ by choosing $\varepsilon$ smaller if needed. In turn, recalling \eqref{trbounds}, we may estimate $$ M\geq \frac1h\mathcal{D}(\overline F_n, E_n)\geq \frac1h \int_{B_r\setminus \overline F_n}\textup{dist}(x, \partial E_n)\, dx\geq \frac{i_n}h\frac{\omega_N}{4^N}r^N\,, $$ from which we easily deduce $$ \textup{dist}(\cdot, \partial E_n)\leq 4^{N}r^{-N}M \omega_N^{-1}h+2r \quad\text{on }B_r\,. $$ Arguing in a similar way also in the case $B_r\subset E_n^c$, we can finally conclude the existence
of a positive constant $C_4=C_4(r)$ (depending also on $N$ and $h$) such that, in all cases, and for $n$ large enough we have
$\textup{dist}(\cdot, \partial E_n)\leq h C_4$ on $B_r\,$ and thus
\begin{equation}\label{Vn1000}
\Big|\frac1h\mathcal{D}(\widetilde F_n, E_n)-\frac1h\mathcal{D}(\overline F_n, E_n)\Big|\leq C_4 |\widetilde F_n\Delta \overline F_n|\leq C_4 C_3\sigma_n M\,, \end{equation} where in last inequality we have used \eqref{difsim}.
Combining \eqref{EFper}, \eqref{EFvol}, and \eqref{Vn1000}, we conclude that for $n$ sufficiently large
\begin{align*}
P(\widetilde F_n)+\frac1h\mathcal{D}(\widetilde F_n, E_n)\leq & P(\overline F_n)+\frac1h\mathcal{D}(\overline F_n, E_n)+n\big||\overline F_n|-m\big|
\\ &+
\sigma_n\bigl[(2^NN+ C_4C_3)M- n r^NC_1\bigr]
\\
<& P(\overline F_n)+\frac1h\mathcal{D}(\overline F_n, E_n)+n\big||\overline F_n|-m\big| \,, \end{align*}
a contradiction to the minimality of $\overline F_n$, since $|\widetilde F_n|=m$.
\end{proof}
As a consequence of the previous proposition, together with some standard arguments from the regularity theory of almost minimal sets, we have the following:
\begin{proposition}[Regularity properties of minimizers]\label{lm:density}
Let $E$ be a bounded set with $|E|=m$ and $P(E)\leq M$ for some $m,\, M>0$.
Then, any solution $F\subset \mathbb{R}^N$ to \eqref{varprob} satisfies the following regularity properties:
\begin{itemize}
\item[i)] There exist $c_0=c_0(N)>0$ and a radius $r_0=r_0(m, M, h, N)>0$ such that
for every $x\in \partial^{*} F$ and $r\in (0, r_0]$ we have
\begin{equation}\label{eq:density}
|B_r(x)\cap F|\geq c_0r^N\qquad\text{and}\qquad |B_r(x)\setminus F|\geq c_0r^N\,.
\end{equation}
In particular, $F$ admits an open representative whose topological boundary coincides with the closure of its reduced boundary, i.e., $\partial F = \overline{ \partial^* F}$. From now on we will always assume that $F$ coincides with its open representative.
\item [ii)]
There exists
$c_1= c_1(m,M,h,N)>0$ such that
\begin{equation}\label{neigh}
\sup_{E\Delta F} \textup{dist}(\cdot, \partial E)\leq c_1 \, .
\end{equation}
\item[iii)] There exists $\overline\Lambda=\overline\Lambda(m, M, h, N)>0$ such that $F$ is a
$\overline\Lambda$-minimizer of the perimeter, that is,
\begin{equation}\label{almost}
P(F)\leq P(F')+\overline\Lambda |F\Delta F'|
\end{equation}
for all measurable set $F'\subset \mathbb{R}^N$ such that diam$(F\Delta F') \le 1$.
\item[iv)] The following Euler-Lagrange condition holds: There exists $\lambda\in \mathbb{R}$ such that for all $X\in C^{1}_c(\mathbb{R}^N; \mathbb{R}^N)$ \begin{equation}\label{micume0} \int_{\partial^* F} \frac{d_E(x)}h X\cdot\nu_F\, d\mathcal{H}^{N-1} +\int_{\partial^* F}\mathrm{div}_\tau X\, d\mathcal{H}^{N-1}=\lambda\int_{\partial^* F}X\cdot\nu_F\, d\mathcal{H}^{N-1}\,. \end{equation} \item[v)] There exists a closed set $\Sigma$ whose Hausdorff dimension is less than or equal to $N-8$, such that $\partial^* F= \partial F\setminus \Sigma$ is an $(N-1)$-submanifold of class $C^{2,\alpha}$ for all $\alpha\in (0,1)$ with
$$
|H_{\partial F} (x)| \le \overline \Lambda \quad \text{ for all } x\in \partial F\setminus \Sigma \, .
$$
\item [vi)] The set $F$ is bounded and more precisely there exist $k_0=k_0(m, M, h, N)\in \mathbb{N}$ and $d_0=d_0(m, M, h, N)>0$ such that $F$ is made up of at most $k_0$ connected components,
each one having diameter bounded from above by $d_0$ (a bound on the diameter from below follows from \textup{i)}). \end{itemize}
\end{proposition}
\begin{proof} By Proposition~\ref{prop:penal}, there exists $\Lambda_0=\Lambda_0(m, M, h, N)$ such that $F$ is a solution to \begin{equation}\label{eq:penalbis}
\left\{P(F)\ +\frac1h\mathcal{D}(F, E)+\Lambda_0\big||F|-m\big|:\, F\subset\mathbb{R}^N\text{ measurable} \right\}\,. \end{equation} The density estimates and \eqref{neigh} follow arguing as in \cite[ Lemma~4.4 and Proposition~3.2]{MSS}, respectively. Now, \eqref{almost} easily follows from the fact that $F$ solves \eqref{eq:penalbis}, taking into account \eqref{neigh}.
Equation \eqref{micume0} can be derived by a standard first variation argument (see for instance \cite{MaggiBook}). In view of (iii) and the classical regularity theory for almost minimizers of the perimeter (see for instance \cite{MaggiBook} and the references therein), we deduce that there exists a closed set $\Sigma$ whose Hausdorff dimension is less than or equal to $N-8$, such that $\partial^* F= \partial F\setminus \Sigma$ is an $(N-1)$-submanifold of class $C^{1,\alpha}$ for all $\alpha\in (0,1)$. The $C^{2,\alpha}$ regularity stated in (v) follows now from the additional elliptic regularity implied by \eqref{micume0}, taking into account that $d_E$ is Lipschitz continuous.
Item (vi) follows rather easily from the density estimates. Indeed, let $r_0=r_0(m, M, h, N)>0$ be the radius given
in i). By an application of Vitali's Covering Lemma, we may find a subset $C\subset F$ such that the closed balls of the family $\{\overline{B_{r_0}(x)}\}_{x\in C}$ are pairwise disjoint and
$$
F\subset \bigcup_{x\in C} \overline{B_{5r_0}(x)}\,.
$$
Since by \eqref{eq:density} we have $|B_{r_0}(x)\cap F|\geq c_0r_0^N$ for every $x\in C$, it follows that $\# C\leq \frac{|F|}{c_0r_0^N}=
\frac{m}{c_0r_0^N}$. In turn, since for each connected component $\hat F$ of $F$ we have $\hat F\subset \cup_{x\in C} \overline{B_{5r_0}(x)}$, we infer that
$$\textrm{diam\,}\hat F\leq 10 r_0 \# C\leq \frac{10 m}{c_0r_0^{N-1}}=:d_0(m, M, h, N)\,.$$
It remains to bound the number of connected components. To this aim, it is enough to show that the measure of any connected component $\hat F$ is bounded below by a positive constant depending only on $m$, $M$, $h$, and $N$.
Set $F':=F\setminus \hat F$. Then, using \eqref{almost} and the Isoperimetric Inequality, we deduce
\begin{equation*}
\overline\Lambda \big|\hat F\big|= \overline\Lambda \big| F\Delta F'\big|+ P(F')-(P(F)-P(\hat F))\geq P(\hat F)\geq N\omega_N^{\frac{1}{N}}\big|\hat F\big|^{\frac{N-1}{N}}\,,
\end{equation*} and thus
$$
\big|\hat F\big|\geq\Big(\frac{N}{\overline\Lambda}\Big)^N\omega_N\,.
$$
This concludes the proof of the proposition. \end{proof}
\begin{remark} Note that \eqref{micume0} and the regularity properties given by Proposition~\ref{lm:density} imply that \begin{equation}\label{micume}
\frac{d_E(x)}h+H_{\partial F}(x)=\lambda \qquad \text{ for every } x\in\partial^* F. \end{equation} \end{remark}
\section{The discrete volume preserving flow}
\subsection{Construction of the discrete flow}
For any bounded set $E\neq \emptyset$ with finite perimeter we let $T_h E$ denote a solution of \eqref{varprob}, with $m=|E|$.
It is convenient to fix a precise representative for $T_h E$; as done for the set $E$, we assume that $T_h E$ coincides with its Lebesgue representative. Now, we construct by induction the discrete-in-time evolution $\{E_h^n\}_{n\in N}$. Let $E_h^1:=T_h E$ be a solution (arbitrarily chosen) to problem \eqref{varprob}; assuming that $E_h^k$ is defined for $k\in \{1,\ldots, n-1\}$, we let $E_h^{n}$ be a solution (arbitrarily chosen) to problem \eqref{varprob} with $E$ replaced by $E_h^{n-1} $.
\subsection{Stationary sets for the discrete flow} In this section we characterize the stationary sets $E$ for the volume preserving discrete flow, i.e., such that the constant sequence $E_h^n\equiv E$ is a discrete volume preserving flow.
\begin{proposition}[Fixed points of the discrete scheme]\label{fpds}
Given $m, \, M, \, h>0$, there exists $s_0= s_0(m,M,h,N)>0$ such that every
stationary bounded set $E$ for the volume preserving discrete flow with time step $h$, with $|E|=m$ and $P(E)\leq M$
is made up by the union of $k$ disjoint balls with mutual distances larger that $s_0$
and equal volume $\frac mk$, for some $k \le k_0$, with $k_0$ as in item \textup{vi)} of Proposition \ref{lm:density}.
Viceversa, if $E$ is given by the union of finitely many disjoint balls with positive mutual distances and equal volumes, then there exists $h^*>0$ such that, for all $h\le h^*$, the volume preserving flow $\{E_h^n\}$ starting from $E$ is unique and given by the constant sequence $E_h^n= E$ for all $n\in \mathbb{N}$. \end{proposition} \begin{proof} By \eqref{micume0} any stationary set $E$ satisfies $$ \int_{\partial^*E}\mathrm{div}_\tau X\, d\mathcal{H}^{N-1}=\lambda\int_{\partial^*E}X\cdot\nu_E\, d\mathcal{H}^{N-1} $$
for all $X\in C^{1}_c(\mathbb{R}^N; \mathbb{R}^N)$ and for some $\lambda\in \mathbb{R}$.
We may then apply \cite[Theorem~1]{DM} to conclude that $E$ is given (up to a negligible set) by a finite union of disjoint (open) balls of equal volume. Moreover, by (vi) of Proposition \ref{lm:density}
the number of such balls is bounded by $k_0$, and therefore, in view of \eqref{eq:density}, their radius is bounded from below by a constant $R$ depending only on $m,M,h,N$.
Now, if two balls of radius larger than or equal to $R$ are at a distance $s$ small enough, then $E$ cannot satisfy the second density estimate in \eqref{eq:density} (for example, \eqref{eq:density} cannot hold for $x\in\partial E$ of minimal distance between the two balls and $r>0$ depending on $R$ and the constants $r_0$, $c_0$ in Proposition \ref{lm:density}-i)). Therefore, \eqref{eq:density} is violated, whenever $s \le s_0$ for a suitable $s_0$ depending only on $m,M,h,N$,
thus establishing the first part of the statement.
Assume now that $E$ is union of finitely many disjoint balls with equal radius $R$ and positive mutual distances. We want to show that, for $h$ small enough, $E$ is the unique volume costrained global minimizer
of the functional
$$J_h(F):= P(F)\ +\frac1h\mathcal{D}(F, E) = P(F) + \frac1h\int_Fd_E(x)\, dx- \frac1h\int_E d_E(x).
$$
We start by showing that for $h$ small enough the second variation of $J_h$ is positive definite with respect to volume preserving variations.
To this purpose, let $X\in C^1_c(\mathbb{R}^N;\mathbb{R}^N)$ be a divergence free vector field and let $\Phi(t,\cdot)$ be the associated flow satisfying $\frac{\partial \Phi}{\partial t} = X(\Phi)$ with initial condition $\Phi(0,x)= x$ for all $x\in\mathbb{R}^N$. Then $|\Phi(t,E)|=|E|= m$ for all $t$, and by standard computations (see for instance \cite{AFM, Cr}) we have
\begin{align*}
\frac{\partial^2 }{\partial t^2} J_h(\Phi(t,E) )
&= \int_{\partial E} |\nabla (X \cdot \nu_E)|^2 - \frac{N-1}{R^2} (X \cdot \nu_E)^2 \, d\mathcal{H}^{N-1}
\\ \nonumber & \quad + \frac 1h \int_{\partial E} \partial_{\nu_E} d_E (X\cdot \nu_E)^2 \, d\mathcal{H}^{N-1} \\ \nonumber
&= \int_{\partial E} |\nabla (X \cdot \nu_E)|^2 + \Big( \frac 1h - \frac{N-1}{R^2} \Big) (X \cdot \nu_E)^2 \, d\mathcal{H}^{N-1} \\ \nonumber &= : \partial^2 J_h(E)[X\cdot\nu_E],
\end{align*}
where in second equality we have used the fact that $\partial_{\nu_E} d_E \equiv 1$ on $\partial E$. From the above expression it is clear that $\partial^2 J_h(E)$ is positive definite on $H^1(\partial E)$, provided that
$h < \frac{R^2}{N-1}$.
Fix $h_0 < \frac{R^2}{N-1}$. Arguing as in \cite{AFM} there exists $\varepsilon>0$ such that $J_{h_0} (E) < J_{h_0} (F)$ for all measurable $F$ such that $|F|=|E|$ and $|E\Delta F|\le \varepsilon$. Now notice that for all $0<h<h_0$ we have \begin{equation}\label{lomi}
J_h(E) = J_{h_0}(E) < J_{h_0} (F) \le J_{h} (F) \qquad \text{ for all $|F|=|E|$ and $0< |E\Delta F|\le \varepsilon$}, \end{equation} i.e., $E$ is an isolated local minimizer for $J_h$ in $L^1$, with minimality neighborhood uniform with respect to $h\le h_0$.
Now, given any sequence $\{h_n\}$ going to zero, let $F_n$ be a volume constrained minimizer of $J_{h_n}$; it is easy to see that $|E\Delta F_n|\to 0$ as $n\to +\infty$, and therefore, for $n$ large enough, $|E\Delta F_n| \le \varepsilon$, so that by \eqref{lomi} $F_n=E$.
\end{proof} \begin{remark} Let us also observe that for $N\leq 7$ we have full regularity of $\partial E$ and thus, in particular, the connected components of $E$ have positive mutual distances and the conclusion of the first part of Proposition \ref{fpds} would also follow by applying the classical Alexandrov's Theorem instead of the more refined results of \cite{DM}. \end{remark}
\subsection{Long-time behaviour}
In the following, we denote by $P_\infty$ the limit of the monotone non-increasing sequence $\{P(E_h^{n})\}_{n\in\mathbb{N}}$: \begin{equation}\label{e:Pinfinito}
P_\infty = \lim_{n\to \infty}P(E_h^{n}). \end{equation}
The following is the main result of the paper on the long time behavior of the discrete-in-time nonlocal mean curvature flow.
\begin{theorem}\label{mainthm}
Let $E$ be a bounded set of finite perimeter with $|E|=m$ and let $h>0$.
Consider any discrete volume constrained mean curvature flow $\{E^{n}_h\}_{n\in\mathbb{N}}$ starting from $E$. Then, setting $L:= N^{-N} \omega_N m^{1-N} P_\infty^{N} \in \mathbb{N}$, where $P_\infty$ is given in \eqref{e:Pinfinito}, there exist $L$ distinct balls $B^1, \ldots, B^L$ with the same radius and at positive distance to each other, such that the sets $E^{n}_h$ converge to $E_\infty:= \bigcup_{i=1}^L B^i$ in $C^k$ for every $k\in\mathbb{N}$. Moreover, the convergence is exponentially fast. \end{theorem}
The following is an immediate corollary.
\begin{corollary}\label{cor:menodidue}
If $P(E)< 2 P(\bar B)$, where $\bar B$ is a ball with volume ${\frac{m}{2}}$, or if $E$ is union of two tangent balls, than $E_\infty$ is a ball. \end{corollary}
\begin{proof} If $P(E)< 2 P(\bar B)$, than by definition $P_\infty<2 P(\bar B)$ and $L<2$. On the other hand, two tangent balls are not stationary for the discrete flow, as shown in Proposition \ref{fpds}. This implies that $P(E_h^n)<P(E)= 2P(\bar B)$, so that one can argue as before. \end{proof}
The rest of this section is devoted to the proof of Theorem \ref{mainthm}.
We start with the following lemma.
\begin{lemma}\label{uniconvd} Let $\{E_h^n\}_{n\in\mathbb{N}}$ be a volume preserving discrete flow starting from $E$ and let $E_h^{k_n}$ be a subsequence such that $\chi_{E_h^{k_n} -\tau_n}\to \chi_F$ in $L^1(\mathbb{R}^N)$ for some set $F$ and a suitable sequence $(\tau_n)_{n\in\mathbb{N}}\subset \mathbb{R}^N$. Then $d_{E_h^{k_n-1} } (\cdot + \tau_n) \to d_F$ locally uniformly in $\mathbb{R}^N$. \end{lemma}
\begin{proof}
We start by observing that for very $n$, by the minimality of $E_h^n$, using $E_h^{n-1}$ as a competitor, we have \begin{equation}\label{triviale}
\frac 1h \mathcal{D}(E_h^n,E_{h}^{n-1}) \le P(E_h^{n-1}) - P(E_h^n). \end{equation} Therefore, summing over $n$ we get $$ \sum_{n=1}^{\infty} \frac 1h \mathcal{D}(E_h^n,E_{h}^{n-1}) \le P(E)\,. $$ In particular, \begin{equation}\label{dissto0} \mathcal{D}(E_h^n,E_{h}^{n-1}) \to 0 \qquad\text{as }n\to\infty\,. \end{equation} Passing to a further (not relabelled) subsequence we may assume that there exists $G$ such that $E_{h}^{k_n-1}-\tau_n\to G$ in $L^1_{loc}(\mathbb{R}^N)$. In fact, by the density estimates \eqref{eq:density} and standard arguments the convergence holds in the Kuratowski sense together with the Kuratowski convergence of their boundaries. Thus, $$ \textup{dist}(\cdot, \partial E_{h}^{k_n-1}-\tau_n)\to \textup{dist}(\cdot, \partial G) \qquad\text{locally uniformly in }\mathbb{R}^N\,. $$ Combining the latter information with \eqref{dissto0}, for every $R>0$ we easily get $$ 0=\lim_{n\to\infty}\int_{(E_h^{k_n}\Delta E_{h}^{k_n-1}-\tau_n)\cap B_R}\textup{dist}(x, \partial E_{h}^{k_n-1}-\tau_n)\, d\mathcal{H}^{N-1}=\int_{(F\Delta G)\cap B_R}\textup{dist}(x, \partial G)\, d\mathcal{H}^{N-1}. $$
The last equality yields $F\Delta G\subset \partial G$. As $G$ still satisfies the density estimates we have $|\partial G|=0$ and the conclusion follows.
\end{proof}
In the next proposition we start by showing that for any discrete volume preserving flow the evolving sets are eventually made up of a constant number $L$ of connected components, each of which converging up to translations to the same ball of volume $m/L$. After establishing such a result, it will remain to show that the magnitude of such translations decays to zero exponentially fast. \begin{proposition}[Long-time behavior up to translations]\label{prop:uptrans} Let $\{E_h^n\}_{n\in\mathbb{N}}$ be a discrete flat flow with time step $h>0$ and prescribed volume $m>0$. Let $P_\infty$ and $L$ be as in the statement of Theorem \ref{mainthm}. Then, for $n$ sufficiently large, $E_h^n$ has $L$ distinct connected components $E_{h,1}^n, \ldots, E_{h,L}^n$, such that $\textup{dist}(E_{h,i}^n, E_{h,j}^n)\geq \frac{s_0}{2}$ for $i\neq j$ (with $s_0$ the constant of Proposition \ref{fpds}), and $E_{h,i}^n-\mathrm{bar} (E_{h,i}^n)$ converge to the ball centered at the origin with volume $\frac{m}{L}$ in $C^k$ for every $k\in \mathbb{N}$. \end{proposition}
\begin{proof} It sufficies to prove that, given any subsequence $\{E_{h}^{k_n}\}_{n\in\mathbb{N}}$, there existe a sub-subsequence (not relabelled) satisfying the conclusions of the proposition.
To this purpose, let $\{E_{h}^{k_n}\}_{n\in\mathbb{N}}$ be a given subsequence. By Propositon \ref{lm:density}-vi), each set $E_h^{k_n}$ is made up of $l_n\leq k_0$ connected components with uniformly bounded diameter less or equal $d_0$. Therefore, there exist $l_n$ balls $\{B_{d_0}(\xi_i^n)\}_{i=1, \ldots, l_n}$ (not necessarily disjoint one from the other), each containing a different component of $E_{h}^{k_n}$ and such that $E_h^{n} \subset \cup_{i=1}^{l_n} B_{d_0} (\xi_i^n)$. Up to passing to a subsequence (not relabelled), we can assume that $l_n= \tilde l$, and for all $1\le i<j\le \tilde l$ the following limits exist $$
\lim_n |\xi_i^n - \xi_j^n| =: d_{i,j}. $$ Now we define the following equivalence classes: $i\equiv j$ if and only if $d_{i,j} < + \infty$. Denote by $l$ the number of such equivalent classes, let $j(i)$ be a representative for each class $i\in \{1, \ldots, l\}$, and set $\sigma_i^n:=\xi_{j(i)}$ for $i=1,\ldots, l$. We have constructed a subsequence $E_h^{k_n}$ satisfying
$E_h^{k_n} \subset \cup_{i=1}^l B_R(\sigma_i^n)$, where $R= d_0 + \max\{d_{i,j} : d_{i,j}<\infty\}+1$, and for all $i\neq j$ there holds $|\sigma_i^n-\sigma_j^n|\to +\infty$ as $n\to +\infty$.
We remark that, while $E_h^{k_n -1}$ are in general not elements of the subsequence $\{E_h^{k_n}\}_{n\in\mathbb{N}}$, they will still play a role in our arguments, since they will be involved in exploiting the minimality properties of $E_h^{k_n}$.
Now, fix $1\le i \le l$, and set $$
F^n_i:= (E_h^{k_n} - \sigma_i^n), \qquad \tilde F^n_i:= (E_h^{k_n} - \sigma_i^n)\cap B_R, \qquad m^n_i:= |\tilde F^n_i|. $$ Up to a subsequence, we have $m^n_i\to m_i$ for some $m_i>0$. Moreover, by Lemma \ref{uniconvd} and the compactness of sets of equi-bounded perimeters, there exist measurable sets $\tilde F_i\subset\subset B_R$ such that (again up to a subsequence) \begin{equation}\label{locconv} \tilde F^n_i \to \tilde F_i \text{ in } L_{}^1, \qquad d_{E_h^{k_n - 1}}(\cdot + \sigma^n_i) \to d_{\tilde F_i}(\cdot) \text{ locally uniformly}. \end{equation} We shall show that $\tilde F_i$ is stationary for the discrete volume preserving flow. To this aim, let
$\tilde G_i$ be any bounded set with $|\tilde G_i|=m_i$. We define the homotetically rescaled sets $\tilde G^n_i :=\left(\frac{m_i^n}{m_i}\right)^{\frac1N} \tilde G_i$ such that $|\tilde G_i^n|=m_i^n$. Note that $\tilde G^n_i \to \tilde G_i$ in $L^1$ and $P(\tilde G^n_i) \to P(\tilde G_i)$ as $n\to +\infty$.
We set now $G_i^n:= F^n_i \cup \tilde G^n_i \setminus \tilde F^n_i$: notice that, since $\tilde G_i^n$ is bounded and the connected components of $F_i^n\setminus \tilde F_i^n$ diverge, it follows that, for sufficiently large $n$, $G_i^n$ is made up by the same connected componets of $F_i^n$ except $\tilde F^n_i$ which is replaced by $\tilde G^n_i$; in particular, $|F_i^n| = |G_i^n|$. By the minimality of $E_h^{k_n}$ we have \begin{equation*} P(F^n_i)\ +\ \frac{1}{h}\int_{F^n_i} d_{E_h^{k_n - 1}}(x + \sigma^n_i)\,dx \le P(G^n_i)\ +\ \frac{1}{h}\int_{G^n_i} d_{E_h^{k_n - 1}}(x + \sigma^n_i)\,dx. \end{equation*} Since
the connected components
of $F_i^n\setminus \tilde F_i^n$ diverge
and the two sets $F_i^n$ and $G_i^n$ differ only for the components $\tilde F_i^n$ and $\tilde G_i^n$, by addiditivity the previous inequality is equivalent to \begin{equation} P(\tilde F^n_i)\ +\ \frac{1}{h}\int_{\tilde F^n_i} d_{E_h^{k_n - 1}}(x + \sigma^n_i)\,dx \le P(\tilde G^n_i)\ +\ \frac{1}{h}\int_{\tilde G^n_i} d_{E_h^{k_n - 1}}(x + \sigma^n_i)\,dx. \end{equation} Passing to the limit as $n\to +\infty$, using \eqref{locconv} and the lower semicontinuity of the perimeter, since the sets $\tilde F_i^n$ and $\tilde G_i^n$ are uniformly bounded, we deduce that \begin{equation} P(\tilde F_i)\ +\ \frac{1}{h}\int_{\tilde F_i} d_{\tilde F_i} (x)\,dx \le P(\tilde G_i)\ +\ \frac{1}{h}\int_{\tilde G_i} d_{\tilde F_i} (x) \,dx. \end{equation} This minimality property extends by density to all competitors $G_i$ with finite perimeter and the same volume $m_i$, so that we deduce that $\tilde F_i$ is a fixed point for the discrete scheme with prescribed volume $m_i$, and whence by Lemma \ref{fpds} $\tilde F_i$ is given by the union of disjoint balls with positive mutual distances and equal volume. Moreover, since $\tilde F_i$ are uniform $\overline{\Lambda}$-minimal by Proposition \ref{lm:density}, from the classical regularity theory (see \cite{MaggiBook}) we also deduce that $\tilde F_i^n$ converge to $\tilde F_i$ in $C^{1, \alpha}$ for every $\alpha\in (0,1)$. In particular, for $n$ large enough, $\tilde F_i^n$ has the same number of connected components of $\tilde F_i$.
Summarizing, we have shown that, for a subsequence (not relabelled) of $E_h^{k_n}$ and for $n$ large enough, $E_h^{k_n}$ is made up by a fixed number $K$ of connected components $E_{h,1}^{k_n},\ldots, E_{h,K}^{k_n}$, each converging to a ball (possibly with different radius, if the components belong to different equivalent classes according to the relation introduced above). Therefore, for $i\leq K$ we have $E_{h,i}^{k_n}- \mathrm{bar}({E_{h,i}^{k_n}} )\to B_{R_i}$, where $B_{R_i}$ is the ball centered at the origin with radius $R_i>0$.
It remains to show that all the radii $R_i$ are equal to $R$: from this and the $C^1$ convergence of the translated components to $B_R$, it follows that $K P(B_{R}) = P_\infty$, i.e., $K=L$. To this aim, we consider the Euler-Lagrange equation \eqref{micume0} for $E_{h}^n$: for every $X\in C_c^1(\mathbb{R}^n;\mathbb{R}^n)$, \[ \int_{\partial E_{h}^{k_n}} \frac{d_{E_{h}^{{k_{n}-1}}}(x)}h X\cdot\nu_{E_{h}^{k_n}}\, d\mathcal{H}^{N-1} +\int_{\partial E_{h}^{k_n}}\mathrm{div}_\tau X\, d\mathcal{H}^{N-1}=\lambda_n\int_{\partial E_{h}^{k_n}}X\cdot\nu_{E_{h}^{k_n}}\, d\mathcal{H}^{N-1}\,. \] By Proposition \ref{lm:density}-ii) and v), we deduce that \[
|\lambda_n|\leq h^{-1} \|d_{E_{h}^{{k_n}-1}}\|_{L^\infty(E_{h}^{{k_n}})} + \|H_{E_{h}^{{k_n}}}\|_{L^\infty(E_{h}^{{k_n}})}\leq h^{-1} c_1 + |\bar{\Lambda}|. \] Therefore, by passing to a further subsequence (not relabelled), we can assume that $\lambda_n\to \lambda$, for some $\lambda\in \mathbb{R}$. Arguing as before, we can localize the Euler-Lagrange equation to each single $F_i^n$: \[ \int_{\partial F_{i}^n} \frac{d_{E_{h}^{{k_n}-1}}(x+\sigma_i^n)}h X\cdot\nu_{F_{i}^n}\, d\mathcal{H}^{N-1} +\int_{\partial F_{i}^n}\mathrm{div}_\tau X\, d\mathcal{H}^{N-1}=\lambda_n\int_{\partial F_{i}^n}X\cdot\nu_{F_{i}^n}\, d\mathcal{H}^{N-1}\,. \] Passing into the limit as $n\to \infty$ and taking into account Lemma \ref{uniconvd}, we deduce that \[ \int_{\partial \tilde F_{i}}\mathrm{div}_\tau X\, d\mathcal{H}^{N-1}=\lambda\int_{\partial \tilde F_{i}}X\cdot\nu_{\tilde F_{i}}\, d\mathcal{H}^{N-1}\,. \] In particular, this shows that $R_i=\frac{N-1}{\lambda}$.
Finally, the $C^k$ convergence follows by a classical bootstrap method. The idea is to describe the boundaries $\partial F_n$ (up to changing the reference frame) locally as graphs of suitable functions $f_n: B'\to \mathbb{R}$, with $B'$ a ball of $\mathbb{R}^{N-1}$. Then we have $$
\mathrm{div}\left(\frac{\nabla f_n}{\sqrt{1+|\nabla f_n|^2}}\right)=H_n\,, $$ with $H_n$ and $\nabla f_n$ uniformly bounded in $C^{0,\alpha}$ for every $\alpha\in (0,1)$. Differentiating the equation in any direction $v$, we get $$ \textup{div}\big(\nabla A(\nabla f_n) \nabla (\partial_v f_n)\big) =\partial_v H_n\,, $$ where the coefficients $A(\nabla f_n) $ are uniformly elliptic and uniformly bounded in the $C^{0,\alpha}$ norm. We may therefore apply \cite[Theorem 5.18]{GM} to conclude that the functions $\partial_v f_n$ are uniformly bounded in the $C^{1,\alpha}$ norm. In order to bound the higher order norms we may now apply a standard bootstrap argument and iteratively use \cite[Theorem 5.18]{GM}. We leave the details to the reader. \end{proof}
We recall the notation introduced before \eqref{schifo}: $B^{(\mu)}$ stands for the ball of volume $\mu$ and $r(\mu)$ denotes its radius.
\begin{lemma}\label{disti0}
Let $\mu>0$ and $\eta>0$. There exists $\bar\delta>0$ with the following property: if $f_1,\, f_2 \in C^1(\partial B^{(\mu)})$ with $\|f_i\|_{C^1} \le \overline\delta$ and $|E_{f_i, \mu}|=\mu$ for $i=1,2$ we have \begin{align}
&r(\mu)^2 (1-\eta) \frac{\|f_1 - f_2\|_{2}^2}{2} \leq \mathcal{D}(E_{f_1, \mu},E_{f_2, \mu}) \leq r(\mu)^2 (1+\eta)\frac{\|f_1 - f_2\|_{2}^2}{2} ,\label{f1menof2} \\ &\frac{1-\eta}2\int_{\partial E_{f_1, \mu}} d^2_{E_{f_2, \mu}} \, d\mathcal H^{N-1}\leq \mathcal{D}(E_{f_1, \mu},E_{f_2, \mu}) \leq \frac{ 1+\eta}2\int_{\partial E_{f_1, \mu}} d^2_{E_{f_2, \mu}} \, d\mathcal H^{N-1}\,,\label{disdis} \\ &
|\mathrm{bar}( E_{f_1, \mu})-\mathrm{bar}( E_{f_2, \mu})|\leq C\sqrt{\mathcal{D}( E_{f_1, \mu}, E_{f_2, \mu})}\,, \label{bari2000} \end{align} where $C>0$ depends only on $N$ and $\mu$. \end{lemma}
\begin{proof}
We start by observing that for any $\eta'>0$, if $\bar\delta$ is sufficiently small, then for every $p_0\in \partial E_{f_2, \mu}$ we have \begin{equation}\label{e:gono} \partial E_{f_2, \mu} \cap B_{4\bar\delta}(p_0) \subset
G:=\left\{y\in \mathbb{R}^N : \left\vert(y-p_0)\cdot \frac{p_0}{|p_0|}\right\vert^2\leq \frac{{\eta'}^2}{1+{\eta'}^2} |y-p_0|^2\right\}. \end{equation}
We divide the rest of the proof into two steps.
\noindent{\bf Step 1.} If $\bar\delta$ is small enough, for every point $p=\lambda p_0\in B_{2\bar\delta}(p_0)$ ($\lambda>0$), we have that \[
\frac1{1+\eta'}{|p-p_0|}\leq \textup{dist}(p, \partial E_{f_2,\mu}) \leq |p-p_0|. \]
The second inequality is, indeed, obvious by definition, given $p_0\in \partial E_{f_2,\mu}$. Concerning the first one, we notice that $\textup{dist}(p, \partial E_{f_2,\mu})\leq |p-p_0|\leq 2\bar\delta$ implies that there exists a point $q\in \partial E_{f_2,\mu}\cap B_{2\bar\delta}(p)$ such that
$\textup{dist}(p, \partial E_{f_2,\mu})= |p-q|$. In particular, from \eqref{e:gono} we infer that $q\in G$ and hence we have \begin{align*}
\textup{dist}(p, \partial E_{f_2,\mu}) \geq \textup{dist}(p, G) = \frac1{\sqrt{1+{\eta'}^2}}|p-p_0|\geq \frac1{1+\eta'}{|p-p_0|}. \end{align*} In particular, if $p_0= (1+f_2(s))s\in \partial E_{f_2, \mu}$ with $s\in \partial B^{(\mu)}$ and \[
p_t:= p_0+t\,\frac{f_1(s)-f_2(s)}{|f_1(s)-f_2(s)|}\frac{s}{|s|} \quad\text{ for all }t\in \big[0,r(\mu)|f_1(s)-f_2(s)|\big], \] we deduce that \begin{equation}\label{e:distanze} \frac1{1+\eta'}\,t\leq \textup{dist}(p_t, \partial E_{f_2,\mu}) \leq t. \end{equation} Then, keeping this same notation and integrating in polar coordinates, we infer that \begin{align} \mathcal{D}(E_{f_1,\mu},E_{f_2,\mu}) &= \int_{E_{f_1,\mu}\Delta E_{f_2,\mu}}\mathrm{dist}(x, \partial E_{f_2,\mu})\, dx \notag\\
&= \int_{\partial B^{(\mu)}} ds \int_{0}^{r(\mu)|f_2(s)-f_1(s)|} \mathrm{dist}(p_t, \partial E_{f_2,\mu}) \left( \frac{|p_t|}{r(\mu)}\right)^{N-1} \, dt. \end{align}
Recalling that $\left\vert\frac{|p_t|}{r(\mu)} - 1\right\vert \leq \bar \delta$ and using \eqref{e:distanze}, we get \begin{equation}\label{split1} \begin{split} \mathcal{D}(E_{f_1,\mu},E_{f_2,\mu}) &\leq (1+\bar\delta)^{N-1}
\int_{\partial B^{(\mu)}} ds \int_{0}^{r(\mu)|f_2(s)-f_1(s)|} t \, dt\\
&= \frac{(1+\bar\delta)^{N-1}}2r(\mu)^2 \int_{\partial B^{(\mu)}} |f_1(s)-f_2(s)|^2 \,ds, \end{split} \end{equation} from which the second inequality in \eqref{f1menof2} follows by taking $\bar \delta$ smaller, if needed. Analogously, \begin{equation}\label{split2} \begin{split} \mathcal{D}(E_{f_1,\mu},E_{f_2,\mu}) &\geq \frac{(1-\bar\delta)^{N-1}}{1+\eta'}
\int_{\partial B^{(\mu)}} ds \int_{0}^{r(\mu)|f_2(s)-f_1(s)|} {t} \, dt\\
&= \frac{(1-\bar\delta)^{N-1}}{2(1+\eta')} r(\mu)^2 \int_{\partial B^{(\mu)}} |f_1(s)-f_2(s)|^2 \,ds\,, \end{split} \end{equation} from which the first inequality in \eqref{f1menof2} follows by taking $\eta'$ and $\bar\delta$ small enough.
\noindent{\bf Step 2.} The inequalities \eqref{disdis} and \eqref{bari2000} are now easy consequences. Indeed, by \eqref{e:distanze} we have that for every $x= (1+f_1(s))s\in \partial E_{f_1,\mu}$ \[
\frac{r(\mu)}{1+\eta'}|f_1(s) - f_2(s)|\leq d_{E_{f_2, \mu}}(x)\leq r(\mu)|f_1(s) - f_2(s)|. \] Therefore, \eqref{disdis} follows from \eqref{split1} and \eqref{split2}, by taking $\eta'$ and $\bar\delta$ smaller if needed, through a simple change of coordinates (recall that the Jacobian of the map $s\mapsto (1+f_1(s))s$ and its inverse are estimated from above by $1+C\bar\delta$ for a suitable dimensional constant $C$).
Finally, note that we can write $$ \mathrm{bar}( E_{f_i, \mu})= \frac{1}{(N+1)r(\mu)^{N}\omega_N} \int_{\partial B^{(\mu)}} (1+f_i(s))^{N+1}s\,d\mathcal{H}^{N-1}(s)\,. $$ Using the fact that $t\mapsto(1+t)^{N+1}$ is $2(N+1)$-Lipschitz for $t$ small, we may estimate
\begin{align*}
&|\mathrm{bar}( E_{f_1, \mu})-\mathrm{bar}( E_{f_2, \mu})| \\ &\leq \frac{1}{(N+1)r(\mu)^{N-1}\omega_N}\left\vert \int_{\partial B^{(\mu)}} \big((1+f_1(s))^{N+1}-(1+f_2(s))^{N+1}\big) d\mathcal{H}^{N-1}(s) \right\vert\\ &\leq\frac{2}{r(\mu)^{N-1}\omega_N}
\int_{\partial B^{(\mu)}} |f_1(s)-f_2(s)| d\mathcal{H}^{N-1}(s)\leq C
\|f_1(s)-f_2(s)\|_2\,, \end{align*} where $C>0$ depends only on $N$ and $\mu$. Thus \eqref{bari2000} follows from the previous inequality combined with \eqref{f1menof2}. \end{proof}
\begin{lemma}\label{disti} Let $h>0$, $\mu>0$. There exists $C=C(h, \mu)>0$ and $\overline\delta=\overline\delta(h, \mu)>0$ with the following property: For any pair of sets $E_{f_1, \mu}$, $E_{f_2, \mu}$ with $f_1 , \, f_2 \in C^2(\partial B^{(\mu)})$,
$\|f_i\|_{C^1} \le \overline\delta$,
and such that $ |E_{f_2, \mu}|=\mu$, $\mathrm{bar}(E_{f_2, \mu})=0$ and \begin{equation}\label{eul} H_{\partial E_{f_2, \mu}}+\frac{d_{E_{f_1, \mu}}}{h}=\lambda\qquad\text{on }\partial E_{f_2, \mu} \end{equation}
for some $\lambda\in \mathbb{R}$, we have \begin{equation}\label{e:crucialdiss} \mathcal{D}\big(B^{(\mu)}, E_{f_2, \mu}\big) \le C \, \mathcal{D}(E_{f_2, \mu} , E_{f_1, \mu})\,. \end{equation} \end{lemma}
\begin{proof}
By Theorem \ref{Aleq} (and Remark~\ref{rm:acdc}), by choosing $\overline\delta$ sufficiently small and using also \eqref{eul}, we have \begin{equation}\label{unoo}
\begin{split}
\|f_2\|^2_{L^2(\partial B^{(\mu )})} & \leq C(\mu)
\| H_{\partial E_{f_2, \mu}} - \overline H_{\partial E_{f_2, \mu}} \|^2_{L^2(\partial B^{(\mu)})}\leq C(\mu) \| H_{\partial E_{f_2, \mu}} - \lambda \|^2_{L^2(\partial B^{(\mu)})}\\
& \leq 2C(\mu) \| H_{\partial E_{f_2, \mu}} - \lambda \|^2_{L^2(\partial E_{f_2, \mu})}= \frac{2C(\mu)}{h^2}\int_{\partial E_{f_2, \mu}}d^2_{E_{f_1, \mu}}\, d\mathcal{H}^{N-1}\,,
\end{split}
\end{equation} where the third inequality follows by bounding the Jacobian of the change of variables by $2$ (which can be done provided $\overline\delta$ is small enough). Note now that by \eqref{f1menof2} (with $f_1$ replaced by $0$) and by \eqref{disdis} (and by taking $\overline\delta$ smaller if needed) we have $$
\mathcal{D}\big(B^{(\mu)}, E_{f_2, \mu}\big)\leq \|f_2\|^2_{L^2(\partial B^{(\mu )})} \qquad\text{and}\qquad \int_{\partial E_{f_2, \mu}}d^2_{E_{f_1, \mu}}\, d\mathcal{H}^{N-1}\leq 4 \mathcal{D}(E_{f_2, \mu} , E_{f_1, \mu})\,. $$ Combining the previous inequalities with \eqref{unoo}, the conclusion follows.
\end{proof}
\begin{remark}\label{rm:purdinonfinire} It is clear that the constants $C$ and $\overline\delta$ in Lemmas~\ref{disti0} and \ref{disti} are uniform with respect to $\mu$ varying on any compact subset of $(0,+\infty)$. \end{remark}
We are now ready to prove the main result of the paper. The main difficulty is in controlling the translations introduced in Proposition~\ref{prop:uptrans} and in proving the convergence of the barycenters. A crucial role in such an argument is played by the dissipation/dissipation inequality \eqref{e:crucialdiss}, which in turn relies on the quantitative Alexandrov type estimate established in Theorem~\ref{Aleq}.
\begin{proof}[Proof of Theorem~\ref{mainthm}] We split the proof into several steps.
\noindent{\bf Step 1.} (Exponential decay of dissipations) Recall that by Proposition~\ref{prop:uptrans} we have $$ P(E^n_h)\to LP\big(B^{\left(\frac{m}L\right)}\big)\,. $$ Thus, summing \eqref{triviale} from $n+1\in \mathbb{N}$ to $+\infty$, we get
\begin{equation}\label{eq1} \sum_{k=n+1}^{+\infty} \frac 1h \mathcal{D}(E_h^k,E_{h}^{k-1}) \le P(E_h^{n}) - LP\Big(B^{\left(\frac{m}L\right)}\Big). \end{equation} Recall that again by Proposition~\ref{prop:uptrans} for $n$ large enough each set $E_h^n$ is made up of $L$ connected components $E_{h, 1}^n, \dots, E_{h, L}^n$, \begin{equation}\label{rec1}
m^n_{i}:=|E_{h, i}^n|\to \frac{m}L \end{equation} and \begin{equation}\label{rec2} E_{h, i}^n-\xi_i^n\to B^{\left(\frac{m}L\right)}\qquad\text{in }C^k \end{equation} as $n\to\infty$, where we set $$ \xi_i^n:=\mathrm{bar}(E_{h, i}^n)\,. $$ With Lemma~\ref{uniconvd} in mind, we also get \begin{equation}\label{rec2.3} E_{h, i}^{n-1}-\xi_i^n\to B^{\left(\frac{m}L\right)}\qquad\text{in }C^k \end{equation} as $n\to\infty$. Combining \eqref{rec2} and \eqref{rec2.3}, we have that for any $k\in \mathbb{N}$ and for $n$ large enough there exist functions $f_{1,i}^n$, $f_{2,i}^{n}\in C^k(\partial B^{(m^n_i)})$ such that (with the notation introduced in \eqref{schifo}) \begin{equation}\label{rec2.5} E_{h, i}^n-\xi_i^n=E_{f_{2,i}^n, m_i^n},\, \, E_{h, i}^{n-1}-\xi_i^n=E_{f_{1,i}^n, m_i^n}
\quad\text{with }\|f_{1,i}^n\|_{C^k},
\|f_{2,i}^n\|_{C^k} \to 0\text{ as }n\to\infty\,. \end{equation} Moreover, again by Proposition~\ref{prop:uptrans} for $n$ large enough we have \begin{equation}\label{rec3} \textup{dist}\left(E_{h, i}^n, E_{h, j}^n\right)\geq\frac{s_0}2\,, \qquad\text{ for }i\neq j\,, \end{equation} with $s_0$ the constant of Proposition~\ref{fpds}. Consider now the the admissible competitor for $E_h^n$ given by $$ \mathfrak{B}_n:=\bigcup_{i=1}^L\left(\xi_{ i}^{n-1}+B^{(m^{n-1}_{i})}\right)\,, $$ and note that by \eqref{rec1} and \eqref{rec2} we also have $$ \textup{dist}\left(\xi_{ i}^{n-1}+B^{(m^{n-1}_{i})}, \xi_{ j}^{n-1}+B^{(m^{n-1}_{j})}\right)\geq\frac{s_0}4 $$ for $n$ large enough and $i\neq j$. The above inequality and \eqref{rec3} in turn yield that \begin{equation}\label{rec4} \begin{array}{lcl} \mathcal{D}(E_h^n,E_{h}^{n-1})&=&\displaystyle\sum_{i=1}^{L}\mathcal{D}(E_{h, i}^n,E_{h, i}^{n-1})\text{ and }\\ \mathcal{D}(\mathfrak{B}_n,E_{h}^{n-1})&=&\displaystyle \sum_{i=1}^{L}\mathcal{D}\left(B^{(m^{n-1}_i)},E_{h, i}^{n-1}-\xi_i^{n-1})\right)\,. \end{array} \end{equation} Testing the minimality of $E_{h, i}^{n}$ with $\mathfrak{B}_n$ and using the second identity in \eqref{rec4}, we have \begin{equation}\label{rec5} P(E_h^n) + \frac 1h \mathcal{D}(E_h^n,E_{h}^{n-1}) \le P(\mathfrak{B_n}) + \frac 1h \sum_{i=1}^{L}\mathcal{D}\left(B^{(m^{n-1}_i)},E_{h, i}^{n-1}-\xi_i^{n-1}\right)\,. \end{equation}
Recall now that by \eqref{rec2.5} and by Lemma~\ref{disti} (see also Remark~\ref{rm:purdinonfinire}) for $n$ large enough we have \begin{multline*} \mathcal{D}\left(B^{(m^{n-1}_i)},E_{h, i}^{n-1}-\xi_i^{n-1}\right)= \mathcal{D}\left(B^{(m^{n-1}_i)},E_{f_{2, m^{n-1}_i}}\right)\\ \leq C \mathcal{D}\left(E_{f_{2, m^{n-1}_i}},E_{f_{1, m^{n-1}_i}}\right)=C\mathcal{D}(E_{h, i}^{n-1}, E_{h, i}^{n-2})\,. \end{multline*} Thus, from \eqref{rec5} and \eqref{rec4} we deduce that \begin{equation}\label{rec6} P(E_h^n)-P(\mathfrak{B_n})\leq \frac Ch \sum_{i=1}^{L}\mathcal{D}(E_{h, i}^{n-1}, E_{h, i}^{n-2})=\frac Ch \mathcal{D}(E_{h}^{n-1}, E_{h}^{n-2})\,. \end{equation} Observe now that by concavity $$ \sum_{i=1}^L m_i^{\frac{N-1}{N}}\leq L \left(\frac mL\right)^{\frac{N-1}{N}}\quad \text{for all } m_1, \dots, m_L\geq 0 \text{ s.t. }\sum_{i=1}^Lm_i=m $$ and thus $$ P(\mathfrak{B_n})\leq LP\big(B^{\left(\frac{m}L\right)}\big)\,. $$ Therefore, from \eqref{rec6} and \eqref{eq1} we get \begin{align*} &\sum_{k=n-1}^{+\infty} \frac 1h \mathcal{D}(E_h^k,E_{h}^{k-1})\\ &= \sum_{k=n+1}^{+\infty} \frac 1h \mathcal{D}(E_h^k,E_{h}^{k-1})+
\frac 1h \mathcal{D}(E_h^{n-1},E_{h}^{n-2})+\frac 1h \mathcal{D}(E_h^{n},E_{h}^{n-1})\\
&\leq P(E_h^{n}) - LP\Big(B^{\left(\frac{m}L\right)}\Big)+\frac 1h \mathcal{D}(E_h^{n-1},E_{h}^{n-2})+\frac 1h \mathcal{D}(E_h^{n},E_{h}^{n-1})\\
&\leq \frac {C+1}h \mathcal{D}(E_{h}^{n-1}, E_{h}^{n-2})+\frac 1h \mathcal{D}(E_h^{n},E_{h}^{n-1})\leq \frac{C+1}{h}\Big(\mathcal{D}(E_{h}^{n-1}, E_{h}^{n-2})+ \mathcal{D}(E_h^{n},E_{h}^{n-1})\Big)\,. \end{align*}
We may now apply Lemma \ref{an1} (with $\ell=2$) below to conclude \begin{equation}\label{finalmente} \mathcal{D}(E_h^n,E_{h}^{n-1})\leq \left(1-\frac{1}{C+1}\right)^{\frac n2}\left(P(E) - LP(B^{(\frac{m}L)})\right)\,. \end{equation}
\noindent {\bf Step 2.} (Exponential convergence of the barycenters) By \eqref{rec2.5}, \eqref{bari2000} and by \eqref{finalmente}, setting \begin{equation}\label{b} b:= \left(1-\frac{1}{C+1}\right)^{\frac 14}\in (0,1)\,, \end{equation} we have for $n$ sufficiently large \begin{align*}
|\xi^{n}_i-\xi_i^{n-1}|&=\big|\mathrm{bar}(E_{f_{2,i}^n, m^i_n})-\mathrm{bar}(E_{f_{1,i}^n, m^i_n})\big|\\ &\leq C\sqrt{\mathcal{D}\big(E_{f_{2,i}^n, m^i_n}, E_{f_{1,i}^n, m^i_n}\big) }= C\sqrt{\mathcal{D}(E^{n}_{h,i}, E_{h,i}^{n-1}) }\\ &\leq C \left(P(E) - LP(B^{(\frac{m}L)})\right)^{\frac12}b^n\,. \end{align*} In turn, the above estimate implies that $\{\xi_i^n\}_n$ satisfies the Cauchy condition and thus there exist $\xi^\infty_i\in \mathbb{R}^N$, $i=1, \dots, L$, such that $\xi_i^n\to \xi_i^{\infty}$ exponentially fast as $n\to\infty$; precisely, $$
|\xi_i^n-\xi_i^{\infty}|\leq \sum_{k=n+1}^{\infty}|\xi_k^n-\xi_{k-1}^n|\leq C \left(P(E) - LP(B^{(\frac{m}L)})\right)^{\frac12}\frac{b^{n}}{1-b}\, $$ for $n$ large enough and for $i=1, \dots, L$. Recalling \eqref{rec2}, we may conclude that for all $k\in \mathbb{N}$ \begin{equation}\label{finalmente2} E^n_{h,i}\to \xi^\infty_i+B^{(\frac mL)}\quad\text{ in }C^k \quad\text{as }n\to\infty \text{ and for }i=1, \dots, L\,. \end{equation}
\noindent{\bf Step 3.} (Exponential convergence of the sets) By \eqref{finalmente2} we can parametrize the boudaries of the sets $E^n_{h,i}-\xi_i^\infty$ as radial graphs over the limiting ball $B^{(\frac mL)}$. Precisely, again with the notation \eqref{schifo}, there exist functions $g_i^n$ such that \begin{equation}\label{tuttob}
E^n_{h,i}-\xi_i^\infty=E_{g_i^n, \frac mL}\qquad\text{and}\qquad \|g_i^n\|_{C^k\big(\partial B^{(\frac mL)}\big)}\to 0\text{ as }n\to\infty\,. \end{equation} In turn, by Lemma~\ref{disti0} (see \eqref{f1menof2}), for $n$ large enough we have that
$\|g_i^n-g_i^{n-1}\|_{L^2(\partial B^{(\frac mL)} )}\leq 2\sqrt{\mathcal{D}(E^n_{h,i}, E^{n-1}_{h,i})}$ and, thus, recalling \eqref{finalmente} and arguing as in Step 2, we get \begin{equation}\label{expl2}
\|g_i^n\|_2\leq \sum_{k=n+1}^{\infty}\|g_i^k-g_i^{k-1}\|_2\leq 2\sum_{k=n+1}^{\infty}\sqrt{\mathcal{D}(E^{k}_{h,i}, E^{k-1}_{h,i})}\leq \left(P(E) - LP(B^{(\frac{m}L)})\right)^{\frac12}\frac{b^n}{1-b}\,, \end{equation} where $b$ is as in \eqref{b}. The above estimate yields the exponential decay of the $L^2$-norms of the radial graphs. We now recall the following well known interpolation inequality: for every $j\in \mathbb{N}$ there exists $C>0$ such that if $g$ sufficiently smooth on
$\partial B^{(\frac mL)}$, then \begin{equation}\label{interp}
\|D^k g\|_{L^2\big(\partial B^{(\frac mL)}\big)}\leq
C \|D^{2k} g\|^{\frac12}_{L^2\big(\partial B^{(\frac mL)}\big)}\|g\|^{\frac12}_{L^2\big(\partial B^{(\frac mL)}\big)}\,, \end{equation} where $D^k$ stands for the collection all $k$-th order (covariant) derivatives of $g$, see for instance \cite{Aubin}. Now, using the fact that from \eqref{tuttob} for every $k$ there exists $n_k\in \mathbb{N}$ such that $$
\sup_{n\geq n_k}\|D^{2k} g^n_i\|_{2}\leq 1\,, $$ we may apply \eqref{interp} to $g^{n}_i$ to deduce from \eqref{expl2} that also
$\|D^{k} g^n_i\|_{2}$ decays exponentially fast for all $k\in \mathbb{N}$. This in turn yields the exponential decay in $C^k$ for every $k$ and concludes the proof of the theorem. \end{proof}
\begin{lemma}\label{an1} Let $\{a_n\}_{n\in \mathbb{N}}$ be a sequence of non-negative numbers. Assume furthermore that there exists $c>1$, $\ell\in \mathbb{N}$ such that $\sum_{n=k}^{+\infty}a_n\leq c\,\sum_{j=k}^{k+\ell-1}a_j$ for every $k\in\mathbb{N}$. Then, $$ a_k\leq \Bigl(1-\frac1c\Bigr)^{\frac{k}{\ell}}S $$ for every $k\in \mathbb{N}$, where $S:=\sum_{n=1}^{+\infty}a_n$. \end{lemma} \begin{proof} We first consider the case $\ell=1$. Set $F(k):=\sum_{n=k}^{+\infty}a_n$ and note that by assumption $F(k)\leq c(F(k)-F(k+1))$ for every $k\in \mathbb{N}$. Hence, by iteration we have $$ a_{k+1}\leq F(k+1)\leq \Bigl(1-\frac1c\Bigr)F(k)\leq\dots\leq \Bigl(1-\frac1c\Bigr)^{k+1}F(0)= \Bigl(1-\frac1c\Bigr)^{k+1} S $$
for every $k\in \mathbb{N}$.
In the case $\ell\ge 2$ it is enough to set $b_k:=\sum_{j=1}^{\ell} a_{\ell(k-1)+j}$ and to observe that the assumption now reads $\sum_{n=k}^{+\infty}b_n\leq c b_k$ so that we may apply the previous case.\end{proof}
\end{document} | arXiv |
Maurer–Cartan form
In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer.
As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the Lie algebra associated to the Lie group G. The Lie algebra is identified with the tangent space of G at the identity, denoted TeG. The Maurer–Cartan form ω is thus a one-form defined globally on G which is a linear mapping of the tangent space TgG at each g ∈ G into TeG. It is given as the pushforward of a vector in TgG along the left-translation in the group:
$\omega (v)=(L_{g^{-1}})_{*}v,\quad v\in T_{g}G.$
Motivation and interpretation
See also: Lie group action
A Lie group acts on itself by multiplication under the mapping
$G\times G\ni (g,h)\mapsto gh\in G.$
A question of importance to Cartan and his contemporaries was how to identify a principal homogeneous space of G. That is, a manifold P identical to the group G, but without a fixed choice of unit element. This motivation came, in part, from Felix Klein's Erlangen programme where one was interested in a notion of symmetry on a space, where the symmetries of the space were transformations forming a Lie group. The geometries of interest were homogeneous spaces G/H, but usually without a fixed choice of origin corresponding to the coset eH.
A principal homogeneous space of G is a manifold P abstractly characterized by having a free and transitive action of G on P. The Maurer–Cartan form[1] gives an appropriate infinitesimal characterization of the principal homogeneous space. It is a one-form defined on P satisfying an integrability condition known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define the exponential map of the Lie algebra and in this way obtain, locally, a group action on P.
Construction
Intrinsic construction
Let g ≅ TeG be the tangent space of a Lie group G at the identity (its Lie algebra). G acts on itself by left translation
$L:G\times G\to G$
such that for a given g ∈ G we have
$L_{g}:G\to G\quad {\mbox{where}}\quad L_{g}(h)=gh,$
and this induces a map of the tangent bundle to itself: $(L_{g})_{*}:T_{h}G\to T_{gh}G.$ A left-invariant vector field is a section X of TG such that [2]
$(L_{g})_{*}X=X\quad \forall g\in G.$
The Maurer–Cartan form ω is a g-valued one-form on G defined on vectors v ∈ TgG by the formula
$\omega _{g}(v)=(L_{g^{-1}})_{*}v.$
Extrinsic construction
If G is embedded in GL(n) by a matrix valued mapping g =(gij), then one can write ω explicitly as
$\omega _{g}=g^{-1}\,dg.$
In this sense, the Maurer–Cartan form is always the left logarithmic derivative of the identity map of G.
Characterization as a connection
If we regard the Lie group G as a principal bundle over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique principal connection on the principal bundle G. Indeed, it is the unique g = TeG valued 1-form on G satisfying
1. $\omega _{e}=\mathrm {id} :T_{e}G\rightarrow {\mathfrak {g}},{\text{ and}}$
2. $\forall g\in G\quad \omega _{g}=\mathrm {Ad} (h)(R_{h}^{*}\omega _{e}),{\text{ where }}h=g^{-1},$
where Rh* is the pullback of forms along the right-translation in the group and Ad(h) is the adjoint action on the Lie algebra.
Properties
If X is a left-invariant vector field on G, then ω(X) is constant on G. Furthermore, if X and Y are both left-invariant, then
$\omega ([X,Y])=[\omega (X),\omega (Y)]$
where the bracket on the left-hand side is the Lie bracket of vector fields, and the bracket on the right-hand side is the bracket on the Lie algebra g. (This may be used as the definition of the bracket on g.) These facts may be used to establish an isomorphism of Lie algebras
${\mathfrak {g}}=T_{e}G\cong \{{\hbox{left-invariant vector fields on G}}\}.$
By the definition of the exterior derivative, if X and Y are arbitrary vector fields then
$d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]).$
Here ω(Y) is the g-valued function obtained by duality from pairing the one-form ω with the vector field Y, and X(ω(Y)) is the Lie derivative of this function along X. Similarly Y(ω(X)) is the Lie derivative along Y of the g-valued function ω(X).
In particular, if X and Y are left-invariant, then
$X(\omega (Y))=Y(\omega (X))=0,$
so
$d\omega (X,Y)+[\omega (X),\omega (Y)]=0$
but the left-invariant fields span the tangent space at any point (the push-forward of a basis in TeG under a diffeomorphism is still a basis), so the equation is true for any pair of vector fields X and Y. This is known as the Maurer–Cartan equation. It is often written as
$d\omega +{\frac {1}{2}}[\omega ,\omega ]=0.$
Here [ω, ω] denotes the bracket of Lie algebra-valued forms.
Maurer–Cartan frame
One can also view the Maurer–Cartan form as being constructed from a Maurer–Cartan frame. Let Ei be a basis of sections of TG consisting of left-invariant vector fields, and θj be the dual basis of sections of T*G such that θj(Ei) = δij, the Kronecker delta. Then Ei is a Maurer–Cartan frame, and θi is a Maurer–Cartan coframe.
Since Ei is left-invariant, applying the Maurer–Cartan form to it simply returns the value of Ei at the identity. Thus ω(Ei) = Ei(e) ∈ g. Thus, the Maurer–Cartan form can be written
$\omega =\sum _{i}E_{i}(e)\otimes \theta ^{i}.$
(1)
Suppose that the Lie brackets of the vector fields Ei are given by
$[E_{i},E_{j}]=\sum _{k}{c_{ij}}^{k}E_{k}.$
The quantities cijk are the structure constants of the Lie algebra (relative to the basis Ei). A simple calculation, using the definition of the exterior derivative d, yields
$d\theta ^{i}(E_{j},E_{k})=-\theta ^{i}([E_{j},E_{k}])=-\sum _{r}{c_{jk}}^{r}\theta ^{i}(E_{r})=-{c_{jk}}^{i}=-{\frac {1}{2}}({c_{jk}}^{i}-{c_{kj}}^{i}),$
so that by duality
$d\theta ^{i}=-{\frac {1}{2}}\sum _{jk}{c_{jk}}^{i}\theta ^{j}\wedge \theta ^{k}.$
(2)
This equation is also often called the Maurer–Cartan equation. To relate it to the previous definition, which only involved the Maurer–Cartan form ω, take the exterior derivative of (1):
$d\omega =\sum _{i}E_{i}(e)\otimes d\theta ^{i}\,=\,-{\frac {1}{2}}\sum _{ijk}{c_{jk}}^{i}E_{i}(e)\otimes \theta ^{j}\wedge \theta ^{k}.$
The frame components are given by
$d\omega (E_{j},E_{k})=-\sum _{i}{c_{jk}}^{i}E_{i}(e)=-[E_{j}(e),E_{k}(e)]=-[\omega (E_{j}),\omega (E_{k})],$
which establishes the equivalence of the two forms of the Maurer–Cartan equation.
On a homogeneous space
Maurer–Cartan forms play an important role in Cartan's method of moving frames. In this context, one may view the Maurer–Cartan form as a 1-form defined on the tautological principal bundle associated with a homogeneous space. If H is a closed subgroup of G, then G/H is a smooth manifold of dimension dim G − dim H. The quotient map G → G/H induces the structure of an H-principal bundle over G/H. The Maurer–Cartan form on the Lie group G yields a flat Cartan connection for this principal bundle. In particular, if H = {e}, then this Cartan connection is an ordinary connection form, and we have
$d\omega +\omega \wedge \omega =0$
which is the condition for the vanishing of the curvature.
In the method of moving frames, one sometimes considers a local section of the tautological bundle, say s : G/H → G. (If working on a submanifold of the homogeneous space, then s need only be a local section over the submanifold.) The pullback of the Maurer–Cartan form along s defines a non-degenerate g-valued 1-form θ = s*ω over the base. The Maurer–Cartan equation implies that
$d\theta +{\frac {1}{2}}[\theta ,\theta ]=0.$
Moreover, if sU and sV are a pair of local sections defined, respectively, over open sets U and V, then they are related by an element of H in each fibre of the bundle:
$h_{UV}(x)=s_{V}\circ s_{U}^{-1}(x),\quad x\in U\cap V.$
The differential of h gives a compatibility condition relating the two sections on the overlap region:
$\theta _{V}=\operatorname {Ad} (h_{UV}^{-1})\theta _{U}+(h_{UV})^{*}\omega _{H}$
where ωH is the Maurer–Cartan form on the group H.
A system of non-degenerate g-valued 1-forms θU defined on open sets in a manifold M, satisfying the Maurer–Cartan structural equations and the compatibility conditions endows the manifold M locally with the structure of the homogeneous space G/H. In other words, there is locally a diffeomorphism of M into the homogeneous space, such that θU is the pullback of the Maurer–Cartan form along some section of the tautological bundle. This is a consequence of the existence of primitives of the Darboux derivative.
Notes
1. Introduced by Cartan (1904).
2. Subtlety: $(L_{g})_{*}X$ gives a vector in $T_{gh}G{\text{ if }}X\in T_{h}G$
References
• Cartan, Élie (1904). "Sur la structure des groupes infinis de transformations" (PDF). Annales Scientifiques de l'École Normale Supérieure. 21: 153–206. doi:10.24033/asens.538.
• R. W. Sharpe (1996). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, Berlin. ISBN 0-387-94732-9.
• Shlomo Sternberg (1964). "Chapter V, Lie Groups. Section 2, Invariant forms and the Lie algebra.". Lectures on differential geometry. Prentice-Hall. LCCN 64-7993.
| Wikipedia |
How big would an amorphous blob have to be to toss part of itself into orbit?
First contact with an alien life form often goes badly. In the case of Biothanata, it always goes badly. The first glimpse of this alien blob is in the form of a falling star, a meteorite. After burning off an ablative layer of rock and juicy outsides, it crash-lands. Once cooled, a red liquid leaks out of what is left of the meteor, quickly consuming any and all bio-matter around it. As it digests the grass, leaves, bugs, and other creatures, it grows, amoeba-like, sending out tendrils, splitting and reforming, but always consuming. It also digests rocks, or at least breaks them down into bite-size pieces to use, though at a much slower pace.
Eventually, if nothing stops it (and nothing has, yet) it consumes all available life on the planet, barring some hardy life forms that are difficult to access. Once it grows large enough, the now enormous blob of red goo begins bunching itself together, then hurling chunks of itself high into the sky. After enough attempts, the giant blob manages to throw one (or more) smaller blobs into space, escaping Earth's gravity. Each blob is packed with rocks and dirt to use as course adjustment. Eventually, over the course of thousands of years, the majority of the space-blob sends itself out to another planet; all that's left is a (relative) handful of indigestible dust and a tiny dried-up blob.
How big would the blob have to be to toss a 10-foot cube of itself out of Earth's orbit? Assume the thrown piece can start larger and accelerate itself by shooting pieces of itself off behind it, form itself into a basic wing or flying disk to catch the wind, and generally behave somewhat intelligently; also assume the "main" blob can lift and hold itself to a height of roughly half its base (higher is possible, but will cause it to fall afterwards). Edit: also assume the blob can be very, staggeringly large, nearing "planet sized" itself - as big as it needs to get before it can hurl blobs into space.
Once a blob breaks free of the world's gravity, it then breaks free of the Sun's gravity by tossing various space debris behind it. Assuming it has all the time in the universe, and manages to accelerate itself as much as possible, how long would it take before it found another planet?
Bonus question: Assuming a starting size of roughly one cubic foot, unchecked growth, and a digestion rate roughly equal to the most aggressive digestion of a creature on Earth, how long would it take for Biothanata to consume the majority of land-based organic life on an Earth-like planet?
space physics aliens science-fiction
ArmanX
ArmanXArmanX
$\begingroup$ How is it attempting to throw chunks into orbit? I'd think burning volatile organics and directing the output through some kind of nozzle could do it. Or take a page from "From the Earth to the Moon" and simply form into a large cannon. $\endgroup$ – Michael Richardson Mar 24 '16 at 19:06
$\begingroup$ How much of the 10 foot cube needs to get to it's destination? How fast will the larger blob shoot it? How fast does the small blob shoot parts of itself off? How quickly should it get to another planet? (no time frame makes it slightly easier, but not much. $\endgroup$ – Lacklub Mar 24 '16 at 19:08
$\begingroup$ @Lacklub - as much as possible, but at least a cubic foot or so; as fast as possible; also as fast as possible; some time within the heat death of the universe. $\endgroup$ – ArmanX Mar 24 '16 at 19:11
$\begingroup$ @ArmanX You might have to be ok with parts of the blob moving at large multiples of the speed of sound. It's hard (but not impossible) to shoot something into space ballistically, but that might be your best bet. The smaller blob can then accelerate with the mass it has by shooting out very small parts at very (very) high speeds. Your target velocity (once you're out of the atmosphere of the earth) is upwards of 40 km/s. $\endgroup$ – Lacklub Mar 24 '16 at 19:48
$\begingroup$ @Lacklub - that's what I had in mind. The main blob would accelerate the smaller blob to escape velocity, or as near as possible, from as high as it can get; the smaller blob would shoot out at very, very high speed - something like "snap the whip", where the end of the "whip" is released into the air. $\endgroup$ – ArmanX Mar 24 '16 at 20:37
Not going to happen
Let's start by assuming the blob is, like most life on Earth, mostly water. We'll also say that it is about the same density as water - 1000 kg/m^3.
Figuring out how hard it would be for the blob to escape the Earth's gravity well will be tricky because we have to take into account things like wind resistance due to the atmosphere. So first we'll ignore the Earth and look at how hard it would be to escape the Sun's gravity well and leave the solar system.
From the Earth, the escape velocity for leaving the solar system is 42km/s. That's dang fast. For reference, the speed of sound in water is 1.48km/s. This is also a hard limit for how fast your blob could throw a chunk of itself - pressure energy can't realistically travel faster than that through water.
So imagine that somehow your blob can throw a chunk of itself at 1.48km/s, then that chunk can throw a chunk of itself at 1.48km/s, and so on until something gets to 42km/s. Simple math tells us the chunks-throwing-chunks needs to happen 29 times.
In order to propel 2/3 of itself forward at 1.48km/s, a chunk would have to propel the other 1/3 backwards at 2.96km/s. As I've already mentioned, that can't happen so the absolute best case scenario would be for the chunk to propel half of its mass forward at each stage.
Unfortunately for your blob, you've got to worry about exponential decay. Cutting itself in half 29 times doesn't leave it much to work with - you'll have $\frac{1}{2^{29}}$ as much left as you started with. So if you took the entire biomass of the Earth (around $4\times 10^{15}$kg), you could get $7.45\times 10^6$kg to escape velocity. That's enough for a 19 meter cube.
Now let's look at the energy densities involved. To keep things simple, consider a chunk as stationary and consider the kinetic energy of a chunk moving at 1.48km/s. This will give us an estimate of how much energy will be required to throw a chunk that fast. $K=\frac{1}{2}mv^2=1.095\times 10^{6}m$ joules, so for a mass to throw an equal mass with that much energy, it must be able to use 1.095 MJ/kg in a very short amount of time. However, that's almost within an order of magnitude of the total energy stored by carbohydrates. So basically the entire chunk has to consist of readily available energy storage and mechanism to propel itself forward.
Already this is very much stretching the bounds of plausibility, but this is the only way that it's going to work. If 2/3 of a chunk propelled 1/3 forward, only $\frac{1}{3^{29}}\approx 1.4\times 10^{-14}$ of the original would remain, so using the entire biomass of the Earth would get 58kg (about two cubic feet) of the cube out of the solar system.
Also, these cubes won't be roaring out of the solar system - by the time they left the solar system they'd be going around 800m/s. So they could potentially get to the next closest star after 50 trillion years. That's long after the destination star will have died.
Another way in which this gets worse for your blob is that 1.48km/s is actually sort of like the speed of light - it would actually require more and more energy to just get closer and closer to that limit. It's likely that getting to half of that, 740m/s, would take as much energy as what my simplification allowed to get to 1.48km/s. So it would require twice as many chunk-throwing-chunk steps, which squares the mass reduction - $\frac{1}{2^{57}}\approx 1.7\times 10^{-18}$ of the original mass could escape the solar system.
Oh, and remember how we completely ignored escaping the Earth's gravity well? Yeah, that problem wouldn't go away even if the blob consumed the entire Earth, rocks and all, because that doesn't somehow destroy the gravity well.
Rob WattsRob Watts
$\begingroup$ Good answer; I appreciate the math. It looks like my world-eater is going to need to acquire some jet fuel to make it back to space... $\endgroup$ – ArmanX Mar 25 '16 at 13:32
$\begingroup$ You don't need to expend all the energy to directly exist the solar system. Maybe ~12kms is enough. Some heliocentric orbits should exist where gravitational assists provide the rest of the acceleration. A solar sail (as proposed in other answers) may help in adjusting course. $\endgroup$ – Innovine Jul 11 '17 at 18:12
$\begingroup$ Having played High Frontier and merely looked at the board and understood how to calculate thrust and fuel consumption...leaving the solar system is Hard. And that's for freaking rockets. The easiest method involves no less than three refueling stops on the rocky bodies of the outer solar system. $\endgroup$ – Draco18s no longer trusts SE Jul 11 '17 at 20:11
$\begingroup$ The Voyager probes have left the solar system, with no refueling stops... Also, what kind of fuel is going to be found on rocky bodies? $\endgroup$ – Innovine Jul 11 '17 at 21:31
$\begingroup$ @Innovine according to this chart on Wikipedia, Voyager 2 had almost enough velocity to escape the solar system before it got any gravity assists. Also, the amount of planning required in order to intentionally get a single gravity assist (let alone multiple ones) is not something an amorphous blob could pull off. $\endgroup$ – Rob Watts Jul 11 '17 at 21:39
Meteorite impacts can splash parts of it into orbit.
This is a real thing. We have identified Martian meteorites which landed on earth, identified by isotope analysis. They were spalled off the surface of Mars by meteorite impacts and launched into orbit. We currently have identified 132 Mars rocks on Earth.
https://en.wikipedia.org/wiki/Martian_meteorite
This is a real and logical way for your blob to take the spacetrain. In fact, it is nearly unavoidable for any blob covered planet. The only factor in the way of this process is a thick atmosphere, which is simply overcome by a bigger hit. Once enough matter is flying about it will surely infect the entire solar system over time.
I do not know if an impact could push it interstellar. If the blob is intelligent enough, stage two could involve forming a thin film, and propelling itself as a solar sail.
Earthworm JimEarthworm Jim
$\begingroup$ Stage two doesn't even need intelligence, just an instinctive reaction to low gravity. $\endgroup$ – Joe Bloggs Dec 20 '16 at 11:41
$\begingroup$ Martian meteorites left Mars through an atmosphere around 1% as thick as Earths. The reverse process isn't very likely. $\endgroup$ – Innovine Jul 11 '17 at 18:07
Uhm... unless this blob is made of rocket fuel, it will not happen.
The reason for that is found in the so called Rocket Equation. One factor here is the "effective exhaust velocity". Without getting too technical — noting that this is actual "rocket science" — that velocity needs to be really high. And you cannot achieve that by "tossing stuff backwards". You need to set something ablaze so that you essentially have an ongoing explosion that you can direct backwards.
If you like you can try this question over at the Space Exploration SE and they can give you all the technical details but in short: it won't happen.
$\begingroup$ If tossing pieces from sea level doesn't work, I expect if the blob gets large enough (read: a significant percentage of the planet), it should be able to "reach" to the edge of space and toss pieces from there; how big is "big enough", as a percentage of the planet's size? $\endgroup$ – ArmanX Mar 24 '16 at 19:17
$\begingroup$ Still will not work, for the same reason as we do not have mountains that reach into space: the ground will crumble and give way. Remember that the Earth's crust is relatively thin and malleable when we are talking dimensions like that. It cannot support any kind of structure that reaches into space unless that structure is very light and extremely strong. $\endgroup$ – MichaelK Mar 24 '16 at 19:21
$\begingroup$ So essentially, the blob would have to actually be a significant portion of the planet, needing to not only absorb the crust, but somehow cool and consume the inner planet as well... hmm. Ok. Looks like my Eater of Planets may need a (jet) boost! $\endgroup$ – ArmanX Mar 24 '16 at 19:24
Instead of a blob, it could be a whispy structure that spreads out, and once parts of it are out of the atmosphere it acts as a solar sail.
I think Fred Hoyle's creature was something like that. Maybe David Gerrold used that too. I don't recall exactly.
JDługoszJDługosz
Others have pointed why it's impossible that the creature propelled itself as a rocked, but it could climb to orbit. If the creature could build a tree-like or reef-like structure tens or hundreds of thousands kilometres high, Earth rotation could give it enough velocity to stay in orbit. Once in orbit, the solar sail proposed by JDługosz could lead it to another planet or even another star.
Since the hard part of the process is building such structure, once built it could be producing solar-sailed offsprings in large amounts to colonise the whole galaxy.
Off course, the mechanical properties of the materials need to build the structure are far beyond anything known, but you know that evolution and natural selection are powerful forces even when faced with such hard problems.
PerePere
$\begingroup$ Seems very unlikely. It would need to climb at least 35,786 kilometers up, which is 3 times the earths diameter. And it'd need to prevent itself turning in to a spiral due to all the suborbital mass causing drag as it rotates. $\endgroup$ – Innovine Jul 11 '17 at 17:56
$\begingroup$ If it had some elastic properties, it might be able to bend itself over to one side, like a catapult, and then straighten out, using earths rotation and its own movement to accelerate the projectile to orbital speeds without needing to reach geosynchronous orbit altitude. Still, Im not sure what kind of energy you can get out of a 20,000km long bendy whippy blob arm, nor how blob structure holds itself up in the first place $\endgroup$ – Innovine Jul 11 '17 at 17:57
$\begingroup$ Answer to first comment: Yes, it's unlikely. However, I since all answers to the question are going to be unlike, our goal can just be to find interesting answers which make sense in spite of being unlikely. $\endgroup$ – Pere Jul 11 '17 at 18:01
$\begingroup$ @Innovine: To second comment: The catapult idea may be the basis for a new answer, although I find it very problematic. $\endgroup$ – Pere Jul 11 '17 at 18:03
$\begingroup$ It would be much easier for the blob to remain in orbit, and extend long tentacles down to the surface, eat up the food, and then retract the tentacles. $\endgroup$ – Innovine Jul 11 '17 at 18:16
Does the creature have to splash down completely in the first place?
Or could the majority of it take position in orbit and extend a pseudopod of some kind down to the planet (and up in the opposite direction). During the consumption of the planet's resources, this acts as giant root for the orbiting mother blob. When the planet is nearly exhausted, the blob climbs back up the pseudopod, space elevator style, and then drifts off to its next interstellar victim.
Evil Dog PieEvil Dog Pie
$\begingroup$ I think it was A.C. Clarke who wrote about a giant spinning creature, which had two long arms. It'd sit in orbit, rotating, with one arm brushing the surface of the planet, and the other stretched far out into space as a counterweight. Creatures would hop onto the arm, get a free lift into space, and let go at the furthest end, slingshotting into much higher orbits $\endgroup$ – Innovine Jul 11 '17 at 18:18
It is not possible to throw something into orbit, and it doesn't matter how fast or how much energy you use.
You don't need any knowledge of orbital velocities or rocket equations to know this cannot work. The simple fact is this: you cannot achieve orbit by using only a single impulse, like a cannonball from a cannon, or a bullet from a gun, or a giant blob throwing bits of itself. The projectile will always go up, around a bit, and back to hit the surface. In practice, it'll immediately burn up when attempting to leaving the atmosphere, and if anything survives that, it'll burn up when re-entering again.
The following diagram may help:
The points where the red orbital line and the surface of the planet intersect are the launch and impact points. No matter what angle or speeds you launch at, this red ellipse always passes through the launch point.*
So after launching, all rockets, bullets and blobs only travel in a large arc. The rocket engine can be (and usually is) turned off slightly after launch, just after getting out of the atmosphere, and the ship, bullet or blob would coast all the way to the highest point. It is here, at the apoapsis, that a second burn needs to be made, accelerating the projectile. This accelerating raises the periapsis (the shortest distance from the planets center to the ellipse), eventually raising the periapsis above the surface. When the periapsis has been raised higher than the atmosphere, the rocket will go around and around with no further input.
The first impulse (or burn) also needs to keep the speed low, to get through the thickest bottom layer of the atmosphere without losing all the energy to friction, or without overheating, or exploding due to aerodynamic stress. The more energy you try to add here, the worse these problems get.
There is only one possible class of orbits achievable by a single impulse. In a pure vacuum (no atmosphere), if you launch exactly horizontally at high enough speed, the launched projectile returns horizontally to the launch point, tangental to the surface. The faster you launch, horizontally, the higher the appapsis will be, at the opposite side of the planet. But at the launch site, the altitude will always be zero. Any mountains near the launch area would be a problem (as is the vacuum to your lifeforms).
The only time anything unpowered leaving the surface with a single impulse can get to an orbit above an atmosphere, is if it is hit by something else when near its apoapsis, providing the second impulse and accelerating it in the prograde direction (so it gets rear ended, speeding it up in the direction it is travelling). It is theorized that a load of melted rocks were blasted out from earth in a gigantic collision, and they bumped each other, forming orbits, which coalesced into the moon, eventually, and anything that didn't get bumped just right rained back down.
TL;DR: You can't get something into orbit by throwing it. Orbital mechanics says no. This is unfortunate, because if you can get to a stable orbit, you have all the time in the universe to deploy a solar sail and eventually float away somewhere else.
You CAN however, break free entirely, with nothing more than brute force. You just need to somehow survive getting through the atmosphere at speeds higher than escape velocity. This will mean burning up, like a shooting star in reverse, but given sufficient ablative protection, it may be possible. Escape velocity, at ground level, is Mach 33 (12km per second), but that speed will decrease rapidly due to friction and drag forces, so the actual launch would need to be much, much faster indeed.
It would require much greater sums of energy than rocket launches, since its very inefficient. But as long as the projectile gets through the atmosphere above 12km/s it will fly off into an orbit around the sun. And that's in theory enough to make it to any point in the solar system and beyond, given aeons of time and the right gravity assists.
InnovineInnovine
$\begingroup$ You second-to-last paragraph is actually a really good point. The whole purpose of leaving the planet is to, eventually, travel to another solar system, so saying "orbit" is not actually what I wanted. Given a sling or trebuchet, it could probably fling chunks high enough to achieve escape velocity... $\endgroup$ – ArmanX Jul 12 '17 at 19:34
$\begingroup$ Actually, getting to orbit is probably the best plan. You can then use solar light pressure to very gradually (and energy efficiently) move further afield. Direct to escape velocity probably requires something like gigatons of an explosive force $\endgroup$ – Innovine Jul 13 '17 at 17:27
$\begingroup$ @ArmanX The centrifugal forces on slings and trebuchet arms would likely cause them to fail long before they get their tips to tens of kilometers per second speeds. Linear acceleration would be best... and high above the atmosphere if at all possible.... maybe if your blob could raise itself up 50-100km, then perhaps some kind of blowdart or cannon mechanism..? spit instead of throw? $\endgroup$ – Innovine Jul 13 '17 at 17:32
$\begingroup$ Something else I thought of; while a single blob arm would crush the surface of the planet before it was long enough to reach into space, if the blob has eaten the entirety of the planet, it could squash itself into a disk, with multiple throwing arms to toss smaller blobs away. And, it may be able to digest the atmosphere, too; some chemical process that binds the various atoms to solids or liquids, or just stores the air in pockets. No atmosphere means a lot less drag. $\endgroup$ – ArmanX Jul 14 '17 at 18:14
$\begingroup$ @armanx if you can digest the atmosphere then things get more interesting alright :) note that you will want to throw the bits from west to east, to take advantage of the planets rotation. $\endgroup$ – Innovine Jul 14 '17 at 19:44
Consuming all life on the target planet is counterproductive - the organism will run out of food, and then it's really up the creek.
Might be better to have a more subtle organism that lives in some form of symbiosis with whatever it encounters on the planet. Then, it can wait (this being a very patient organism) for the inhabitants to develop space travel, and just hitch a ride. Why do the hard work, when you can get the native organisms to do it for you?
A truly imaginative organism might even guide the evolution of native creatures in a specific direction, with the goal of just getting back into space.
tj1000tj1000
$\begingroup$ The creature's entire point is to consume all life; it eats everything from organic life to plain ol' rocks and sea water. Its life cycle is spending hundreds, thousands, even millions of years travelling from one solar system to another, splashing down, eating everything, and leaving again. It's fairly mindless, and probably left over from some alien evil genius... $\endgroup$ – ArmanX Jul 12 '17 at 19:08
$\begingroup$ It would be interesting to estimate if the calorie content of the entire planets biomass is enough to overcome the friction of getting from the surface through the atmosphere and emerginging above escape velocity. $\endgroup$ – Innovine Jul 13 '17 at 17:36
Not the answer you're looking for? Browse other questions tagged space physics aliens science-fiction or ask your own question.
How could a creature get off a planet without technology?
How could an amorphous blob create rocket fuel?
A planet made of trash?
How could the air be stopped from falling off of a flat world?
Is space piracy orbitally practical?
Terraforming an Earth-like Planet
How fast would a Martian space elevator travel?
Is a coral-based planet possible? | CommonCrawl |
Journal of Environmental Health Sciences (한국환경보건학회지)
Korean Society of Environmental Health (한국환경보건학회)
Environment > Environmental Health
Journal of Environmental Health Sciences, JEHS, is an official journal of the Korean Society of Environmental Health. The mission of the journal is to promote research, policy, education, and practice in the field of environmental health by publishing papers of high scientific quality. Main research and development interests of the journal include but not limited to: exposure sciences, environmental monitoring, environmental epidemiology, toxicology and biomonitoring, risk assessment, environmental engineering, and environmental health policy in both general environments and workplaces. Categories of submission papers are original articles, reviews, brief reports, case reports, special topics, editorials, letters, meeting reports, news and book reviews.
Climate Change and Health - A Systemic Review of Low and High Temperature Effects on Mortality
Lim, Youn-Hee;Kim, Ho 397
https://doi.org/10.5668/JEHS.2011.37.6.397 PDF KSCI
Objectives: The impact of climate change on the health has been of increasing concern due to a recent temperature increase and weather abnormality, and the research results of the impact varied depending on regions. We synthesized risk estimates of the overall health effects of low and high temperature taking account of the heterogeneity. Methods: A comprehensive literature search was conducted using PUBMED to identify journal articles of low and/or high temperature effects on mortality. The search was limited to the English language and epidemiological studies using time-series analysis and/or case-crossover design. Random-effect models in meta analysis were used to estimate the percent increase in mortality with $1^{\circ}C$ temperature decrease or increase with 95% confidence intervals (CI) in cold or hot days. Results: Twenty three studies were presented in two tables: 1) low temperature effects; 2) high temperature effects on mortality. The combined effects of low and high temperatures on total mortality were 2% (95% CI, 1-4%) per $1^{\circ}C$ decrease and 4% (95% CI, 2-5%) per $1^{\circ}C$ increase of temperature, respectively. Conclusions: This meta analysis found that both low and high temperatures affected mortality, and the magnitude of high temperature appeared to be stronger than that of low temperature.
Evaluation of Atopy and Its Possible Association with Indoor Bioaerosol Concentrations and Other Factors at the Residence of Children
Ha, Jin-Sil;Jung, Hea-Jung;Byun, Hyae-Jeong;Yoon, Chung-Sik;Kim, Yang-Ho;Oh, In-Bo;Lee, Ji-Ho;Ha, Kwon-Chul 406
Objectives: Exposure to bioaerosols in the indoor environment could be associated with a variety adverse health effects, including allergic disease such atopy. The objectives of this study were to assess children's exposure to bioaerosol in home indoor environments and to evaluate the association between atopy and bioaerosol, environmental, and social factors in Ulsan, Korea. Methods: Samples of viable airborne bacteria and fungi were collected by impaction onto agar plates using a Quick Take TM 30 and were counted as colony forming units per cubic meter of air (CFU/$m^3$). Bioaerosols were identified using standard microbial techniques by differential stains and/or microscopy. The environmental factors and possible causes of atopy based on ISAAC (International Study of Allergy and Asthma in Childhood) were collected by questionnaire. Results: The bioaerosol concentrations in indoor environments showed log-normal distribution (p < 0.01). Geometric mean (GM) and geometric standard deviation (GSD) of airborne bacteria and fungi in homes were 189.0 (2.5), 346.1(2.0) CFU/$m^3$, respectively. Indoor fungal levels were significantly higher than those of bacteria (p < 0.001). The concentration of airborne bacteria exceeded the limit recommended by the Korean Ministry of Environment, 800 CFU/$m^3$, in three out of 92 samples (3.3%) from 52 homes. The means of indoor to outdoor ratio (I/O) for airborne bacteria and fungi were 8.15 and 1.13, respectively. The source of airborne bacteria was not outdoors but indoors. GM of airborne bacteria and fungi were 217.6, 291.8 CFU/$m^3$ in the case's home and 162.0, 415.2 CFU/$m^3$ in the control's home respectively. The difference in fungal distributions between case and control were significant (p = 0.004) and the odds ratio was 0.996 (p = 0.027). Atopy was significantly associated with type of house (odds ratio = 1.723, p = 0.047) and income (odds ratio = 1.891, p = 0.041). Some of the potential allergic fungal genera isolated in homes were Cladosporium spp., Botrytis spp., Aspergillus spp., Penicillium spp., and Alternatia spp. Conclusions: These results suggest that there this should be either 'was little' meaning 'basically no significant association was found' or 'was a small negative' mean that an association was found but it was minor. It's a very improtant distinction. Association between airborne fungal concentrations and atopy and certain socioeconomic factors may affect the prevalence of childhood atopy.
Blood Lead Concentration and Hypertension in Korean Adults Aged 40 and Over According to KNHANES IV (2008)
Kim, Sun-Young;Lee, Duk-Hee 418
Objectives: The purpose of this study was to examine the cross-sectional relationship between low blood lead levels and increasing blood pressure among Korean adults using a nationally representative sample of the Korean population: the Korea National Health and Nutrition Examination Survey (KNHANES) 2008. Methods: A total of 918 subjects aged 40 and older and not currently being treated for hypertension participated in this study. Information about age, gender, smoking status, alcohol consumption, education level, and the use of anti-hypertensive medication was collected. The blood pressure was defined as the mean of the second and the third measurements after three time measurements. Lead levels were determined by an analysis of blood samples. Multiple linear and logistic regression analyses were implemented after adjusting for covariates including age, gender, educational level, smoking status, alcohol consumption, and BMI. Results: This study showed that the average differences in systolic and diastolic blood pressure comparing the lowest to highest quintile of blood lead were 4.33 mmHg (95% CI, 0.66-8.00; p for trend = 0.027) and 2.66 mmHg (95% CI, 0.26-5.06; p for trend = 0.021), respectively. After multivariate adjustment for covariates, the prevalence odds ratio (POR) of subjects in the highest quintile was associated with a 1.70-fold increase in the risks of hypertension (95% CI, 0.83-3.49; p for trend test = 0.112) over those in the lowest quintile of blood lead concentration, However, it was not statistically significant. Conclusions: This study provided evidence for an association between low- levels of blood lead and elevations in blood pressure and risk for hypertension in the general population of Korea.
Analysis of Micronuclei and Its Association with Genetic Polymorphisms in Hospital Workers Exposed to Ethylene Oxide
Lee, Sun-Yeong;Kim, Yang-Jee;Choi, Young-Joo;Lee, Joong-Won;Lee, Young-Hyun;Shin, Mi-Yeon;Kim, Won;Yoon, Chung-Sik;Kim, Sung-Kyoon;Chung, Hai-Won 429
Objectives: Ethylene oxide (EtO) is classified as a human carcinogen, but EtO is still widely used to sterilize heat-sensitive materials in hospitals. Employees working around sterilizers are exposed to EtO after sterilization. The aim of the present study was to assess the exposure of EtO level, coupled with occupationally induced micronuclei from hospital workers. The influence of genetic polymorphisms of detoxifying genes (GSTT1 and GSTM1) and DNA repair genes (XRCC1 and XRCC3) on the frequencies of micronuclei in relation to exposure of EtO was also investigated. Methods: The study population was composed of 35 occupationally exposed workers to EtO, 18 student controls and 44 unexposed hospital controls in Korea. Exposure to EtO is measured by passive personal samplers. We analyzed the frequencies of micronuclei by performing cytokinesis-block micronucleus assay (CBMN assay) and GSTM1, GSTT1, XRCC1, and XRCC3 were also genotyped by performing polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP). Results: The frequencies of micronuclei in EtO exposure group, student controls and hospital controls were $18.00{\pm}7.73$, $10.47{\pm}7.96$ and $13.86{\pm}6.35$ respectively and their differences were statistically significant, but no significant differences according to the level of EtO were observed. There was a dose-response relationship between the frequencies of micronuclei and cumulative dose of EtO, but no significantly differences were observed. We also investigated the influence of genetic polymorphisms (GSTM1, GSTT1, XRCC1, and XRCC3) on the frequencies of micronuclei, but there were no differences in the frequencies of micronuclei by genetic polymorphisms. Conclusions: The frequencies of micronuclei in EtO exposure group was significantly higher than control groups. A dose-response relationship was found between the level of EtO exposure and the frequencies of micronuclei, but no statistically differences were observed. We also found that the frequencies of micronuclei were increased according to cumulative EtO level. There was no association of the genetic GSTM1, GSTT1, XRCC1, and XRCC3 state with the frequency of micronuclei induced by EtO exposure.
Concentrations of PBDE Congeners in Breast Milk and Predictors of Exposure in Seoul Residents
We, Sung-Ug;Yoon, Cho-Hee;Min, Byung-Yoon 440
Objectives: This study was designed to determine the levels of polybrominated diphenyl ethers (PBDEs) in breast milk and to evaluate the relations with factors affecting these levels. Methods: The congener levels of PBDE in 22 samples of breast milk were analyzed using a high resolution gas chromatograph with a high resolution mass detector. In accordance with our standard operating procedures, the recoveries of internal standards had to range between 68% and 118%. Since the distribution of PBDE concentrations is close to log-normal, the data were logarithmically transformed before analysis. Test subjects were healthy primipara and multipara mothers with a mean age of 32 (SD = 2.7) in 2006. Results: Seven PBDE congeners (BDE-28, 47, 99, 100, 153, 154, and 183) were detected and identified in all of the pooled breast milk samples, indicating widespread contamination from PBDEs in the environment in Korea. Residue levels of total PBDEs (sum PBDEs from tri- to hepta-BDE) ranged from 0.84-13.1 ng/g lipid with median and geometric mean levels of 2.6 ng/g lipid and 2.74 ng/g lipid, respectively. PBDE congeners 47, 99 and 153 markedly predominated and accounted for about 75% of the amount of the PBDE congeners analyzed. BDE-47 was the dominant congener in most samples, whereas BDE-153 was predominant in a few (n = 7/22). BDE-47 was highly correlated with total PBDEs (r = 0.987, p < 0.01). In analyses of the differences of the means of log transformed breast milk PBDE levels for groups of potential covariates, only breast milk BDE-47 and BDE-99 levels were significantly associated with fish (p < 0.05) and meat consumption (p < 0.01). However, we did not find significant correlations between PBDE levels and maternal age, body mass index (BMI), parity, job presence and smoking status. Conclusions: Our findings are mainly limited due to the small sampling size and low doses of PBDEs exposure. Background and human exposure data of PBDEs is lacking, and longitudinal investigations into the environment and biota are encouraged to determine the health impact on future populations in Korea.
A Comparison of the Adjustment Methods for Assessing Urinary Concentrations of Cadmium and Arsenic: Creatinine vs. Specific Gravity
Kim, Dong-Kyeong;Song, Ji-Won;Park, Jung-Duck;Choi, Byung-Sun 450
Objectives: Biomarkers in urine are important in assessing exposures to environmental or occupational chemicals and for evaluateing renal function by exposure from these chemicals. Spot urine samples are needed to adjust the concentration of these biomarkers for variations in urine dilution. This study was conducted to evaluate the suitability of adjusting the urinary concentration of cadmium (uCd) and arsenic (uAs) by specific gravity (SG) and urine creatinine (uCr). Methods: We measured the concentrations of blood cadmium (bCd), uCd, uAs, uCr, SG and N-acetyl-${\beta}$-D-glucosaminidase (NAG) activity, which is a sensitive marker of tubular damage by low dose Cd exposure, in spot urine samples collected from 536 individuals. The value of uCd, uAs and NAG were adjusted by SG and uCr. Results: The uCr levels were affected by gender (p < 0.01) and muscle mass (p < 0.01), while SG levels were affected by gender (p < 0.05). Unadjusted uCd and uAs were correlated with SG (uCd: r = 0.365, p < 0.01; uAs: r = 0.488, p < 0.01), uCr (uCd: r = 0.399, p < 0.01; uAs: r = 0.484, p < 0.01). uCd and uAs adjusted by SG were still correlated with SG (uCd: r = 0.360, p < 0.01, uAs: r = 0.483, p < 0.01). uCd and uAs adjusted by uCr and modified uCr ($M_{Cr}$) led to a significant negative correlation with uCr (uCd: r = -0.367, p < 0.01; uAs: r = -0.319, p < 0.01) and $M_{Cr}$ (uCd: r = -0.292, p < 0.01; uAs: r = -0.206, p < 0.01). However, uCd and uAs adjusted by conventional SG ($C_{SG}$) were disappeared from these urinary dilution effects (uCd: r = -0.081; uAs: r = 0.077). Conclusions: $C_{SG}$ adjustment appears to be more appropriate for variations in cadmium and arsenic in spot urine.
A Study on Installation of Washstands in Bathrooms of Elementary School
Kwon, Woo-Taeg;Lee, Woo-Sik 460
Objectives: Students in elementary schools usually wash their hands in a washstand. However, little attention is paid to the washstand itself. Today, the importance of personal sanitation and hygiene is greatly emphasized. Therefore students' parents and the public are growing increasingly interested in accessibility to washstands by elementary school students in their schools. Methods: With respect to this study, a survey of students and teachers inelementary schools was performed on the installation of washstands in order to determine the proper number of washstands per school. Results: The results show that 1.1 boys (per class) need a washstand, while 1.8 girls (per class) do so in order to maintain a 50% level of crowdedness. By of the regression equation, to maintain 50% congestion (50% of all students feel congestion) there should be 18.5 boys, and the 15.76 girls per washstand. Table 3 is based on the above results, the number of students per washstand (x) and congestion (y), separated by gender according to the results of regression analysis, the correlation of male models in the linear regression analysis and correlation of girls in the regression equation can be obtained. The linear regression fit of less than 0.7 determines that the coefficients of determination are 0.5399 and 0.4195, respectively. Significance was much smaller. Also, according to the simulation using the diffusion model, with 29 students per class more than one washstand should be provided in a school. Girls (per class) need 0.7 more washstands than boys (per class). Conclusions: More washstand facilities for girls than boys are needed. If the target is based on school class size two washstands should be installed. Finally, guidelines and/or standards in the Schools Health Act of Korea forin elementary school washstands is considerably needed.
Conditioning and Dewatering Properties of Digested and Thickened Sludge with Inorganic Conditioner
Kim, Jeong-Ho;Nam, Se-Yong 467
Objectives: Wastewater treatment plants typically produce a large volume of waste sludge. In this study, the conditioning and dewatering properties of a digested and thickened sludge from an industrial wastewater treatment plant were investigated in order to improve the dewaterbility of the sludge. Methods: Jar-tests and Buchner funnel tests were carried out to assess the conditioning and dewatering properties of a waste sludge. TTF (Time to Filter Test) and SRF (Specific Resistance to Filtration) were adopted as the indices of sludge dewaterbility. Results: The valuation indices influencing the dewaterbility of the waste sludge, including TTF, SRF, water contents, VS/TS ratio and turbidity, were measured. The TTF and SRF of the digested and thickened sludge were decreased to 40 sec, $3.43{\times}10^{12}$ m/kg, and 39 sec, $1.09{\times}10^{12}$ m/kg, respectively. Conclusions: The conditioner composed of natural inorganic materials turned out to be effective in the reduction of sludge water contents.
Comparison of Concentration of Urinary Metabolites of PAHs from Smokers and Nonsmokers
Kho, Young-Lim;Lee, Eun-Hee 474
This study investigated urinary metabolites of polycyclic aromatic hydrocarbons (PAHs) in the urine of smokers and non-smokers by liquid chromatography triple quordrupole tandem mass spectroscopy (LC/MS/MS). Compounds analyzed for urinary biomarkers of PAHs were five mono-hydroxylated PAHs metabolites; 1-naphthol, 2-naphthol, 1-hydroxypyrene(1-OHP), 3-phenanthrol, 2-fluorenol. Urine samples were pretreated by enzymatic hydrolysis and solid phase extraction method. Smokers were composed of 17 men and five women; non-smokers 17 men and 16 women. Smoking increased urinary concentrations of five PAHs metabolites significantly higher than those of nonsmokers. Statistically significant correlations among the five PAHs metabolites were shown. The results suggest that LC/MS/MS technology should be useful in the environmental health discipline.
Antimicrobial Efficacies of Citra-Kill®, Disinfectant Solution against Salmonella Typhimurium and Brucella Ovis
Cha, Chun-Nam;Lee, Yeo-Eun;Son, Song-Ee;Yoo, Chang-Yeul;Kim, Suk;Lee, Hu-Jang 482
Salmonellosis and brucellosis have caused a considerable danger of farmed animals and economic loss in animal farming industry. In this study, the disinfection efficacy of Citra-Kill$^{(R)}$, a commercial disinfectant, composed to quaternary ammonium chloride and citric acid was evaluated against S. typhimurium and Brucella ovis. A bactericidal efficacy test by broth dilution method was used to determine the lowest effective dilution of the disinfectant following exposure to test bacteria for 30 min at $4^{\circ}C$. Citra-Kill$^{(R)}$ and test bacteria were diluted with distilled water (DW), hard water (HW) or organic matter suspension (OM) according to treatment condition. On OM condition, the bactericidal activity of Citra-Kill$^{(R)}$ against S. typhimurium and Brucella ovis was lowered compared to that on HW condition. As Citra-Kill$^{(R)}$ possesses bactericidal efficacy against animal pathogenic bacteria such as S. typhimurium and Brucella ovis, this disinfectant solution can be used to control the spread of animal bacterial diseases. | CommonCrawl |
Sandwich problems on orientations
Olivier Durand de Gevigney1,
Sulamita Klein2,
Viet-Hang Nguyen1 &
Zoltán Szigeti1
Journal of the Brazilian Computer Society volume 18, pages 85–93 (2012)Cite this article
The graph sandwich problem for property Π is defined as follows: Given two graphs G1=(V,E1) and G2=(V,E2) such that E1⊆E2, is there a graph G=(V,E) such that E1⊆E⊆E2 which satisfies property Π? We propose to study sandwich problems for properties Π concerning orientations, such as Eulerian orientation of a mixed graph and orientation with given in-degrees of a graph. We present a characterization and a polynomial-time algorithm for solving the m-orientation sandwich problem.
Given two graphs G1=(V,E1) and G2=(V,E2) with the same vertex set V and E1⊆E2, a graph G=(V,E) is called a sandwich graph for the pair G1,G2 if E1⊆E⊆E2. The graph sandwich problem for property Π is defined as follows [12]:
Graph Sandwich Problem for Property Π Instance: Given undirected graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2. Question: Is there a graph G=(V,E) such that E1⊆E⊆E2 and G satisfies property Π?
We call E1 the mandatory edge set, E0=E2∖E1 the optional edge set and E3 the forbidden edge set, where E3 denotes the set of edges of the complementary graph \(\overline{G}_{2}\) of G2. Thus any sandwich graph G=(V,E) for the pair G1,G2 must contain all mandatory edges, no forbidden edges and may contain a subset of the optional edges. Graph sandwich problems have attracted much attention lately arising from many applications and as a natural generalization of recognition problems [1–3, 7, 22, 24]. The recognition problem for a class of graphs \(\mathcal{C}\) is equivalent to the graph sandwich problem in which G1=G2=G, where G is the graph we want to recognize and property Π is "to belong to class \(\mathcal{C}\)".
In this paper we propose to study sandwich problems for properties Π concerning orientations, such as Eulerian orientation of a mixed graph and orientation with given in-degrees of a graph, or more generally of a mixed graph.
The paper is organized as follows: Sect. 2 contains some basic definitions, notations and results. Section 3 contains some known results on degree constrained sandwich problems. We consider the undirected version and the directed version, the complexity, the characterization and the related optimization problems. We also define a simultaneous version and discuss its complexity. Section 4 focuses on Eulerian sandwich problems. We consider first undirected graphs and then directed graphs. These problems were already solved in [12], here we point out that the undirected case reduces to T-joins, while the directed case to circulations. We discuss the complexity of the problems and their characterizations and we also propose some mixed versions. In Sect. 5 we consider sandwich problems regarding an m-orientation, i.e., given undirected graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2 and a non-negative integer vector m on V, we show that it is polynomial to decide whether there exists a sandwich graph G=(V,E) (E1⊆E⊆E2) that has an orientation \(\vec{G}\) whose in-degree vector is m that is \(d^{-}_{\vec{G}}(v)=m(v)\) for all v∈V. This result stands in contrast to the strongly connected m-orientation sandwich problem which we show is NP-complete. Section 6 is devoted to a new kind of sandwich problem where we may contract (and not delete) optional edges and property Π is being bipartite.
Undirected graphs :
Let G=(V,E) be an undirected graph. For vertex sets X and Y, the cut induced by X is defined to be the set of edges of G having exactly one end-vertex in X and is denoted by . The degree (or d E (X)) of X is the cardinality of the cut induced by X, that is, d G (X)=|δ G (X)|. The number of edges between X∖Y and Y∖X is denoted by . The number of edges of G having both (resp. at least one) end-vertices in X is denoted by or i E (X) or simply i(X) (resp. ). It is well-known that (1) is satisfied for all X,Y⊆V,
We say that a vector m on V is the degree vector of G if d G (v)=m(v) for all v∈V. For a vector m on V, we consider m as a modular function, that is, we use the notation . Let us recall that d G (X) is the degree function of G. We define as the modular function defined by the degree vector d G (v) of G. Note that \(\hat{d}_{G}(X)=d_{G}(X)+2i_{G}(X)\forall X\subseteq V\).
We denote by the set of vertices of G of odd degree. For an edge set F of G, the subgraph induced by F, that is, (V,F), is denoted by . We say that G is Eulerian if the degree of each vertex is even, that is, if T G =∅. Note that we do not suppose the graph to be connected.
Let T be a vertex set in G. An edge set F of G is called T-join if the set of odd degree vertices in the subgraph induced by F coincide with T, that is if TG(F)=T. Given a cost vector on the edge set of G, a minimum cost T-join can be found in polynomial time by Edmonds and Johnson's algorithm [5].
Let f be a non-negative integer vector on V. An edge set F of G is called an f-factor of G if f is the degree vector of G(F), that is, d F (v)=f(v) for all v∈V. If f(v)=1 for all v∈V, then we say that F is a 1-factor or a perfect matching. An f-factor—if it exists—can be found in polynomial time, see [20]. The graph G is called 3-regular if each vertex is of degree 3. Note that for a 3-regular graph, the existence of two edge-disjoint perfect matchings is equivalent to the existence of three edge-disjoint perfect matchings which is equivalent to the 3-edge-colorability of the graph.
Directed graphs :
Let D=(V,A) be a directed graph. For a vertex set X, the set of arcs of D entering (resp. leaving) X is denoted by (resp. ). The in-degree (resp. out-degree ) of X is the number of arcs of D entering (resp. leaving) X, that is \(d^{-}_{D}(X)=|\varrho _{D}(X)|\) (resp. \(d^{+}_{D}(X)=|\delta_{D}(X)|\)). The set of arcs of G having both end-vertices in X is denoted by . The following equality will be used frequently without reference:
We say that a vector m on V is the in-degree vector of D if \(d^{-}_{D}(v)=m(v)\) for all v∈V. Let us recall that \(d^{-}_{D}(X)\) is the in-degree function of D. Let f be a non-negative integer vector on V. An arc set F of D is called a directedf-factor of D if f is the in-degree vector of D(F), that is, \(d^{-}_{F}(v)=f(v)\) for all v∈V.
We say that D is Eulerian if the in-degree of v is equal to the out-degree of v for all v∈V, that is, \(d^{-}_{D}(v)=d^{+}_{D}(v)\) for all v∈V. Note that we do not suppose the graph to be connected.
Let f and g be two vectors on the arcs of D such that f(e)≤g(e) for all e∈A. A vector x on the arcs of D is a circulation if (3) and (4) are satisfied.
Note that if f(e)=g(e)=1 for all e∈A, then D is Eulerian if and only if f is a circulation. We will use the following characterization when a circulation exists.
(Hoffmann [15])
LetD=(V,A) be a directed graph andfandgtwo vectors onAsuch thatf(e)≤g(e)∀e∈A. There exists a circulation inDif and only if
We say that H=(V,E∪A) is a mixed graph if E is an edge set and A is an arc set on V. For an undirected graph G=(V,E), if we replace each edge uv by the arc uv or vu, then we get the directed graph . We say that \(\vec{G}\) is an orientation of G.
Mixed graphs having Eulerian orientations are characterized as follows.
(Ford, Fulkerson [8])
A mixed graphH=(V,E∪A) has an Eulerian orientation if and only if
The following theorem characterizes graphs having an orientation with a given in-degree vector.
(Hakimi [13])
Given an undirected graphG=(V,E) and a non-negative integer vectormonV, there exists an orientation\(\vec{G}\)ofGwhose in-degree vector ismif and only if
Functions :
Let b be a set function on the subsets of V. We say that b is submodular if for all X,Y⊆V,
The function b is called supermodular if −b is submodular. A function is modular if it is supermodular and submodular. We will use frequently in this paper the following facts.
Claim 1
The degree functiond G (Z) of an undirected graphGand the in-degree function\(d^{-}_{D}(Z)\)of a directed graphDare submodular and the functioni(Z) is supermodular.
The minimum value of a submodular function can be found in polynomial time.
(Frank [9])
Letbandpbe a submodular and a supermodular set function onVsuch thatp(X)≤b(X) for allX⊆V. Then there exists a modular functionmonVsuch thatp(X)≤m(X)≤b(X) for allX⊆V. Ifbandpare integer valued thenmcan also be chosen integer valued.
A pair (p,b) of set functions on 2V is a strong pair if p (resp. b) is supermodular (submodular) and they are compliant, that is, for every pairwise disjoint triple X,Y,Z,
$$b(X\cup Z) - p(Y\cup Z) \geq b(X) -p(Y).$$
Note that a pair (α,β) of modular functions is a strong pair if and only if α≤β. If (p,b) is a strong pair then the polyhedron
is called a generalized polymatroid (or a g-polymatroid). When α≤β are modular, we also call the g-polymatroid Q(α,β) a box.
(Frank, Tardos [11])
The intersection of an integral g-polymatroidQ(p,b) and an integral boxQ(α,β) is an integral g-polymatroid. It is nonempty if and only ifα≤bandp≤β.
Matroids :
A set system \(M=(V,{\mathcal{F}})\) is called a matroid if \({\mathcal{F}}\) satisfies the following three conditions:
(I1)
\(\emptyset\in{\mathcal{F}}\),
if \(F\in{\mathcal{F}}\) and F′⊆F, then \(F'\in{\mathcal{F}}\),
if \(F, F'\in{\mathcal{F}}\) and |F|>|F′|, then there exists f∈F∖F′ such that \(F'\cup f\in{\mathcal{F}}\).
A subset X of V is called independent in M if \(X\in{\mathcal{F}}\), otherwise it is called dependent. The maximal independent sets of V are the basis of M. Let \({\mathcal{B}}\) be the set of basis of M. Then \({\mathcal{B}}\) satisfies the following two conditions:
\({\mathcal{B}}\neq\emptyset\),
if \(B, B'\in{\mathcal{B}}\) and b∈B∖B′, then there exists b′∈B′∖B such that \((B-b)\cup b'\in{\mathcal{B}}\).
Conversely, if a set system \((V,{\mathcal{B}})\) satisfies (B1) and (B2), then \(M=(V,{\mathcal{F}})\) is a matroid, where \({\mathcal{F}}=\{F\subseteq V:\exists B\in{\mathcal{B}}, F\subseteq B\}\).
For S⊂V, the matroid M∖S obtained from M by deletingS is defined as \(M\setminus S=(V\setminus S, {\mathcal{F}}|_{V\setminus S})\), where X⊆V∖S belongs to \({\mathcal{F}}|_{V\setminus S}\) if and only if \(X\in{\mathcal{F}}\). For \(S\in{\mathcal{F}}\), the matroid M/S obtained from M by contractingS is defined as \(M/S=(V\setminus S, {\mathcal{F}}_{S})\), where X⊆V∖S belongs to \({\mathcal{F}}_{S}\) if and only if \(X\cup S\in{\mathcal{F}}\). Let {V1,…,V l } be a partition of V and a1,…,a l a set of non-negative integers. Then \(M=(V,{\mathcal{F}})\) is a matroid, where \({\mathcal{F}}=\{F\subseteq V: |F\cap V_{i}|\leq a_{i}\}\), we call it partition matroid. The dual matroid M∗ of M is defined as follows: the basis of M∗ are the complements of the basis of M.
Let \(M=(V,{\mathcal{F}})\) be a matroid and c a cost vector on V={v1,…,v n }. We can find a minimum cost basis F n of M in polynomial time by the greedy algorithm: take a non-decreasing order of the elements of V:c(v1)≤…≤c(v n ). Let F0 be empty and for i=1,…,n, let F i =Fi−1+v i if \(F_{i-1}+v_{i}\in {\mathcal{F}}\), otherwise let F i =Fi−1.
If M1 and M2 are two matroids on the same ground set V, then we can find a common basis of M1 and M2 in polynomial time (if there exists one) by the matroid intersection algorithm of Edmonds [4].
(Edmonds, Rota [18])
For an integer-valued, non-decreasing, submodular functionbdefined on a ground setS, the set {F⊆S;|F′|≤b(F′) for all ∅≠F′⊆F} forms the set of independent sets of a matroidM b whose rank functionr b is given by
$$r_b(Z) = \min\bigl\{b(X) + |Z-X|, X \subseteq Z\bigr\}.$$
Given an undirected graph G=(V,E) and a non-negative integer vector m on V, let \(\bar{m}^{G}=\bar{m}\) be the set function defined on E by \(\bar{m}(F) = m(V(F))\) where V(F) is the set of vertices covered by F. One can easily check that \(\bar{m}\) is integer valued, non-decreasing and submodular. Thus, by Theorem 7, \(\bar {m}\) defines a matroid \(M_{\bar{m}}\). The following claim is straightforward.
The set {F⊆E:m(X)≥i F (X),∀X⊆V} is the set of independent sets of the matroid\(M_{\bar{m}}\).
Degree constrained sandwich problems
Before studying sandwich problems on orientations of given in-degrees, let us start as a warming up by considering sandwich problems for undirected and directed graphs of given degrees. These problems reduce to the undirected and directed f-factor problems. We mention that the directed case is much easier than the undirected case because the addition of an arc in a directed graph contributes only to the in-degree of the head and not of the tail, while the addition of an edge in an undirected graph contributes to the degree of both end-vertices. This section contains no new results, we added it for the sake of completeness.
Undirected graphs
Undirected Degree Constrained Sandwich ProblemInstance: Given undirected graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2 and a non-negative integer vector f on V. Question: Does there exist a sandwich graph G=(V,E) (E1⊆E⊆E2) such that d G (v)=f(v) for all v∈V?
Complexity: It is in P because the answer is Yes if and only if there exists an \((f(v)-d_{G_{1}}(v))\)-factor in the optional graph G0=(V,E0).
Characterization: The general f-factor theorem due to Tutte [25] can be applied to get a characterization.
Optimization: The minimum cost f-factor problem in undirected graphs can be solved in polynomial time, see Schrijver [20].
Simultaneous Undirected Degree Constrained Sandwich ProblemInstance: Given two edge-disjoint graphs G1=(V,E1) and G2=(V,E2) in G3=(V,E3) and two non-negative integer vectors f1 and f2 on V. Question: Do there exist simultaneously sandwich graphs \(\hat{G}_{1}=(V,\hat{E}_{1})\)\((E_{1}\subseteq\hat{E}_{1}\subseteq E_{3})\) and \(\hat{G}_{2}=(V,\hat{E}_{2})\)\((E_{2}\subseteq\hat{E}_{2}\subseteq E_{3})\) such that \(\hat{E}_{1}\cap\hat{E}_{2}=\emptyset\) and \(d_{\hat{G}_{1}}(v)=f_{1}(v)\) and \(d_{\hat{G}_{2}}(v)=f_{2}(v)\) for all v∈V?
Complexity: It is NP-complete because it contains as a special case whether there exist two edge-disjoint perfect matchings so 3-edge-colorability of 3-regular graphs. Indeed, let G=(V,E) be an arbitrary 3-regular graph. Let G1 and G2 be the edgeless graph on V, G3=G and f1(v)=f2(v)=1 for all v∈V. Then the sandwich graphs \(\hat{G}_{1}\) and \(\hat{G}_{2}\) exist if and only if \(\hat{E}_{1}\) and \(\hat{E}_{2}\) are edge-disjoint perfect matchings of G or equivalently, if there exists a 3-edge-coloring of G. Since the problem of 3-edge-colorability of 3-regular graphs is NP-complete [16], so is our problem.
Directed graphs
Directed Degree Constrained Sandwich ProblemInstance: Given directed graphs D1=(V,A1) and D2=(V,A2) with A1⊆A2 and a non-negative integer vector f on V. Question: Does there exist a sandwich graph D=(V,A) (A1⊆A⊆A2) such that \(d^{-}_{D}(v)=f(v)\) for all v∈V?
Complexity+Characterization: It is in P because the answer is Yes if and only if there exists a directed \((f(v)-d^{-}_{D_{1}}(v))\)-factor in the optional directed graph D0=(V,A0), hence we have the following.
TheDirected Degree ConstrainedSandwich Problemhas aYesanswer if and only if\(d^{-}_{D_{2}}(v)\geq f(v)\geq d^{-}_{D_{1}}(v)\)for allv∈V.
Optimization: The feasible arc sets form the basis of a partition matroid, so the greedy algorithm provides a minimum cost solution.
Simultaneous Directed Degree ConstrainedSandwich Problem 1Instance: Given two arc-disjoint directed graphs D1=(V,A1) and D2=(V,A2) in D3=(V,A3) and two non-negative integer vectors f1 and f2 on V. Question: Do there exist simultaneously sandwich graphs \(\hat{D}_{1}=(V,\hat{A}_{1})\)\((A_{1}\subseteq\hat{A}_{1}\subseteq A_{3})\) and \(\hat{D}_{2}=(V,\hat{A}_{2})\)\((A_{2}\subseteq\hat{A}_{2}\subseteq A_{3})\) such that \(\hat{A}_{1}\cap\hat{A}_{2}=\emptyset\) and \(d^{-}_{\hat{D}_{1}}(v)=f_{1}(v)\) and \(d^{-}_{\hat{D}_{2}}(v)=f_{2}(v)\) for all v∈V?
Complexity: It is in P because the answer is Yes if and only if \(d^{-}_{D_{3}}(v)\geq f_{1}(v)+f_{2}(v)\), \(f_{1}(v)\geq d^{-}_{D_{1}}(v)\) and \(f_{2}(v)\geq d^{-}_{D_{2}}(v)\) for all v∈V.
Simultaneous Directed Degree ConstrainedSandwich Problem 2Instance: Given directed graphs D1=(V,A1) and D2=(V,A2) with A1⊆A2 and two non-negative integer vectors f and g on V. Question: Does there exist a sandwich graph D=(V,A) (A1⊆A⊆A2) such that \(d^{-}_{D}(v)=f(v)\) and \(d^{+}_{D}(v)=g(v)\) for all v∈V.
Complexity: The feasible arc sets for the in-degree constraint form the basis of a partition matroid and the feasible arc sets for the out-degree constraint form the basis of a partition matroid. The answer is Yes if and only if there exists a common basis in these two matroids. Thus it is in P by the matroid intersection algorithm of Edmonds [4].
Eulerian sandwich problems
In this section we consider first two problems that were already solved in [12]: Eulerian sandwich problems for undirected and directed graphs. We point out that the undirected case reduces to T-joins, while the directed case to circulations. We show that in both cases the simultaneous versions are NP-complete.
Then we propose to study the problem in mixed graphs. We show two cases that can be solved. The first case will be solved by the Discrete Separation Theorem 5 of Frank [9], while the second case reduces to the Directed Eulerian Sandwich Problem. The general case, however, remains open.
Undirected Eulerian Sandwich ProblemInstance: Given undirected graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2. Question: Does there exist a sandwich graph G=(V,E) (E1⊆E⊆E2) that is Eulerian?
Complexity: It is in P because the answer is Yes if and only if there exists a \(T_{G_{1}}\)-join in the optional graph G0.
Characterization: The answer is Yes if and only if each connected component of G0 contains an even number of vertices of \(T_{G_{1}}\).
Optimization: The minimum cost T-join problem can be solved in polynomial time [5].
Simultaneous Undirected Eulerian Sandwich ProblemInstance: Given two edge-disjoint graphs G1=(V,E1) and G2=(V,E2) in G3=(V,E3). Question: Do there exist simultaneously Eulerian sandwich graphs \(\hat{G}_{1}=(V,\hat{E}_{1})\)\((E_{1}\subseteq\hat{E}_{1}\subseteq E_{3})\) and \(\hat{G}_{2}=(V,\hat{E}_{2})\)\((E_{2}\subseteq\hat{E}_{2}\subseteq E_{3})\) such that \(\hat{E}_{1}\cap\hat{E}_{2}=\emptyset\)?
Complexity: It is NP-complete because it contains as a special case whether there exist two edge-disjoint perfect matchings so 3-colorability of 3-regular graphs. Indeed, let G=(V,E) be an arbitrary 3-regular graph. Let G3 be obtained from G by adding 2 edge-disjoint perfect matchings M1 and M2 to G, let G1=(V,M1) and G2=(V,M2). Then the Eulerian sandwich graphs \(\hat{G}_{1}\) and \(\hat{G}_{2}\) exist if and only if \(\hat{E}_{1}\setminus M_{1}\) and \(\hat{E}_{2}\setminus M_{2}\) are edge-disjoint perfect matchings of G or equivalently, if there exists a 3-edge-coloring of G. Since the problem of 3-edge-colorability of 3-regular graphs is NP-complete [16], so is our problem.
Directed Eulerian Sandwich ProblemInstance: Given directed graphs D1=(V,A1) and D2=(V,A2) with A1⊆A2. Question: Does there exist a sandwich graph D=(V,A) (A1⊆A⊆A2) that is Eulerian?
Complexity: It is in P because it can be reformulated as a circulation problem: let f(e)=1,g(e)=1 if e∈A1 and f(e)=0,g(e)=1 if e∈A0. This way the arcs of A1 are forced and the arcs of A0 can be chosen if necessary.
Characterization: The answer is Yes if and only if \(d^{-}_{D_{1}}(X)\leq d^{+}_{D_{2}}(X)\) for all X⊆V by Theorem 1.
Optimization: The minimum cost circulation problem can be solved in polynomial time, see Tardos [23].
Simultaneous Directed Eulerian SandwichProblemInstance: Given two arc-disjoint directed graphs D1=(V,A1) and D2=(V,A2) in D3=(V,A3). Question: Do there exist simultaneously Eulerian sandwich graphs \(\hat{D}_{1}=(V,\hat{A}_{1})\)\((A_{1}\subseteq\hat{A}_{1}\subseteq A_{3})\) and \(\hat{D}_{2}=(V,\hat{A}_{2})\)\((A_{2}\subseteq\hat{A}_{2}\subseteq A_{3})\) such that \(\hat{A}_{1}\cap\hat{A}_{2}=\emptyset\)?
Complexity: It is NP-complete, it contains as a special case (D1=(V,t1s1), D2=(V,t2s2) and D3=D) the following directed 2-commodity integral flow problem that is NP-complete [6]: Given a directed graph D and two pairs of vertices, s1,t1 and s2,t2, decide whether there exist a path from s1 to t1 and a path from s2 to t2 that are arc-disjoint.
Mixed graphs
Mixed Eulerian Sandwich ProblemInstance: Given mixed graphs H1=(V,E1∪A1) and H2=(V,E2∪A2) with E1⊆E2,A1⊆A2. Question: Does there exist a sandwich mixed graph H=(V,E∪A) (E1⊆E⊆E2,A1⊆A⊆A2) that has an Eulerian orientation?
Complexity: We provide two special cases that can be treated, while the general problem remains open.
Special Case 1: E1=E2=E and \(d^{+}_{A_{2}}(X)-d^{-}_{A_{1}}(X)+\hat{d}_{E}(X)\) is even for all X⊆V.
Characterization+Complexity: We show that the problem is in P and we provide a characterization.
TheMixed Eulerian Sandwich ProblemwithE1=E2=Eand\(d^{+}_{A_{2}}(X)-d^{-}_{A_{1}}(X)+\hat{d}_{E}(X)\)is even for allX⊆Vhas aYesanswer if and only if
In particular, this problem is in P.
By the result of Sect. 4.2, the answer is Yes if and only if there exists an orientation \(\vec{E}\) of E such that, ∀X⊆V, \(d^{-}_{A_{1}\cup\vec{E}}(X)\leq d^{+}_{A_{2}\cup\vec{E}}(X)\) or equivalently
Let m be the in-degree vector of \(\vec{E}\). Then \(d^{-}_{\vec{E}}(X)-d^{+}_{\vec{E}}(X)=\sum_{v\in X}(d^{-}_{\vec{E}}(v)-d^{+}_{\vec{E}}(v))=\sum_{v\in X}(2d^{-}_{\vec{E}}(v)-d_{E}(v))= 2m(X)-\hat{d}_{E}(X)\), and (12) becomes
Let \(b(X)=\frac{1}{2}(d^{+}_{A_{2}}(X)-d^{-}_{A_{1}}(X)+\hat{d}_{E}(X))\). Then b, being the sum of a modular function and a submodular function \((b(X)=\frac{1}{2}\sum_{v\in X}(d^{+}_{A_{1}}(v)-d^{-}_{A_{1}}(v)+d_{E}(v))+d^{+}_{A_{0}}(X))\), is a submodular function and, by the assumption, it is integer valued. By Theorem 3, an orientation \(\vec{E}\) satisfying (12) exists if and only if there exists a vector m such that i E (X)≤m(X)≤b(X), that is, by Claim 1 and Theorem 5, if and only if i E (X)≤b(X). This is equivalent to (11) and can be decided in polynomial time by Theorem 4, namely the submodular function b′(X)=b(X)−i E (X) must have minimum value 0. □
Special Case 2: E1=∅.
Characterization+Complexity: It is in P because it can be reformulated as the following problem: We create two copies of each edge in E2 and orient them in opposite directions. Denote this arc set by \(\overrightarrow{E^{2}_{2}}\). It is not difficult to see that the graph (V,E2∪A2) has a subgraph containing (V,A1) with an Eulerian orientation if and only if the graph \((V,\overrightarrow{E^{2}_{2}} \cup A_{2})\) has a directed Eulerian subgraph containing (V,A1). Indeed, in such a graph, if every edge of E2 is used at most once, we are done. If some edge of E2 is used twice, as two arcs in opposite directions, we can just remove these two arcs, the obtained graph remaining Eulerian and containing (V,A1). Now applying the result for Directed Eulerian Sandwich Problem we have
TheMixed Eulerian Sandwich ProblemwithE1=∅ has aYesanswer if and only if
Let D1=(V,A1) and \(D_{2} = (V, A_{2} \cup\overrightarrow{E^{2}_{2}})\). By the arguments above, the Mixed Eulerian Sandwich Problem with E1=∅ has a solution if and only if there is an Eulerian sandwich graph for D1 and D2 or equivalently, \(d^{-}_{D_{1}}(X) \leq d^{+}_{D_{2}}(X)\) for all X⊆V. By \(d^{+}_{D_{2}}(X) = d^{+}_{A_{2}}(X)+ d_{E_{2}}(X)\), we have \(d^{-}_{A_{1}}(X) -d^{+}_{A_{2}}(X) \leq d_{E_{2}}(X)\) for all X⊆V. Note that \(d_{E_{2}}(X)+d^{+}_{A_{2}}(X)-d^{-}_{A_{1}}(X)\) is a submodular function, and hence by Theorem 4, (14) can be verified in polynomial time. □
m-orientation sandwich problems
In this section we consider the sandwich problem where the property Π is to have an orientation of given in-degrees.
m-Orientation
m-Orientation Sandwich ProblemInstance: Given undirected graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2 and a non-negative integer vector m on V. Question: Does there exist a sandwich graph G=(V,E) (E1⊆E⊆E2) that has an orientation \(\vec{G}\) whose in-degree vector is m that is \(d^{-}_{\vec{G}}(v)=m(v)\) for all v∈V?
Characterization: We prove the following theorem.
The following assertions are equivalent.
Them-Orientation Sandwich Problemhas aYesanswer.
E1is independent in\(M_{\bar{m}}\)and\(M_{\bar{m}}\)has an independent set of sizem(V).
\(r_{\bar{m}}(E_{1}) = |E_{1}|\)and\(r_{\bar{m}}(E_{2})\geq m(V)\).
\(i_{E_{1}}(X) \leq m(X) \leq e_{E_{2}}(X)\)for allX⊆V.
(a) Implies (d). Let X⊆V. Since each edge of G1 in X contributes 1 to m(X), we have \(i_{E_{1}}(X) \leq m(X)\). On the other hand, the edges of G2 that have no end-vertex in X cannot contribute 1 to m(X), so we have \(m(X) \leq e_{E_{2}}(X)\).
(d) Implies (c). Let F be a subset of E1 and X=V(F). The condition \(i_{E_{1}}(X) \leq m(X)\) implies \(|F| \leq m(V(F)) = \bar{m}(F)\), that is, \(|E_{1}| \leq\bar{m}(F) + |E_{1}\setminus F|\). By Theorem 7, \(r_{\bar{m}}(E_{1}) \geq|E_{1}|\), or equivalently \(r_{\bar{m}}(E_{1}) = |E_{1}|\). Let now F be a subset of E2 and X=V∖V(F). The condition \(m(X) \leq e_{E_{2}}(X)\) implies that \(m(V) \leq m(V(F)) +e_{E_{2}}(V-V(F))\leq\bar{m}(F) + |E_{2}\setminus F|\). By Theorem 7, \(r_{\bar{m}}(E_{2}) \geq m(V)\).
(c) Implies (b). By definition.
(b) Implies (a). By (b), E1 is independent in \(M_{\bar{m}}\) and there exists an independent in \(M_{\bar{m}}\) of size m(V). Therefore, by (I3), there exists an independent set E of size m(V) that contains E1. By Theorem 3 and Claim 2, E is a solution of the m-orientation Sandwich Problem. □
We say that a subset F of E0 is feasible if (V,F∪E1) has an m-orientation. The next corollary of Theorem 11 characterizes the feasible sets.
If them-Orientation Sandwich Problemhas aYesanswer, then a subsetFofE0is feasible if and only ifFis a base of the matroid\(M_{\bar{m}}/E_{1}\).
Complexity: The condition (d) of Theorem 11 can be verified in polynomial time by Theorem 4, so the m-Orientation Sandwich Problem is in P.
Optimization: The minimum cost version of the problem can be solved in polynomial time. First, we find an optimal feasible subset F by greedy algorithm. Then we can orient the edges of F∪E1 using a known algorithm. (See [10] for example.)
Corollary 1 and the matroid intersection algorithm of Edmonds [4] imply that the two following simultaneous versions of the m-orientation Sandwich Problem are also in P.
Simultaneousm-Orientation Sandwich Problem 1Instance: Given two edge-disjoint undirected subgraphs G1=(V,E1) and G2=(V,E2) of an undirected graph G3=(V,E3) and two non-negative integer vectors m1 and m2 on V. Question: Do there exist simultaneously edge-disjoint sandwich graphs \(\hat{G}_{1}=(V,\hat{E}_{1})\)\((E_{1}\subseteq\hat{E}_{1}\subseteq E_{3})\) and \(\hat{G}_{2}=(V,\hat{E}_{2})\)\((E_{2}\subseteq\hat{E}_{2}\subseteq E_{3})\) such that \(\hat{G}_{i}\) has an orientation whose in-degree vector is m i for i∈{1,2}?
Note that the two input matroids for the matroid intersection algorithm must be taken as \((M^{G_{1}}_{\bar{m}_{1}}/E_{1})\setminus E_{2}\) and the dual matroid of \((M^{G_{2}}_{\bar{m}_{2}}/E_{2})\setminus E_{1}\).
Simultaneousm-Orientation Sandwich Problem 2Instance: Given two undirected subgraphs G1=(V,E1) and G2=(V,E2) of an undirected graph G3=(V,E3) and two non-negative integer vectors m1 and m2 on V. Question: Does there exist an edge set F in E3∖(E1∪E2) such that the graph G i =(V,E i ∪F) admits an orientation whose in-degree vector is m i for i∈{1,2}?
Strongly connected m-orientation
Strongly Connectedm-Orientation Sandwich ProblemInstance: Given undirected graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2 and a non-negative integer vector m on V. Question: Does there exist a sandwich graph G=(V,E) (E1⊆E⊆E2) that has a strongly connected orientation \(\vec{G}\) whose in-degree function is m?
Complexity: It is NP-complete because the special case E1=∅,m(v)=1∀v∈V is equivalent to decide if G2 has a Hamiltonian cycle.
(m1,m2)-orientation
(m1,m2)-Orientation Sandwich ProblemInstance: Given undirected graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2 and non-negative integer vectors m1 and m2 on V. Question: Does there exist a sandwich graph G=(V,E) (E1⊆E⊆E2) that has an orientation \(\vec{G}\) whose in-degree vector is m1 and whose out-degree vector is m2?
Complexity: The problem is NP-complete since it contains as a special case (E1=∅) the NP-complete problem of [19].
Mixed m-orientation
Mixedm-Orientation Sandwich ProblemInstance: Given mixed graphs G1=(V,E1∪A1) and G2=(V,E2∪A2) with E1⊆E2, A1⊆A2 and an non-negative integer vector m on V. Question: Does there exist a sandwich mixed graph G=(V,E∪A) with E1⊆E⊆E2 and A1⊆A⊆A2 that has an orientation \(\overrightarrow {G}=(V,\overrightarrow{E} \cup A)\) whose in-degree vector is m?
Characterization: Suppose that E1⊆E⊆E2 has been chosen and oriented, then the problem is reduced to the Directed Degree Constrained Sandwich Problem with \(m_{1}(v)=m(v) - d^{-}_{\overrightarrow{E}}(v)\) which, by Theorem 8, has a solution if and only if \(d^{-}_{A_{2}}(v) \geq m(v) -d^{-}_{\overrightarrow{E}}(v) \geq d^{-}_{A_{1}}(v)\) for all v∈V. Hence the Mixedm-orientation Sandwich Problem has a solution if and only if there exists E1⊆E⊆E2 which admits an orientation \(\overrightarrow{E}\) with \(m(v)-d^{-}_{A_{1}}(v) \geq d^{-}_{\overrightarrow{E}}(v) \geq m(v)-d^{-}_{A_{2}}(v)\) for all v∈V. Let m2:V→ℤ satisfy \(m(v)-d^{-}_{A_{2}}(v)\leq m_{2}(v)\leq m(v)-d^{-}_{A_{1}}(v)\). By Theorem 11, there exists E1⊆E⊆E2 which admits an orientation \(\overrightarrow {E}\) with \(d^{-}_{\overrightarrow{E}}(v) = m_{2}(v)\) if and only if \(i_{E_{1}}(X) \leq m_{2}(X) \leq e_{E_{2}}(X)\) for all X⊆V. Therefore we have
TheMixedm-orientation Sandwich Problemhas aYesanswer if and only if there exists an integer-valued functionm2:V→ℤ such that, ∀v∈Vand ∀X⊆V,
The pair\((i_{E_{1}}, e_{E_{2}})\)is a strong pair.
Let X,Y,Z be three pairwise disjoint subset of V. We show that \(e_{E_{2}}(X\cup Z)-i_{E_{1}}(Y\cup Z) \geq e_{E_{2}}(X) - i_{E_{1}}(Y)\). In fact, we have \(i_{E_{1}}(Y\cup Z) - i_{E_{1}}(Y) = i_{E_{1}}(Z) +d_{E_{1}}(Y,Z) \leq i_{E_{2}}(Z) + d_{E_{2}}(Y,Z)\), and \(e_{E_{2}}(X\cup Z) -e_{E_{2}}(X) = i_{E_{2}}(Z) + d_{E_{2}}(Z) - d_{E_{2}}(X,Z)\). As X,Y,Z are pairwise disjoint, \(d_{E_{2}}(Y,Z) + d_{E_{2}}(X,Z) \leq d_{E_{2}}(Z)\). The claim follows by Claim 1. □
By Claim 3, 4 and Theorem 6 applied for \(\alpha(v)= m(v)-d^{-}_{A_{2}}(v), \beta (v)=m(v)-d^{-}_{A_{1}}(v), p=i_{E_{1}}, b=e_{E_{2}}\), we have
TheMixedm-orientation Sandwich Problemhas aYesanswer if and only if
for every subsetXofV.
Note that Theorem 12 implies Theorems 8 and 11.
Complexity: The condition (15) can be verified in polynomial time by Theorem 4. If it is satisfied, then a vector m2 satisfying the conditions in Claim 3 can be found using a greedy algorithm for g-polymatroids. Then we find and orient an edge set E (E1⊆E⊆E2) with in-degree m2 (m-orientation Sandwich Problem). Last, we choose an arc set A (A1⊆A⊆A2) such that \(d^{-}_{A}(v)=m_{1}(v)=m(v)-m_{2}(v)\), for all v∈V (Directed Degree Constrained Sandwich Problem).
Contracting sandwich problems
In this section, we propose to consider a new type of sandwich problem. Instead of deleting edges from the optional graph, we are interested in contracting edges. We solve the problem for the property Π being a bipartite graph.
Contracting Sandwich ProblemInstance: Given an undirected graph G=(V,E) and E0⊆E. Question: Does there exist F⊆E0 such that contracting F results in a bipartite graph?
Complexity: Since a graph is bipartite if and only if all its cycles have an even length, the problem is equivalent to finding F⊆E0 such that, for all cycles C, |C∩F|≡|C| mod 2.
Fix a spanning forest T of G. For e∈E∖T, denote C(T,e) the unique cycle contained in T∪e. By [18, Theorem 9.1.2], if C is a cycle of G then C=Δe∈CC(T,e), where Δ denotes the symmetric difference of sets. Therefore, |C∩F|≡∑e∈C|C(T,e)∩F| mod 2. Let \(\mathcal{C}_{T}\) denote the collection of cycles C(T,e) of G. The problem is reduced to finding F⊆E0 such that, for all \(C\in\mathcal{C}_{T}\), |C∩F|≡|C| mod 2, or equivalently, finding an F′(=E∖F)⊇E1=E∖E0 such that |F′∩C|≡0 mod 2, for all \(C\in\mathcal{C}_{T}\).
Consider now the matrix M defined as the following. The rows of M correspond to \(C \in\mathcal{C}_{T}\) and the columns correspond to the edges of G; the entry M Ce is 1 if e∈C and is 0 otherwise. For X⊆E, let χ X denote the characteristic vector of X. For a vector x∈{0,1}E, let x|X denote the projection of x on X. Let 1 be the all-one vector in {0,1}E. A subset F′⊆E satisfies |F′∩C|≡0 mod 2, for all \(C\in\mathcal{C}_{T}\), if and only if χF′∈KerM in \(\mathbb{F}_{2}\). Such an F′ is the solution of the Contracting Sandwich Problem if and only if \(\chi_{F'|E_{1}} = \mathbf {1}_{|E_{1}}\).
Let B be a basis of the kernel of M in \(\mathbb{F}_{2}\). (This can be computed in polynomial time using the Gauss elimination.) Consider the projections B′ of B on E1. Then the Contracting Sandwich Problem has a solution if and only if \(\mathbf{1}_{|E_{1}}\) is in the subspace of \(\{0,1\}^{E_{1}}\) spanned by B′, that is, \(\mathrm{rank} B'= \mathrm{rank} B'\cup\mathbf{1}_{|E_{1}}\). This can be decided in polynomial time using the Gauss elimination. We conclude that the Contracting Sandwich Problem is in P.
We finish with a related problem. For a fixed integer k, solving the Contracting Sandwich Problem when E0=E with extra requirement |F|≤k is known to be tractable in polynomial time [14]. However, the authors mention that finding a solution of minimum cardinality is NP-complete.
Dantas S, Klein S, Mello CP, Morgana A (2009) The graph sandwich problem for P4-sparse graphs. Discrete Math 309:3664–3673
Dantas S, Figueiredo CMH, Golumbic MC, Klein S, Maffray F (2011) The chain graph sandwich problem. Ann Oper Res 188:133–139
Dourado M, Petito P, Teixeira RB, Figueiredo CMH (2008) Helly property, clique graphs, complementary graph classes, and sandwich problems. J Braz Comput Soc 14:45–52
Edmonds J (1977) Matroid intersection. Discrete Optim 4:39–49 (Proc Adv Res Inst Discrete Optimization and Systems Appl, Banff, Alta)
Edmonds J, EL Johnson (1973) Matching, Euler tours and the Chinese postman. Math Program 5:88–124
Article MATH Google Scholar
Even S, Itai A, Shamir A (1976) On the complexity of timetable and multicommodity flow problems. SIAM J Comput 5:691–703
Figueiredo CMH, Faria L, Klein S, Sritharan R (2007) On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs. Theor Comput Sci 38:57–67
Ford LR, Fulkerson DR (1962) Flows in Networks. Princeton University Press, Princeton
MATH Google Scholar
Frank A (1982) An algorithm for submodular functions on graphs. Ann Discrete Math 16:97–120
Frank A, Gyárfás A (1976) How to orient the edges of a graph. Colloq Math Soc János Bolyai 18:353–364
Frank A, Tardos E (1988) Generalized polymatroids and submodular flows. Submodular optimization. Math Program 42(3):489–563
Golumbic MC, Kaplan H, Shamir R (1995) Graph sandwich problems. J Algorithms 19(3):449–473
Hakimi SL (1965) On the degrees of the vertices of a directed graph. J Franklin Inst 279(4):290–308
Heggernes P, van't Hof P, Lokshtanov D, Paul C (2011) Obtaining a bipartite graph by contracting few edges. In: Proceedings of IARCS annual conference on foundations of software technology and theoretical computer science, pp 217–228
Hoffman AJ (1960) Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. Proc Symp Appl Math 10:113–127
Holyer I (1981) The NP-completeness of edge-coloring. SIAM J Comput 10(4):718–720
Iwata S, Fleischer L, Fujishige S (2001) A combinatorial strongly polynomial algorithm for minimizing submodular functions. J ACM 48(4):761–777
Oxley JG (1992) Matroid theory. Oxford University Press, New York
Pálvölgyi D (2009) Deciding soccer scores and partial orientations of graphs. Acta Univ Sapientiae, Math 1(1):35–42
Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Algorithms and combinatorics, vol 24. Springer, Berlin
Schrijver A (2000) A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J Comb Theory, Ser B 80(2):346–355
Sritharan R (2008) Chordal bipartite completion of colored graphs. Discrete Math 308:2581–2588
Tardos E (1985) A strongly polynomial minimum cost circulation algorithm. Combinatorica 5(3):247–255
Teixeira RB, Dantas S, Figueiredo CMH (2010) The polynomial dichotomy for three nonempty part sandwich problems. Discrete Appl Math 158:1286–1304
Tutte WT (1952) The factors of graphs. Can J Math 4:314–328
This research was partially supported by CNPq, CAPES (Brazil)/COFECUB (France), FAPERJ.
Laboratoire G-SCOP, CNRS, Grenoble INP, UJF, 46, Avenue Félix Viallet, Grenoble, 38000, France
Olivier Durand de Gevigney, Viet-Hang Nguyen & Zoltán Szigeti
Instituto de Matemática and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
Olivier Durand de Gevigney
Viet-Hang Nguyen
Zoltán Szigeti
Correspondence to Olivier Durand de Gevigney.
Durand de Gevigney, O., Klein, S., Nguyen, VH. et al. Sandwich problems on orientations. J Braz Comput Soc 18, 85–93 (2012). https://doi.org/10.1007/s13173-012-0065-7
Sandwich problems
Orientation of graphs
Submodular flows | CommonCrawl |
\begin{document}
\vspace*{0mm}
\title[Filtrations of the knot concordance group]{Amenable signatures, algebraic solutions, and filtrations of the knot concordance group}
\author{Taehee Kim} \address{
Department of Mathematics\\
Konkuk University \\
Seoul 05029\\
Korea } \email {[email protected]} \protect\thanks@warning{This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (no.~2011-0030044(SRC-GAIA) and no.~2015R1D1A1A01056634).}
\dedicatory{Dedicated to the memory of Tim D. Cochran}
\def\textup{2010} Mathematics Subject Classification{\textup{2010} Mathematics Subject Classification} \expandafter\let\csname subjclassname@1991\endcsname=\textup{2010} Mathematics Subject Classification \expandafter\let\csname subjclassname@2000\endcsname=\textup{2010} Mathematics Subject Classification \subjclass{57M25, 57N70
} \keywords{Knot, Concordance, Grope, $n$-solution, Algebraic $n$-solution, Amenable signature}
\begin{abstract} It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling operators. In this paper, for each of the successive quotients of the filtrations we give a new infinite rank subgroup which trivially intersects the previously known infinite rank subgroups. Instead of iterated doubling operators, the generating knots of these new subgroups are constructed using the notion of algebraic $n$-solutions, which was introduced by Cochran and Teichner. Moreover, for each slice knot $K$ whose Alexander polynomial has degree greater than 2, we construct the generating knots such that they have the same derived quotients and higher-order Alexander invariants up to a certain order.
In the proof, we use an $L^2$-theoretic obstruction for a knot to being $n.5$-solvable given by Cha, which is based on $L^2$-theoretic techniques developed by Cha and Orr. We also generalize the notion of algebraic $n$-solutions to the notion of $R$-algebraic $n$-solutions where $R$ is either rationals or the field of $p$ elements for a prime $p$.
\end{abstract}
\maketitle
\section{Introduction}
In this paper, we address the structure of the grope and solvable filtrations of the knot concordance group. Two oriented knots $K_0$ and $K_1$ in the 3-sphere $S^3$ are {\it concordant} if there is a smoothly and properly embedded annulus in $S^3\times [0,1]$ whose boundary is the union of $K_0\times\{0\}$ and $-K_1\times \{1\}$. It is known that $K_0$ and $K_1$ are concordant if and only if the connected sum $K_0\# (-K_1)$ bounds a smoothly embedded disk in the 4-ball $D^4$, namely, $K_0\# (-K_1)$ is a {\it slice knot}. Concordance classes form an abelian group under connected sum, which is called {\it the knot concordance group}, and we denote it by $\mathcal{C}$.
The notion of concordance on knots was introduced by Fox and Milnor in the 1950's. In the 1960's Levine classified the algebraic concordance group \cite{Levine:1969-1,Levine:1969-2}, and in the 1970's Casson and Gordon showed that the surjection from $\mathcal{C}$ to the algebraic concordance group is not injective \cite{Casson-Gordon:1986-1}.
In the late 1990's, Cochran, Orr, and Teichner \cite{Cochran-Orr-Teichner:1999-1} introduced the grope and solvable filtrations of the knot concordance group denoted by $\{\mathcal{G}_n\}$ and $\{\mathcal{F}_n\}$, respectively, which are indexed by nonnegative half-integers. The subgroup $\mathcal{G}_n$ consists of knots bounding a {\it grope} of height $n$ in $D^4$, where a grope is a certain 2-complex constructed by attaching surfaces along their boundaries. See Definition~\ref{def:grope} for a precise definition of a grope. Similarly, $\mathcal{F}_n$ is the subgroup of knots such that the zero-framed surgery on the knot bounds an {\it $n$-solution}, where an $n$-solution is a 4-manifold satisfying certain conditions on the (equivariant) intersection form with twisted coefficients (see Definition~\ref{def:n-solution}). For all $n$ it is known that $\mathcal{G}_{n+2}\subset \mathcal{F}_n$ \cite[Theorem~8.11]{Cochran-Orr-Teichner:1999-1}, and one may consider an $n$-solution as an order $n$ approximation of the exterior of a slice disk in $D^4$. These filtrations reflect classical invariants at low levels. For instance, a knot $K$ has vanishing Arf invariant if and only if $K\in \mathcal{F}_0$, and $K$ is algebraically slice if and only if $K\in \mathcal{F}_{0.5}$ \cite{Cochran-Orr-Teichner:1999-1}. Furthermore, a knot in $\mathcal{F}_{1.5}$ has vanishing Casson--Gordon invariants \cite{Cochran-Orr-Teichner:1999-1}, but it is known that there exists a knot with vanishing Cassson--Gordon invariants which is not in $\mathcal{F}_{1.5}$ \cite{Kim:2004-1}.
In this paper, for each of the successive quotients of the grope and solvable filtrations we give a new infinite rank subgroup which trivially intersects the previously known infinite rank subgroups.
We recall results on finding infinite rank subgroups of the successive quotients $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ and $\mathcal{F}_n/\mathcal{F}_{n.5}$ for $n\ge 2$. We are interested in the cases for $n\ge 2$ since the cases for $n\le 2$ were well-known from classical invariants. For finding infinite rank subgroups of the quotients, there are three different approaches using the following: 1) rationally universal solvable representations; 2) algebraic $n$-solutions; 3) iterated doubling operators. In this paper, by an iterated doubling operator we mean any of iterated (generalized) doubling operators used in \cite{Cochran-Harvey-Leidy:2009-1, Cochran-Harvey-Leidy:2009-2,Cha:2010-1,Horn:2010-1}. For a more precise definition of an iterated doubling operator, see the proof of Theorem~\ref{thm:refined_main_theorem-2}.
First, in their seminal papers \cite{Cochran-Orr-Teichner:1999-1,Cochran-Orr-Teichner:2002-1}, using the von Neumann--Cheeger--Gromov $\rho^{(2)}$-invariants associated to {\it rationally universal solvable representations}, Cochran, Orr, and Teichner showed that $\mathcal{F}_2/\mathcal{F}_{2.5}$ has an infinite rank subgroup, giving the first example of nonslice knots with vanishing Casson--Gordon invariants. For each integer $n\ge 2$, Cochran and Teichner showed that $\mathcal{F}_n/\mathcal{F}_{n.5}$ and $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ are infinite and have positive rank using Cheeger--Gromov's universal bound on $\rho^{(2)}$-invariants and their new notion of an {\it algebraic $n$-solution} \cite{Cochran-Teichner:2003-1}. Later Cochran--Teichner's work was refined further by Cochran and the author \cite{Cochran-Kim:2004-1}. Also, in \cite{Cochran-Kim:2004-1} the notion of an algebraic $n$-solution was generalized.
Instead of algebraic $n$-solutions, using {\it iterated doubling operators} and unlocalized higher-order Blanchfield linking forms Cochran, Harvey, and Leidy obtained the first example of an infinite rank subgroup of $\mathcal{F}_n/\mathcal{F}_{n.5}$ for each integer $n\ge 2$ \cite{Cochran-Harvey-Leidy:2009-1}. They extended their work further and showed that the solvable filtration $\{\mathcal{F}_n\}$ has refined filtrations related to primary decomposition whose successive quotient groups have infinite rank \cite{Cochran-Harvey-Leidy:2009-2}. For the grope filtration, Horn \cite{Horn:2010-1} proved that $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ has an infinite rank subgroup for each integer $n\ge 2$, where the generating knots of the subgroups were constructed using iterated doubling operators.
In all of the above work, knots were obstructed to being in $\mathcal{F}_{n.5}$ (hence not in $\mathcal{G}_{n+2.5}$) using the $\rho^{(2)}$-invariants associated to a representation mapping to a poly-torsion-free-abelian (henceforth PTFA) group. By a PTFA group, we mean a group which allows a subnormal series whose successive quotient groups are torsion-free and abelian. These obstructions are essentially based on the vanishing criterion of the $\rho^{(2)}$-invariants associated to a PTFA representation given by Cochran--Orr--Teichner \cite[Theorem 4.2]{Cochran-Orr-Teichner:1999-1}. For the reader's convenience, we review the $\rho^{(2)}$-invariant in Subsection~\ref{subsec:amenable signature}, where the $\rho^{(2)}$-invariant is defined to be an $L^2$-signature defect.
In \cite{Cha-Orr:2009-1}, Cha and Orr developed $L^2$-theoretic methods: for a representation mapping to a group which is amenable and lies in Strebel's class $D(R)$ for some commutative ring $R$, they proved the homology cobordism invariance of the $L^2$-Betti numbers and the $\rho^{(2)}$-invariants, and presented a method for controlling the $L^2$-dimension of homology with $L^2$-coefficients. (The reader may refer to \cite{Cha-Orr:2009-1} for definitions of amenable groups and Strebel's class $D(R)$, but the definitions will not be needed in this paper.)
Based on the work in \cite{Cha-Orr:2009-1}, Cha found a vanishing criterion of $\rho^{(2)}$-invariants for $n.5$-solvable knots in \cite[Theorem~4.2]{Cochran-Orr-Teichner:1999-1} to include the vanishing of the $\rho^{(2)}$-invariants associated to a representation mapping to a group which is amenable and lies in Strebel's class $D(R)$ where $R=\mathbb{Q}$ or $\mathbb{Z}_p$ for a prime $p$ \cite[Therorem 1.3]{Cha:2010-1}. In this paper, we call this extended vanishing criterion {\it Amenable Signature Theorem (for $n.5$-solvability)} (see Theorem~\ref{thm:obstruction}). We note that the above class of groups includes PTFA groups and some groups with torsion elements (see Lemma~\ref{lem:amenable and D(R)}). Using Amenable Signature Theorem~\ref{thm:obstruction}, Cha constructed an infinite rank subgroup of $\mathcal{F}_n/\mathcal{F}_{n.5}$ for each integer $n\ge 2$ for which the $\rho^{(2)}$-invariants associated to a PTFA representation vanish \cite[Theorem 1.4]{Cha:2010-1}. The generating knots of the infinite rank subgroups in \cite{Cha:2010-1} were constructed also using iterated doubling operators.
In this paper, we combine algebraic $n$-solutions and Amenable Signature Theorem to find infinite rank subgroups of the grope and solvable filtrations, and give Theorem~\ref{thm:main} below. In the following, $\Delta_K(t)$ denotes the Alexander polynomial of a knot $K$. For a group $G$, let $G^{(0)}:=G$, and inductively for $n\ge 0$ define $G^{(n+1)}:=[G^{(n)},G^{(n)}]$. The group $G^{(n)}$ is called the {\it $n$-th derived group} of $G$. Also recall that $\mathcal{G}_{n+2}\subset \mathcal{F}_n$ for all~$n$.
\begin{theorem}\label{thm:main} Let $n\ge 2$ be an integer. Let $K$ be a slice knot with $\deg \Delta_K(t) > 2$. Then there is a sequence of knots $K_1,K_2,\ldots$ satisfying the following. \begin{enumerate}
\item For each $i$, there is an isomorphism
\[
\pi_1(S^3\smallsetminus K_i)/\pi_1(S^3\smallsetminus K_i)^{(n+1)} \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(S^3\smallsetminus K)/\pi_1(S^3\smallsetminus K)^{(n+1)}
\]
which preserves the peripheral structures.
\item The $K_i$ are in $\mathcal{G}_{n+2}$ and they are linearly independent modulo $\mathcal{F}_{n.5}$. In particular, the knots $K_i$ generate an infinite rank subgroup in each of $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ and $\mathcal{F}_n/\mathcal{F}_{n.5}$.
\item In $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$, the infinite rank subgroup generated by the $K_i$ trivially intersects the infinite rank subgroup in \cite{Horn:2010-1}.
\item In $\mathcal{F}_n/\mathcal{F}_{n.5}$, the infinite rank subgroup generated by the $K_i$ trivially intersects the infinite rank subgroups in \cite{Cochran-Harvey-Leidy:2009-1, Cochran-Harvey-Leidy:2009-2,Cha:2010-1}. \end{enumerate}
\end{theorem} In Theorem~\ref{thm:main}, the condition $\deg \Delta_K(t)>2$ is best possible; by the work of Friedl and Teichner \cite{Friedl-Teichner:2005-1}; Theorem~\ref{thm:main} does not hold for any $n\ge 2$ for a certain slice knot $K$ with $\deg \Delta_K(t)=2$ (also refer to \cite[Proposition~5.10]{Cochran-Kim:2004-1}). Theorem~\ref{thm:main} is an immediate consequence of Theorems~\ref{thm:refined main theorem-1} and \ref{thm:refined_main_theorem-2}. As in Theorems~\ref{thm:main}(3) and (4), the infinite rank subgroups generated by the $K_i$ trivially intersect the previously known infinite rank subgroups, which were constructed using iterated doubling operators.
An advantage of constructing the $K_i$ using not iterated doubling operators but algebraic $n$-solutions is that Theorem~\ref{thm:main}(1) is obtained for \emph{any} slice knot with $\deg \Delta_K(t)> 2$. Theorem~\ref{thm:main}(1) is related to studying the structure on knot concordance under a fixed Seifert form. It is well-known by the work of Freedman \cite{Freedman:1982-1,Freedman-Quinn:1990-1} that a knot with trivial Alexander polynomial is topologically slice, and hence determines a unique topological concordance class, namely, the class of topologically slice knots. It was asked if there is any other Alexander polynomial or a Seifert form which determines a unique topological concordance class. This question was answered negative that for each Seifert form of a knot $K$ with nontrivial Alexander polynomial, there exists infinitely many mutually topologically nonconcordant knots $K_i$ having the Seifert form \cite{Livingston:2002-1,Kim:2005-1}. (Also refer to \cite{Kim:2017-1}.) This result was refined further that in \cite[Theorem~5.1]{Cochran-Kim:2004-1} it was shown under the condition $\deg \Delta_K(t)>2$ that for each $n\ge 2$ those mutually nonconcordant knots $K_i$ can be constructed using algebraic $n$-solutions such that the $K_i$ are mutually distinct in $\mathcal{F}_n/\mathcal{F}_{n.5}$ and satisfies the property in Theorem~\ref{thm:main}(1). Furthermore, in \cite{Cochran-Kim:2004-1} it was shown that the $K_i$ have the same \emph{$m$th order Seifert presentation} (see \cite[Definitions~5.5 and 5.9]{Cochran-Kim:2004-1}) as $K$ for $m=0,1,\ldots, n-1$, and hence have the same Seifert form. In this paper, in Theorem~\ref{thm:main}, we refine it further and construct the $K_i$ such that they are linearly independent in $\mathcal{F}_n/\mathcal{F}_{n.5}$, still satisfying the condition in Theorem~\ref{thm:main}(1). We also note that the $K_i$ in Theorem~\ref{thm:main} also have the same $m$th order Seifert presentation for $m=0,1,\ldots, n-1$. (This can be easily shown using the same arguments in the proof of \cite[Theorem~5.1]{Cochran-Kim:2004-1} and will not be discussed in this paper.)
We also refer the reader to \cite{Kim:2006-1} and \cite{Cha-Kim:2008-1} for more applications of algebraic $n$-solutions to doubly slice knots and the solvable filtration of the \emph{rational} knot concordance group.
We construct the $K_i$ in Theorem~\ref{thm:main} using a well-known process called {\it infection} or {\it satellite construction} which involve a {\it seed knot}, {\it axes (knots)}, and {\it infection knots} (see Section~\ref{sec:construction}). Then, we show that the $K_i$ are linearly independent modulo $\mathcal{F}_{n.5}$ using Amenable Signature Theorem~\ref{thm:obstruction} following the ideas in \cite{Cha:2010-1}. We give more details: let $J$ be a nontrivial linear combination of the $K_i$. For an $n$-solution $V$ for~$J$, as obstructions for $V$ to being an $n.5$-solution, we use the $\rho^{(2)}$-invariants associated to a representation factoring through $\pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1W/\mathcal{P}^{n+1}\pi_1W$ where $W$ is a certain 4-manifold containing $V$. Here, $\mathcal{P}^{n+1}\pi_1W$ is a subgroup of $\pi_1W$ obtained from a certain {\it mixed-coefficient commutator series} of $\pi_1W$ which is defined depending on the choice of a prime $p$. See Definition~\ref{def:commutator series} for a precise definition of $\mathcal{P}^{n+1}\pi_1W$. We remark that as can be seen in the proof of Theorem~\ref{thm:refined main theorem-1} the choice of a prime $p$ is specific to the linear combination $J$. Namely, we can choose a representation specific to $J$ and use the corresponding $\rho^{(2)}$-invariant, and this makes easier to show that $J$ is not trivial modulo $\mathcal{F}_{n.5}$. In the representation, the quotient group $\pi_1W/\mathcal{P}^{n+1}\pi_1W$ has torsion elements, hence is not a PTFA group. But it is amenable and lies in Strebel's class $D(\mathbb{Z}_p)$, and this fact enables us to use Amenable Signature Theorem~\ref{thm:obstruction} and obtain desired computations of $\rho^{(2)}$-invariants.
When we construct the $K_i$ using infection, we need to choose the axes in a subtle way: for a given slice knot $K$ with $\deg \Delta_K(t) >2$, we use $K$ as a seed knot and choose axes $\eta_i$, $1\le i \le m$, such that for each homomorphism on $\pi_1$ induced from the inclusion $M(K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} W$ where $M(K)$ is the 0-surgery on $K$ and $W$ is an $n$-solution for $K$, the axes $\eta_i$ should satisfy a certain nontriviality property under the homomorphism. The desired nontriviality property and existence of such axes are given in Theorem~\ref{thm:eta_i}, which is a key technical theorem in this paper. Roughly speaking, the nontriviality property requires that for a given $n$-solution $W$ and $\mathcal{P}^{n+1}\pi_1W $, there exists some $\eta_i$ such that $\eta_i\notin \mathcal{P}^{n+1}\pi_1W $. To prove Theorem~\ref{thm:eta_i} we generalize the notion of an algebraic $n$-solution in \cite{Cochran-Teichner:2003-1,Cochran-Kim:2004-1} and define the notion of an {\it $R$-algebraic $n$-solution} where $R=\mathbb{Q}$ or $\mathbb{Z}_p$ for a prime $p$ (see Definition~\ref{def:algebraic n-solution}). We also use a nontriviality theorem on homology with twisted coefficients which is obtained using higher-order Blanchfield linking forms (see Theorem~\ref{thm:nontrivial} and the proof of Theorem~\ref{thm:eta_i}).
We prove Theorems~\ref{thm:main}(3) and (4) in Theorem~\ref{thm:refined_main_theorem-2} following the ideas in \cite[Section 9]{Cochran-Harvey-Leidy:2009-2}. For each positive half-integer $n$, a prime $p$, and a group $G$, we define a subgroup $G^{(n)}_{cot,p}$ of $G$ using a localization of a group ring (see Subsection~\ref{subsec:distinction}). Then, using it we define the notion of \emph{$(n,p)$-solvable knots} (Definition~\ref{def:p-F_n}). We note that $G^{(n)}\subset G^{(n)}_{cot,p}$ for all prime $p$, and an $n$-solvable knot is $(n,p)$-solvable for all prime $p$. Finally, we show that a nontrivial linear combination of the $K_i$ in Theorem~\ref{thm:main} is not $(n.5,p)$-solvable for some prime $p$, but a knot concordant to a linear combination of the knots in \cite{Horn:2010-1,Cochran-Harvey-Leidy:2009-1, Cochran-Harvey-Leidy:2009-2,Cha:2010-1} is $(n.5,p)$-solvable for all prime $p$.
Other than finding infinite rank subgroups of $\mathcal{F}_n/\mathcal{F}_{n.5}$ and $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$, there are many interesting results on the grope and solvable filtrations \cite{Kim-Kim:2008-1,Cochran-Harvey-Leidy:2009-2,Cochran-Harvey-Leidy:2009-3,Burke:2014-1,Davis:2014-1,Kim-Kim:2014-1,Jang:2015-1}. For instance, it is known that for each integer $n\ge 2$ and for each of $\mathcal{F}_n/\mathcal{F}_{n.5}$ and $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ there exists a subgroup infinitely generated by knots of order 2 \cite{Cochran-Harvey-Leidy:2009-3,Jang:2015-1}. These knots are constructed using iterated doubling operators, and it is unknown whether or not one can construct a subgroup infinitely generated by knots of order 2 using algebraic $n$-solutions.
We can also define the grope and solvable filtrations $\{\mathcal{G}_n^{top}\}$ and $\{F_n^{top}\}$ of the topological knot concordance group \cite{Cochran-Orr-Teichner:1999-1}. We note that all the results in this paper also hold under this topological setting; in this paper all the examples of manifolds and gropes are constructed smoothly, and the obstructions obtained from Amenable Signature Theorem~\ref{thm:obstruction} are topological obstructions. Moreover, by the work of Freedman and Quinn \cite{Freedman-Quinn:1990-1}, it is known that in fact the smooth and topological filtrations are equivalent, that is, $K\in \mathcal{G}_n$ (resp. $K\in \mathcal{F}_n$) if and only $K\in \mathcal{G}_n^{top}$ (resp. $K\in \mathcal{F}_n^{top}$), see \cite[Remark~2.19]{Cha:2012-1} and \cite[p.454]{Cochran-Harvey-Leidy:2009-2}.
This paper is organized as follows. In Section~\ref{sec:preliminaries}, we introduce the grope and solvable filtrations of the knot concordance group and give Amenable Signature Theorem~\ref{thm:obstruction}. In Section~\ref{sec:construction}, we discuss the construction of examples using infection (or satellite construction). We prove Theorem~\ref{thm:main} in Section~\ref{sec:infinite rank subgroup}, and discuss higher-order Blanchfield linking forms and the notion of $R$-algebraic $n$-solutions in Section~\ref{sec:Blanchfield linking form and algebraic n-solutions}.
In this paper, manifolds are assumed to be smooth, compact, and oriented, and $\mathbb{Z}_p$ denotes the field of $p$ elements for a prime $p$. By abuse of notation we use the same symbol for a knot and its homotopy class and homology classes. Homology groups come with integer coefficients unless specified otherwise. For a knot $K$, we denote by $E(K)$ and $M(K)$ the exterior of $K$ in $S^3$ and the 0-surgery on $K$ in $S^3$, respectively.
\section{Preliminaries}\label{sec:preliminaries} In this section, we review the grope and solvable filtrations of the knot concordance group, and recall necessary results on amenable signatures and mixed-coefficient commutator series of a group in \cite{Cha:2010-1}.
\subsection{The grope and solvable filtrations}\label{subsec:filtration} In this subsection we review the notions of a grope, an $n$-solution, an $n$-cylinder and the grope and solvable filtrations of the knot concordance group.
\begin{definition}\cite{Freedman-Teichner:1995-1}\cite[Definition 7.9]{Cochran-Orr-Teichner:1999-1}\label{def:grope}
A {\it grope of height 1} is a compact connected surface with a single boundary component. This boundary component is called {\it the base circle}. Let $\Sigma$ be a grope of height 1 of genus $g$. Let $\{\alpha_i, \beta_i\}_{1\le i\le g}$ be a standard symplectic basis of circles on $\Sigma$ such that $\alpha_i$ and $\beta_i$ are dual to each other. For an integer $n\ge 1$, a {\it grope of height $n+1$} is a 2-complex obtained by attaching gropes of height $n$ to each $\alpha_i$ and $\beta_i$ along the base circles. A {\it grope of height $n.5$} is a 2-complex obtained by attaching gropes of height $n$ to each $\alpha_i$ and gropes of height $n-1$ to each $\beta_i$ along the base circles. Here, the surface $\Sigma$ is called {\it the bottom stage} of the grope, and a grope of height 0 is understood to be the empty set. For a grope embedded in a 4-manifold, we require the grope has a neighborhood which is diffeomorphic to the product of $\mathbb{R}$ and the standard neighborhood of the (abstract) grope in $\mathbb{R}^3$. In this paper, a grope in a 4-manifold means a grope smoothly embedded in the 4-manifold in this way. We also note that in the literature a grope defined in this way is called a {\it symmetric grope}. \end{definition}
Let $\mathbb{N}_0 = \mathbb{N}\cup \{0\}$. For each $n\in \frac12 \mathbb{N}_0$, we denote by $\mathcal{G}_n$ the subset of $\mathcal{C}$ which consists of knots bounding a grope of height $n$ in $D^4$. It is known that each $\mathcal{G}_n$ is a subgroup of $\mathcal{C}$ and $\mathcal{G}_m\subset \mathcal{G}_n$ for $m\ge n$, and hence $\{\mathcal{G}_n\}$ is a filtration of $\mathcal{C}$. We call it the {\it grope filtration} of $\mathcal{C}$.
For a group $G$, recall that $G^{(n)}$ denotes the $n$-th derived group of $G$. Now for a 4-manifold $W$, let $\pi:=\pi_1W$ and let $R:=\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{Z}_p$. Then for each $n\ge 0$, there is the equivariant intersection form \[ \lambda_n^R\colon H_2(W;R[\pi/\pi^{(n)}]) \times H_2(W;R[\pi/\pi^{(n)}]) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} R[\pi/\pi^{(n)}] . \] We drop the decoration $R$ from $\lambda_n^R$ when it is understood from the context. Below, we generalize the notions of an $n$-cylinder and a rational $n$-cylinder in \cite{Cochran-Kim:2004-1}.
\begin{definition}(\cite[Section 2]{Cochran-Kim:2004-1} for $R=\mathbb{Z}$ and $\mathbb{Q}$) \label{def:n-cylinder}
Let $R=\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{Z}_p$. Let $n$ be a nonnegative integer. Let $W$ be a compact connected 4-manifold with $\partial W=\coprod_{i=1}^\ell M_i$ where each $M_i$ is a connected component with $H_1(M_i)\cong R$. Let $\pi:=\pi_1W$ and $r=\frac12 \operatorname{rank}_R\operatorname{Coker}\{H_2(\partial W;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(W;R)\}$.
\begin{enumerate}
\item $W$ is an {\it $R$-coefficient $n$-cylinder} if each inclusion from $M_i$ to $W$ induces an isomorphism on $H_1(M_i;R)$ and there exist $x_1, x_2, \ldots, x_r$ and $y_1, y_2, \ldots, y_r$ in $H_2(W;R[\pi/\pi^{(n)}])$ such that $\lambda_n(x_i,x_j)=0$ and $\lambda_n(x_i,y_j)=\delta_{ij}$ for $1\le i,j\le r$.
\item $W$ is an {\it $R$-coefficient $n.5$-cylinder} if $W$ satisfies (1), and furthermore there exist lifts $\tilde{x}_1,\tilde{x}_2,\ldots, \tilde{x}_r$ of $x_1,x_2,\ldots, x_r$ in $H_2(W;R[\pi/\pi^{(n+1)}])$ such that $\lambda_{n+1}(\tilde{x}_i, \tilde{x}_j)=0$ for $1\le i,j\le r$.
\end{enumerate}
We also require $W$ to be spin when $R=\mathbb{Z}$. A $\mathbb{Z}$-coefficient $n$-cylinder is also called an {\it $n$-cylinder}. The submodule generated by $x_1, x_2,\ldots, x_r$ (resp. $\tilde{x}_1,\tilde{x}_2,\ldots, \tilde{x}_r$) is called an {\it $n$-Lagrangian} (resp. {\it $(n+1)$-Lagrangian}), and the submodule generated by $y_1,y_2, \ldots, y_r$ is called its {\it $n$-dual}. \end{definition}
It is obvious that an $n$-cylinder is a $\mathbb{Q}$-coefficient $n$-cylinder. We also have the following proposition.
\begin{proposition}\label{prop:implication of cylinder}
An $n$-cylinder is a $\mathbb{Z}_p$-coefficient $n$-cylinder. \end{proposition} \begin{proof}
Let $W$ be an $n$-cylinder with connected boundary components $M_i$, $1\le i\le \ell$. Since $H_1(M_i)\cong H_1(W)\cong \mathbb{Z}$, the map $H_1(M_i;\mathbb{Z}_p)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W;\mathbb{Z}_p)$ is an isomorphism for each $i$. For $R=\mathbb{Z}$ or $\mathbb{Z}_p$, let $r(R)=\operatorname{rank}_R\operatorname{Coker}\{H_2(\partial W;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(W;R)\}$. By naturality of intersection forms, it suffices to show that $r(\mathbb{Z})=r(\mathbb{Z}_p)$. From the long exact sequence
\begin{multline*}
0\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \operatorname{Coker}\{H_2(\partial W;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(W;R)\} \\
\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(W,\partial W;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\partial W;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} 0
\end{multline*}
and the fact that $\operatorname{rank}_R H_2(W,\partial W;R) = \operatorname{rank}_R H_2(W;R)$, we obtain that \[ r(R) = \operatorname{rank}_R H_2(W;R) -(\ell -1). \] Since $H_1(M_i)\cong H_1(W)\cong \mathbb{Z}$, the groups $H_1(W)$ and $\operatorname{Coker}\{H_1(\partial W)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W)\}$ have no $p$-torsion. Therefore, by \cite[Lemma~3.14]{Cha:2010-1} we have $\operatorname{rank}_\mathbb{Z} H_2(W)=\operatorname{rank}_{\mathbb{Z}_p} H_2(W;\mathbb{Z}_p)$. Therefore $r(\mathbb{Z})=r(\mathbb{Z}_p)$. \end{proof}
An $n$-cylinder in \cite{Cochran-Kim:2004-1} appeared as a generalization of an $n$-solution in \cite{Cochran-Orr-Teichner:1999-1}. Below we generalize the notion of an $n$-solution to the notion of an $R$-coefficient $n$-solution.
\begin{definition}(\cite[Section 8]{Cochran-Orr-Teichner:1999-1} for $R=\mathbb{Z}$ and $\mathbb{Q}$) \label{def:n-solution}
An $R$-coefficient $n$-cylinder with a single boundary component is also called an {\it $R$-coefficient $n$-solution}. A $\mathbb{Z}$-coefficient $n$-solution is called an {\it $n$-solution}. A closed 3-manifold $M$ with $H_1(M)\cong \mathbb{Z}$ is {\it $n$-solvable via $W$} if there exists an $n$-solution $W$ with boundary $M$. A knot $K$ in $S^3$ is {\it $n$-solvable via $W$} and $W$ is an {\it $n$-solution for $K$} if $M(K)$ is $n$-solvable via $W$. \end{definition}
For each $n\in \frac12 \mathbb{N}_0$, we denote by $\mathcal{F}_n$ the subset of $n$-solvable knots in $\mathcal{C}$. It is known that $\mathcal{F}_n$ is a subgroup of $\mathcal{C}$ and $\mathcal{F}_m\subset \mathcal{F}_n$ if $m\ge n$ \cite{Cochran-Orr-Teichner:1999-1}. Therefore, $\{\mathcal{F}_n\}$ is a filtration of $\mathcal{C}$, and we call it the {\it solvable filtration} of $\mathcal{C}$. The filtrations $\{\mathcal{G}_n\}$ and $\{\mathcal{F}_n\}$ have the following relationship.
\begin{theorem}\label{thm:G_n and F_n}\cite[Theorem 8.11]{Cochran-Orr-Teichner:1999-1}
Let $n\in \frac12 \mathbb{N}_0$. If a knot $K$ bounds a grope of height $n+2$ in $D^4$, then $K$ is $n$-solvable. That is, $\mathcal{G}_{n+2}\subset \mathcal{F}_n$. \end{theorem} It is unknown whether or not the converse $\mathcal{F}_n\subset \mathcal{G}_{n+2}$ holds in general, but it is known that $\mathcal{F}_n=\mathcal{G}_{n+2}$ when $n=0,\,0.5$ \cite[Theorem~8.13 and Remark~8.14]{Cochran-Orr-Teichner:1999-1}.
\subsection{Amenable signatures}\label{subsec:amenable signature}
In this subsection, we briefly review the von Neumann--Cheeger--Gromov $\rho^{(2)}$-invariants \cite{Cheeger-Gromov:1985-1}, and introduce Amenable Signature Theorem~\ref{thm:obstruction}. For a closed 3-manifold $M$, let $\Gamma$ be a countable group and $\psi\colon \pi_1M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Gamma$ a group homomorphism. Then Chang and Weinberger \cite{Chang-Weinberger:2003-1} showed that there exists a group $G$ containing $\Gamma$ and a 4-manifold $W$ such that $\phi\colon \pi_1M\xrightarrow{\psi} \Gamma\hookrightarrow G$ extends to $ \pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$. Then, for $\mathcal{N} G$, which is the group von Neumann algebra of $G$, using $\phi$ we obtain a homomorphism $\mathbb{Z}[\pi_1W]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{N} G$ and the corresponding equivariant hermitian intersection form \[ \lambda\colon H_2(W;\mathcal{N} G)\times H_2(W;\mathcal{N} G)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{N} G \] and the $L^2$-signature $\sign^{(2)}_G(W) \in \mathbb{R}$. Let $\operatorname{sign}(W)$ be the ordinary signature of $W$ and let $S_G(W):=\sign^{(2)}_G(W) - \operatorname{sign}(W)$, an $L^2$-signature defect. Then the {\it von Neumann--Cheeger--Gromov $\rho^{(2)}$-invariant associated to $(M,\psi)$} is defined to be \[ \rho^{(2)}(M,\psi) := S_G(W). \] It is known that it is independent of the choices of a group $G$ and a 4-manifold of $W$. For more details on the von Neumann--Cheeger--Gromov $\rho^{(2)}$-invariants, refer to \cite{Cochran-Teichner:2003-1, Cha:2010-1}.
The following lemma gives the existence of a universal upper bound on the $\rho^{(2)}$-invariants for a fixed closed 3-manifold. \begin{lemma}\label{lem:universal bound}
\begin{enumerate}
\item \cite{Cheeger-Gromov:1985-1,Ramachandran:1993-1} For a closed 3-manifold $M$, there exists a constant $C_M$ such that $|\rho^{(2)}(M,\phi)|\le C_M$ for every homomorphism $\phi\colon \pi_1M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ where $G$ is a group.
\item \cite{Cha:2014-1} Furthermore, if $M=M(K)$ for a knot $K$ with crossing number $c(K)$, then we can take $C_M=69713280\cdot c(K)$.
\end{enumerate} \end{lemma}
Using $L^2$-theoretic techniques and the amenable signature theorem on homology bordism developed in \cite{Cha-Orr:2009-1}, Cha obtained the following obstruction for a knot to being $n.5$-solvable, which generalizes the obstruction from the $\rho^{(2)}$-invariant associated with a PTFA representation in \cite{Cochran-Orr-Teichner:1999-1}.
\begin{theorem}[Amenable Signature Theorem for $n.5$-solvability] \label{thm:obstruction} \cite[Theorem 1.3]{Cha:2010-1}
Let $K$ be an $n.5$-solvable knot. Let $G$ be an amenable group lying in Strebel's class $D(R)$ for $R=\mathbb{Q}$ or $\mathbb{Z}_p$ such that $G^{(n+1)}=\{e\}$. Let $\phi\colon \pi_1M(K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ be a homomorphism which sends the meridian of $K$ to an infinite order element in $G$. Suppose $\phi$ extends to an $n.5$-solution for $M(K)$. Then, $\rho^{(2)}(M(K),\phi)=0$. \end{theorem} The only amenable groups in Strebel's class $D(R)$ which we will use in this paper are the groups given in Lemma~\ref{lem:amenable and D(R)} below, and we will not need the definitions of amenable group and Strebel's class $D(R)$. (One may find the definitions in \cite{Cha-Orr:2009-1}.)
The computation of $\rho^{(2)}$-invariants is not easy in general, but for the $\rho^{(2)}$-invariants associated to an abelian representation, we have the lemma below. For a knot $K$, let $\sigma_K$ be the Levine-Tristram signature function. That is, for $\omega\in S^1\subset \mathbb{C}$, $\sigma_K(\omega)$ is the signature of the hermitian matrix $(1-\omega)A+(1-\bar{\omega})A^T$ where $A$ is a Seifert matrix for $K$.
\begin{lemma}\label{lem:computation of rho-invariant}\cite[Proposition 5.1]{Cochran-Orr-Teichner:2002-1}\cite[Corollary 4.3]{Friedl:2003-5}
Let $K$ be a knot and let $\mu$ be the meridian of $K$. If $\phi\colon \pi_1M(K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is a homomorphism whose image is abelian, then
\[
\rho^{(2)}(M(K),\phi)=
\begin{cases}
\int_{S^1} \sigma_K(\omega)\,d\omega & \mbox{if } \phi(\mu)\in G \mbox{ has infinite order} \\[1ex]
\sum_{r=0}^{d-1}\sigma_K(e^{2\pi r\sqrt{-1}/d}) & \mbox{if } \phi(\mu)\in G \mbox{ has finite order } d.
\end{cases}
\]
\end{lemma}
\subsection{Mixed-coefficient commutator series}\label{subsec:commutator series} To show linear independence of knots modulo $n.5$-solvability, we will use Amenable Signature Theorem~\ref{thm:obstruction}, which is available for the $\rho^{(2)}$-invariants associated to a representation to a group which is amenable and in $D(R)$. In this subsection, we give examples of groups which are amenable and in $D(R)$. Namely, for a group $G$, we will construct a certain subnormal series $\{\mathcal{P}^kG\}$ of $G$ such that $G/\mathcal{P}^kG$ are amenable and in $D(R)$. \begin{definition}\cite[Definition 4.1]{Cha:2010-1}\label{def:commutator series} Let $G$ be a group and $\mathcal{P}=(R_0,R_1,\ldots)$ be a sequence of rings with unity. The {\it $\mathcal{P}$-mixed-coefficient commutator series $\{\mathcal{P}^kG\}$} of $G$ is defined inductively as follows: let $\mathcal{P}^0G:=G$. For a nonnegative integer $k$, we define \[ \mathcal{P}^{k+1}G := \operatorname{Ker}\left\{ \mathcal{P}^kG\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \frac{\mathcal{P}^kG}{[\mathcal{P}^kG,\mathcal{P}^kG]}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \frac{\mathcal{P}^kG}{[\mathcal{P}^kG,\mathcal{P}^kG]}\otimesover\mathbb{Z} R_k \right\}. \]
\end{definition}
We note that $ \mathcal{P}^kG/[\mathcal{P}^kG,\mathcal{P}^kG]\cong H_1(G;\mathbb{Z}[G/\mathcal{P}^kG])$ and $(\mathcal{P}^kG/[\mathcal{P}^kG,\mathcal{P}^kG])\otimes_\mathbb{Z} R_k\cong H_1(G;R_k[G/\mathcal{P}^kG])$. For example, if $R_k=\mathbb{Z}$ for all $k$, then $\mathcal{P}^kG = G^{(k)}$, the $k$-th derived group of $G$. Also if $R_k=\mathbb{Q}$ for all $k$, then $\mathcal{P}^kG=G^{(k)}_r$, the $k$-th rational derived group of $G$. Note that $G^{(k)}\subset \mathcal{P}^kG$ for all $\mathcal{P}$ and $k$, and the group $\mathcal{P}^kG/\mathcal{P}^{k+1}G$ injects into $H_1(G;R_k[G/\mathcal{P}^kG])$.
Using $\mathcal{P}$-mixed-coefficient commutator series, we obtain groups which are amenable and in $D(R)$ as below.
\begin{lemma}\label{lem:amenable and D(R)}\cite[Lemma 4.3]{Cha:2010-1} Let $G$ be a group and $n$ a nonnegative integer. Let $\mathcal{P}=(R_0, R_1, \ldots)$ be a sequence of rings with unity such that for each $k< n$, every integer relatively prime to $p$ is invertible in $R_k$. Then, for each $k\le n$, the group $G/\mathcal{P}^kG$ is amenable and lies in $D(\mathbb{Z}_p)$. \end{lemma}
Later in the proof of Theorem~\ref{thm:main}, we will use a $\mathcal{P}$-mixed-coefficient commutator series where $\mathcal{P}=(R_0.R_1,\ldots, R_n)$ with $R_i=\mathbb{Q}$ for $0\le i\le n-1$ and $R_n=\mathbb{Z}_p$ for some prime $p$ and a representation to the group $G/\mathcal{P}^{n+1}G$ for some group $G$. In this case, $G/\mathcal{P}^{n+1}G$ has a subgroup $\mathcal{P}^nG/\mathcal{P}^{n+1}G$ which injects into a $\mathbb{Z}_p$-vector space $H_1(G;\mathbb{Z}_p[G/\mathcal{P}^nG])$. Therefore $G/\mathcal{P}^{n+1}G$ has $p$-torsion elements and it is not a PTFA group.
\section{Construction of examples}\label{sec:construction} In this section, we discuss how to construct a knot bounding a grope of height $n+2$ in $D^4$. We will construct such a knot using a process called {\it infection} or {\it satellite construction}. Let $K$ be a knot in $S^3$ and $n$ a positive integer. Let $\eta_1,\eta_2,\ldots, \eta_m$ be disjoint simple closed curves in $E(K)$ such that $\eta_i\in\pi_1E(K) ^{(n)}$ for all $i$. Suppose $\eta_i$ form an unlink in $S^3$. Let $J_1, \ldots, J_m$ be knots. For each $1\le i\le m$, remove the open tubular neighborhood of $\eta_i$ in $S^3$ and glue in the exterior of $J_i$ by identifying their common boundaries using an orientation-reversing homeomorphism in such a way that the meridian (resp. the longitude) of $\eta_i$ is identified with the longitude (resp. the meridian) of $J_i$. The resulting 3-manifold is homeomorphic to $S^3$, and now the knot $K$ becomes a new knot in this $S^3$. We denote this knot by $K(\eta_i;J_i)$ or $K(\eta_1,\ldots, \eta_m; J_1,\ldots, J_m)$ and say that it is obtained by {\it infecting $K$ by $J_i$ along $\eta_i$}. In the case that $J_i=J$ for some knot $J$ for all $i$, we simply write $K(\eta_1,\ldots, \eta_m;J)$ or $K(\eta_i;J)$. We call $K$, $\eta_i$, and $J_i$ the {\it seed knot}, the {\it axes}, and the {\it auxiliary knots} (or {\it infection knots}), respectively. Fore more details, we refer the reader to \cite{Cochran-Orr-Teichner:2002-1}.
Roughly speaking, a grope of height $n+2$ bounded by $K(\eta_i;J_i)$ is constructed by `stacking' gropes of height 2 bounded by $J_i$ and gropes of height $n$ bounded by $\eta_i$. Construction of a grope under infection was investigated in \cite{Cochran-Teichner:2003-1} and later in \cite{Horn:2010-1}. Afterwards, a more systematic way of construction was given in \cite{Cha:2012-1,Cha-Kim:2016-1}. For instance, the following theorem is implicitly proved in \cite[Definition~4.4]{Cha-Kim:2016-1}. In the following, a {\it capped grope} is a grope with disks attached along symplectic basis curve on the top stage surfaces of the grope.
\begin{theorem}\label{thm:grope of height n+2} \cite{Cha-Kim:2016-1} Let $K$ be a knot and let $\eta_i$, $1\le i\le m$, be curves in $S^3\smallsetminus K$ which form an unlink in $S^3$. Suppose that $\eta_i$ bound disjoint capped gropes of height $n$ in $S^3$ which do not meet $K$ except for the caps. For each $i$ with $1\le i\le m$, suppose that $J_i$ is a knot which bounds a grope of height 2 in $D^4$. Then the knot $K(\eta_i;J_i)$ bounds a grope of height $n+2$ in $D^4$. \end{theorem}
Therefore we need to find axes $\eta_i$ bounding gropes of height $n$. The following lemma shows how to find such axes.
\begin{lemma}\cite[Lemma~3.9]{Cochran-Teichner:2003-1}\label{lem:axes bounding grope of height n} Let $K$ be a knot and $\eta_i$, $1\le i\le m$, curves in $S^3\smallsetminus K$ which form an unlink in $S^3$. Suppose $\eta_i\in \pi_1(S^3\smallsetminus K)^{(n)}$. Then, there exist capped gropes $G_i$ of height $n$ disjointly embedded in $S^3$ such that $G_i$ do not meet $K$ except for the caps and for each $i$ the grope $G_i$ is bounded by a knot in the homotopy class of $\eta_i$. \end{lemma}
Theorem~\ref{thm:grope of height n+2} and Lemma~\ref{lem:axes bounding grope of height n} yield the following corollary immediately.
\begin{corollary}\label{cor:knot bounding a grope of height n+2} Let $K$ be a knot and let $\eta_i$, $1\le i\le m$, be curves in $S^3\smallsetminus K$ which form an unlink in $S^3$. Suppose $\eta_i\in \pi_1(S^3\smallsetminus K)^{(n)}$. For each $i$ with $1\le i\le m$, suppose $J_i$ is a knot bounding a grope of height 2 in $D^4$. Then we can homotope $\eta_i$ such that the knot $K(\eta_i;J_i)$ bounds a grope of height $n+2$ in $D^4$. \end{corollary}
We discuss further how to choose $\eta_i$ and $J_i$. Namely, to prove Theorem~\ref{thm:main} we will need to construct infinitely many knots bounding a grope of height $n+2$ in $D^4$ which are linearly independent modulo $\mathcal{F}_{n.5}$. For that purpose, we need to make specific choices for $\eta_i$ and $J_i$.
The following theorem is a generalization of \cite[Theorem 5.13]{Cochran-Kim:2004-1}, and in Section~\ref{sec:infinite rank subgroup} it will be used for the choice of axes $\eta_i$ in the construction of the generating knots of the subgroups in Theorem~\ref{thm:main}.
\begin{theorem}\label{thm:eta_i} Let $n\ge 1$ be an integer. Let $K$ be a knot with nontrivial Alexander polynomial $\Delta_K(t)$. Suppose $\deg \Delta_K(t) > 2$ if $n> 1$. Let $\Sigma$ be a Seifert surface for $K$. Then there exists an unlink $\{\eta_1,\ldots, \eta_m\}$ in $S^3$ which does not meet $\Sigma$ and satisfies the following:
\begin{enumerate}
\item For all $i$, $\eta_i\in \pi_1M(K)^{(n)}$ and the $\eta_i$ bound capped gropes of height $n$ which are disjointly embedded in $S^3\smallsetminus K$. Here, the caps are allowed to intersect $K$.
\item Let $\mathcal{P}=(R_0,R_1,\ldots, R_n)$ where $R_i=\mathbb{Q}$ for $0\le i\le n-1$ and $R_n=\mathbb{Z}_p$ where $p$ is a prime greater than the top coffeicient of $\Delta_K(t)$. Then, for each $n$-cylinder $W$ with $M(K)$ as one of its boundary components, there exists some $\eta_i$ such that $j_*(\eta_i)\notin \mathcal{P}^{n+1}\pi_1W$ where $j_*\colon \pi_1M(K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1W$ is the inclusion-induced homomorphism.
\end{enumerate} \end{theorem} \noindent The above theorem is the most technical part of the paper, and its proof is postponed to the end of Subsection~\ref{subsec:algebraic n-solution}. Theorem~\ref{thm:eta_i} is significant since it shows that there is a finite set of $\eta_i$ in $S^3\smallsetminus K$ (in fact, in $S^3\smallsetminus \Sigma$) which satisfies the nontriviality property in Theorem~\ref{thm:eta_i}~(2) for {\it every $n$-cylinder} for $M(K)$: if one needs to find a finite set of $\eta_i$ satisfying the nontriviality property for {\it a specific choice of an $n$-cylinder}, it can be more easily done (see Theorem~\ref{thm:nontrivial}.) \begin{figure}
\caption{A clasper surgery description of the knot $P_m$}
\label{figure:Knot_P_m}
\end{figure}
We will need the following lemma for the choice of infection knots $J_i$ when we show linear independence of knots $K(\eta_1,\ldots, \eta_m;J_i)$, $i\ge 1$, in the proof of Theorem~\ref{thm:main}.
\begin{lemma}\label{lemma:J_0^i}\cite[Proposition 3.4]{Jang:2015-1}
For an arbitrary constant $C$, there exists a sequence of knots $J_1, J_2, \ldots,$ and an increasing sequence of odd primes $p_1,p_2,\ldots, $ which satisfy the following: let $\omega_i := e^{2\pi\sqrt{-1}/p_i}$.
\begin{enumerate}
\item Each $J_i$ bounds a grope of height 2 in $D^4$,
\item $\sum_{r=0}^{p_i-1}\sigma_{J_i}(\omega_i^r) > C$,
\item $\sum_{r=0}^{p_j-1}\sigma_{J_i}(\omega_j^r) =0$ for $j<i$.
\end{enumerate} \end{lemma} In \cite[Propositoin~3.4]{Jang:2015-1}, the knots $J_i$ were constructed to satisfy more conditions such as $\int_{S^1} \sigma_{J_i}(\omega)\,d\omega=0$ to obtain the knots $K(\eta_1,\ldots, \eta_m;J_i)$ for which the $\rho^{(2)}$-invariants associated to a PTFA representation vanish. We do not need this property in this paper, and for our purpose we can use simpler $J_i$. That is, the proof of \cite[Propositoin~3.4]{Jang:2015-1} tells us that in Lemma~\ref{lemma:J_0^i}, for each $i$ we can take $J_i$ to be a connected sum of $N$ copies of $P_{m_{i+1}}\# (-P_{m_i})$ where $N$ is an integer bigger than $C/2$, $P_m$ is a knot whose clasper surgery description is given in Figure~\ref{figure:Knot_P_m}, and $m_1,m_2,\ldots,$ is a certain increasing sequence of positive integers. In Figure~\ref{figure:Knot_P_m}, $P_m$ is obtained from the unknot $U$ by performing clasper surgery along the tree $T$. The knot $P_m$ for $m=1$ was given in \cite[Figure~3.7]{Cochran-Teichner:2003-1}, and $P_m$ for $m>1$ were given in \cite{Horn:2010-1}. The surgery descriptions of $P_m$ are given in \cite[Figure~3.6]{Cochran-Teichner:2003-1} and \cite[Figure~3]{Horn:2010-1}.
\section{Infinite rank subgroups of filtrations}\label{sec:infinite rank subgroup}
In this section, we prove Theorem~\ref{thm:main}, which will be a direct consequence of Theorems~\ref{thm:refined main theorem-1} and \ref{thm:refined_main_theorem-2}. First, we construct the generating knots of the desired subgroups in Theorem~\ref{thm:main}. Let $n> 1$. Let $K$ be a slice knot with $\deg \Delta_K(t)>2$. Let $\{\eta_1,\ldots \eta_m\}$ be an unlink in $S^3$ lying in $E(K)$ given by Theorem~\ref{thm:eta_i}. By Corollary~\ref{cor:knot bounding a grope of height n+2}, we can homotope $\eta_i$ such that $K(\eta_i;J)$ bounds a grope of height $n+2$ in $D^4$ for every knot $J$ which bounds a grope of height 2 in $D^4$. By Lemma~\ref{lem:universal bound}, there exists a constant $C$ such that $|\rho^{(2)}(M(K),\phi)|<C$ for all homomorphisms $\phi\colon \pi_1M(K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ for every group $G$. Choose a sequence of knots $J_1, J_2, \ldots ,$ using Lemma~\ref{lemma:J_0^i} with this constant $C$ such that the prime $p_1$ is greater than the top coefficient of $\Delta_K(t)$. Finally, for each $i\ge 1$ define $K_i:=K(\eta_1,\ldots, \eta_m;J_i)$. (Therefore, the choice of $\eta_1,\ldots,\eta_m$ is independent of $K_i$.)
\subsection{Linear independence of examples}\label{subset:linear independence}
In this subsection, in Theorem~\ref{thm:refined main theorem-1} we show that the knots $K_i$ defined as above satisfy Theorems~\ref{thm:main}(1) and (2). To prove Theorem~\ref{thm:refined main theorem-1}, we will need the following lemma. Recall that for a 4-manifold $W$ with a homomorphism $\phi\colon \pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ , we let $S_G(W):=\sign^{(2)}(W)-\operatorname{sign}(W)$. \begin{lemma}\label{lem:building block} Let $K$ be a slice knot. Let $\{\eta_1.\eta_2,\ldots,\eta_m\}$ be an unlink in $E(K)$ such that $\eta_i \in \pi_1E(K)^{(n)}$ for all $1\le i\le m$. Let $M:=M(K(\eta_i;J_i))$. Suppose $J_i$ is a knot with vanishing Arf invariant for $1\le i\le m$. Then, letting $\mathbb{Z}_\infty:=\mathbb{Z}$, we have the following.
\begin{enumerate}
\item There exists an $n$-solution $W$ for $M$ satisfying the following: suppose $\phi\colon \pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is a homomorphism where $G$ is an amenable group lying in Strebel's class $D(R)$ for some ring $R$. Then, $S_G(W)=\sum_{i=1}^m\rho^{(2)}(M(J_i), \phi_i)$ where for each $i$ the map $\phi_i\colon \pi_1M(J_i)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}_{d_i}$ is a surjective homomorphism sending the meridian of $J_i$ to $1\in \mathbb{Z}_{d_i}$ with $d_i$ the order of $\phi(\eta_i)$ in $G$.
\item There exists an $n$-cylinder $V$ with $\partial V=M(K)\coprod (-M)$ satisfying the following: suppose $\phi\colon \pi_1V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is a homomorphism where $G$ is an amenable group lying in Strebel's class $D(R)$ for some ring $R$. Then, $S_G(V)=-\sum_{i=1}^m\rho^{(2)}(M(J_i), \phi_i)$ where for each $i$ the map $\phi_i\colon \pi_1M(J_i)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}_{d_i}$ is a surjective homomorphism sending the meridian of $J_i$ to $1\in \mathbb{Z}_{d_i}$ with $d_i$ the order of $\phi(\eta_i)$ in $G$.
\end{enumerate}
\end{lemma} \begin{proof} Part~(1) is essentially due to \cite[Proposition 4.4]{Cha:2010-1} and its proof, noticing that \[ \rho^{(2)}(M,\phi) = \rho^{(2)}(M(K), \phi) + \sum_{i=1}^m\rho^{(2)}(M(J_i), \phi_i) \] where, by abuse of notation, $\phi$ also denotes the restriction of the map $\phi\colon \pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ to the corresponding subspace.
We prove Part~(2). Since each $J_i$ has vanishing Arf invariant, it is $0$-solvable. Let $W_i$ be a $0$-solution for $J_i$. By doing surgery along $\pi_1W_i^{(1)}$ if necessary, we may assume that $\pi_1W_i\cong \mathbb{Z}$, generated by the meridian of $J_i$.
Let $V=M(K)\times [0,1]\cup (\coprod_{i=1}^m-W_i)$ where each $-W_i$ is attached to $M(K)\times [0,1]$ by identifying the solid torus $M(J_i)-E(J_i)\subset \partial W_i$ with the tubular neighborhood of $\eta_i\times 0\subset M(K)\times 0$ in such a way that the 0-linking longitude of $\eta_i$ is identified with the meridian of $J_i$ and the meridian of $\eta_i$ is identified with the 0-linking longitude of $J_i$. Then $\partial V=M(K)\amalg (-M)$ and an inclusion from a boundary component $M(K)$ or $M$ to $V$ induces an isomorphism on first homology. Using Mayer-Vietoris sequences, one can also show that $0$-Lagrangians and $0$-duals of $W_i$ give rise to an $n$-Lagrangian and its $n$-dual of $V$. This shows that $V$ is an $n$-cylinder.
As in \cite[Lemma 2.3]{Cochran-Harvey-Leidy:2009-1}, one can obtain that
\[
\rho^{(2)}(M(K),\phi_K) - \rho^{(2)}(M,\phi_M) = -\sum_{i=1}^m \rho^{(2)}(M(J_i),\phi_i)
\]
where $\phi_K$, $\phi_M$, and $\phi_i$ are the restrictions of $\phi $ to the corresponding subspaces. By the definition of $\rho^{(2)}$-invariants, we have $\rho^{(2)}(M(K),\phi_K) - \rho^{(2)}(M,\phi_M) = S_G(V)$. Note that since $\phi_i$ factors through $\pi_1W_i\cong\mathbb{Z}$, the image of $\phi_i$ is the abelian subgroup in $G$ generated by the image of the meridian of $J_i$. Since the meridian of $J_i$ is identified with the 0-linking longitude of $\eta_i$ in $M(K)$, by the subgroup property of $\rho^{(2)}$-invariants (see\cite[p.108]{Cochran-Orr-Teichner:2002-1}) we may assume that the map $\phi_i\colon \pi_1M(J_i)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}_{d_i}$ is a surjective homomorphism sending the meridian of $J_i$ to $1\in \mathbb{Z}_{d_i}$ with $d_i$ the order of $\phi(\eta_i)$ in $G$. \end{proof}
\begin{theorem}\label{thm:refined main theorem-1} Let $n\ge 2$ be an integer and let $K$ be a slice knot with $\deg \Delta_K(t) > 2$. Let $K_i$ be the knots obtained from $K$ as in the beginning of Section~\ref{sec:infinite rank subgroup}. Then, $K_i\in \mathcal{G}_{n+2}$ for all $i$, and $K_i$ are linearly independent modulo $\mathcal{F}_{n.5}$. Moreover, letting $G:=\pi_1(S^3\smallsetminus K)$ and $G_i:=\pi_1(S^3\smallsetminus K_i)$, we can construct $K_i$ such that for each $i$ there is an isomorphism $G_i/G_i^{(n+1)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G/G^{(n+1)}$ preserving peripheral structures. \end{theorem}
\begin{proof}
First, we prove the last part. Since there is a degree 1 map from $E(J_i)$ to $E(\mbox{unknot})$, for each $i$, there is a degree 1 map $f_i\colon E(K_i) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} E(K)$ relative to the boundary such that $f_i$ is the identity outside the copies of $E(J_i)$. Since $\eta_\ell\in \pi_1M(K)^{(n)}$ for all $\ell$, by \cite[Theorem~8.1]{Cochran:2002-1} the $f_i$ induces an isomorphism $G_i/G_i^{(n+1)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G/G^{(n+1)}$ which preserves peripheral structures.
The $K_i$ bound a grope of height $n+2$ in $D^4$ by Corollary~\ref{cor:knot bounding a grope of height n+2} and Lemma~\ref{lemma:J_0^i}, hence $K_i\in \mathcal{G}_{n+2}$. Let $J:=\#_ia_iK_i$ where $a_i\in \mathbb{Z}$, a nontrivial connected sum of finitely many copies of $K_i$ and their inverses $-K_i$. To prove the theorem it suffices to show that $J$ is not $n.5$-solvable.
Suppose $J$ is $n.5$-solvable. We may assume $a_1\ne 0$, and furthermore, by taking the inverse of $K_1$ if necessary, we may assume $a_1>0$. Note that the $J_i$ have vanishing Arf invariant since they bound a grope of height 2 in $D^4$ and hence $0$-solvable. We construct building blocks for a certain 4-manifold as follows.
\begin{enumerate}
\item Let $V$ be an $n.5$-solution for $M(J)$.
\item Let $E$ be the standard cobordism between $M(J)$ and $\amalg_ia_iM(K_i)$ as constructed (with the name $C$) in \cite[p.113]{Cochran-Orr-Teichner:2002-1}. We may assume $\partial E = (\amalg_ia_iM(K_i))\amalg (-M(J))$.
\item Let $V_1$ be an $n$-cylinder with $\partial V=M(K) \amalg(- M(K_1))$ as given in Lemma~\ref{lem:building block}~(2).
\item Let $W_i$ be an $n$-solution for $M(K_i)$ as given in Lemma~\ref{lem:building block}~(1).
\end{enumerate}
Let $b_1:=a_1-1$ and $b_i=|a_i|$ for $i\ge 2$. For $1\le r\le b_i$, let $W_i^r$ be a copy of $-\epsilon W_i$ where $\epsilon_i=a_i/|a_i|$. Now we define
\[
W=V\bigcup_{\partial_-E} E \bigcup_{\partial_+ E} \left(V_1\coprod\left(\coprod_{i}\coprod_{r=1}^{b_i}W_i^r\right)\right)
\]
where $\partial_+E = \coprod_ia_iM(K_i)$ and $\partial_-E=M(J)$. See Figure~\ref{figure:Cobordism_W}. Note that $\partial W=M(K)$. Using Mayer-Vietoris sequences, one can easily show that the $n$-Lagrangians and $n$-duals of $V_1$, $W_i^r$, and $V$ form an $n$-Lagrangian and its $n$-dual for $W$, and therefore $W$ is an $n$-solution for $M(K)$.
\begin{figure}
\caption{Cobordism $W$}
\label{figure:Cobordism_W}
\end{figure}
Let $\mathcal{P}:=(R_0,R_1,\ldots, R_n)$ where $R_i=\mathbb{Q}$ for $i\le n-1$ and $R_n=\mathbb{Z}_{p_1}$. Using Definition~\ref{def:commutator series} we obtain $\mathcal{P}^j\pi_1 W$ for $1\le j\le n+1$, which are subgroups of $\pi_1W$. Let $G:=\pi_1W/\mathcal{P}^{n+1}\pi_1W$, which is amenable and lies in $D(\mathbb{Z}_{p_1})$ by Lemma~\ref{lem:amenable and D(R)}, and let $\phi\colon \pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ be the quotient map. For convenience, we denote a restriction of $\phi$ by $\phi$ as well.
Since $\partial W=M(K)$, we have $S_G(W)=\rho^{(2)}(M(K),\phi)$. On the other hand, by Novikov additivity, we have
\[
S_G(W)=S_G(V) + S_G(E) + S_G(V_1) + \sum_i\sum_{r=1}^{b_i}S_G(W_i^r).
\]
Here, $S_G(V)=0$ by Amenable Signature Theorem~\ref{thm:obstruction}.
Also, $S_G(E)=0$ since $\operatorname{Coker}\{H_2(\partial_-E)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(E)\}=0$ (see \cite[Lemma~4.2]{Cochran-Orr-Teichner:2002-1}\cite[Lemma~2.4]{Cochran-Harvey-Leidy:2009-1}).
By Lemma~\ref{lem:building block}(2),
\[
S_G(V_1)=-\sum_{i=1}^m\rho^{(2)}(M(J_1), \phi_i)
\]
where $\phi_i\colon \pi_1M(J_1)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}_{d_i}$ is a surjective homomorphism such that $d_i$ is the order of $\phi(\eta_i)$ in $G$ and $\phi_i$ sends the meridian of $J_1$, which is the 0-linking longitude of $\eta_i$, to $1\in \mathbb{Z}_{d_i}$. Since $\eta_i\in \pi_1M(K)^{(n)}$, one can see that $\phi(\eta_i)$ lies in $\mathcal{P}^n\pi_1W/\mathcal{P}^{n+1}\pi_1W$. Since $\mathcal{P}^n\pi_1W/\mathcal{P}^{n+1}\pi_1W$ injects into $H_1(\pi_1W;\mathbb{Z}_{p_1}[\pi_1W/\mathcal{P}^n\pi_1W])$, which is a $\mathbb{Z}_{p_1}$-vector space, we have $d_i=1\mbox{ or } p_1$. Then, by Lemma~\ref{lem:computation of rho-invariant}, $\rho^{(2)}(M(J_1),\phi_i)=0$ if $d_i=1$, and $\rho^{(2)}(M(J_1),\phi_i)=\sum_{r=0}^{p_1-1}\sigma_{J_1}(e^{2\pi r\sqrt{-1}/p_1})$ if $d_i=p_1$. Furthermore, since the $\eta_\ell$ $(1\le \ell\le m)$ were chosen using Theorem~\ref{thm:eta_i}, there exists some $\eta_i$ such that $\phi(\eta_i)\ne e$ in $G$. Therefore $d_i=p_1$ for some $i$ and we have
\[
S_G(V_1) \le -\sum_{r=0}^{p_1-1}\sigma_{J_1}(e^{2\pi r\sqrt{-1}/p_1}).
\]
We compute $S_G(W_i^r)$. By Lemma~\ref{lem:building block}(1), we have
\[
S_G(W_i)= \sum_{j=1}^m\rho^{(2)}(M(J_i), \phi_j).
\]
When $i=1$, $W_1^r=-W_1$ and hence
\[
S_G(W_1^r)= - \sum_{r=0}^{p_1-1}\sigma_{J_1}(e^{2\pi r\sqrt{-1}/p_1}) \mbox{ or } 0,
\]
computed as above. When $i\ge 2$, similarly to the case $i=1$, we have
\[
S_G(W_i^r)= \pm\sum_{r=0}^{p_1-1}\sigma_{J_i}(e^{2\pi r\sqrt{-1}/p_1}) \mbox{ or } 0.
\]
But by Lemma~\ref{lemma:J_0^i}(3), we have $S_G(W_i^r)=0$ for $i\ge 2$.
Summing up the above computations, we obtain that
\[
S_G(W) \le -\sum_{r=0}^{p_1-1}\sigma_{J_1}(e^{2\pi r\sqrt{-1}/p_1})
\]
and therefore we have $|S_G(W)|> C$ by our choice of $J_1$ and Lemma~\ref{lemma:J_0^i}(2). But from $S_G(W)=\rho^{(2)}(M(K),\phi)$ and our choice of $C$, we have $|S_G(W)|< C$, which is a contradiction. \end{proof}
\subsection{Distinction from the knots constructed via iterated doubling operators}\label{subsec:distinction}
The purpose of this subsection is to prove Theorem~\ref{thm:refined_main_theorem-2} below. We note that in \cite{Horn:2010-1} it was shown that a nontrivial combination of the knots generating the infinite rank subgroup of $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ in \cite{Horn:2010-1} is nontrivial modulo $\mathcal{F}_{n.5}$. That is, the infinite rank subgroup of $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ in \cite{Horn:2010-1} injects into $\mathcal{F}_n/\mathcal{F}_{n.5}$ under the quotient map $\mathcal{C}/\mathcal{G}_{n+2.5}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{C}/\mathcal{F}_{n.5}$. \begin{theorem}\label{thm:refined_main_theorem-2} Let $n\ge 2$ be an integer and let $K_i$ be the knots in Theorem~\ref{thm:refined main theorem-1}. In $\mathcal{F}_n/\mathcal{F}_{n.5}$, the infinite rank subgroup generated by the $K_i$ trivially intersects the infinite rank subgroups of $\mathcal{F}_n/\mathcal{F}_{n.5}$ in \cite{Cochran-Harvey-Leidy:2009-1}, \cite{Horn:2010-1}, \cite{Cochran-Harvey-Leidy:2009-2}, and \cite{Cha:2010-1}. In particular, in $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$, the infinite rank subgroup generated by $K_i$ trivially intersects the infinite rank subgroup of $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ in \cite{Horn:2010-1}. \end{theorem}
We give a proof at the end of this subsection. The proof of the above theorem is based on ideas in \cite[Section 9]{Cochran-Harvey-Leidy:2009-2}. Using the terms in \cite[Section 9]{Cochran-Harvey-Leidy:2009-2}, our knots $K_i$ can be considered as {\it generalized COT knots}, and on the other hand the knots in \cite{Cochran-Harvey-Leidy:2009-1}, \cite{Horn:2010-1}, \cite{Cochran-Harvey-Leidy:2009-2}, and \cite{Cha:2010-1} are called {\it CHL knots}. Therefore, to prove Theorem~\ref{thm:refined_main_theorem-2}, we need to extend the results in \cite[Section 9]{Cochran-Harvey-Leidy:2009-2}, which are about distinguishing COT knots from CHL knots, to the case of generalized COT knots. We will do this by showing the following: 1) For each $n>0$ and a prime $p$, we define a subset $\mathcal{F}^{cot,p}_n$ of the set of (isotopy classes of) knots in $S^3$; 2) We show that if $K$ is a nontrivial linear combination of the $K_i$ in Theorem~\ref{thm:refined main theorem-1} which generate an infinite rank subgroup in $\mathcal{F}_n/\mathcal{F}_{n.5}$ (and in $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$), we have $K\notin \mathcal{F}^{cot,p}_{n.5}$ for some prime $p$; 3) We show that if a knot $K$ is concordant to a nontrivial linear combination of CHL knots, then $K\in \mathcal{F}^{cot,p}_{n.5}$ for all prime $p$.
First, for each $n> 0$, we define a subset $\mathcal{F}^{cot,p}_n$, which generalizes the subset $\mathcal{F}^{cot}_n$ defined in \cite{Cochran-Harvey-Leidy:2009-2}. Fix an integer $n\ge 0$ and a prime $p$. Let $G$ be a group such that $H_1(G)\cong \mathbb{Z}$. Since $G^{(1)}/G^{(n)}_r$ is a PTFA group \cite[Proposition~2.1]{Harvey:2006-1}, $\mathbb{Z}_p[G^{(1)}/G^{(n)}_r]$ embeds into its skew quotient field, say $\mathbb{K}$ (see Lemma~\ref{lem:quotient field}). Since $H_1(G)=G/G^{(1)}\cong \mathbb{Z}$, the group ring $\mathbb{Z}_p[G/G^{(n)}_r]$ can be embedded into the noncommutative Laurent polynomial ring $\mathbb{K}[t^{\pm 1}]$, which is a noncommutative PID. Now we define
\[ G^{(n+1)}_{cot,p} := \operatorname{Ker}\left\{ G^{(n)}_r\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \frac{G^{(n)}_r}{[G^{(n)}_r,G^{(n)}_r]}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \frac{G^{(n)}_r}{[G^{(n)}_r,G^{(n)}_r]}\otimesover{\mathbb{Z}[G/G^{(n)}_r]} \mathbb{K}[t^{\pm 1}] \right\}. \] Note that $G^{(n+1)}\subset G^{(n+1)}_{cot,p}$ for all prime $p$ and \[ \frac{G^{(n)}_r}{[G^{(n)}_r,G^{(n)}_r]}\otimesover{\mathbb{Z}[G/G^{(n)}_r]} \mathbb{K}[t^{\pm 1}] \cong H_1(G;\mathbb{K}[t^{\pm 1}]) \] since $\mathbb{K}[t^{\pm 1}]$ is a flat left module over $\mathbb{Z}[G/G^{(n)}_r]$ by \cite[Proposition II.3.5]{Stenstrom:1975}.
For a 4-manifold $W$ with $H_1(W)\cong \mathbb{Z}$, let $\pi:=\pi_1W$. For each $n>0$ and a prime $p$, there is the equivariant intersection form \[ \lambda_n^p\colon H_2(W;\mathbb{Z}[\pi/\pi^{(n)}_{cot,p}]) \times H_2(W;\mathbb{Z}[\pi/\pi^{(n)}_{cot,p}]) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z}[\pi/\pi^{(n)}_{cot,p}]. \] \begin{definition}\label{def:p-F_n} Let $n$ be a positive integer and $p$ a prime. A knot $K$ is {\it $(n,p)$-solvable via $W$} if there exists a compact connected 4-manifold $W$ with boundary $M(K)$ satisfying the following: let $\pi:=\pi_1W$ and $r:=\frac12 \operatorname{rank}_\mathbb{Z} H_2(W)$. \begin{enumerate}
\item The inclusion $M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} W$ induces an isomorphism $H_1(M)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W)$.
\item There exist $x_1, x_2, \ldots, x_r$ and $y_1, y_2, \ldots, y_r$ in $H_2(W;\mathbb{Z}[\pi/\pi^{(n)}_{cot,p}])$ such that $\lambda_n^p(x_i,x_j)=0$ and $\lambda_n^p(x_i,y_j)=\delta_{ij}$ for $1\le i,j\le r$. \end{enumerate} We say $K$ is {\it $(n.5,p)$-solvable via $W$} if the following additional condition is satisfied: \begin{enumerate}
\item[(3)] There exist $\tilde{x}_1,\tilde{x}_2,\ldots, \tilde{x}_r$ in $H_2(W;\mathbb{Z}[\pi/\pi^{(n+1)}_{cot,p}])$ such that $\lambda_{n+1}^p(\tilde{x}_i, \tilde{x}_j)=0$ for $1\le i,j\le r$ and $\tilde{x}_i$ and $x_i$ are represented by the same surface for each $i$. \end{enumerate} We denote the set of $(n,p)$-solvable knots and $(n.5,p)$-solvable knots by $\mathcal{F}^{cot,p}_n$ and $\mathcal{F}^{cot,p}_{n.5}$, respectively. The submodules generated by $x_i$ and $y_i$ are called an {\it $(n,p)$-Lagrangian} and an {\it $(n,p)$-dual}, respectively. The submodule generated by $\tilde{x}_i$ is called an {\it $(n+1,p)$-Lagrangian.} \end{definition} Since $\pi^{(n)}\subset \pi^{(n)}_{cot,p}$, if a knot $K$ is $n$-solvable, then it is $(n,p)$-solvable for all prime $p$. We note that the notion of $G^{(n)}_{cot,p}$ is not functorial. That is, for a group homomorphism $\phi\colon G\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H$, in general $\phi(G^{(n)}_{cot,p})\nsubseteq H^{(n)}_{cot,p}$. Due to this fact, the subset $\mathcal{F}^{cot,p}_n$ does not descend to a subgroup of $\mathcal{C}$.
The following theorem generalizes \cite[Theorem 5.2]{Cochran-Harvey-Leidy:2009-2} and \cite[Theorem 3.2]{Cha:2010-1}, and it can be proved using the same arguments as in the proof of \cite[Theorem 3.2]{Cha:2010-1}. Therefore, we give only a sketch of the proof. \begin{theorem} \label{thm:vanishing rho-invariant-(n,p)} Let $p$ be a prime and $n$ a positive integer. Let $K$ be a knot which is $(n.5, p)$-solvable via $W$. Let $G$ be an amenable group lying in Strebel's class $D(\mathbb{Z}_p)$. Let $\pi:=\pi_1W$ and suppose we are given a homomorphism
\[
\phi\colon \pi_1M(K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi/\pi^{(n+1)}_{cot,p}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G
\]
which sends the meridian of $K$ to an infinite order element in $G$. Then $\rho^{(2)}(M(K),\phi)=0$. \end{theorem} \begin{proof}[Sketch of proof] We have $\rho^{(2)}(M(K),\phi)=S_G(W)=\sign^{(2)}_G(W)-\operatorname{sign}(W)$, and we will show that $S_G(W)=0$. From Definition~\ref{def:p-F_n}, we have an $(n+1,p)$-Lagrangian in $H_2(W;\mathbb{Z}[\pi/\pi^{(n+1)}_{cot,p}])$ generated by $\tilde{x}_1,\tilde{x}_2,\ldots, \tilde{x}_r$ and its $n$-dual in $H_2(W;\mathbb{Z}[\pi/\pi^{(n)}_{cot,p}])$ generated by $y_1,y_2,\ldots, y_r$ where $r=\frac12 \operatorname{rank}_\mathbb{Z} H_2(W)$. The images of $\tilde{x_i}$ and $y_i$ consist of a 0-Lagrangian and its 0-dual in $H_2(W;\mathbb{Q})$, and therefore $\operatorname{sign}(W)=0$.
We show that $\sign^{(2)}_G(W)=0$. Let $\mathcal{N} G$ be the group von Neumann algebra of $G$ and \[ \lambda\colon H_2(W;\mathcal{N} G)\times H_2(W;\mathcal{N} G)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathcal{N} G \] be the corresponding intersection form where the coefficients of the homology groups are twisted via $\phi$. Since $\phi\colon \pi_1M(K)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ factors through $\pi/\pi^{(n+1)}_{cot,p}$, we have the induced map $H_2(W;\mathbb{Z}[\pi/\pi^{(n+1)}_{cot,p}])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_2(W;\mathcal{N} G)$.
Let $H$ be the submodule of $H_2(W;\mathcal{N} G)$ generated by the images of $\tilde{x_i}$ under this map, and $\bar{H}$ be the submodule of $H_2(W;\mathbb{Z}_p)$ generated by the images of $\tilde{x_i}$. Since the images of $\tilde{x_i}$ in $H_2(W;\mathbb{Z}_p)$ has a dual, which is the images of $y_i$, we have $\dim_{\mathbb{Z}_p}\bar{H}=r$. By Lemmas 3.13 and 3.14 in \cite{Cha:2010-1}, we have $\dim^{(2)} H_2(W;\mathcal{N} G)=\dim_{\mathbb{Z}_p} H_2(W;\mathbb{Z}_p)=2r$ where $\dim^{(2)}$ denotes the $L^2$-dimension function on $\mathcal{N} G$-modules. By \cite[Theorem~3.11]{Cha:2010-1}, we have \[ \dim^{(2)} H\ge \dim^{(2)} H_2(W;\mathcal{N} G) - \dim_{\mathbb{Z}_2}H_2(W;\mathbb{Z}_p)+\dim_{\mathbb{Z}_p}\bar{H}=r. \] Now by \cite[Proposition 3.7]{Cha:2010-1}, we have $\sign^{(2)}_G(W)=0$. \end{proof} Now we give a proof of Theorem~\ref{thm:refined_main_theorem-2}.
\begin{proof}[Proof of Theorem~\ref{thm:refined_main_theorem-2}] Since $\mathcal{G}_{n+2.5}\subset \mathcal{F}_{n.5}$, the last part follows from the fact that the basis knots of the infinite rank subgroup $\mathcal{G}_{n+2}/\mathcal{G}_{n+2.5}$ in \cite{Horn:2010-1} are linearly independent modulo $\mathcal{F}_{n.5}$, which is shown in the proof of \cite[Theorem~5.2]{Horn:2010-1}.
Let $J$ be a nontrivial linear combination of the $K_i$ and let $L$ be a knot which is concordant to a linear combination of the $n$-solvable knots in \cite{Cochran-Harvey-Leidy:2009-1}, \cite{Horn:2010-1}, \cite{Cochran-Harvey-Leidy:2009-2}, or \cite{Cha:2010-1}, which are constructed using iterated doubling operators and generate an infinite rank subgroup of $\mathcal{F}_n/\mathcal{F}_{n.5}$. We will show $J\notin \mathcal{F}_{n.5}^{cot,p}$ for some prime $p$, but $L\in \mathcal{F}_{n.5}^{cot,p}$ for all prime $p$, and this will prove the theorem.
First, suppose $J=\#_ia_iK_i$ where $a_i\in \mathbb{Z}$. As in the proof of Theorem~\ref{thm:refined main theorem-1}, we may assume $a_1>0$. We will show $J\notin \mathcal{F}_{n.5}^{cot,p_1}$. Suppose to the contrary that $J$ is $(n.5,p_1)$-solvable via $V$. We follow the proof of Theorem~\ref{thm:refined main theorem-1} with the changes and observations below: \begin{enumerate}
\item Now $V$ is not an $n.5$-solution; it is only an $(n.5,p_1)$-solution.
\item Use $\pi_1W^{(n+1)}_{cot,p_1}$ instead of $\mathcal{P}^{n+1}\pi_1W$. We keep $\mathcal{P}^n\pi_1W$ as it is, noting that $\mathcal{P}^n\pi_1W = \pi_1W^{(n)}_r$.
\item $\mathcal{P}^n\pi_1W/\pi_1W^{(n+1)}_{cot,p_1}$ is a $\mathbb{Z}_{p_1}$-vector space: from the
definitions, one can see that $\mathcal{P}^{n+1}\pi_1W\subset \pi_1W^{(n+1)}_{cot,p_1}$, and therefore $\mathcal{P}^n\pi_1W/\pi_1W^{(n+1)}_{cot,p_1}$ is a quotient group of $\mathcal{P}^n\pi_1W/\mathcal{P}^{n+1}\pi_1W$, which is a $\mathbb{Z}_{p_1}$-vector space.
\item We change the definition of $G$: let $G:=\pi_1W/\pi_1W^{(n+1)}_{cot,p_1}$. Then $G$ admits a subnormal series
\[
\{e\} \subset G^{(n)}_r\subset G^{(n-1)}_r\subset \cdots G^{(1)}_r\subset G
\]
whose successive quotients are abelian and have no torsion coprime to $p_1$. Therefore, the group $G$ is amenable and lies in Strebel's class $D(\mathbb{Z}_{p_1})$ (see \cite[Lemma~6.8]{Cha-Orr:2009-1}).
\item Use Theorem~\ref{thm:vanishing rho-invariant-(n,p)} instead of Theorem~\ref{thm:obstruction} to show $S_G(V)=0$.
\item The crucial part is to show that there is some $\eta_i$ such that $\phi(\eta_i)\ne e$ for the homomorphism $\phi\colon \pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$. In the proof of Theorem~\ref{thm:refined main theorem-1}, we use Theorem~\ref{thm:eta_i} to show this when $G=\pi_1W/\mathcal{P}^{n+1}\pi_1W$. Theorem~\ref{thm:eta_i} is proved in the end of Subsection~\ref{subsec:algebraic n-solution} in which we show that there exists some $\eta_i$ such that $\eta_i\notin \mathcal{P}^{n+1}\pi_1W$. But in that proof a stronger fact is proved for $\eta_i$: this $\eta_i$ maps to a nontrivial element in $H_1(W;\mathbb{K}[t^{\pm 1}])$ where $\mathbb{K}$ is the skew quotient field of $\mathbb{Z}_p[\pi_1W^{(1)}/\pi_1W^{(n)}_r]$ (in our case $p=p_1$). This implies that this $\eta_i$ maps to a nontrivial element even in $\mathcal{P}^n\pi_1W/\pi_1W^{(n+1)}_{cot,p_1}$. Therefore for this $\eta_i$, we have $\phi(\eta_i)\ne e$. \end{enumerate}
Then, we obtain $|S_G(W)|>C$ as in the proof of Theorem~\ref{thm:refined main theorem-1}, which is a contradiction. This completes the proof that $J\notin \mathcal{F}_{n.5}^{cot,p_1}$.
Next, we show $L\in \mathcal{F}_{n.5}^{cot,p}$ for all prime $p$. The proof is similar to the one of Proposition~9.2 in \cite{Cochran-Harvey-Leidy:2009-2}. Suppose $L$ is concordant to a knot $J:=\#_i a_i J_i$ where $a_i\in \mathbb{Z}$ and $J_i$ are $n$-solvable knots in \cite{Cochran-Harvey-Leidy:2009-1}, \cite{Horn:2010-1}, \cite{Cochran-Harvey-Leidy:2009-2}, or \cite{Cha:2010-1}, which are constructed using iterated doubling operators and generate an infinite rank subgroup of $\mathcal{F}_n/\mathcal{F}_{n.5}$.
We explain iterated doubling operators in more detail. Let $R$ be a slice knot and $\alpha$ be a finite set of simple closed curves $\{\eta_1,\eta_2,\ldots, \eta_m\}$ in $S^3\smallsetminus R$ such that $\eta_i$ form an unlink in $S^3$ and $\eta_i\in \pi_1(S^3\smallsetminus R)^{(1)}$ for all $i$. For a given knot $J$, we write $R_\alpha(J)$ for the knot $R(\eta_1,\ldots,\eta_m;J)$, the knot obtained by infecting $R$ by $J$ along $\eta_\ell$ $(1\le \ell\le m)$ as defined in Section~\ref{sec:construction}, and we say $R_\alpha$ is a {\it doubling operator}. Then, an {\it iterated doubling operator} (at level $n$) is obtained by applying doubling operators $n$ times: $R^n_{\alpha_n}\circ R^{n-1}_{\alpha_{n-1}}\circ \cdots \circ R^1_{\alpha_1}$. In this paper, by a knot constructed using iterated doubling operators we mean a knot $(R^n_{\alpha_n}\circ R^{n-1}_{\alpha_{n-1}}\circ \cdots \circ R^1_{\alpha_1})(J)$ for some knot $J$ with vanishing Arf invariant. Here, we need $J$ to have vanishing Arf invariant to make the resulting knot $n$-solvable (see \cite[Proposition~3.1]{Cochran-Orr-Teichner:2002-1}). Furthermore, in this paper to simplify notations we consider only the iterated doubling operators where each $\alpha_i$ for $R^i$ is a single curve. Our proof can be easily and straightforwardly adapted to the general case of iterated doubling operators with multiple curves at each level.
Now since $L$ is concordant to $J=\#a_iJ_i$, there exists a homology cobordism $V$ between $M(L)$ and $M(J)$. Let $C$ be the standard cobordism between $M(J)$ and $\coprod a_i M(J_i)$. Then $V\cup C$, the union along the common boundary $M(J)$, has boundary $\partial (V\cup C) = \coprod (-a_i M(J_i)) \coprod M(L)$.
Let us write $J_i=(R^n_{\alpha_n}\circ R^{n-1}_{\alpha_{n-1}}\circ \cdots \circ R^1_{\alpha_1})(J_i^0)$ for some knot $J_i^0$. Let $\mu_0$ be the meridian of $J_i^0$. Let $W_i$ be an $n$-solution for $M(J_i)$ such that $H_2(W_i)\cong H_2(V_i)$ for a 0-solution $V_i$ for $J_i^0$. We may assume $V_i\subset W_i$ and $\pi_1V_i\cong \mathbb{Z}$, which is generated by the meridian $\mu_0$ of $J_i^0$. The existence of such $W_i$ is well-known in the literature (see the proofs of \cite[Theorem~6.2]{Cochran-Harvey-Leidy:2009-2} and \cite[Proposition~4.4]{Cha:2010-1}).
Let $W$ be the union of $V\cup C$ and $\coprod a_iW_i$ along their common boundary $\coprod a_iM(J_i)$. Then $\partial W=M(L)$. We will show that $W$ is an $(n.5, p)$-solution for $L$ for all prime $p$.
Using Mayer-Vietoris sequences, one can see that $H_2(W)\cong \oplus H_2(W_i)^{|a_i|}$, and therefore $H_2(W)\cong \oplus H_2(V_i)^{|a_i|}$. Since $\alpha_j\in \pi_1(S^3\smallsetminus R^j)^{(1)}$ for $j=1,2,\ldots, n$, one can easily see that $\mu_0\in \pi_1M(J_i)^{(n)}$, and hence $\mu_0\in \pi_1W^{(n)}$. Since $\pi_1V_i$ is generated by the meridian $\mu_0$, the 0-Lagrangians and 0-duals of $V_i$ for all $i$ form an $n$-Lagrangian, say $[L_j]$, and $n$-dual, say $[D_j]$, of $W$. Since $\pi^{(n)}\subset \pi^{(n)}_{cot,p}$, $[L_j]$ and $[D_j]$ are an $(n,p)$-Lagrangian and its $(n,p)$-dual, respectively, for all prime $p$.
We will show that $[L_j]$ form an $(n+1,p)$-Lagrangian of $W$ for all prime $p$, which implies that $W$ is an $(n.5, p)$-solution for all prime $p$. Since each of $L_j$ is a surface in $V_i$ for some $i$ and $V_i\subset W$ for all $i$, it suffices to show that $\pi_1V_i$ maps to $\pi_1W^{(n+1)}_{cot,p}$ by the inclusion-induced homomorphism. Since $\pi_1V_i\cong \mathbb{Z}$ is generated by the meridian $\mu_0$, we only need to show $\mu_0\in \pi_1W^{(n+1)}_{cot,p}$.
Fix a prime $p$ and let $G:=\pi_1W$. Since $\mu_0\in G^{(n)}$, by the definition of $G^{(n+1)}_{cot,p}$ we only need to show $\mu_0\in G^{(n)}_r/[G^{(n)}_r,G^{(n)}_r]$ is $\mathbb{K}[t^{\pm 1}]$-torsion where $\mathbb{K}$ is the skew quotient field of the group ring $\mathbb{Z}_p[G^{(1)}/G^{(n)}_r]$. From the iterated doubling operator construction of $J_i$, the meridian $\mu_0$ is identified with the curve $\alpha_1$. We show that $\alpha_1$ is $\mathbb{K}[t^{\pm 1}]$-torsion following the arguments in the proof of \cite[Proposition~9.2]{Cochran-Harvey-Leidy:2009-2}. Let $J_i^1:=R^1_{\alpha_1}(J_i^0)$ and let $\mu_1$ be the meridian of $J_i^1$. Again since $\alpha_j\in \pi_1(S^3\smallsetminus R^j)^{(1)}$ for all $j$, one can easily prove $\mu_1\in G^{(n-1)}$. Therefore $\pi_1M(J_i^1)\subset G^{(n-1)}$, and hence $\pi_1M(J_i^1)^{(1)}\subset G^{(n)}_r$ and $\pi_1M(J_i^1)^{(2)}\subset [G^{(n)}_r, G^{(n)}_r]$. Now let $\Delta(t)$ be the Alexander polynomial of $J_i^1$, which is the same as that of $R^1$. Since $\alpha_1\in \pi_1(S^3\smallsetminus R^1)^{(1)}$, the polynomial $\Delta(t)$ annihilates $\alpha_1$ in the module $\pi_1M(J_i^1)^{(1)}/\pi_1M(J_i^1)^{(2)}$, and therefore $\Delta(\mu_1)$ annihilates $\alpha_1$ in the module $G^{(n)}_r/[G^{(n)}_r,G^{(n)}_r]$. Since $n\ge 2$ and $\mu_1\in G^{(n-1)}$, we have $\mu_1\in G^{(1)}$ and $\Delta(\mu_1)\in \mathbb{Z}_p[G^{(1)}/G^{(n)}_r]\subset \mathbb{K}\subset \mathbb{K}[t^{\pm 1}]$. Furthermore, since $\Delta(\mu_1)$ augments to 1, i.e., $\Delta(1)=1$, $\Delta(\mu_1)\ne 0$ in $\mathbb{K}[t^{\pm 1}]$. This implies that $\alpha_1$ is $\mathbb{K}[t^{\pm 1}]$-torsion. \end{proof}
\section{Modulo $p$ Blanchfield linking forms and algebraic $n$-solutions}\label{sec:Blanchfield linking form and algebraic n-solutions}
This section is devoted to proving Theorem~\ref{thm:eta_i}. To this end, in Subsection~\ref{subsec:Blanchfield form} we prove Theorem~\ref{thm:nontrivial} which asserts the nontriviality of some homomorphisms on first homology with twisted coefficients induced from the inclusion from $M(K)$ to an $n$-cylinder one of whose boundary components is $M(K)$. Then, in Subsection~\ref{subsec:algebraic n-solution} we introduce the notion of $\mathbb{Z}_p$-coefficient algebraic $n$-solutions, which generalizes the notion of algebraic $n$-solutions in \cite{Cochran-Teichner:2003-1, Cochran-Kim:2004-1}, and relevant theorems. Finally, we give a proof of Theorem~\ref{thm:eta_i} at the end of Section~\ref{sec:Blanchfield linking form and algebraic n-solutions}.
\subsection{Modulo $p$ Blanchfield linking forms}\label{subsec:Blanchfield form} The purpose of this subsection is to prove Theorem~\ref{thm:nontrivial} below, which plays a key role in the proof of Theorem~\ref{thm:eta_i}. Theorem~\ref{thm:nontrivial} generalizes \cite[Theorem~3.8]{Cochran-Kim:2004-1} to the case of a homomorphism on first homology whose coefficient group is a localization of a group ring with $\mathbb{Z}_p$ coefficients.
First, we need the following lemma. \begin{lemma}[{\cite[Proposition 2.5]{Cochran-Orr-Teichner:1999-1} for $R=\mathbb{Q}$ and \cite[Lemma 5.2]{Cha:2010-1} for $R=\mathbb{Z}_p$}] \label{lem:quotient field}
Let $R=\mathbb{Q}$ or $\mathbb{Z}_p$. If $\Gamma$ is a PTFA group, then $R\Gamma$ is an Ore domain. That is, $R\Gamma$ embeds in the (skew) quotient filed $\mathcal{K}=R\Gamma(R\Gamma-\{0\})^{-1}$. \end{lemma} Let $R=\mathbb{Q}$ or $\mathbb{Z}_p$. Let $\Gamma$ be a PTFA group such that $H_1(\Gamma)\cong \Gamma/[\Gamma,\Gamma]\cong\mathbb{Z}$. Let $\mathcal{K}$ be the (skew) quotient field of $R\Gamma$ obtained by Lemma~\ref{lem:quotient field}. Since a subgroup of a PTFA group is also PTFA, the group $[\Gamma,\Gamma]$ is PTFA. By Lemma~\ref{lem:quotient field}, the group ring $R[\Gamma, \Gamma]$ embeds into the (skew) quotient field, say $\mathbb{K}$. Since $H_1(\Gamma)\cong\mathbb{Z}=\langle t\rangle$, we have a (noncommutative) PID $\mathbb{K}[t^{\pm 1}]$ such that $R\Gamma\subset \mathbb{K}[t^{\pm 1}]\subset \mathcal{K}$.
\begin{theorem}[{\cite[Theorem 3.8]{Cochran-Kim:2004-1} for $R=\mathbb{Q}$}] \label{thm:nontrivial}
Let $n\ge 1$ be an integer. Let $R=\mathbb{Q}$ or $\mathbb{Z}_p$. For a knot $K$, let $W$ be an $n$-cylinder with $M(K)$ as one of its boundary components. Let $\Gamma$ be an PTFA group such that $H_1(\Gamma)\cong \mathbb{Z}=\langle t\rangle$ and $\Gamma^{(n)}=\{e\}$. Let $\phi\colon \pi_1W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Gamma$ be a homomorphism which induces an isomorphism $H_1(W)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\Gamma)$. Let $d:=\operatorname{rank}_R H_1(M_\infty;R)$ where $M_\infty$ is the infinite cyclic cover of $M(K)$. Then we have
\[
\operatorname{rank}_\mathbb{K} \operatorname{Im}\{i_*\colon H_1(M(K);\mathbb{K}[t^{\pm 1}]) \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W;\mathbb{K}[t^{\pm 1}])\} \ge \left\{
\begin{array}{cl}
(d-2)/2 & \mbox{if } n>1,\\
d/2 & \mbox{if } n=1,
\end{array}\right.
\] where $i_*$ is the inclusion-induced homomorphism. \end{theorem} We give a proof of Theorem~\ref{thm:nontrivial} at the end of this subsection, after showing needed materials.
We note that the image of $i_*$ is nontrivial if $d>2$ when $n>1$ and if $d>0$ when $n=1$. In the above setting, when $R=\mathbb{Q}$, the rank $d$ is equal to the degree of $\Delta_K(t)$, the Alexander polynomial of $K$. When $R=\mathbb{Z}_p$, the rank $d$ is still equal to the degree of $\Delta_K(t)$ if the prime $p$ is bigger than the top coefficient of $\Delta_K(t)$.
To prove Theorem~\ref{thm:nontrivial}, we need to generalize the various results which were used for the proof of \cite[Theorem~3.8]{Cochran-Kim:2004-1} to the case of homology with twisted coefficients which are obtained as a localization of a group ring with $\mathbb{Z}_p$ coefficients. A key ingredient of the proof of Theorem~\ref{thm:nontrivial} is higher-order Blanchfield linking forms which are adapted to homology with such coefficients.
We briefly review higher-order Blanchfield linking forms. The Blanchfield linking form in a noncommutative setting was first defined by Duval \cite{Duval:1986-1} on boundary links over a group ring of a free group. Then, Cochran--Orr--Teichner introduced the noncommutative (higher-order) Blanchfield linking form for a knot over a group ring $\mathbb{Z}\Gamma$ of a PTFA group $\Gamma$ and its locallizations \cite{Cochran-Orr-Teichner:1999-1}. This was generalized to the Blanchfield linking form for a knot over a group ring $\mathbb{Z}_p\Gamma$ for a PTFA group $\Gamma$ by Cha \cite{Cha:2010-1}. In this paper, we need the Blanchfield linking form for a knot over a (PID) localization of a group ring $\mathbb{Z}_p \Gamma$ for a PTFA group $\Gamma$. It will be defined in Theorem~\ref{thm:Blanchfield}, and will be used to prove Theorem~\ref{thm:nontrivial}.
Let $\Gamma$ be a PTFA group and $R=\mathbb{Q}$ or $\mathbb{Z}_p$. Then by Lemma~\ref{lem:quotient field} $R\Gamma$ has the (skew) quotient field of $\mathcal{K}$. If $\mathcal{R}$ is a ring such that $R\Gamma\subset \mathcal{R}\subset \mathcal{K}$ and $M$ is a closed 3-manifold such that $H_1(M)\cong \mathbb{Z}$ and $\pi_1M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Gamma$ is a nontrivial representation, then we have the composition of maps \[ H_1(M;\mathcal{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \overline{H^2(M;\mathcal{R})}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \overline{H^1(M;\mathcal{K}/\mathcal{R})}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \overline{\operatorname{Hom}_R(H_1(M;\mathcal{R}), \mathcal{K}/\mathcal{R})}. \] where the maps are Poincar\'{e} duality, the inverse of a Bockstein homomorphism, and the Kronecker evaluation map. Here, the inverse of a Bockstein homomorphism exists since $H^1(M;\mathcal{K})=H^2(M;\mathcal{K})=0$ and hence the Bockstein homomorphism is in fact an isomorphism (see \cite[Proposition~2.11]{Cochran-Orr-Teichner:1999-1}). Also, $\overline{H^*(M;\mathcal{R})}$ are made into right $\mathcal{R}$-modules using the involution of $\mathcal{R}$.
The following theorem shows that under this setting, if $\mathcal{R}$ is a PID, then the composition gives rise to a nonsingular symmetric linking form.
\begin{theorem}[{\cite[Theorem 2.13]{Cochran-Orr-Teichner:1999-1} for $R=\mathbb{Q}$ and essentially due to \cite[Section 5]{Cha:2010-1} for $R=\mathbb{Z}_p$}] \label{thm:Blanchfield}
Let $R=\mathbb{Q}$ or $\mathbb{Z}_p$. Let $M$ be a closed 3-manifold with $H_1(M)\cong \mathbb{Z}$. Let $\phi\colon \pi_1M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Gamma$ be a nontrivial PTFA coefficient system. Let $\mathcal{R}$ be a (noncommutative) PID such that $R\Gamma\subset \mathcal{R}\subset \mathcal{K}$ where $\mathcal{K}$ is the (skew) quotient field of $R\Gamma$. Then there exists a nonsingular symmetric linking form (called the {\it Blanchfield linking form}) \[ B\ell\colon H_1(M;\mathcal{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \overline{H^2(M;\mathcal{R})}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \overline{H^1(M;\mathcal{K}/\mathcal{R})}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \overline{\operatorname{Hom}_R(H_1(M;\mathcal{R}), \mathcal{K}/\mathcal{R})}. \] \end{theorem}
\begin{proof}
When $R=\mathbb{Z}_p$, the proof is identical to that of \cite[Theorem 2.13]{Cochran-Orr-Teichner:1999-1} (for the case $R=\mathbb{Q})$ except for the following change: in the proof, replace \cite[Proposition 2.11]{Cochran-Orr-Teichner:1999-1} by \cite[Lemma 5.3]{Cha:2010-1}. \end{proof}
Lemma~\ref{lem:exact sequence} below is needed to prove Proposition~\ref{prop:self-annihilating}. When $R=\mathbb{Q}$ it was proved in \cite{Cochran-Kim:2004-1}. The proof for the case $R=\mathbb{Z}_p$ is essentially the same as that for the case $R=\mathbb{Q}$, and hence it is omitted.
\begin{lemma}[{\cite[Lemma 3.5]{Cochran-Kim:2004-1} for $R=\mathbb{Q}$}] \label{lem:exact sequence}
Let $R=\mathbb{Q}$ or $\mathbb{Z}_p$. Let $W$ be an $R$-coefficient $n$-cylinder with $M$ as one of its boundary components. Let $\Gamma$ be a PTFA group such that $\Gamma^{(n)}=\{e\}$, and let $\phi\colon \pi_1M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \Gamma$ be a nontrivial coefficient system which extends to $\pi_1W$. Let $\mathcal{R}$ be a (noncommutative) PID such that $R\Gamma\subset \mathcal{R}\subset \mathcal{K}$ where $\mathcal{K}$ is the (skew) quotient field of $R\Gamma$. Then the sequence of maps
\[
TH_2(W,M;\mathcal{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(M;\mathcal{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W;\mathcal{R})
\]
is exact. (Here for an $\mathcal{R}$-module $N$, $TN$ denotes the $\mathcal{R}$-torsion submodule of $N$.) \end{lemma}
For an $\mathcal{R}$-submodule $P$ of $H_1(M;\mathcal{R})$, we define \[ P^\perp := \{x\in H_1(M;\mathcal{R})\,\mid\, B\ell(x)(y)=0 \mbox{ for all } y\in P\}. \] We say that a submodule $P$ of $H_1(M;\mathcal{R})$ is {\it self-annihilating with respect to $B\ell$} if $P=P^\perp$. The following proposition generalizes \cite[Theorem~4.4]{Cochran-Orr-Teichner:1999-1} and \cite[Proposition~3.6]{Cochran-Kim:2004-1}. \begin{proposition}[{\cite[Proposition 3.6]{Cochran-Kim:2004-1} for $R=\mathbb{Q}$}] \label{prop:self-annihilating}
Suppose the same hypotheses as in Lemma~\ref{lem:exact sequence}. If $P=\operatorname{Ker}\{i_*\colon H_1(M;\mathcal{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W;\mathcal{R})\}$ where $i_*$ is the inclusion-induced homomorphism, then $P\subset P^\perp$. Moreover, if $W$ is an $R$-coefficient $n$-solution for $M$, then $P=P^\perp$. \end{proposition}
\begin{proof} For $R=\mathbb{Z}_p$, make the following changes in the proof of \cite[Proposition 3.6]{Cochran-Kim:2004-1}: replace Corollary~3.3 and Lemma~3.5 in \cite{Cochran-Kim:2004-1} by \cite[Lemma 5.3]{Cha:2010-1} and Lemma~\ref{lem:exact sequence} above, respectively. \end{proof}
The proof of the following lemma is based on the ideas in \cite[Sections 3 and 4]{Cochran:2002-1}. \begin{lemma}\label{lem:rank}
\begin{enumerate}
\item Let $G$ be a PTFA group and $R=\mathbb{Z}_p$. Let $A$ be a wedge of $m$ circles. Suppose we have a nontrivial coefficient system $\pi_1A\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$. Then $\operatorname{rank}_{RG}H_1( A;RG)=m-1$.
\item Suppose the same hypotheses as in Theorem~\ref{thm:nontrivial}. Suppose $n>1$ and $R=\mathbb{Z}_p$. Then, $\operatorname{rank}_\mathbb{K} H_1(E(K);\mathbb{K}[t^{\pm 1}])\ge d-1$.
\end{enumerate}
\end{lemma} \begin{proof}
Note that $A$ is a finite 1-complex with the Euler characteristic $1-m$. By the invariance of the Euler characteristic, we have $\operatorname{rank}_{RG} H_0(A;RG)-\operatorname{rank}_{RG}H_1(A;RG)=1-m$. Since the homomorphism $\pi_1A\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is nontrivial, for the (skew) quotient field $\mathbb{K}$ of $RG$ and the map $\phi\colon \pi_1A\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{K}$ we have
\[
H_0(A;\mathbb{K})=\mathbb{K}/\{\phi(g)a-a\,\mid\,g\in \pi_1A, a\in \mathbb{K}\}=0.
\]
Now we have $\operatorname{rank}_{RG} H_0(A;RG)=0$ and $\operatorname{rank}_{RG}H_1(A;RG)=m-1$. This proves (1).
Let $\tilde{E}$ be the infinite cyclic cover of $E(K)$ and let $G:=[\Gamma,\Gamma]$ (therefore $\mathbb{K}$ is the quotient field of $R[\Gamma,\Gamma]$). Then $H_1(E(K);\mathbb{K}[t^{\pm 1}]) \cong H_1(\tilde{E};\mathbb{K})$ and $d=\operatorname{rank}_RH_1(\tilde{E};R)$. Since $\mathbb{K}$ is the (skew) quotient field of $RG$, we have $\operatorname{rank}_\mathbb{K} H_1(\tilde{E};\mathbb{K}) = \operatorname{rank}_{RG} H_1(\tilde{E};RG)$, and it suffices to show that $\operatorname{rank}_{RG} H_1(\tilde{E};RG)\ge d-1$.
Let $A$ be a wedge of $d$ circles and let $j\colon A\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \tilde{E}$ be a map which induces a monomorphism on $H_1(A;R)$. For convenience we identify $A$ with $j(A)$. Since $\tilde{E}$ has the homotopy type of a 2-complex, we may assume $(\tilde{E}, A)$ is a 2-complex. Let $C_2\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} C_1\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} C_0$ be the free $RG$-chain complex obtained from the cell structure of the $G$-cover of $(\tilde{E},A)$.
Since $H_2(\tilde{E};R)=0$ and $H_1(A;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\tilde{E};R)$ is injective, we have $H_2(\tilde{E},A;R)=0$. Therefore $C_2\otimes_{RG}R\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} C_1\otimes_{RG}R$ is injective. Since PTFA groups lie in Strebel's class $D(R)$ \cite{Strebel:1974-1}, the group $G$ is in $D(R)$. Therefore $C_2\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} C_1$ is also injective, and hence $H_2(\tilde{E},A;RG)=0$. Then, it follows that the map $H_1(A;RG)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(\tilde{E};RG)$ is injective. Now $\operatorname{rank}_{RG}H_1(\tilde{E};RG)\ge \operatorname{rank}_{RG}H_1(A;RG)=d-1$ where the equality holds by Part (1). \end{proof}
Now we are ready to give a proof of Theorem~\ref{thm:nontrivial}.
\begin{proof}[Proof of Theorem~\ref{thm:nontrivial}]
For $R=\mathbb{Z}_p$, the proof is the same as the proof of \cite[Theorem 3.8]{Cochran-Kim:2004-1} (for the case $R=\mathbb{Q}$) except for the following changes: noticing that an $n$-cylinder is a $\mathbb{Z}_p$-coefficient $n$-cylinder by Proposition~\ref{prop:implication of cylinder}, replace \cite[Theorem 2.13]{Cochran-Orr-Teichner:1999-1}, \cite[Proposition 3.6]{Cochran-Kim:2004-1}, and the coefficients $\mathbb{Q}$ by Theorem~\ref{thm:Blanchfield}, Proposition~\ref{prop:self-annihilating}, and the coefficients $\mathbb{Z}_p$, respectively. One also needs to replace \cite[Corollary 4.8]{Cochran:2002-1} by Lemma~\ref{lem:rank}~(2). \end{proof}
\subsection{Algebraic $n$-solutions}\label{subsec:algebraic n-solution} In this section, we introduce the notion of {\it $R$-algebraic $n$-solution} where $R=\mathbb{Q}$ or $\mathbb{Z}_p$, and using it we prove Theorem~\ref{thm:eta_i}. The notion of an algebraic $n$-solutoin was introduced by Cochran and Teichner in \cite{Cochran-Teichner:2003-1} and later generalized by Cochran and the author in \cite[Section 6]{Cochran-Kim:2004-1}. The new notion of an $R$-algebraic $n$-solution generalizes an algebraic $n$-solution to the case of $\mathbb{Z}_p$ coefficients. An $R$-algebraic $n$-solution may be considered as an algebraic abstraction of an $n$-cylinder.
All the results in this section are based on the ideas and results in \cite{Cochran-Teichner:2003-1} and \cite{Cochran-Kim:2004-1} which deal with the case $R=\mathbb{Q}$. In fact, the results in this section are mainly a $\mathbb{Z}_p$-coefficient version of the corresponding result with $\mathbb{Q}$ coefficients in \cite[Section 6]{Cochran-Kim:2004-1}. But since Section~6 in \cite{Cochran-Kim:2004-1} is quite technical, we rewrite the theorems with $\mathbb{Z}_p$ coefficients carefully and completely, and clarify how one should modify the proofs of the corresponding theorems in \cite{Cochran-Kim:2004-1}.
For a group $G$, let $G_k:=G/G^{(k)}_r$ where $G^{(k)}_r$ is the $k$-th rational derived group of $G$. Since $G_k$ is a PTFA group, $\mathbb{Z} G_k$ (and $\mathbb{Q} G_k$) embeds into the (skew) quotient field which we denote by $\mathbb{K}(G_k)$. By Lemma~\ref{lem:quotient field}, the group ring $\mathbb{Z}_pG_k$ also embeds into the (skew) quotient field, and we also denote it by $\mathbb{K}(G_k)$, or $\mathbb{K}_p(G_K)$ to emphasize the coefficients $\mathbb{Z}_p$. We write $\mathbb{K}$ for $\mathbb{K}(G_k)$ when it is understood from the context.
The following is a generalization of an algebraic $n$-solution defined in \cite[Definition 6.1]{Cochran-Teichner:2003-1} and \cite[Definition 6.1]{Cochran-Kim:2004-1}.
\begin{definition}\label{def:algebraic n-solution}(\cite[Definition 6.1]{Cochran-Kim:2004-1} for $R=\mathbb{Q}$) Let $R=\mathbb{Q}$ or $\mathbb{Z}_p$ and let $\mathbb{K}(G_k)$ be the (skew) quotient field of $RG_k$. Let $S$ be a group with $H_1(S;R)\ne 0$. Let $F$ be a free group of rank $2g$ and let $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ be a homomorphism. A nontrivial homomorphism $r\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is an {\it $R$-algebraic $n$-solution} for $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ if the following hold:
\begin{enumerate}
\item For each $0\le k\le n-1$, the map $r_*\colon H_1(S;\mathbb{K}(G_k))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(G;\mathbb{K}(G_k))$ is nontrivial.
\item For each $0\le k\le n-1$, the map $i_*\colon H_1(F;\mathbb{K}(G_k))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S;\mathbb{K}(G_k))$ is surjective.
\end{enumerate}
A $\mathbb{Q}$-algebraic $n$-solution is also called an {\it algebraic $n$-solution}. \end{definition}
\begin{remark}\label{rem:algebraic n-solution}
\begin{enumerate}
\item For $k<n$, an $R$-algebraic $n$-solution is an $R$-algebraic $k$-solution.
\item Since $\mathbb{K}(G_k)$ is a flat $RG_k$-module \cite[Proposition II.3.5]{Stenstrom:1975}, we have $H_1(-;\mathbb{K}(G_k))\cong H_1(-;RG_k)\otimes_{RG_k}\mathbb{K}(G_k)$.
\item We have $H_1(G;RG_k)\cong H_1(G;\mathbb{Z} G_k)\otimes_\mathbb{Z} R\cong G^{(k)}_r/[G^{(k)}_r,G^{(k)}_r]\otimes_\mathbb{Z} R$.
\item The case $k=0$ in the condition (2) implies that $H_1(F;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S;R)$ is surjective.
\end{enumerate} \end{remark}
The following proposition shows the relationship between $n$-cylinders and $R$-algebraic $n$-solutions. Roughly speaking, for a knot $K$ and an $n$-cylinder $W$ for $M(K)$, the homomorphism on the fundamental groups of the infinite cyclic covers of $M(K)$ and $W$ induced from the inclusion gives rise to an $R$-algebraic $n$-solution.
\begin{proposition}[{\cite[Proposition 6.3]{Cochran-Kim:2004-1} for $R=\mathbb{Q}$}] \label{prop:n-cylinder and n-solution}
Let $n\ge 1$. Let $K$ be a knot with nontrivial Alexander polynomial $\Delta_K(t)$. Suppose the degree of $\Delta_K(t)$ is greater than 2 if $n>1$. Let $W$ be an $n$-cylinder with $M(K)$ as one of its boundary components. Let $\Sigma$ be a capped-off Seifert surface for $K$. Let $S:=\pi_1M(K)^{(1)}$ and $G:=\pi_1W^{(1)}$. Let $R:=\mathbb{Q}$ or $\mathbb{Z}_p$ where $p$ is a prime greater than the top coefficient of $\Delta_K(t)$. Let $F$ be a free group of rank $2g$ where $g$ is the genus of $\Sigma$ and let $F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(M(K)\smallsetminus \Sigma)$ be a homomorphism inducing an isomorphism on $H_1(F;R)$. Let $i$ be the composition $F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(M(K)\smallsetminus \Sigma)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$. Then, the inclusion-induced map $j\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is an $R$-algebraic $n$-solution for $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$. \end{proposition} \begin{proof}
For $R=\mathbb{Z}_p$, modify the proof of Proposition~6.3 in \cite{Cochran-Kim:2004-1} (for the case $R=\mathbb{Q}$) as follows:
\begin{enumerate}
\item Replace the coefficients $\mathbb{Z}$ by $\mathbb{Z}_p$ and let $\mathbb{K}$ be the (skew) quotient field of $\mathbb{Z}_p\Gamma^{(1)}=\mathbb{Z}_pG_k$.
\item Replace \cite[Theorem 3.8]{Cochran-Kim:2004-1} by Theorem~3.5 in this paper.
\item In the proof of \cite[Proposition~6.3]{Cochran-Kim:2004-1}, it was given that $d:=\operatorname{rank}_\mathbb{Q} H_1(M_\infty;\mathbb{Q})$ where $M_\infty$ is the infinite cyclic cover of $M(K)$, and it was used that $d$ is equal to the degree of $\Delta_K(t)$. In our case with $\mathbb{Z}_p$ coefficients, we set $d:=\operatorname{rank}_{\mathbb{Z}_p} H_1(M_\infty;\mathbb{Z}_p)$. Note that $d$ is still equal to the degree of $\Delta_K(t)$ since $p$ is greater than the top coefficient of $\Delta_K(t)$ by our hypothesis.
\item To establish Property (2) in Definition~\ref{def:algebraic n-solution}, choose a map $W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} Y=M(K)\smallsetminus \Sigma$ inducing $F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1Y$ that is 1-connected on homology with $\mathbb{Z}_p$ coefficients.
\item Use the $\mathbb{Z}_p$-coefficient version of \cite[Proposition 2.10]{Cochran-Orr-Teichner:1999-1} to show that the map $F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1Y$ induces a 1-connected map on homology with $\mathbb{K}(G_k)$ coefficients: Proposition~2.10 in \cite{Cochran-Orr-Teichner:1999-1} holds with $\mathbb{Z}_p$ coefficients since $G_k$ lies in Strebel's class $D(\mathbb{Z}_p)$ (see Lemma~\ref{lem:amenable and D(R)}).
\item The proof of \cite[Proposition~6.3]{Cochran-Kim:2004-1} uses Harvey's work in \cite{Harvey:2005-1} with the (skew) quotient field $\mathbb{K}$ of $\mathbb{Z} G_k$. We can still use Harvey's work in the same way for the case of $\mathbb{Z}_p$ coefficients with our $\mathbb{K}=\mathbb{K}_p(G_k)$, the (skew) quotient field of $\mathbb{Z}_p G_k$. \end{enumerate} \end{proof} \noindent We note that in the proofs of Proposition~\ref{prop:n-cylinder and n-solution} and \cite[Proposition 6.3]{Cochran-Kim:2004-1}, it is implicitly proved that the map $i_*\colon H_1(F;\mathbb{K}(G_k))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S;\mathbb{K}(G_k))$ is surjective for all $k\ge 0$.
By Remark~\ref{rem:algebraic n-solution}~(3), for a group $S$ and $R=\mathbb{Q}$ or $\mathbb{Z}_p$, an element of $S^{(n)}$ can be considered as an element of $H_1(S;RS_n)$.
\begin{theorem}[{\cite[Theorem 6.4]{Cochran-Kim:2004-1} for $R=\mathbb{Q}$ only}] \label{thm:special tuple}
Let $F$ be a free group of rank $2g$ and let $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ be a group homomorphism. For each $n\ge 0$, there exists a finite collection $\mathcal{P}_n$ of sets of $2g-1$ (if $n>0$) or $2g$ (if $n=0$) elements of $F^{(n)}$ which satisfies the following:
If $r\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is an $R$-algebraic $n$-solution for $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ for both $R=\mathbb{Q}$ and $R=\mathbb{Z}_p$, then there is an element in $\mathcal{P}_n$ which maps to a generating set of $H_1(S;\mathbb{K}(G_n))$ under the composition
\[
F^{(n)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S^{(n)}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S;RS_n)\xrightarrow{r_*}H_1(S;RG_n)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S;\mathbb{K}(G_n)).
\]
for both $R=\mathbb{Q}$ and $R=\mathbb{Z}_p$. \end{theorem}
An element of $\mathcal{P}_n$ is called an (unordered) {\it tuple}, and a tuple mapped to a generating set of $H_1(S;\mathbb{K}(G_n))$ in the above theorem is called a {\it special tuple}. In the proof of Theorem~\ref{thm:eta_i}, the $\eta_i$ will be defined as the image of the elements of the tuples in $\mathcal{P}_{n-1}$ under the homomorphism $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$, which is a finite set, and the existence of a special tuple will give us the existence of the desired curve $\eta_i$.
\begin{proof}[Proof of Theorem~\ref{thm:special tuple}]
Suppose $x_1,x_2,\ldots, x_{2g}$ generate $F$. Let $\mathcal{P}_0=\{\{x_1,x_2,\ldots, x_{2g}\}\}$. When $n=0$, the above composition becomes $F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S;R)$ and the theorem follows since $H_1(F;R)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S;R)$ is surjective by Remark~\ref{rem:algebraic n-solution}~(4) and $x_i$ generate $H_1(F;R)\cong R^{2g}$.
Now assume $n\ge 1$. We define $\mathcal{P}_n$ inductively as in the proof of \cite[Theorem 6.4]{Cochran-Kim:2004-1}. Define
\[\mathcal{P}_1:=\{\{[x_i,x_1],\dots,
[x_i,x_{i-1}],[x_i,x_{i+1}],\ldots,[x_i,x_{2g}]\}\,\,|\,\,1\le i
\le 2g\},
\]
which is a set of $(2g-1)$-tuples. Suppose $\mathcal{P}_k$ has been constructed for $k\ge 1$. We define $\mathcal{P}_{k+1}$ as follows: a $(2g-1)$-tuple $\{z_1,...,z_{2g-1}\}$ is in $\mathcal{P}_{k+1}$ if and only if there is $\{w_1,...,w_{2g-1}\}\in \mathcal{P}_k$ such that for each $1\le i\le 2g-1$, $z_i=[w_i,w_i^{x_j}]$ for some $j$ with $1\le j\ne i\le 2g$ or $z_i=[w_i,w_k]$ for some $k$ with $1\le k\ne i\le 2g-1$. Here $w_i^{x_j}$ denotes $x_j^{-1}w_ix_j$.
By \cite[Theorem 6.4]{Cochran-Kim:2004-1} the conclusion of the theorem holds when $R=\mathbb{Q}$. That is, there is an element in $\mathcal{P}_n$ mapping to a generating set of $H_1(S;\mathbb{K}(G_n))$ where $\mathbb{K}(G_n)$ is the (skew) quotient field of $\mathbb{Q} G_n$.
We assert that the above element in $\mathcal{P}_n$ also maps to a generating set of $H_1(S;\mathbb{K}(G_n))$ when $\mathbb{K}(G_n)=\mathbb{K}_p(G_n)$, the (skew) quotient field of $\mathbb{Z}_pG_n$. This is proved by modifying the proof of Theorem~6.4 in \cite{Cochran-Kim:2004-1} as follows:
\begin{enumerate}
\item Replace the group rings over $\mathbb{Z}$ coefficients by the corresponding group rings over $\mathbb{Z}_p$ coefficients. For example, change $\mathbb{Z} F_n$ to $\mathbb{Z}_p F_n$.
\item Let $\mathbb{K}(G_n)$ denote the (skew) quotient field of $\mathbb{Z}_pG_n$.
\item Use Lemma~\ref{lem:rank}~(1) in this paper instead of \cite[Lemma 3.9]{Cochran:2002-1}.
\item Use Lemma~\ref{lem:good tuple} below instead of \cite[Lemma 6.5]{Cochran-Kim:2004-1}. Note that we need the hypothesis that $r\colon S \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is a ($\mathbb{Q}$-)algebraic $n$-solution in Lemma~\ref{lem:good tuple}.
\item In the last part of the proof, one needs to use that $H_1(F;\mathbb{Z}_pG_n)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(S:\mathbb{Z}_pG_n)$ is surjective after tensoring with $\mathbb{K}_p(G_n)$. Note that this is where we use the assumption that $r\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is a $\mathbb{Z}_p$-algebraic $n$-solution.
\end{enumerate} \end{proof}
Let $R$ be a ring. For each $1\le i\le 2g$, let $\partial_i;F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbb{Z} F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} RF$ be the Fox free differential calculus defined by $\partial_i (x_j)=\delta_{ij}$ and $\partial _i(gh) = \partial_i g + (\partial_i h)g^{-1}$. For $k\ge 0$, denote by $\pi_k$ the quotient map $RF\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} RF_k$. By abuse of notation, denote by $r$ the map $RF_k\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} RG_k$ induced from $r\circ i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$. We denote by $d_i^k$ the composition $r\circ\pi_k\circ \partial_i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} RG_k$ for $1\le i\le 2g$. We simply write $d_i$ for $d_i^k$ when the superscript is understood from the context.
\begin{lemma}[{\cite[Lemma 6.5]{Cochran-Kim:2004-1}) for $R=\mathbb{Q}$}] \label{lem:good tuple}
Let $n\ge 1$. Suppose we are given an algebraic $n$-solution $r\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ for $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$. Then, there exists a $(2g-1)$-tuple $\{w_1,\ldots,w_{2g-1}\}\in \mathcal{P}_n$ such that the $2g-1$ vectors $\{(d_1(w_i),d_2(w_i),\ldots, d_{2g-1}(w_i))\,\mid\, 1\le i\le 2g-1\}$ in $(RG_n)^{2g-1}$ are right linearly independent over $RG_n$ for $R=\mathbb{Q}$ and $R=\mathbb{Z}_p$ for all prime $p$. \end{lemma} \begin{proof}
In the proof of Lemma ~6.5 in \cite{Cochran-Kim:2004-1}, for each $1\le k\le n$, a tuple $\{w_1^k,\ldots, w_{2g-1}^k\}\in \mathcal{P}_k$ was constructed inductively such that $\{(d_1(w_i^k),d_2(w_i^k),\ldots, d_{2g-1}(w_i^k))\,\mid\, 1\le i\le 2g-1\}$ in $(\mathbb{Q} G_k)^{2g-1}$ are right linearly independent over $\mathbb{Q} G_k$ using the fact that $r\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is an algebraic $n$-solution.
We will show that for each $k$, the vectors $\{(d_1(w_i^k),d_2(w_i^k),\ldots, d_{2g-1}(w_i^k))\,\mid\, 1\le i\le 2g-1\}$ are also right linearly independent over $\mathbb{Z}_pG_k$, and it will complete the proof by taking $\{w_1,\ldots,w_{2g-1}\}=\{w_1^n,\ldots,w_{2g-1}^n\}$. The proof follows the lines of the proof of \cite[Lemma 6.5]{Cochran-Kim:2004-1}, and we explain the needed modifications below.
For $k=1$, in the proof of \cite[Lemma 6.5]{Cochran-Kim:2004-1} we assume that $(r\circ \pi_1 )(x_{2g})$ is nontrivial in $G_1$ and take $\{w_1^1,\ldots, w_{2g-1}^1\}=\{[x_{2g},x_1],\ldots, [x_{2g},x_{2g-1}]\}$. Then, it was shown that the vectors $\{(d_1(w_i^1),d_2(w_i^1),\ldots, d_{2g-1}(w_i^1))\,\mid\, 1\le i\le 2g-1\}$ are linearly independent over $\mathbb{Q} G_1$ (in the first paragraph of \cite[p.1428]{Cochran-Kim:2004-1}). Using the same argument, it can be seen that $\{w_1^1,\ldots, w_{2g-1}^1\}$ are also linearly independent over $\mathbb{Z}_pG_1$.
We use an induction argument. Suppose that for $k<n$ it has been shown that the vectors $\{(d_1(w_i^k),d_2(w_i^k),\ldots, d_{2g-1}(w_i^k))\,\mid\, 1\le i\le 2g-1\}$ are linearly independent over $\mathbb{Z}_pG_k$. In \cite{Cochran-Kim:2004-1} it was shown that we may assume $(r\circ \pi_{k+1})(w_1^k)\ne e$ in $G_{k+1}$. Then $\{w_1^{k+1},\ldots, w_{2g-1}^{k+1}\}$ was defined in \cite{Cochran-Kim:2004-1} as follows: we take $w_i^{k+1}=[w_i^k,w_i^{x_{2g}}]$ if $(r\circ \pi_{k+1}) (w_i^k)\ne e$ in $G_{k+1}$ and $w_i^{k+1}=[w_i^k, w_1^k]$, otherwise.
Then it was shown in \cite{Cochran-Kim:2004-1} that in $\mathbb{Z} G_{k+1}$, for some $t_i\in \mathbb{Z} F$
\[
(d_1^{k+1}(w_i^{k+1}),d_2^{k+1}(w_i^{k+1}),\ldots, d_{2g-1}^{k+1}(w_i^{k+1})) = (d_1^{k+1}(w_i^k),d_2^{k+1}(w_i^k),\ldots, d_{2g-1}^{k+1}(w_i^k))\cdot (r\circ \pi_{k+1})(t_i)
\]
where $(r\circ \pi_{k+1})(t_i)\ne e$ in $\mathbb{Z} G_{k+1}$. We note that using the same argument in \cite{Cochran-Kim:2004-1} one can show that $(r\circ \pi_{k+1})(t_i)\ne e$ in $\mathbb{Z}_p G_{k+1}$ as well. This implies that it suffices to show that the vectors $\{(d_1^{k+1}(w_i^k),d_2^{k+1}(w_i^k),\ldots, d_{2g-1}^{k+1}(w_i^k))\,\mid\, 1\le i\le 2g-1\}$ are linearly independent over $\mathbb{Z}_p G_{k+1}$.
For simplicity, for $1\le i\le 2g-1$, let $\mathbf{v_i^{k+1}}=(d_1^{k+1}(w_i^k),d_2^{k+1}(w_i^k),\ldots, d_{2g-1}^{k+1}(w_i^k))$ and $\mathbf{v_i^k}=(d_1^k(w_i^k),d_2^k(w_i^k),\ldots, d_{2g-1}^k(w_i^k))$. By the induction hypothesis, the vectors $\mathbf{v_i^k}$, $1\le i\le 2g-1$, are linearly independent over $\mathbb{Z}_p G_{k+1}$ (and over $\mathbb{Q} G_{k+1}$).
Let $H=G_r^{(k)}/G_{r+1}^{(k)} = \operatorname{Ker} \{G_{k+1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G_k\}$. Then $H$ is a torsion-free abelian group, and hence lies in Strebel's class $D(\mathbb{Z}_p)$ \cite{Strebel:1974-1}. In the last paragraph of the poof of \cite[Lemma 6.5]{Cochran-Kim:2004-1} it was shown that linear independence of the vectors $\mathbf{v_i^k}$ over $\mathbb{Z} G_k$ implies linear independence of the vectors $\mathbf{v_i^{k+1}}$ over $\mathbb{Z} G_{k+1}$. Using the same argument and the fact that the group $H$ lies in $D(\mathbb{Z}_p)$, changing the coefficient group for group rings from $\mathbb{Z}$ to $\mathbb{Z}_p$ in the proof, one can show that linear independence of the vectors $\mathbf{v_i^k}$ over $\mathbb{Z}_p G_k$ implies linear independence of the vectors $\mathbf{v_i^{k+1}}$ over $\mathbb{Z}_p G_{k+1}$. This completes the proof. \end{proof}
Now we give a proof of Theorem~\ref{thm:eta_i}. \begin{proof}[proof of Theorem~\ref{thm:eta_i}] Let $p$ be a prime greater than the top coefficient of $\Delta_K(t)$ and $W$ an $n$-cylinder one of whose boundary components is $M(K)$. Let $\mathcal{P}=(R_0, \ldots, R_n)$ where $R_i=\mathbb{Q}$ for $i\le n-1$ and $R_n=\mathbb{Z}_p$. With this $\mathcal{P}$, recall that for a group $G$ and $k\le n$, $\mathcal{P}^kG=G^{(k)}_r$, the $k$-th rational derived group of $G$. Recall that for a group $G$, we denote $G/G^{(k)}_r$ by $G_k$.
For convenience, we also denote by $\Sigma$ the capped-off Seifert surface for $K$. Supose $\Sigma$ has genus $g$. Let $M:=M(K)$, $S:=\pi_1M^{(1)}$, and $G:=\pi_1W^{(1)}$. Let $F$ be a free group of rank $2g$ and $F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(M\smallsetminus\Sigma)$ a homomorphism inducing an isomorphism on $H_1(F;R)$ for $R=\mathbb{Q}$ and $\mathbb{Z}_p$. Let $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \pi_1(M\smallsetminus\Sigma)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ be a map induced from this map. Since $G_{n-1}$ is PTFA \cite[Proposition 2.1]{Harvey:2006-1}, $\mathbb{Z}_pG_{n-1}$ embeds into the (skew) quotient field, say $\mathbb{K}$, by Lemma~\ref{lem:quotient field}.
Let $M_\infty$ and $W_\infty$ denote the infinite cyclic covers of $M$ and $W$, respectively. Let $\Gamma:=\pi_1W/\mathcal{P}^n\pi_1W=\pi_1W/\pi_1W^{(n)}_r$. Then $\Gamma$ is a PTFA group such that $\Gamma^{(n)}=\{e\}$. Since $H_1(\Gamma)\cong \mathbb{Z}=\langle t\rangle$ and
\[
[\Gamma,\Gamma]=\pi_1(W)^{(1)}/\pi_1W^{(n)}_r=G/G^{(n-1)}_r=G_{n-1},
\]
we have $\Gamma\cong G_{n-1}\rtimes \langle t\rangle$. Therefore, we have
\begin{align*} H_1(W_\infty;\mathbb{K}) & \cong H_1(W_\infty;\mathbb{Z}_pG_{n-1})\otimesover{\mathbb{Z}_pG_{n-1}}\mathbb{K} \\
& \cong H_1(W;\mathbb{Z}_p[G_{n-1}\rtimes \langle t\rangle])\otimesover{\mathbb{Z}_pG_{n-1}}\mathbb{K} \\
& \cong H_1(W;\mathbb{K}[t^{\pm 1}]).
\end{align*}
Similarly, we have $H_1(M_\infty;\mathbb{K})\cong H_1(M;\mathbb{K}[t^{\pm 1}])$.
Now by Proposition~\ref{prop:n-cylinder and n-solution}, the inclusion-induced homomorphism $j\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is an $R$-algebraic $n$-solution for $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ for $R=\mathbb{Q}$ and $\mathbb{Z}_p$, and hence an $R$-algebraic $(n-1)$-solution for both $R=\mathbb{Q}$ and $\mathbb{Z}_p$ (see Remark~\ref{rem:algebraic n-solution}~(1)). Let $\mathcal{P}_{n-1}$ be the finite collection of tuples of $2g-1$ (if $n>1$) or $2g$ (if $n=1$) elements of $F^{(n-1)}$ obtained using Theorem~\ref{thm:special tuple} with the map $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ above. The image of the elements of the tuples in $\mathcal{P}_{n-1}$ under the homomorphism $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$ is a finite subset of $S^{(n-1)}=\pi_1M^{(n)}$, which we denote by $\eta_1, \eta_2, \ldots \eta_m$. Since the map $i$ factors through $\pi_1(M\smallsetminus\Sigma)$, we can find the representatives $\eta_i$ in $S^3 \smallsetminus\Sigma$, and by crossing change we can make $\eta_i$ form an unlink. Now since $j\colon S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} G$ is an $R$-algebraic $(n-1)$-solution for $i\colon F\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} S$, by Theorem~\ref{thm:special tuple} there exists a $(2g-1)$-tuple $\{w_1,w_2,\ldots, w_{2g-1}\} \subset F^{(n-1)}$ (or a $2g$-tuple $\{w_1,\ldots, w_{2g}\}\subset F$ if $n=1$) which maps to a generating set of $H_1(S;\mathbb{K})$. Therefore, since $H_1(S;\mathbb{K})\cong H_1(M_\infty;\mathbb{K})\cong H_1(M;\mathbb{K}[t^{\pm 1}])$, the $\eta_i$ can be regarded as a generating set of $H_1(M;\mathbb{K}[t^{\pm 1}])$ (as a $\mathbb{K}$-module).
Let $d:=\deg \Delta_K(t)=\operatorname{rank}_\mathbb{Q} H_1(M_\infty;\mathbb{Q})$. Since $p$ is greater than the top coefficient of $\Delta_K(t)$, we have $d=\operatorname{rank}_{\mathbb{Z}_p}H_1(M_\infty;\mathbb{Z}_p)$. Note that $d\ge 4 $ if $n\ge 2$ and $d\ge 2$ if $n=1$ by assumption. Since $\Gamma$ is a PTFA group such that $\Gamma^{(n)}=\{e\}$, by Theorem~\ref{thm:nontrivial} this implies that the map $H_1(M;\mathbb{K}[t^{\pm 1}])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} H_1(W;\mathbb{K}[t^{\pm 1}])$ is nontrivial. Therefore, since the $\eta_i$ generate $H_1(M;\mathbb{K}[t^{\pm 1}])$ as a $\mathbb{K}$-module, there exists some $\eta_i$ which maps to a nontrivial element in $H_1(W;\mathbb{K}[t^{\pm 1}])$.
Note that $H_1(W;\mathbb{K}[t^{\pm 1}])\cong H_1(W;\mathbb{Z}_p\Gamma)\otimesover{\mathbb{Z}_pG_{n-1}}\mathbb{K}$. By the definition of $\mathcal{P}^{n+1}\pi_1W$, the group $\mathcal{P}^n\pi_1W/\mathcal{P}^{n+1}\pi_1W$ injects to $H_1(\pi_1W;\mathbb{Z}_p\Gamma)\cong H_1(W;\mathbb{Z}_p\Gamma)$. Since $\eta_i\in \pi_1M^{(n)}$, it maps into $\pi_1W^{(n)}$, hence into $\mathcal{P}^n\pi_1W$. From these observations, one can deduce that the $\eta_i$, which maps to a nontrivial element in $H_1(W;\mathbb{K}[t^{\pm 1}])$, also maps nontrivially to $\mathcal{P}^n\pi_1W/\mathcal{P}^{n+1}\pi_1W$. In particular, we see that $\eta_i\notin \mathcal{P}^{n+1}\pi_1W$.
Finally, by Lemma~\ref{lem:axes bounding grope of height n} we can homotope all $\eta_i$ such that all of $\eta_i$ bound capped gropes of height $n$ which are disjointly embedded in $S^3\smallsetminus K$ as desired. \end{proof}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2}
\end{document} | arXiv |
B6 polytope
In 6-dimensional geometry, there are 64 uniform polytopes with B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices respectively. The 6-demicube is added with half the symmetry.
Orthographic projections in the B6 Coxeter plane
6-cube
6-orthoplex
6-demicube
They can be visualized as symmetric orthographic projections in Coxeter planes of the B6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 64 polytopes can be made in the B6, B5, B4, B3, B2, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.
These 64 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
# Coxeter plane graphs Coxeter-Dynkin diagram
Schläfli symbol
Names
B6
[12]
B5 / D4 / A4
[10]
B4
[8]
B3 / A2
[6]
B2
[4]
A5
[6]
A3
[4]
1
{3,3,3,3,4}
6-orthoplex
Hexacontatetrapeton (gee)
2
t1{3,3,3,3,4}
Rectified 6-orthoplex
Rectified hexacontatetrapeton (rag)
3
t2{3,3,3,3,4}
Birectified 6-orthoplex
Birectified hexacontatetrapeton (brag)
4
t2{4,3,3,3,3}
Birectified 6-cube
Birectified hexeract (brox)
5
t1{4,3,3,3,3}
Rectified 6-cube
Rectified hexeract (rax)
6
{4,3,3,3,3}
6-cube
Hexeract (ax)
64
h{4,3,3,3,3}
6-demicube
Hemihexeract
7
t0,1{3,3,3,3,4}
Truncated 6-orthoplex
Truncated hexacontatetrapeton (tag)
8
t0,2{3,3,3,3,4}
Cantellated 6-orthoplex
Small rhombated hexacontatetrapeton (srog)
9
t1,2{3,3,3,3,4}
Bitruncated 6-orthoplex
Bitruncated hexacontatetrapeton (botag)
10
t0,3{3,3,3,3,4}
Runcinated 6-orthoplex
Small prismated hexacontatetrapeton (spog)
11
t1,3{3,3,3,3,4}
Bicantellated 6-orthoplex
Small birhombated hexacontatetrapeton (siborg)
12
t2,3{4,3,3,3,3}
Tritruncated 6-cube
Hexeractihexacontitetrapeton (xog)
13
t0,4{3,3,3,3,4}
Stericated 6-orthoplex
Small cellated hexacontatetrapeton (scag)
14
t1,4{4,3,3,3,3}
Biruncinated 6-cube
Small biprismato-hexeractihexacontitetrapeton (sobpoxog)
15
t1,3{4,3,3,3,3}
Bicantellated 6-cube
Small birhombated hexeract (saborx)
16
t1,2{4,3,3,3,3}
Bitruncated 6-cube
Bitruncated hexeract (botox)
17
t0,5{4,3,3,3,3}
Pentellated 6-cube
Small teri-hexeractihexacontitetrapeton (stoxog)
18
t0,4{4,3,3,3,3}
Stericated 6-cube
Small cellated hexeract (scox)
19
t0,3{4,3,3,3,3}
Runcinated 6-cube
Small prismated hexeract (spox)
20
t0,2{4,3,3,3,3}
Cantellated 6-cube
Small rhombated hexeract (srox)
21
t0,1{4,3,3,3,3}
Truncated 6-cube
Truncated hexeract (tox)
22
t0,1,2{3,3,3,3,4}
Cantitruncated 6-orthoplex
Great rhombated hexacontatetrapeton (grog)
23
t0,1,3{3,3,3,3,4}
Runcitruncated 6-orthoplex
Prismatotruncated hexacontatetrapeton (potag)
24
t0,2,3{3,3,3,3,4}
Runcicantellated 6-orthoplex
Prismatorhombated hexacontatetrapeton (prog)
25
t1,2,3{3,3,3,3,4}
Bicantitruncated 6-orthoplex
Great birhombated hexacontatetrapeton (gaborg)
26
t0,1,4{3,3,3,3,4}
Steritruncated 6-orthoplex
Cellitruncated hexacontatetrapeton (catog)
27
t0,2,4{3,3,3,3,4}
Stericantellated 6-orthoplex
Cellirhombated hexacontatetrapeton (crag)
28
t1,2,4{3,3,3,3,4}
Biruncitruncated 6-orthoplex
Biprismatotruncated hexacontatetrapeton (boprax)
29
t0,3,4{3,3,3,3,4}
Steriruncinated 6-orthoplex
Celliprismated hexacontatetrapeton (copog)
30
t1,2,4{4,3,3,3,3}
Biruncitruncated 6-cube
Biprismatotruncated hexeract (boprag)
31
t1,2,3{4,3,3,3,3}
Bicantitruncated 6-cube
Great birhombated hexeract (gaborx)
32
t0,1,5{3,3,3,3,4}
Pentitruncated 6-orthoplex
Teritruncated hexacontatetrapeton (tacox)
33
t0,2,5{3,3,3,3,4}
Penticantellated 6-orthoplex
Terirhombated hexacontatetrapeton (tapox)
34
t0,3,4{4,3,3,3,3}
Steriruncinated 6-cube
Celliprismated hexeract (copox)
35
t0,2,5{4,3,3,3,3}
Penticantellated 6-cube
Terirhombated hexeract (topag)
36
t0,2,4{4,3,3,3,3}
Stericantellated 6-cube
Cellirhombated hexeract (crax)
37
t0,2,3{4,3,3,3,3}
Runcicantellated 6-cube
Prismatorhombated hexeract (prox)
38
t0,1,5{4,3,3,3,3}
Pentitruncated 6-cube
Teritruncated hexeract (tacog)
39
t0,1,4{4,3,3,3,3}
Steritruncated 6-cube
Cellitruncated hexeract (catax)
40
t0,1,3{4,3,3,3,3}
Runcitruncated 6-cube
Prismatotruncated hexeract (potax)
41
t0,1,2{4,3,3,3,3}
Cantitruncated 6-cube
Great rhombated hexeract (grox)
42
t0,1,2,3{3,3,3,3,4}
Runcicantitruncated 6-orthoplex
Great prismated hexacontatetrapeton (gopog)
43
t0,1,2,4{3,3,3,3,4}
Stericantitruncated 6-orthoplex
Celligreatorhombated hexacontatetrapeton (cagorg)
44
t0,1,3,4{3,3,3,3,4}
Steriruncitruncated 6-orthoplex
Celliprismatotruncated hexacontatetrapeton (captog)
45
t0,2,3,4{3,3,3,3,4}
Steriruncicantellated 6-orthoplex
Celliprismatorhombated hexacontatetrapeton (coprag)
46
t1,2,3,4{4,3,3,3,3}
Biruncicantitruncated 6-cube
Great biprismato-hexeractihexacontitetrapeton (gobpoxog)
47
t0,1,2,5{3,3,3,3,4}
Penticantitruncated 6-orthoplex
Terigreatorhombated hexacontatetrapeton (togrig)
48
t0,1,3,5{3,3,3,3,4}
Pentiruncitruncated 6-orthoplex
Teriprismatotruncated hexacontatetrapeton (tocrax)
49
t0,2,3,5{4,3,3,3,3}
Pentiruncicantellated 6-cube
Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)
50
t0,2,3,4{4,3,3,3,3}
Steriruncicantellated 6-cube
Celliprismatorhombated hexeract (coprix)
51
t0,1,4,5{4,3,3,3,3}
Pentisteritruncated 6-cube
Tericelli-hexeractihexacontitetrapeton (tactaxog)
52
t0,1,3,5{4,3,3,3,3}
Pentiruncitruncated 6-cube
Teriprismatotruncated hexeract (tocrag)
53
t0,1,3,4{4,3,3,3,3}
Steriruncitruncated 6-cube
Celliprismatotruncated hexeract (captix)
54
t0,1,2,5{4,3,3,3,3}
Penticantitruncated 6-cube
Terigreatorhombated hexeract (togrix)
55
t0,1,2,4{4,3,3,3,3}
Stericantitruncated 6-cube
Celligreatorhombated hexeract (cagorx)
56
t0,1,2,3{4,3,3,3,3}
Runcicantitruncated 6-cube
Great prismated hexeract (gippox)
57
t0,1,2,3,4{3,3,3,3,4}
Steriruncicantitruncated 6-orthoplex
Great cellated hexacontatetrapeton (gocog)
58
t0,1,2,3,5{3,3,3,3,4}
Pentiruncicantitruncated 6-orthoplex
Terigreatoprismated hexacontatetrapeton (tagpog)
59
t0,1,2,4,5{3,3,3,3,4}
Pentistericantitruncated 6-orthoplex
Tericelligreatorhombated hexacontatetrapeton (tecagorg)
60
t0,1,2,4,5{4,3,3,3,3}
Pentistericantitruncated 6-cube
Tericelligreatorhombated hexeract (tocagrax)
61
t0,1,2,3,5{4,3,3,3,3}
Pentiruncicantitruncated 6-cube
Terigreatoprismated hexeract (tagpox)
62
t0,1,2,3,4{4,3,3,3,3}
Steriruncicantitruncated 6-cube
Great cellated hexeract (gocax)
63
t0,1,2,3,4,5{4,3,3,3,3}
Omnitruncated 6-cube
Great teri-hexeractihexacontitetrapeton (gotaxog)
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Klitzing, Richard. "6D uniform polytopes (polypeta)".
Notes
1. Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equationswith time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12(3): 1163-1182. doi: 10.3934\/cpaa.2013.12.1163.
L. Bakker. The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows. Communications on Pure & Applied Analysis, 2013, 12(3): 1183-1200. doi: 10.3934\/cpaa.2013.12.1183.
Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure & Applied Analysis, 2013, 12(3): 1201-1220. doi: 10.3934\/cpaa.2013.12.1201.
M. Carme Leseduarte, Ramon Quintanilla. Phragm\u00E9n-Lindel\u00F6f alternative for an exact heat conduction equation with delay. Communications on Pure & Applied Analysis, 2013, 12(3): 1221-1235. doi: 10.3934\/cpaa.2013.12.1221.
Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional H\u00E9non equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12(3): 1237-1241. doi: 10.3934\/cpaa.2013.12.1237.
Liping Wang, Juncheng Wei. Infinite multiplicity for an inhomogeneous supercritical problem inentire space. Communications on Pure & Applied Analysis, 2013, 12(3): 1243-1257. doi: 10.3934\/cpaa.2013.12.1243.
Seunghyeok Kim. On vector solutions for coupled nonlinear Schr\u00F6dinger equations with critical exponents. Communications on Pure & Applied Analysis, 2013, 12(3): 1259-1277. doi: 10.3934\/cpaa.2013.12.1259.
Hua Nie, Wenhao Xie, Jianhua Wu. Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor. Communications on Pure & Applied Analysis, 2013, 12(3): 1279-1297. doi: 10.3934\/cpaa.2013.12.1279.
Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12(3): 1299-1306. doi: 10.3934\/cpaa.2013.12.1299.
Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure & Applied Analysis, 2013, 12(3): 1307-1319. doi: 10.3934\/cpaa.2013.12.1307.
Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12(3): 1321-1339. doi: 10.3934\/cpaa.2013.12.1321.
Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure & Applied Analysis, 2013, 12(3): 1341-1347. doi: 10.3934\/cpaa.2013.12.1341.
Julie Lee, J. C. Song. Spatial decay bounds in a linearized magnetohydrodynamic channel flow. Communications on Pure & Applied Analysis, 2013, 12(3): 1349-1361. doi: 10.3934\/cpaa.2013.12.1349.
Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singularterm. Communications on Pure & Applied Analysis, 2013, 12(3): 1363-1380. doi: 10.3934\/cpaa.2013.12.1363.
Zhijun Zhang. Large solutions of semilinear elliptic equationswith a gradient term: existence and boundary behavior. Communications on Pure & Applied Analysis, 2013, 12(3): 1381-1392. doi: 10.3934\/cpaa.2013.12.1381.
John R. Graef, Shapour Heidarkhani, Lingju Kong. Multiple solutions for a class of $(p_1, \\ldots, p_n)$-biharmonic systems. Communications on Pure & Applied Analysis, 2013, 12(3): 1393-1406. doi: 10.3934\/cpaa.2013.12.1393.
Fengping Yao. Optimal regularity for parabolic Schr\u00F6dinger operators. Communications on Pure & Applied Analysis, 2013, 12(3): 1407-1414. doi: 10.3934\/cpaa.2013.12.1407.
Morteza Fotouhi, Leila Salimi. Controllability results for a class ofone dimensional degenerate\/singular parabolic equations. Communications on Pure & Applied Analysis, 2013, 12(3): 1415-1430. doi: 10.3934\/cpaa.2013.12.1415.
Hayk Mikayelyan, Henrik Shahgholian. Convexity of the freeboundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12(3): 1431-1443. doi: 10.3934\/cpaa.2013.12.1431.
Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure & Applied Analysis, 2013, 12(3): 1445-1468. doi: 10.3934\/cpaa.2013.12.1445.
Ji\u0159\u00ED Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure & Applied Analysis, 2013, 12(3): 1469-1486. doi: 10.3934\/cpaa.2013.12.1469.
Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure & Applied Analysis, 2013, 12(3): 1487-1499. doi: 10.3934\/cpaa.2013.12.1487.
John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12(3): 1501-1526. doi: 10.3934\/cpaa.2013.12.1501. | CommonCrawl |
Only show content I have access to (161)
Life Sciences (205)
Microscopy and Microanalysis (44)
The Journal of Agricultural Science (32)
Proceedings of the International Astronomical Union (23)
Psychological Medicine (17)
Powder Diffraction (15)
The Journal of Laryngology & Otology (15)
Journal of Mechanics (13)
Parasitology (12)
The Aeronautical Journal (9)
Journal of Plasma Physics (8)
Animal Science (6)
Laser and Particle Beams (6)
BSAS (26)
AMMS - Australian Microscopy and Microanalysis Society (15)
EAAP (15)
MSC - Microscopical Society of Canada (15)
Nestle Foundation - enLINK (12)
Australian Mathematical Society Inc (10)
Royal Aeronautical Society (9)
AMA Mexican Society of Microscopy MMS (4)
MSA - Microscopy Society of America (4)
Malaysian Society of Otorhinolaryngologists Head and Neck Surgeons (4)
Testing Membership Number Upload (4)
JLO (1984) Ltd (3)
The smart morphing winglet driven by the piezoelectric Macro Fiber Composite actuator
X. Chen, J. Liu, Q. Li
Journal: The Aeronautical Journal , First View
Published online by Cambridge University Press: 12 January 2022, pp. 1-18
A smart morphing winglet driven by piezoelectric Macro Fiber Composite (MFC) is designed to adjust cant angle autonomously for various flight conditions. The smart morphing winglet is composed of the MFC actuator, DC-DC converter, power supply, winglet part and wing part. A hinge is designed to transfer the bending deformation of intelligent MFC bending actuator to rotation of the winglet structure so as to achieve the adaptive cant angle. Experimental and numerical work are conducted to evaluate the performance of smart morphing winglet. It is demonstrated that the proposed intelligent MFC bending actuator has an excellent bending performance and load resistance. This smart morphing winglet exhibits the excellent characteristic of flexibility on large deformation and lightweight. Moreover, a series of wind tunnel tests are performed, which demonstrate that the winglet driven by intelligent MFC bending actuator produces sufficient deformation in various wind speed. At high wind speed, the cant angle of the winglet can reach 16 degrees, which is still considered to be very useful for improving the aerodynamic performance of the aircraft. The aerodynamic characteristics are investigated by wind tunnel tests with various attack angles. As a result, when the morphing winglet is actuated, the lift-to-drag ratio could vary up to 11.9% and 6.4%, respectively, under wind speeds of 5.4 and 8.5m/s. Meanwhile, different flight phases such as take-off, cruise and landing are considered to improve aerodynamic performance by adjusting the cant angle of winglet. The smart morphing winglet varies the aerofoil autonomously by controlling the low winglet device input voltage to remain optimal aerodynamic performance during the flight process. It demonstrates the feasibility of piezoelectric composites driving intelligent aircraft.
Some generating functions and inequalities for the andrews–stanley partition functions
Additive number theory; partitions
Na Chen, Shane Chern, Yan Fan, Ernest X. W. Xia
Journal: Proceedings of the Edinburgh Mathematical Society , First View
Published online by Cambridge University Press: 27 December 2021, pp. 1-16
Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$ . In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$ , where $\pi '$ is the conjugate of $\pi$ . Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$ . Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$ with $m=16$ and $24$ . These results are refinements of some inequalities due to Swisher.
Research on parameter matching characteristics of pneumatic launch systems based on co-simulation
Z. Zhang, Y. Peng, X. Wei, H. Nie, H. Chen, L. Li
Journal: The Aeronautical Journal / Volume 126 / Issue 1296 / February 2022
Pneumatic launch systems for Unmanned Aerial Vehicles (UAVs), including mechanical and pneumatic systems, are complex and non-linear. They are subjected to system parameters during launch, which leads to difficulty in engineering research analysis. For example, the mismatch between the UAV parameters and the parameter design indices of the launch system as well as the unclear design indices of the launching speed and overload of UAVs have a great impact on launch safety. Considering this situation, some studies are presented in this paper. Taking the pneumatic launch system as a research object, a pneumatic launcher dynamic simulation model is built based on co-simulation considering the coupling characteristics of the mechanical structure and transmission system. Its accuracy was verified by laboratory test results. Based on this model, the paper shows the effects of the key parameters, including the mass of the UAV, cylinder volume, pressure and moment of inertia of the pulley block, on the performance of the dynamic characteristics of the launch process. Then, a method for matching the parameter characteristics between the UAV and launch system based on batch simulation is proposed. The set of matching parameter values of the UAV and launch system that satisfy the launch take-off safety criteria are calculated. Finally, the influence of the system parameters of the launch process on the launch performance was analysed in detail, and the design optimised. Meaningful conclusions were obtained. The analysis method and its results can provide a reference for engineering and theoretical research and development of pneumatic launch systems.
Velocity obstacle–based conflict resolution and recovery method
F. Sun, Y. Chen, X. Xu, Y. Mu, Z. Wang
Considering the shortcomings of current methods for real-time resolution of two-aircraft flight conflicts, a geometric optimal conflict resolution and recovery method based on the velocity obstacle method for two aircraft and a cooperative conflict resolution method for multiple aircraft are proposed. The conflict type was determined according to the relative position and velocity of the aircraft, and a corresponding conflict mitigation strategy was selected. A resolution manoeuvre and a recovery manoeuvre were performed. On the basis of a two-aircraft conflict resolution model, a multi-aircraft cooperative conflict resolution game was constructed to identify an optimal solution for maximising group welfare. The solution and recovery method is simple and effective, and no new flight conflicts are introduced during track recovery. For multi-aircraft conflict resolution, an equilibrium point that maximises the welfare function of the group was identified, and thus, an optimal strategy for multi-aircraft conflict resolution was obtained.
Parameter optimisation of a carrier-based UAV drawbar based on strain fatigue analysis
H. Chen, X. Fang, Z. Zhang, X. Xie, H. Nie, X. Wei
Journal: The Aeronautical Journal / Volume 125 / Issue 1288 / June 2021
Carrier-based unmanned aerial aircraft (UAV) structure is subjected to severe tensile load during takeoff, especially the drawbar, which affects its fatigue performance and structural safety. However, the complex structural features pose great challenges for the engineering design. Considering this situation, a structural design, fatigue analysis, and parameters optimisation joint working platform are urgently needed to solve this problem. In this study, numerical analysis of strain fatigue is carried out based on the laboratory fatigue failure of the carrier-based aircraft drawbar. Taking the sensitivity of drawbar parameters to stress and life into account and optimum design of drawbar with fatigue life as a target using the parametric method, this study also includes cutting-edge parameters of milling cutters, structural details of the drawbar and so on. Then an experimental design is applied using the Latin hypercube sampling method, and a surrogate model based on RBF neural network is established. Lastly, a multi-island genetic algorithm is introduced for optimisation. The results show that the error between the obtained optimal solution and simulation is 0.26%, while the optimised stress level is reduced by 15.7%, and the life of the drawbar is increased by 122%.
Effects of riboflavin supplementation on performance, nutrient digestion, rumen microbiota composition and activities of Holstein bulls
H. M. Wu, J. Zhang, C. Wang, Q. Liu, G. Guo, W. J. Huo, L. Chen, Y. L. Zhang, C. X. Pei, S. L. Zhang
Journal: British Journal of Nutrition / Volume 126 / Issue 9 / 14 November 2021
Print publication: 14 November 2021
To investigate the influences of dietary riboflavin (RF) addition on nutrient digestion and rumen fermentation, eight rumen cannulated Holstein bulls were randomly allocated into four treatments in a repeated 4 × 4 Latin square design. Daily addition level of RF for each bull in control, low RF, medium RF and high RF was 0, 300, 600 and 900 mg, respectively. Increasing the addition level of RF, DM intake was not affected, average daily gain tended to be increased linearly and feed conversion ratio decreased linearly. Total tract digestibilities of DM, organic matter, crude protein (CP) and neutral-detergent fibre (NDF) increased linearly. Rumen pH decreased quadratically, and total volatile fatty acids (VFA) increased quadratically. Acetate molar percentage and acetate:propionate ratio increased linearly, but propionate molar percentage and ammonia-N content decreased linearly. Rumen effective degradability of DM increased linearly, NDF increased quadratically but CP was unaltered. Activity of cellulase and populations of total bacteria, protozoa, fungi, dominant cellulolytic bacteria, Prevotella ruminicola and Ruminobacter amylophilus increased linearly. Linear increase was observed for urinary total purine derivatives excretion. The data suggested that dietary RF addition was essential for rumen microbial growth, and no further increase in performance and rumen total VFA concentration was observed when increasing RF level from 600 to 900 mg/d in dairy bulls.
Cholesterol esterification enzymes promote cancer growth and are potential therapeutic targets for repurposed drugs: a systematic review and meta-analysis of pre-clinical evidence
A. Websdale, P. Chalmers, X. Chen, Y. Kiew, X. Luo, R. Mwarazi, R. Wu, G. Cioccoloni, H. Røberg-Larsen, T.A. Hughes, M.A. Zulyniak, J.L. Thorne
Journal: Proceedings of the Nutrition Society / Volume 80 / Issue OCE5 / 2021
Characterization of a hospital-based gastroenteritis outbreak caused by GII.6 norovirus in Jinshan, China
X. F. Zhang, J. R. Chen, C. L. Song, D. J. Xie, M. Tan, L. Wang, M. M. Koroma, Y. Z. Hou, Z. P. Dong, J. R. Yu, W. T. Duan, D. D. Zhao, J. R. Du, L. Zhu, Y. C. Dai
Published online by Cambridge University Press: 09 December 2020, e289
An acute gastroenteritis (AGE) outbreak caused by a norovirus occurred at a hospital in Shanghai, China, was studied for molecular epidemiology, host susceptibility and serological roles. Rectal and environmental swabs, paired serum samples and saliva specimens were collected. Pathogens were detected by real-time polymerase chain reaction and DNA sequencing. Histo-blood group antigens (HBGA) phenotypes of saliva samples and their binding to norovirus protruding proteins were determined by enzyme-linked immunosorbent assay. The HBGA-binding interfaces and the surrounding region were analysed by the MegAlign program of DNAstar 7.1. Twenty-seven individuals in two care units were attacked with AGE at attack rates of 9.02 and 11.68%. Eighteen (78.2%) symptomatic and five (38.4%) asymptomatic individuals were GII.6/b norovirus positive. Saliva-based HBGA phenotyping showed that all symptomatic and asymptomatic cases belonged to A, B, AB or O secretors. Only four (16.7%) out of the 24 tested serum samples showed low blockade activity against HBGA-norovirus binding at the acute phase, whereas 11 (45.8%) samples at the convalescence stage showed seroconversion of such blockade. Specific blockade antibody in the population played an essential role in this norovirus epidemic. A wide HBGA-binding spectrum of GII.6 supports a need for continuous health attention and surveillance in different settings.
The benefits of betahistine or vestibular rehabilitation (Tetrax biofeedback) on the quality of life and fall risk in patients with Ménière's disease
J L Liu, J G Liu, X B Chen, Y H Liu
Journal: The Journal of Laryngology & Otology / Volume 134 / Issue 12 / December 2020
This study aimed to evaluate the benefits of betahistine or vestibular rehabilitation (Tetrax biofeedback) on the quality of life and fall risk in patients with Ménière's disease.
Sixty-six patients with Ménière's disease were randomly divided into three groups: betahistine, Tetrax and control groups. Patients' Dizziness Handicap Index and Tetrax fall index scores were obtained before and after treatment.
Patients in the betahistine and Tetrax groups showed significant improvements in Dizziness Handicap Index and fall index scores after treatment versus before treatment (p < 0.05). The improvements in the Tetrax group were significantly greater than those in the betahistine group (p < 0.05).
Betahistine and vestibular rehabilitation (Tetrax biofeedback) improve the quality of life and reduce the risk of falling in patients with Ménière's disease. Vestibular rehabilitation (Tetrax biofeedback) is an effective management method for Ménière's disease.
Negotiating French Wine and European Identities at the European Community
Maria X. Chen
Journal: Contemporary European History / Volume 29 / Issue 4 / November 2020
This article examines the French role in creating an integrated wine policy at the European level and demonstrates that political negotiations over the policy revealed competing European conceptions of agriculture and identity. Drawing on research in EEC and French historical archives, this article argues that in spite of the risks involved in relinquishing sovereignty over a key national industry with deep cultural resonance, the French government was determined to transfer responsibility for much of the sector to the European Community due to continued domestic pressure. Further, it suggests that common values around the centrality of agriculture in the European project meant that countries were persistent in realising a wine policy even though wine was not a natural fit in the pantheon of other goods for which common markets were created.
Magnetic energy dissipation during the current quench of disruption in EAST
Focus on Fusion
T. Tang, L. Zeng, D. L. Chen, R. S. Granetz, S. T. Mao, Y. M. Duan, L. Zhang, H. D. Zhuang, X. Zhu, H. Q. Liu, B. Shen, Y. X. Jie, X. Gao
Journal: Journal of Plasma Physics / Volume 86 / Issue 5 / October 2020
Published online by Cambridge University Press: 08 October 2020, 905860509
A disruption database characterizing the current quench of disruptions with ITER-like tungsten divertor has been developed on EAST. It provides a large number of plasma parameters describing the predisruptive plasma, current quench time, eddy current, and mitigation by massive impurity injection, which shows that the current quench time strongly depends on magnetic energy and post-disruption electron temperature. Further, the energy balance and magnetic energy dissipation during the current quench phase has been well analysed. Magnetic energy is also demonstrated to be dissipated mainly by ohmic reheating and inductive coupling, and both of the two channels have great effects on current quench time. Also, massive gas injection is an efficient method to speed up the current quench and increase the fraction of impurity radiation.
Optimization of mechanical properties and electrical conductivity in Al–Mg–Si 6201 alloys with different Mg/Si ratios
Siamak Nikzad Khangholi, Mousa Javidani, Alexandre Maltais, X.-Grant Chen
Journal: Journal of Materials Research / Volume 35 / Issue 20 / 28 October 2020
The effects of the Mg/Si ratio and aging treatment on the strength and electrical conductivity of Al–Mg–Si 6201 conductor alloys were investigated. Four experimental alloys with different Mg/Si ratios of 2, 1.5, 1, and 0.86 and with a constant Mg level of 0.65 wt% were prepared. It was revealed that excessive Si (a low Mg/Si ratio) increased the peak strength, while the corresponding electrical conductivity decreased. To fulfill the minimum required electrical conductivity (52.5% IACS), the alloys with low Mg/Si ratios required a longer aging time after peak aging to improve electrical conductivity. The alloy with an Mg/Si ratio of ~1 was the best candidate, exhibiting the highest strength up to 54% IACS. On the high end of electrical conductivity (54–56% IACS), the alloy with an Mg/Si ratio of ~1.5 provides a better compromise between strength and electrical conductivity. Furthermore, the strengthening mechanisms and the factors influencing electrical conductivity were discussed for further optimization.
Experimental study on low-speed streaks in a turbulent boundary layer at low Reynolds number
X. Y. Jiang, C. B. Lee, C. R. Smith, J. W. Chen, P. F. Linden
Journal: Journal of Fluid Mechanics / Volume 903 / 25 November 2020
Published online by Cambridge University Press: 18 September 2020, A6
A study of low-speed streaks (LSSs) embedded in the near-wall region of a turbulent boundary layer is performed using selective visualization and analysis of time-resolved tomographic particle image velocimetry (tomo-PIV). First, a three-dimensional velocity field database is acquired using time-resolved tomo-PIV for an early turbulent boundary layer. Second, detailed time-line flow patterns are obtained from the low-order reconstructed database using 'tomographic visualizations' by Lagrangian tracking. These time-line patterns compare remarkably well with previously observed patterns using hydrogen bubble flow visualization, and allow local identification of LSSs within the database. Third, the flow behaviour in proximity to selected LSSs is examined at varying wall distances ( $10 < y^+ < 100$) and assessed using time-line and material surface evolution, to reveal the flow structure and evolution of a streak, and the flow structure evolving from streak development. It is observed that three-dimensional wave behaviour of the detected LSSs appears to develop into associated near-wall vortex flow structures, in a process somewhat similar to transitional boundary layer behaviour. Fourth, the presence of Lagrangian coherent structures is assessed in proximity to the LSSs using a Lagrangian-averaged vorticity deviation process. It is observed that quasi-streamwise vortices, adjacent to the sides of the streak-associated three-dimensional wave, precipitate an interaction with the streak. Finally, a hypothesis based on the behaviour of soliton-like coherent structures is made which explains the process of LSS formation, bursting behaviour and the generation of hairpin vortices. Comparison with other models is also discussed.
The aerodynamic optimisation of a low-Reynolds paper plane with adjoint method
International Symposium on Smart Aircraft 2019 Collection
Y. Zhang, X. Zhang, G. Chen
Journal: The Aeronautical Journal / Volume 125 / Issue 1285 / March 2021
The aerodynamic performance of a deployable and low-cost unmanned aerial vehicle (UAV) is investigated and improved in present work. The parameters of configuration, such as airfoil and winglet, are determined via an optimising process based on a discrete adjoint method. The optimised target is locked on an increasing lift-to-drag ratio with a limited variation of pitching moments. The separation that will lead to a stall is delayed after optimisation. Up to 128 design variables are used by the optimised solver to give enough flexibility of the geometrical transformation. As much as 20% enhancement of lift-to-drag ratio is gained at the cruise angle-of-attack, that is, a significant improvement in the lift-to-drag ratio adhering to the preferred configuration is obtained with increasing lift and decreasing drag coefficients, essentially entailing an improved aerodynamic performance.
Characterisation of Bordetella bronchiseptica isolated from rabbits in Fujian, China
J. Wang, S. Sun, Y. Chen, D. Chen, L. Sang, X. Xie
Bordetella bronchiseptica is a potential zoonotic pathogen, which mainly causes respiratory diseases in humans and a variety of animal species. B. bronchiseptica is one of the important pathogens isolated from rabbits in Fujian Province. However, the knowledge of the epidemiology and characteristics of the B. bronchiseptica in rabbits in Fujian Province is largely unknown. In this study, 219 B. bronchiseptica isolates recovered from lung samples of dead rabbits with respiratory diseases in Fujian Province were characterised by multi-locus sequencing typing, screening virulence genes and testing antimicrobial susceptibility. The results showed that the 219 isolates were typed into 11 sequence types (STs) including five known STs (ST6, ST10, ST12, ST14 and ST33) and six new STs (ST88, ST89, ST90, ST91, ST92 and ST93) and the ST33 (30.14%, 66/219), ST14 (26.94%, 59/219) and ST12 (16.44%, 36/219) were the three most prevalent STs. Surprisingly, all the 219 isolates carried the five virulence genes (fhaB, prn, cyaA, dnt and bteA) in the polymerase chain reaction screening. Moreover, the isolates were resistant to cefixime, ceftizoxime, cefatriaxone and ampicillin at rates of 33.33%, 31.05%, 11.87% and 3.20%, respectively. This study showed the genetic diversity of B. bronchiseptica in rabbits in Fujian Province, and the colonisation of the human-associated ST12 strain in rabbits in Fujian Province. The results might be useful for monitoring the epidemic strains, developing preventive methods and preventing the transmission of epidemic strains from rabbits to humans.
Validation and calibration of the Eating Assessment in Toddlers FFQ (EAT FFQ) for children, used in the Growing Up Milk – Lite (GUMLi) randomised controlled trial
Amy L. Lovell, Peter S. W. Davies, Rebecca J. Hill, Tania Milne, Misa Matsuyama, Yannan Jiang, Rachel X. Chen, Anne-Louise M. Heath, Cameron C. Grant, Clare R. Wall
Journal: British Journal of Nutrition / Volume 125 / Issue 2 / 28 January 2021
Print publication: 28 January 2021
The Eating Assessment in Toddlers FFQ (EAT FFQ) has been shown to have good reliability and comparative validity for ranking nutrient intakes in young children. With the addition of food items (n 4), we aimed to re-assess the validity of the EAT FFQ and estimate calibration factors in a sub-sample of children (n 97) participating in the Growing Up Milk – Lite (GUMLi) randomised control trial (2015–2017). Participants completed the ninety-nine-item GUMLi EAT FFQ and record-assisted 24-h recalls (24HR) on two occasions. Energy and nutrient intakes were assessed at months 9 and 12 post-randomisation and calibration factors calculated to determine predicted estimates from the GUMLi EAT FFQ. Validity was assessed using Pearson correlation coefficients, weighted kappa (κ) and exact quartile categorisation. Calibration was calculated using linear regression models on 24HR, adjusted for sex and treatment group. Nutrient intakes were significantly correlated between the GUMLi EAT FFQ and 24HR at both time points. Energy-adjusted, de-attenuated Pearson correlations ranged from 0·3 (fibre) to 0·8 (Fe) at 9 months and from 0·3 (Ca) to 0·7 (Fe) at 12 months. Weighted κ for the quartiles ranged from 0·2 (Zn) to 0·6 (Fe) at 9 months and from 0·1 (total fat) to 0·5 (Fe) at 12 months. Exact agreement ranged from 30 to 74 %. Calibration factors predicted up to 56 % of the variation in the 24HR at 9 months and 44 % at 12 months. The GUMLi EAT FFQ remained a useful tool for ranking nutrient intakes with similar estimated validity compared with other FFQ used in children under 2 years.
Atomic-scale insights on the plate-shaped γ″ phase in Mg–Gd–Y–Ag–Zr alloy
Zhenyang Liu, Zongrui Pei, Bin Chen, X. Q. Zeng
Journal: Journal of Materials Research / Volume 35 / Issue 14 / 28 July 2020
The γ″ phase (hexagonal structure with space group ${ P\bar{6}}2{ m}$) plays an important role in the strengthening of Mg–Gd–Y–Ag–Zr alloy. In this study, Cs-corrected high-angle annular dark-field scanning transmission electron microscopy was applied to characterize the Mg–Gd–Y–Ag–Zr alloy in different conditions (as-cast, solution-treated, and isothermally aged at 200 °C). The nucleation, growing process, and transformation behavior of the plate-shaped γ″ phase were systematically investigated on the atomic scale. We found that the nucleation sites of the γ″ phase were separated by close-packed planes of the Mg matrix and the γ″ phase developed in two perpendicular directions of $\langle 10\bar{1}0 \rangle$ and ⟨0001⟩. The growing process of the γ″ phase on the atomic scale was captured. The γ″ phase was thermodynamically stable at room temperature, and no transformation behavior of the γ″ phase was observed up to 200 h during isothermal aging at 200 °C.
Effects of guanidinoacetic acid supplementation on growth performance, nutrient digestion, rumen fermentation and blood metabolites in Angus bulls
S. Y. Li, C. Wang, Z. Z. Wu, Q. Liu, G. Guo, W. J. Huo, J. Zhang, L. Chen, Y. L. Zhang, C. X. Pei, S. L. Zhang
Journal: animal / Volume 14 / Issue 12 / December 2020
Guanidinoacetic acid (GAA) can improve the growth performance of bulls. This study investigated the influences of GAA addition on growth, nutrient digestion, ruminal fermentation and serum metabolites in bulls. Forty-eight Angus bulls were randomly allocated to experimental treatments, that is, control, low-GAA (LGAA), medium-GAA (MGAA) and high-GAA (HGAA), with GAA supplementation at 0, 0.3, 0.6 and 0.9 g/kg DM, respectively. Bulls were fed a basal diet containing 500 g/kg DM concentrate and 500 g/kg DM roughage. The experimental period was 104 days, with 14 days for adaptation and 90 days for data collection. Bulls in the MGAA and HGAA groups had higher DM intake and average daily gain than bulls in the LGAA and control groups. The feed conversion ratio was lowest in MGAA and highest in the control. Bulls receiving 0.9 g/kg DM GAA addition had higher digestibility of DM, organic matter, NDF and ADF than bulls in other groups. The digestibility of CP was higher for HGAA than for LGAA and control. The ruminal pH was lower for MGAA, and the total volatile fatty acid concentration was greater for MGAA and HGAA than for the control. The acetate proportion and acetate-to-propionate ratio were lower for MGAA than for LGAA and control. The propionate proportion was higher for MGAA than for control. Bulls receiving GAA addition showed decreased ruminal ammonia N. Bulls in MGAA and HGAA had higher cellobiase, pectinase and protease activities and Butyrivibrio fibrisolvens, Prevotella ruminicola and Ruminobacter amylophilus populations than bulls in LGAA and control. However, the total protozoan population was lower for MGAA and HGAA than for LGAA and control. The total bacterial and Ruminococcus flavefaciens populations increased with GAA addition. The blood level of creatine was higher for HGAA, and the activity of l-arginine glycine amidine transferase was lower for MGAA and HGAA, than for control. The blood activity of guanidine acetate N-methyltransferase and the level of folate decreased in the GAA addition groups. The results indicated that dietary addition of 0.6 or 0.9 g/kg DM GAA improved growth performance, nutrient digestion and ruminal fermentation in bulls.
Effect of inorganic phosphate supplementation on egg production in Hy-Line Brown layers fed 2000 FTU/kg phytase
X. Cheng, J. K. Yan, W. Q. Sun, Z. Y. Chen, S. R. Wu, Z. Z. Ren, X. J. Yang
Journal: animal / Volume 14 / Issue 11 / November 2020
Phytase has long been used to decrease the inorganic phosphorus (Pi) input in poultry diet. The current study was conducted to investigate the effects of Pi supplementation on laying performance, egg quality and phosphate–calcium metabolism in Hy-Line Brown laying hens fed phytase. Layers (n = 504, 29 weeks old) were randomly assigned to seven treatments with six replicates of 12 birds. The corn–soybean meal-based diet contained 0.12% non-phytate phosphorus (nPP), 3.8% calcium, 2415 IU/kg vitamin D3 and 2000 FTU/kg phytase. Inorganic phosphorus (in the form of mono-dicalcium phosphate) was added into the basal diet to construct seven experimental diets; the final dietary nPP levels were 0.12%, 0.17%, 0.22%, 0.27%, 0.32%, 0.37% and 0.42%. The feeding trial lasted 12 weeks (hens from 29 to 40 weeks of age). Laying performance (housed laying rate, egg weight, egg mass, daily feed intake and feed conversion ratio) was weekly calculated. Egg quality (egg shape index, shell strength, shell thickness, albumen height, yolk colour and Haugh units), serum parameters (calcium, phosphorus, parathyroid hormone, calcitonin and 1,25-dihydroxyvitamin D), tibia quality (breaking strength, and calcium, phosphorus and ash contents), intestinal gene expression (type IIb sodium-dependent phosphate cotransporter, NaPi-IIb) and phosphorus excretion were determined at the end of the trial. No differences were observed on laying performance, egg quality, serum parameters and tibia quality. Hens fed 0.17% nPP had increased (P < 0.01) duodenum NaPi-IIb expression compared to all other treatments. Phosphorus excretion linearly increased with an increase in dietary nPP (phosphorus excretion = 1.7916 × nPP + 0.2157; R2 = 0.9609, P = 0.001). In conclusion, corn–soybean meal-based diets containing 0.12% nPP, 3.8% calcium, 2415 IU/kg vitamin D3 and 2000 FTU/kg phytase would meet the requirements for egg production in Hy-Line Brown laying hens (29 to 40 weeks of age). | CommonCrawl |
Trigonometric Fourier Series – Definition and Explanation
Signals and SystemsElectronics & ElectricalDigital Electronics
A periodic signal can be represented over a certain interval of time in terms of the linear combination of orthogonal functions, if these orthogonal functions are trigonometric functions, then the Fourier series representation is known as trigonometric Fourier series.
Consider a sinusoidal signal $x(t)=A\:sin\:\omega_{0}t$ which is periodic with time period $T$ such that $T=2\pi/ \omega_{0}$. If the frequencies of two sinusoids are integral multiples of a fundamental frequency $(\omega_{0})$, then the sum of these two sinusoids is also periodic.
We can prove that a signal $x(t)$ that is a sum of sine and cosine functions whose frequencies are integral multiples of the fundamental frequency $(\omega_{0})$, is a periodic signal.
Let the signal $x(t)$ is given by,
$$\mathrm{x(t)=a_{0}+a_{1}\:cos\:\omega_{0}t+a_{2}\:cos\:2\omega_{0}t+a_{3}\:cos\:3\omega_{0}t+....+a_{k}\:cos\:k\omega_{0}t}$$
$$\mathrm{\:\:\:\:\:\:\:\:\:\:+b_{1}\:sin\:\omega_{0}t+b_{2}\:sin\:2\omega_{0}t+b_{3}\:sin\:3\omega_{0}t+...+b_{k}\:sin\:k\omega_{0}t}$$
$$\mathrm{\Rightarrow\:x(t)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos\:n\omega_{0}t+b_{n}\:sin\:n\omega_{0}t… (1)}$$
Where, $a_{0},a_{1},a_{2}....a_{k}$ and $b_{0},b_{1},b_{2}....b_{k}$ are the constants and $\omega_{0}$ is the fundamental frequency.
Again, if a signal $x(t)$ is a periodic signal, then it must satisfy the following condition −
$$\mathrm{x(t)=x(t+T);\:\:for\:all\:t}$$
$$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos\:n\omega_{0}(t+T)+b_{n}\:sin\:n\omega_{0}(t+T)}$$
$$\mathrm{\because\:Time\:period,T=\left ( \frac{2\pi}{\omega_{0}}\right )}$$
$$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos\:n\omega_{0}(t+\frac{2\pi}{\omega_{0}})+b_{n}\:sin\:n\omega_{0}(t+\frac{2\pi}{\omega_{0}})}$$
$$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos(n\omega_{0} t+2n\pi)+b_{n}\:sin(n\omega_{0} t+2n\pi)}$$
$$\mathrm{\because\:cos(2n\pi+\theta )=cos\:\theta \:\:and\:\:sin(2n\pi+\theta )=sin\:\theta}$$
Using these trigonometric identities, we get,
$$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos(n\omega_{0} t)+b_{n}\:sin(n\omega_{0} t)=x(t)… (2)}$$
From equation (2) it is clear that the signal $x(t)$ which is a sum of sine and cosine functions of frequencies 0,$\omega_{0},2\omega_{0},...k\omega_{0}$ is a periodic signal with a time period T. If in the expression of $x(t),k\rightarrow \infty$ then we can obtain the Fourier series representation of any periodic signal $x(t)$.
Therefore, any periodic signal can be represented as an infinite sum of sine and cosine functions which themselves are periodic signals of angular frequencies 0,$\omega_{0},2\omega_{0},...k\omega_{0}$ . This set of harmonically related sine and cosine functions form a complete set of orthogonal functions over the time interval $t$ to $(t+T)$
Hence, the trigonometric form of Fourier series can be defined as under −
The infinite series of sine and cosine terms of frequencies 0,$\omega_{0},2\omega_{0},...k\omega_{0}$ is called the trigonometric form of Fourier series and can be represented as,
$$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\: n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t… (3)}$$
Where, $a_{0},a_{n}$ and $b_{n}$ are called trigonometric Fourier series confidents.
$$\mathrm{a_{0}=\frac{1}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)\:dt… (4)}$$
$$\mathrm{a_{n}=\frac{2}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)\:cos\:n\omega_{0}t\:dt… (5)}$$
$$\mathrm{b_{n}=\frac{2}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)\:sin\:n\omega_{0}t\:dt… (6)}$$
The coefficient $a_{0}$ is known as the DC component.
$(a_{1}\:cos\:\omega_{0}t+b_{1}\:sin\:\omega_{0}t)$ is called the first harmonic term.
$(a_{2}\:cos\:\omega_{0}t+b_{2}\:sin\:2\omega_{0}t)$is called the second harmonic term.
Similarly, $(a_{n}\:cos\:n\omega_{0}t+b_{n}\:sin\:n\omega_{0}t)$ is called the nth harmonic term.
Numerical Example
Find the trigonometric Fourier series for the waveform shown below.
As we can see the given waveform is periodic with a time period $T= 2\pi$.
Mathematically, the given waveform can be described as,
$$\mathrm{x(t)=\begin{cases}(\frac{A}{\pi})t & for\:0 ≤ t ≤\:\pi\0 & for\:\pi≤ t ≤2\pi\end{cases}}$$
$$\mathrm{t_{0}=0\:\:and\:\:(t_{0}+T)= 2\pi}$$
Then, the fundamental frequency of the given function is,
$$\mathrm{\omega_{0}=\frac{2\pi}{T}=\frac{2\pi}{2\pi}=1}$$
Thus, the coefficient $a_{0}$ is given by,
$$\mathrm{a_{0}=\frac{1}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)dt}$$
$$\mathrm{\Rightarrow\:a_{0}=\frac{1}{2\pi}\int_{0}^{2\pi}x(t)\:dt=\frac{1}{2\pi}\int_{0}^{\pi}(\frac{A}{\pi})t\:dt+\frac{1}{2\pi}\int_{0}^{2\pi}0\:dt=\frac{A}{2\pi^{2}}\left [ \frac{t^{2}}{2}\right ]_{0}^{\pi}=\frac{A}{4}}$$
The coefficient $a_{n}$ is given by,
$$\mathrm{a_{n}=\frac{2}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)cos\:n\omega_{0}t\:\:dt}$$
$$\mathrm{\Rightarrow\:a_{n}=\frac{2}{2\pi} \int_{0}^{\pi}(\frac{A}{\pi})t\:cos\:nt\:dt=\frac{A}{\pi^{2}}\int_{0}^{\pi}t\:cos\:nt\:dt}$$
By solving the above integration, we get,
$$\mathrm{\Rightarrow\:a_{n}=\frac{A}{\pi^{2}n^{2}}[cos\:n\pi]}$$
$$\mathrm{\therefore\:a_{n}=\begin{cases}-(\frac{2A}{\pi^{2}n^{2}}) & for\:odd\:n\0 & for\:even \:n\end{cases}}$$
Similarly, the coefficient $b_{n}$ is given by,
$$\mathrm{b_{n}=\frac{2}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)sin\:n\omega_{0}t\:dt}$$
$$\mathrm{\Rightarrow\:b_{n}=\frac{2}{2\pi}\int_{0}^{\pi}(\frac{A}{\pi})t\:sin\:nt\:dt=\frac{A}{\pi^{2}}\int_{0}^{\pi}t\:sin\:nt\:dt}$$
On solving this integration, we have,
$$\mathrm{b_{n}=\frac{A}{\pi^{2}}\left [-\frac{\pi\:cos\:n\pi}{n} +\left (\frac{sin\:nt}{n^{2}} \right )_{0}^{\pi} \right ]}$$
$$\mathrm{\Rightarrow\:b_{n}=-\frac{A}{n\pi}cos\:n\pi=\frac{A}{n\pi}(-1)^{n+1}}$$
$$\mathrm{\therefore\:b_{n}=\begin{cases}(\frac{A}{n\pi}) & for\:odd\:n\(-\frac{A}{n\pi}) & for\:even\:n\end{cases}}$$
Therefore, the trigonometric Fourier series is,
$$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:n\:\omega_{0}t+b_{n}\:sin\:n\omega_{0}t}$$
$$\mathrm{\Rightarrow\:x(t)=\frac{A}{4}-\frac{2A}{\pi^{2}}\sum_{n=odd}^{\infty}\frac{cos\:nt}{n^{2}}+\frac{A}{\pi}\sum_{n=1}^{\infty}(-1)^{n+1}\cdot \frac{sin\:nt}{n}}$$
Manish Kumar Saini
Fourier Cosine Series – Explanation and Examples
Relation between Trigonometric & Exponential Fourier Series
Expressions for the Trigonometric Fourier Series Coefficients
Difference between Fourier Series and Fourier Transform
Fourier Series – Representation and Properties
Derivation of Fourier Transform from Fourier Series
Time Series Analysis: Definition and Components
Series-Parallel Circuit: Definition and Examples
GIBBS Phenomenon for Fourier Series
Linearity and Conjugation Property of Continuous-Time Fourier Series
Signals & Systems – Complex Exponential Fourier Series
Expression for Exponential Fourier Series Coefficients
Fourier Series Representation of Periodic Signals
Time Differentiation and Integration Properties of Continuous-Time Fourier Series
Convolution Property of Continuous-Time Fourier Series | CommonCrawl |
Moral graph
In graph theory, a moral graph is used to find the equivalent undirected form of a directed acyclic graph. It is a key step of the junction tree algorithm, used in belief propagation on graphical models.
A directed acyclic graph
The corresponding moral graph, with newly added arcs shown in red
The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of non-adjacent nodes that have a common child, and then making all edges in the graph undirected. Equivalently, a moral graph of a directed acyclic graph G is an undirected graph in which each node of the original G is now connected to its Markov blanket. The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be married by sharing an edge.[1]
Moralization may also be applied to mixed graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two vertices that both have outgoing edges to the same chain, and then forgets the orientation of the directed edges of the graph.
Weakly recursively simplicial
A graph is weakly recursively simplicial if it has a simplicial vertex and the subgraph after removing a simplicial vertex and some edges (possibly none) between its neighbours is weakly recursively simplicial. A graph is moral if and only if it is weakly recursively simplicial.
A chordal graph (a.k.a., recursive simplicial) is a special case of weakly recursively simplicial when no edge is removed during the elimination process. Therefore, a chordal graph is also moral. But a moral graph is not necessarily chordal.[2]
Recognising moral graphs
Unlike chordal graphs that can be recognised in polynomial time, Verma & Pearl (1993) proved that deciding whether or not a graph is moral is NP-complete.
See also
• D-separation
• Tree decomposition
References
1. Cowell, Robert G.; Dawid, A. Philip; Lauritzen, Steffen L.; Spiegelhalter, David J. (1999). "3.2.1 Moralization". Probabilistic Networks and Expert Systems: Exact Computational Methods for Bayesian Networks. Springer-Verlag New York. pp. 31–33. doi:10.1007/0-387-22630-3_3. ISBN 0-387-98767-3.
2. Cowell et al. (1999), p. 50.
• Verma, T. S.; Pearl, J. (1993), "Deciding morality of graphs is NP-complete", Uncertainty in Artificial Intelligence: 391–399, arXiv:1303.1501, doi:10.1016/B978-1-4832-1451-1.50052-4, ISBN 9781483214511, S2CID 14690613.
External links
• M. Studeny: On mathematical description of probabilistic conditional independence structures
| Wikipedia |
\begin{document}
\begin{titlepage}
\title{Generalizing the Hypergraph Laplacian via
a Diffusion Process with Mediators}
\begin{abstract}
In a recent breakthrough STOC~2015 paper, a continuous diffusion process was considered on hypergraphs (which has been refined in a recent JACM 2018 paper) to define a Laplacian operator, whose spectral properties satisfy the celebrated Cheeger's inequality. However, one peculiar aspect of this diffusion process is that each hyperedge directs flow only from vertices with the maximum density to those with the minimum density, while ignoring vertices having strict in-beween densities.
In this work, we consider a generalized diffusion process, in which vertices in a hyperedge can act as mediators to receive flow from vertices with maximum density and deliver flow to those with minimum density. We show that the resulting Laplacian operator still has a second eigenvalue satsifying the Cheeger's inequality.
Our generalized diffusion model shows that there is a family of operators whose spectral properties are related to hypergraph conductance, and provides a powerful tool to enhance the development of spectral hypergraph theory. Moreover, since every vertex can participate in the new diffusion model at every instant, this can potentially have wider practical applications.
\end{abstract}
\thispagestyle{empty} \end{titlepage}
\section{Introduction} \label{sec:intro}
Spectral graph theory, and specifically, the well-known Cheeger's inequality give a relationship between the edge expansion properties of a graph and the eigenvalues of some appropriately defined matrix~\cite{alon1986eigenvalues,alon1985lambda1}. Loosely speaking, for a given graph, its edge expansion or \emph{conductance} gives a lower bound on the ratio of the number of edges leaving a subset~$S$ of vertices to the sum of vertex degrees in $S$. It is natural that graph conductance is studied in the context of graph partitioning or clustering~\cite{jacm/KannanVV04,colt/MakarychevMV15,PengSZ15}, whose goal is to minimize the weight of edges crossing different clusters with respect to intra-cluster edges. The reader can refer to the standard references~\cite{chung1997spectral,hoory2006expander} for an introduction to spectral graph theory.
\noindent \textbf{Recent Generalization to Hypergraphs.} In an edge-weighted hypergraph $H = (V,E,w)$, an edge $e \in E$ is a non-empty subset of $V$. The edges have positive weights indicated by $w : E \rightarrow \mathbb{R}_+$. The weight of each vertex $v \in V$ is its weighted degree $w_v := \sum_{e \in E: v \in e} w_e$. A subset $S$ of vertices has weight $w(S) := \sum_{v \in S} w_v$, and the edges it cuts is $\partial S := \{e \in E : $ $e$ intersects both $S$ and $V \setminus S \}$.
The conductance of $S \subseteq V$ is defined as $\phi(S) := \frac{w(\partial S)}{w(S)}$. The conductance of $H$ is defined as:
\begin{equation} \phi_H := \min_{\emptyset \subsetneq S \subsetneq V} \max\{\phi(S), \phi(V \setminus S)\}. \label{eq:hyper_exp} \end{equation}
Until recently, it was an open problem to define a spectral model for hypergraphs. In a breakthrough STOC~2015 paper, Louis~\cite{louis2015hypergraph} considered a continuous diffusion process on hypergraphs (which has been refined in a recent JACM paper~\cite{chan2018jacm}), and defined an operator $\ensuremath{\mathsf{L}\xspace}_w f := - \frac{df}{dt}$, where $f \in \mathbb{R}^V$ is some appropriate vector associated with the diffusion process. As in classical spectral graph theory, $\ensuremath{\mathsf{L}\xspace}_w$ has non-negative eigenvalues, and the all-ones vector $\ensuremath{\mathbf{1}}$ is an eigenvector with eigenvalue 0. Moreover, the operator $\ensuremath{\mathsf{L}\xspace}_w$ has a second eigenvalue~$\gamma_2$, and the Cheeger's inequality can be recovered\footnote{ In fact, as shown in this work, a stronger upper bound holds: $\phi_H \leq \sqrt{2 \gamma_2}$.} for hypergraphs:
\centerline{$
\frac{\gamma_2}{2} \leq \phi_H \leq 2 \sqrt{\gamma_2}. $}
\noindent \textbf{Limitation of the Existing Diffusion Model~\cite{chan2018jacm,louis2015hypergraph}.} Suppose at some instant, each vertex has some \emph{measure} that is given by a measure vector $\varphi \in \mathbb{R}^V$. A corresponding \emph{density} vector $f \in \mathbb{R}^V$ is defined by $f_u := \frac{\varphi_u}{w_u}$, for each $u \in V$. Then, at this instant, each edge $e \in E$ will cause measure to flow from vertices $S_e(f) := {\sf argmax}_{s \in e} f_s$ having the maximum density to vertices $I_e(f) := {\sf argmin}_{i \in e} f_i$ having the minimum density, at a rate of $w_e \cdot \max_{s,i \in e} (f_s - f_i)$. Observe that there can be more than one vertex achieving the maximum or the minimum density in an edge, and a vertex can be involved with multiple number of edges. As shown in~\cite{chan2018jacm}, it is non-trivial to determine the net rate of incoming measure for each vertex.
One peculiar aspect of this diffusion process is that each edge~$e$ only concerns its vertices having the maximum or the minimum density, and ignores the vertices having strict in-between densities. Even though this diffusion process leads to a theoretical treatment of spectral hypergraph properties, its practical use is somehow limited, because it would be considered more natural if vertices having intermediate densities in an edge also take part in the diffusion process.
For instance, in a recent work on semi-supervised learning on hypergraphs~\cite{ZhangHTC17}, the diffusion operator is used to construct an update vector that changes only the solution values of vertices attaining the maximum or the minimum in hyperedges. Therefore, we consider the following open problem in this work:
\emph{Is there a diffusion process on hypergraphs that involves all vertices in every edge at every instant such that the resulting operator still retains desirable spectral properties?}
\subsection{Our Contribution and Results.}
\noindent \textbf{Generalized Diffusion Process with Mediators.} We consider a diffusion process where for each edge~$e$, a vertex~$j \in e$ can act as a \emph{mediator} that receives flow from vertices in $S_e(f)$ and delivers flow to~$I_e(f)$. Formally, we denote $[e] := e \cup \{0\}$, where $0$ is a special index that does not refer to any vertex. Each edge~$e$ is equipped with non-negative constants $(\beta^e_j : j \in [e])$ such that $\sum_{j \in [e]} \beta^e_j = 1$. Intuitively, for $j =0$, $\beta^e_0$ refers to the effect of flow going directly from $S_e(f)$ to $I_e(f)$; for each vertex~$j \in e$, $\beta^e_j$ refers to the significance of~$j$ as a mediator between $S_e(f)$ and $I_e(f)$. The complete description of the diffusion rules is in Definition~\ref{defn:rules}. Here are some interesting special cases captured by the new diffusion model.
\begin{compactitem} \item For each $e \in E$, $\beta^e_0 = 1$. This is the existing model in~\cite{chan2018jacm,louis2015hypergraph}.
\item For each $e \in E$, there is some $j_e \in e$ such that $\beta^e_{j_e} = 1$, i.e., each edge has one special vertex that acts as its mediator who regulates all flow within the edge.
\item For each $e \in E$, for each $j \in e$, $\beta^e_j = \frac{1}{|e|}$, i.e., every vertex in an edge are equally important as mediators. \end{compactitem}
\begin{theorem}[Recovering Cheeger's Inequality via Diffusion Process with Mediators] \label{th:main} Given a hypergraph $H = (V, E, w)$ and mediator constants $(\beta^e_j: e \in E, j \in [e])$, the diffusion process in Definition~\ref{defn:rules} defines an operator $\ensuremath{\mathsf{L}\xspace}_w f := - \frac{df}{dt}$ that has a second eigenvalue~$\gamma_2$ satisfying $ \frac{\gamma_2}{2} \leq \phi_H \leq 2 \sqrt{ \gamma_2} $, where $\phi_H$ is the hypergraph conductance defined in~(\ref{eq:hyper_exp}). \end{theorem}
\noindent \textbf{Impacts of New Diffusion Model.} Our generalized diffusion model shows that there is a family of operators whose spectral properties are related to hypergraph conductance. On the theoretical aspect, this provides a powerful tool to enhance the development of spectral hypergraph theory.
On the practical aspect, as mentioned earlier, in the context of semi-supervised learning~\cite{hein2013total,ZhangHTC17}, the following minimization convex program is considered: the objective function is $\ensuremath{\mathsf{Q}\xspace}(f) := \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w$, and the $f$ values of labeled vertices are fixed. For an iterative method to solve the convex program, our new diffusion model can possibly lead to an update vector that modifies every coordinate in the current solution, thereby potentially improving the performance of the solver.
\subsection{Related Work}
\noindent \emph{Other Works on Diffusion Process and Spectral Graph Theory.} Apart from the most related aforementioned works~\cite{chan2018jacm,louis2015hypergraph} that we have already mentioned, similar diffusion models (without mediators) have been considered for directed normal graphs~\cite{yoshida2016nonlinear} and directed hypergraphs~\cite{CTWZ2017} to define operators whose spectral properties are analyzed.
\noindent \emph{Higher-Order Cheeger Inequalities.} For normal graphs, Cheeger-like inequalities to relate higher-order spectral properties with multi-way edge expansion have been investigated~\cite{KwokLL16,kwok2013improved,lee2014multiway,louis2014approximation,louis2011algorithmic,louis2012many}. On the other hand, for hypergraphs, the higher-order spectral properties of the diffusion operator are still unknown. However, Cheeger-like inequalities have been derived in terms of the discrepancy ratio~\cite{chan2018jacm}, but not related to the spectral properties of the diffusion operator.
\section{Preliminaries} \label{sec:hyper-notation}
We consider an edge-weighted hypergraph $H = (V,E,w)$. Without loss of generality, we assume that the weight $w_i := \sum_{e \in E: i \in e} w_e$ of each vertex~$i \in V$ is positive, since any vertex with zero weight can be removed. We use $\ensuremath{\mathsf{W}}\xspace \in \mathbb{R}^{V \times V}$ to denote the diagonal matrix whose $(i,i)$-th entry is the vertex weight~$w_i$; we let $\ensuremath{\mathsf{I}_n\xspace}$ denote the identity matrix.
We use $\mathbb{R}^V$ to denote the set of column vectors. Given $f \in \mathbb{R}^V$, we use $f_u$ or $f(u)$ to indicate the coordinate corresponding to $u \in V$. We use $A^\ensuremath{\mathsf{T}}\xspace$ to denote the transpose of a matrix $A$.
We use $\ensuremath{\mathbf{1}} \in \mathbb{R}^V$ to denote the vector having $1$ in every coordinate. For a vector $x \in \mathbb{R}^V$, we define its support as the set of coordinates at which $x$ is non-zero, i.e. ${\sf supp}(x) := \set{i : x_i \neq 0}$.
We use $\chi_S \in \{0,1\}^V$ to denote the indicator vector of the set $S \subset V$, i.e., $\chi_S(v) = 1 $ \emph{iff} $v \in S$.
Recall that the conductance $\phi_H$ of a hypergraph $H$ is defined in (\ref{eq:hyper_exp}). We drop the subscript whenever the hypergraph is clear from the context.
\noindent \textbf{Generalized Quadratic Form.} For each edge $e \in E$, we denote $[e] := e \cup \{0\}$, where $0$ is a special index that does not correspond to any vertex. Then, each edge~$e$ is associated with non-negative constants $(\beta^e_j: j \in [e])$ such that $\sum_{j \in [e]} \beta^e_j = 1$. The \emph{generalized quadratic form} is defined for each $f \in \mathbb{R}^V$ as:
\begin{equation} \textstyle \ensuremath{\mathsf{Q}\xspace}(f) := \sum_{e\in E}\w{e}\{ \beta^e_0\max_{s,i\in e}\left(\f{s}-\f{i}\right)^2+\sum_{j\in e}\beta^e_{j}[(\max_{s\in e}\f{s}-\f{j})^2 +(\f{j}-\min_{i\in e}\f{i})^2 ] \}. \label{eq:quad} \end{equation}
\noindent For each non-zero $f \in \mathbb{R}^V$, its \emph{discrepancy ratio} is defined as $\ensuremath{\mathsf{D}\xspace}_w(f) := \frac{\ensuremath{\mathsf{Q}\xspace}(f)}{\sum_{u \in V} \w{u} \f{u}^2}$.
\noindent \emph{Remark.} Observe that for each $S \subseteq V$, the corresponding indicator vector $\chi(S) \in \{0,1\}^V$ satisfies $\ensuremath{\mathsf{Q}\xspace}(\chi(S)) = w(\partial S)$. Hence, we have $\ensuremath{\mathsf{D}\xspace}_w(\chi(S)) = \phi(S)$.
\noindent \textbf{Special Case.} We denote $\ensuremath{\mathsf{Q}\xspace}^0(f) := \sum_{e \in E} w_e \max_{s,i\in e}\left(\f{s}-\f{i}\right)^2$ for the case when $\beta^e_0 =1$ for all $e$, which was considered in~\cite{chan2018jacm}. As we shall see later, for $j \in e$, the weight $\beta^e_j$ denotes the significance of vertex~$j$ as a ``mediator'' in the diffusion process to direct measure from vertices of maximum density to those with minimum density. As in~\cite{chan2018jacm}, we consider three isomorphic spaces as follows.
\noindent \textbf{Density Space.} This is the space associated with the quadratic form $\ensuremath{\mathsf{Q}\xspace}$. For $f,g \in \mathbb{R}^V$, the inner product is defined as $\langle f, g \rangle_w := f^\ensuremath{\mathsf{T}}\xspace \ensuremath{\mathsf{W}}\xspace g$, and the associated norm is $\| f \|_w := \sqrt{\langle f, f \rangle_w}$. We use $f \perp_w g$ to denote $\langle f, g \rangle_w = 0$.
\noindent \textbf{Normalized Space.} Given $f \in \mathbb{R}^V$ in the density space, the corresponding vector in the normalized space is $x := \Wh f$. The normalized discrepancy ratio is $\ensuremath{\mathcal{D}}\xspace(x) := \ensuremath{\mathsf{D}\xspace}_w(\Wmh x) = \ensuremath{\mathsf{D}\xspace}_w(f)$.
In the normalized space, the usual $\ell_2$ inner product and norm are used. Observe that if $x$ and $y$ are the corresponding normalized vectors for $f$ and $g$ in the density space, then $\langle x, y \rangle = \langle f, g \rangle_w$.
\noindent \emph{Towards Cheeger's Inquality.} Using the inequality $a^2 + b^2 \leq (a+b)^2 \leq 2(a^2 + b^2)$ for non-negative $a$ and $b$, we conclude that $\ensuremath{\mathsf{Q}\xspace}(f) \leq \ensuremath{\mathsf{Q}\xspace}^0(f) \leq 2 \ensuremath{\mathsf{Q}\xspace}(f)$ for all $f \in \mathbb{R}^V$. This immediately gives a partial result of Theorem~\ref{th:main}.
\begin{lemma}[Cheeger's Inequality for Quadratic Form] \label{lemma:cheeger}
Suppose $\gamma_2 := \min_{\ensuremath{\mathbf{0}} \neq f \perp_w \ensuremath{\mathbf{1}}} \frac{\ensuremath{\mathsf{Q}\xspace}(f)}{\|f\|_w^2}$. Then, we have $\frac{\gamma_2}{2} \leq \phi_H \leq 2 \sqrt{\gamma_2}$, where $\phi_H$ is the hypergraph conductance defined in (\ref{eq:hyper_exp}). \end{lemma}
\begin{proof}
Denote $\gamma^0_2 := \min_{\ensuremath{\mathbf{0}} \neq f \perp_w \ensuremath{\mathbf{1}}} \frac{\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}$. Then, the result from~\cite{chan2018jacm} and an improved upper bound in Appendix~\ref{sec:up_bound_cheeger} gives: $\frac{\gamma^0_2}{2} \leq \phi_H \leq \sqrt{2 \gamma^0_2}$. Finally, $\ensuremath{\mathsf{Q}\xspace} \leq \ensuremath{\mathsf{Q}\xspace}^0 \leq 2 \ensuremath{\mathsf{Q}\xspace}$ implies that $\gamma_2 \leq \gamma^0_2 \leq 2 \gamma_2$. Hence, the result follows.
$\Box$ \end{proof}
\noindent \textbf{Goal of This Paper.} In view of Lemma~\ref{lemma:cheeger}, the most technical part of the paper is to define an operator\footnote{In the literature, the weighted Laplacian is actually $\ensuremath{\mathsf{W}}\xspace \ensuremath{\mathsf{L}\xspace}_w$ in our notation. Hence, to avoid confusion, we restrict the term Laplacian to the normalized space.} $\ensuremath{\mathsf{L}\xspace}_w: \mathbb{R}^V \rightarrow \mathbb{R}^V$ such that $\langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w = \ensuremath{\mathsf{Q}\xspace}(f)$, and show that $\gamma_2$ defined in Lemma~\ref{lemma:cheeger} is indeed an eigenvalue of $\ensuremath{\mathsf{L}\xspace}_w$. To achieve this, we shall consider a diffusion process in the following measure space.
\noindent \textbf{Measure Space.} Given a density vector $f \in \mathbb{R}^V$, multiplying each coordinate with its corresponding weight gives the measure vector $\varphi := \ensuremath{\mathsf{W}}\xspace f$. Observe that a vector in the measure space can have negative coordinates. We do not consider inner product explicitly in this space, and so there is no special notation for it.
\noindent \textbf{Transformation between Different Spaces.} We use the Roman letter $f$ for vectors in the density space, $x$ for vectors in the normalized space, and Greek letter $\varphi$ for vectors in the measure space. Observe that an operator defined on one space induces operators on the other two spaces. For instance, if $\ensuremath{\mathsf{L}\xspace}$ is an operator defined on the measure space, then $\ensuremath{\mathsf{L}\xspace}_w := \ensuremath{\mathsf{W}^{-1}}\xspace \ensuremath{\mathsf{L}\xspace} \ensuremath{\mathsf{W}}\xspace$ is the corresponding operator on the density space and $\ensuremath{\mathcal{L}}\xspace := \Wmh \ensuremath{\mathsf{L}\xspace} \Wh$ is the one on the normalized space. Moreover, all three operators have the same eigenvalues. Recall that the Rayleigh quotients are defined as $\ensuremath{\mathsf{R}}\xspace_w(f) := \frac{\langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w}{\langle f, f \rangle_w}$ and $\ensuremath{\mathcal{R}}\xspace(x) := \frac{\langle x, \ensuremath{\mathcal{L}}\xspace x \rangle}{\langle x, x \rangle}$. For $\Wh f = x$, we have $\ensuremath{\mathsf{R}}\xspace_w(f) = \ensuremath{\mathcal{R}}\xspace(x)$.
\section{Diffusion Process with Mediators} \label{sec:diffusion}
\noindent \textbf{Intuition.} Given an edge-weighted hypergraph $H=(V,E,w)$, suppose at some instant, each vertex has some measure given by the vector $\varphi \in \mathbb{R}^V$, whose corresponding density vector is $f = \ensuremath{\mathsf{W}^{-1}}\xspace \varphi$. The idea of a diffusion process is that within each edge~$e \in E$, measure should flow from vertices with higher densities to those with lower densities, and the rate of flow has a positive correlation with the difference in densities and the strength of the edge~$e$ given by $w_e$. If the diffusion process is well-defined, then an operator on the density space can be defined as $\ensuremath{\mathsf{L}\xspace}_w f := - \frac{df}{dt}$. This induces the Laplacian operator $\ensuremath{\mathcal{L}}\xspace := \Wh \ensuremath{\mathsf{L}\xspace}_w \Wmh$ on the normalized space.
In previous work~\cite{chan2018jacm}, within an edge, measure only flows from vertices $S_e(f) := {\sf argmax}_{s \in e} f_s \subseteq e$ having the maximum density to those $I_e(f) := {\sf argmin}_{i \in e} f_i$ having minimum densities, where the rate of flow is $w_e \cdot \max_{s,i \in e} (f_s - f_i)$. If all $f_u$'s for an edge $e$ are equal, then we use the convention that $I_e(f) = S_e(f) = e$. Note that vertices $j \in e \setminus (S_e(f) \cup I_e(f))$ with strict in-between densities do not participate due to edge~$e$ at this instant.
\noindent \textbf{Generalized Diffusion Process with Mediators.} In some applications as mentioned in Section~\ref{sec:intro}, it might be more natural if every vertex in an edge~$e$ plays some role in diverting flow from $S_e(f)$ to $I_e(f)$. In our new diffusion model, each edge~$e$ is associated with constants $(\beta^e_j: j \in [e])$ such that $\sum_{j \in [e]} \beta^e_j = 1$.
Here, $0$ is a special index and the parameter $\beta^e_0$ corresponds to the significance of measure flowing directly from $S_e(f)$ to $I_e(f)$. For $j \in e$, $\beta^e_j$ indicates the significance of vertex~$j$ as a ``mediator'' to receive measure from $S_e(f)$ and deliver measure to $I_e(f)$. The formal rules are given as follows.
\begin{definition}[Rules of Diffusion Process] \label{defn:rules} Suppose at some instant the system is in a state given by the density vector $f \in \mathbb{R}^V$, with measure vector $\varphi =\ensuremath{\mathsf{W}}\xspace f$. Then, at this instant, measure is transferred between vertices according to the following rules. For $u \in e$ and $j \in [e]$, the pair $(e,j)$ imposes some rules on the diffusion process; let $\varphi'_u(e,j)$ be the net rate of measure flowing into vertex~$u$ due to the pair~$(e,j)$.
\begin{compactitem}
\item [\textsf{R(0)}] For each vertex $u \in V$, the density changes according to the net rate of incoming measure divided by its weight:
\centerline{$
w_u \frac{d f_u}{d t} = \varphi'_u := \sum_{e \in E: u \in e} \sum_{j \in [e]} \varphi'_u(e,j).
$}
\item [\textsf{R(1)}]
We have $\varphi'_u(e,j) < 0$ and $u \neq j$ implies that $u \in S_e(f)$.
Similarly, $\varphi'_u(e,j) > 0$ and $u \neq j$ implies that $u \in I_e(f)$.
\item [\textsf{R(2)}] Each edge $e \in E$ and $j \in [e]$,
the rates of flow satisfy the following.
For $j = 0$, the rate of flow from $S_e(f)$ to $I_e(f)$ due to $(e,0)$ is:
\centerline{$
- \sum_{u \in S_e(f)} \varphi'_u(e,0) = w_e \cdot \beta^e_0 \cdot \max_{s,i \in e}(f_{s}-f_{i})
= \sum_{u \in I_e(f)} \varphi'_u(e,0).
$}
For $j \in e$, the rate of flow from $S_e(f)$ to $j$ due to $(e,j)$ is:
\centerline{$
- \sum_{u \in S_e(f)} \varphi'_u(e,j) = w_e \cdot \beta^e_j \cdot (\max_{s \in e} f_{s} -f_{j});
$}
the rate of flow from $j$ to $I_e(f)$ due to $(e,j)$ is:
\centerline{$
\sum_{u \in I_e(f)} \varphi'_u(e,j) = w_e \cdot \beta^e_j \cdot (f_{j}- \min_{i \in e}f_{i}).
$}
Then the net rate of flow received by $j$ due to $(e,j)$ is:
\centerline{$
w_e \cdot \beta^e_j \cdot (\max_{s \in e} f_s + \min_{i \in e} f_i - 2 f_j)
= \varphi'_j(e,j).
$}
\end{compactitem} \end{definition}
\noindent \textbf{Existence of Diffusion Process.} The diffusion rules in Definition~\ref{defn:rules} are much more complicated than those in~\cite{chan2018jacm}. It is not immediately obvious whether such a process is well-defined. However, the techniques in~\cite{CTWZ2017} can be employed. Intuitively, by repeatedly applying the procedure described in Section~\ref{sec:disp}, all higher-order derivatives of the density vector can be determined, which induce an equivalence relation on~$V$ such that vertices in the same equivalence class will have the same density in infinitesimal time. This means the hypergraph can be reduced to a simple graph, in which the diffusion process is known to be well-defined. However, to argue this formally is non-trivial, and the reader can refer to the details in~\cite{CTWZ2017}.
As in~\cite{chan2018jacm}, if we define an operator using the diffusion process in Definition~\ref{defn:rules}, then the resulting Rayleigh quotient coincides with the discrepancy ratio. The proof of the following lemma is deferred to Appendix~\ref{sec:ray_disc}.
\begin{lemma} [Rayleigh Quotient Coincides with Discrepancy Ratio] \label{lemma:ray_disc} Suppose $\ensuremath{\mathsf{L}\xspace}_w$ on the density space is defined as $\ensuremath{\mathsf{L}\xspace}_w f := - \frac{df}{dt}$ by the rules in Definition~\ref{defn:rules}. Then, the Rayleigh quotient associated with $\ensuremath{\mathsf{L}\xspace}_w$ satisfies that for any $f$ in the density space, $\ensuremath{\mathsf{R}}\xspace_w(f) = \ensuremath{\mathsf{D}\xspace}_w(f)$. By considering the isomorphic normalized space, we have for each $x$, $\ensuremath{\mathcal{R}}\xspace(x) = \ensuremath{\mathcal{D}}\xspace(x)$. \end{lemma}
\section{Computing the First Order Derivative in the Diffusion Process} \label{sec:disp}
In Section~\ref{sec:diffusion}, we define a diffusion process, whose purpose is to define an operator $\ensuremath{\mathsf{L}\xspace}_w f := - \frac{df}{dt}$, where $f \in \mathbb{R}^V$ is in the density space. In this section, we show that the diffusion rules uniquely determine the first order derivative vector $\frac{df}{dt}$; moreover, we give an algorithm to compute it.
\noindent \textbf{Infinitesimal Considerations.} In Definition~\ref{defn:rules}, if a vertex $u$ is losing measure due to the pair $(e, j)$ and $u \neq j$, then $u$ must be in $S_e(f)$. However, $u$ must also continue to stay in $S_e(f)$ in infinitesimal time; otherwise, if $u$ is about to leave $S_e(f)$, then $u$ should no longer lose measure due to $(e,j)$. Hence, the vertex~$u$ should have the maximum first-order derivative of $f_u$ among vertices in $S_e(f)$. A similar rule should hold when $u$ is gaining measure due to $(e,j)$ and $u \neq j$. This is formalized as the first-order variant of \textsf{(R1)}:
Rule \textsf{(R3)} First-Order Derivative Constraints:
If $\varphi'_u(e,j)<0$ and $u \neq j$, then $u \in {\sf argmax}_{s \in S_e(f)} \frac{d f_s}{dt}$.
If $\varphi'_u(e, j)>0$ and $u \neq j$, then $u \in {\sf argmin}_{i \in I_e(f)} \frac{d f_i}{dt}$.
\noindent \textbf{Considering Each Equivalence Class $U$ Independently.} As in~\cite{chan2018jacm}, we consider the equivalence relation induced by $f \in \mathbb{R}^V$, where two vertices $u$ and $v$ are in the same equivalence class \emph{iff} $f_u = f_v$. For vertices in some equivalence class~$U$, their current $f$ values are the same, but their values could be about to be separated because their first derivatives might be different.
\noindent \textbf{Subset with the Largest First Derivative: Densest Subset.} Suppose $X \subseteq U$ are the vertices having the largest derivative in $U$. Then, these vertices should receive or contribute rates of measure in each of the following cases.
\begin{compactitem}
\item[1.] The subset $X$ receives measure due to edges $I_X := \{e \in E: I_e(f) \subseteq X\}$, because the corresponding vertices in $X$ continue to have minimum $f$ values in these edges; we let $c^I_e \geq 0$ be the rate of measure received by $I_e(f)$ due to $(e,j)$ for $j \notin I_e(f)$.
\item[2.] The subset $X$ contributes measure due to edges $S_X := \{e \in E: S_e(f) \cap X \neq \emptyset\}$, because the corresponding vertices in $X$ continue to have maximum $f$ values in these edges; we let $c^S_e \geq 0$ be the rate of measure delivered by $S_e(f)$ due to $(e,j)$ for $j \notin S_e(f)$.
\item[3.] Each $j \in X$ receives or contributes measure due to all $(e,j)$'s such that $e \in E$ and $j \in e$; we let $c_j \in \mathbb{R}$ be the net rate of measure received by vertex~$j$ due to $(e,j)$ for all $e \in E$ such that $j \in e$.
\end{compactitem}
Hence, the net rate of measure received by $X$ is
\centerline{ $\mathfrak{C}(X) := \sum_{e \in I_X} c^I_e - \sum_{e \in S_X} c^S_e + \sum_{j \in X} c_j$. }
Therefore, given an instance $(U, I_U, S_U)$, the problem is to find a maximal subset~$P\subseteq U$ with the largest density $\delta(P) := \frac{\mathfrak{C}(P)}{w(P)}$, which will be the $\frac{df}{dt}$ values for the vertices in $P$. For the remaining vertices in $U$, the sub-instance $(U \setminus P, I_U \setminus I_P, S_U \setminus S_P)$ is solved recursively. The procedure and the precise parameters are given in Fig.~\ref{fig:define_r}. Efficient algorithms for this densest subset problem are described in~\cite{DanischCS17,chan2018jacm}.
\begin{figure}
\caption{Procedure to compute $r=\frac{df}{dt}$}
\label{fig:define_r}
\end{figure}
The next lemma shows that the procedure in Fig.~\ref{fig:define_r} returns a vector $r \in \mathbb{R}^V$ that coincides with the first-order derivative $\frac{df}{dt}$ of the density vector obeying rules~\textsf{(R0)} to \textsf{(R3)}. This implies that these rules uniquely determine the first-order derivative. Given $f \in \mathbb{R}^V$ and $r = \frac{df}{dt}$, we denote $r_S(e) := \max_{u \in S_e(f)} r_u$ and $r_I(e) := \min_{u \in I_e(f)} r_u$.
\begin{lemma}[Densest Subset Problem Determines First-Order Deriative] \label{lemma:define_lap} Given a density vector $f \in \mathbb{R}^V$, rules~\textsf{(R0)} to \textsf{(R3)} uniquely determine $r = \frac{df}{dt} \in \mathbb{R}^V$, which can be found by the procedure described in Fig.~\ref{fig:define_r}. Moreover, $\sum_{e \in E} c^I_e\cdot r_I(e) -\sum_{e \in E} c^S_e\cdot r_S(e)
+\sum_{j\in V}c_j\cdot r_j= \sum_{u \in V} \varphi'_u r_u = \|r\|^2_w$. \end{lemma}
\begin{proof} Using the same approach as in~\cite{chan2018jacm}, we consider each equivalence class $U$ in Fig.~\ref{fig:define_r}, where all vertices in a class have the same $f$ values.
For each such equivalence class $U \subset V$, define $I_U := \{e \in E: U\cap I_e(f)\not=\emptyset\}$, $S_U := \{e \in E: U\cap S_e(f)\not=\emptyset\}$. Notice that each $e$ can only be in exactly one of $I_U$ and $S_U$.
\noindent \textbf{Considering Each Equivalence Class $U$.} Suppose $T$ is the set of vertices within $U$ that have the maximum first-order derivative $r = \frac{df}{dt}$. It suffices to show that $T$ is the maximal densest subset in the densest subset instance $(U, I_U \cup S_U)$ defined in Fig.~\ref{fig:define_r}.
Because of rule~\textsf{(R3)}, the rate of net measure received by~$T$ is $\mathfrak{C}(T)$. Hence, all vertices~$u \in T$ have $r_u = \frac{\mathfrak{C}(T)}{w(T)}$.
Next, suppose~$P$ is the maximal densest subset found in Fig.~\ref{fig:define_r}. Observe that the net rate of measure entering $P$ is at least $\mathfrak{C}(P)$. Hence, there exists some vertex~$v \in P$ such that $\frac{\mathfrak{C}(P)}{w(P)} \leq r_v \leq \frac{\mathfrak{C}(T)}{w(T)}$, where the last inequality follows from the definition of $T$.
Since $P$ is the maximal densest subset, it follows that in the above inequality, actually all equalities hold and all vertices in $P$ have the same $r$ value. In general, the maximal densest subset contains all densest subsets, and it follows that $T \subseteq P$. Since all vertices in $P$ have the maximum $r$ value within $U$, we conclude that $P = T$.
\noindent \emph{Recursive Argument.} Hence, it follows that the set $T$ can be uniquely identified in Fig.~\ref{fig:define_r} as the set of vertices having maximum $r$ values, which is also the unique maximal densest subset. Then, the argument can be applied recursively for the smaller instance with $U' := U \setminus T$, $I_{U'} := I_U \setminus I_T$, $S_{U'} := S_U \setminus S_T$.
\noindent \textbf{Claim.} $ \sum_{e \in E} c^I_e\cdot r_I(e) -\sum_{e \in E} c^S_e\cdot r_S(e) +\sum_{j\in V}c_j\cdot r_j
= \sum_{u \in V} \varphi'_u r_u = \|r\|^2_w. $
Consider some $T$ defined above with $\delta := \delta(T) = r_u$, for $u \in T$.
Observe that \begin{align} \begin{split} \textstyle \sum_{u \in T} \varphi'_u r_u &=\left( \textstyle \sum_{e\in I_T} c^I_e -\sum_{e\in S_T}c^S_e +\sum_{j \in T} c_j \right) \cdot \delta \\ &\textstyle = \sum_{e \in I_T} c^I_e \cdot\min_{i\in I_e}r_i - \sum_{e \in S_T} c^S_e \cdot \max_{s\in S_e}r_s +\sum_{j\in T}c_j\cdot r_j \end{split} \nonumber \end{align} where the last equality is due to rule~\textsf{(R3)}.
Observe that every $u \in V$ will be in exactly one such $T$, and every $e \in E$ will be accounted for exactly once in each of $I_T$ and $S_T$, ranging over all $T$'s. Hence, summing over all $T$'s gives the result.
$\Box$ \end{proof}
\section{Spectral Properties of Laplacian} \label{sec:eigen}
A classical result in spectral graph theory is that for a $2$-graph whose edge weights are given by the adjacency matrix $A$, the parameter $\gamma_2 := \min_{\ensuremath{\mathbf{0}} \neq x \perp \Wh \ensuremath{\mathbf{1}}} \ensuremath{\mathcal{D}}\xspace(x)$ is an eigenvalue of the normalized Laplacian \mbox{$\ensuremath{\mathcal{L}}\xspace := \ensuremath{\mathsf{I}_n\xspace} - \Wmh A \Wmh$}, where a corresponding minimizer $x_2$ is an eigenvector of $\ensuremath{\mathcal{L}}\xspace$. Observe that $\gamma_2$ is also an eigenvalue on the operator $\ensuremath{\mathsf{L}\xspace}_w := \ensuremath{\mathsf{I}_n\xspace} - \ensuremath{\mathsf{W}^{-1}}\xspace A$ induced on the density space.
In this section, we generalize the result to hypergraphs. Observe that any result for the normalized space has an equivalent counterpart in the density space, and vice versa.
\begin{theorem}[Eigenvalue of Hypergraph Laplacian]
\label{th:hyper_lap}
For a hypergraph with edge weights $w$,
there exists a normalized Laplacian $\ensuremath{\mathcal{L}}\xspace$ such that the
normalized discrepancy ratio $\ensuremath{\mathcal{D}}\xspace(x)$ coincides
with the corresponding Rayleigh quotient $\ensuremath{\mathcal{R}}\xspace(x)$.
Moreover,
the parameter $\gamma_2 := \min_{\ensuremath{\mathbf{0}} \neq x \perp \Wh \ensuremath{\mathbf{1}}} \ensuremath{\mathcal{D}}\xspace(x)$ is an eigenvalue of $\ensuremath{\mathcal{L}}\xspace$,
where any minimizer $x_2$ is a corresponding eigenvector. \end{theorem}
Before proving Theorem~\ref{th:hyper_lap}, we first consider the spectral properties of the normalized Laplacian $\ensuremath{\mathcal{L}}\xspace$ induced by the diffusion process defined in Section~\ref{sec:disp}.
\begin{lemma}[First-Order Derivatives]
\label{lemma:deriv}
Consider the diffusion process satisfying rules~\textsf{(R0)}
to~\textsf{(R3)} on
the measure space with $\varphi \in \mathbb{R}^V$, which
corresponds to $f = \ensuremath{\mathsf{W}^{-1}}\xspace \varphi$ in the density space.
Suppose
$\ensuremath{\mathsf{L}\xspace}_w$ is the induced operator on the density space such that
$\frac{d f}{d t} = - \ensuremath{\mathsf{L}\xspace}_w f$.
Then, we have the following derivatives.
\begin{compactitem}
\item[1.] $\frac{d \|f\|^2_w}{dt} = - 2 \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w$.
\item[2.] $\frac{d \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w}{dt}
= - 2 \|\ensuremath{\mathsf{L}\xspace}_w f \|^2_w$.
\item[3.] Suppose $\ensuremath{\mathsf{R}}\xspace_w(f)$ is the Rayleigh quotient
with respect to the operator $\ensuremath{\mathsf{L}\xspace}_w$ on the density space.
Then, for $f \neq \ensuremath{\mathbf{0}}$, $\frac{d \ensuremath{\mathsf{R}}\xspace_w(f)}{dt} = -\frac{2}{\|f\|^4_w} \cdot
(\|f\|^2_w \cdot \|\ensuremath{\mathsf{L}\xspace}_w f\|^2_w - \langle f , \ensuremath{\mathsf{L}\xspace}_w f \rangle^2_w) \leq 0$,
by the Cauchy-Schwarz inequality
on the $\langle \cdot , \cdot \rangle_w$ inner product, where equality
holds \emph{iff} $\ensuremath{\mathsf{L}\xspace}_w f \in \operatorname{span}(f)$.
By considering a transformation to the normalized space, for any $x \neq \ensuremath{\mathbf{0}}$,
$\frac{d \ensuremath{\mathcal{R}}\xspace(x)}{dt} \leq 0$, where equality holds \emph{iff} $\ensuremath{\mathcal{L}}\xspace x \in \operatorname{span}(x)$.
\end{compactitem}
\end{lemma}
\begin{proof}
For the first statement,
$\frac{d \|f\|_w^2}{d t} = 2 \langle f, \frac{d f}{d t} \rangle_w
= - 2 \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w$.
For the second statement,
from the proof of Lemma~\ref{lemma:ray_disc}
we have
$$\langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w
=\textstyle
\sum_{e\in E}\w{e}\{\beta^e_0\max_{s,i\in e}\left(\f{s}-\f{i}\right)^2+\sum_{j\in e}\beta^e_{j}[\left(\max_{s\in e}\f{s}-\f{j}\right)^2
+\left(\f{j}-\min_{i\in e}\f{i}\right)^2]\}.$$
Hence, by the Envelope Theorem,
\begin{align}
\begin{split}
\textstyle
\frac{d \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w}{dt}
=&
\textstyle
2 \sum_{e \in E} \w{e}
\left[\vphantom{\sum_{j\in e}}
\beta^e_0\max_{s,i\in e}\left(\f{s}-\f{i}\right)
\left(\max_{s\in S_e}\frac{df_s}{dt}-\min_{i\in I_e}\frac{df_i}{dt}\right)
\right]
\\
&\textstyle
+\sum_{j\in e}\beta^e_{j}\left(\max_{s\in e}\f{s}-\f{j}\right)
\left(\max_{s\in S_e}\frac{df_s}{dt}-\frac{df_j}{dt}\right)
\\
&\textstyle
\left.
+\sum_{j\in e}\beta^e_{j}\left(\f{j}-\min_{i\in e}\f{i}\right)
\left(\frac{df_j}{dt}-\min_{i\in I_e}\frac{df_i}{dt}\right)
\right]
\\
=&\textstyle
2 \sum_{e \in E} \w{e}\left\{
\left[
\beta^e_0\max_{s,i\in e}\left(\f{s}-\f{i}\right)
+\sum_{j\in e}\beta^e_{j}\left(\max_{s\in e}\f{s}-\f{j}\right)
\right]
\max_{s\in S_e}\frac{df_s}{dt}
\right.
\\
&\textstyle
-\left[
\beta^e_0\max_{s,i\in e}\left(\f{s}-\f{i}\right)
+\sum_{j\in e}\beta^e_{j}\left(\f{j}-\min_{i\in e}\f{i}\right)
\right]
\min_{i\in I_e}\frac{df_i}{dt}
\\
&\textstyle
\left.
+\sum_{j\in e}\beta^e_{j}\left(2\f{j}-\max_{s\in e}\f{s}-\min_{i\in e}\f{i}\right)
\frac{df_j}{dt}
\right\}.
\\
=&\textstyle
2 \left(
\sum_{e \in E}
c^I_e\cdot\max_{s\in S_e}r_s
-\sum_{e \in E}
c^S_e\cdot\max_{i\in I_e}r_i
-\sum_{j\in V}c_j\cdot r_j
\right)
\end{split}
\nonumber
\end{align}
\noindent where $c^I_e,c^S_e,c_j$ are defined in Fig.~\ref{fig:define_r}.
From Lemma~\ref{lemma:define_lap},
this equals $- 2 \|r\|^2_w = - 2 \|\ensuremath{\mathsf{L}\xspace}_w f\|^2_w$.
Finally, for the third statement, we have
$\frac{d}{dt} \frac{\langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w}{\langle f, f \rangle_w} = \frac{1}{\|f\|^4_w} (\| f \|^2_w \cdot \frac{d \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w}{ dt} - \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w \cdot \frac{d \|f\|^2_w}{dt})
= -\frac{2}{\|f\|^4_w} \cdot
(\|f\|^2_w \cdot \|\ensuremath{\mathsf{L}\xspace}_w f\|^2_w - \langle f , \ensuremath{\mathsf{L}\xspace}_w f \rangle^2_w)$,
where the last equality follows from the first two statements.
$\Box$ \end{proof}
We next prove some properties of the normalized Laplacian $\ensuremath{\mathcal{L}}\xspace$ with respect to orthogonal projection in the normalized space.
\begin{lemma}[Laplacian and Orthogonal Projection]
\label{lemma:lap_proj}
Suppose $\ensuremath{\mathcal{L}}\xspace$ is the normalized Laplacian.
Moreover, denote $x_1 := \Wh \ensuremath{\mathbf{1}}$, and
let $\Pi$ denote the orthogonal projection
into the subspace that is orthogonal to $x_1$.
Then, for all $x$, we have the following:
\begin{compactitem}
\item[1.] $\ensuremath{\mathcal{L}}\xspace(x) \perp x_1$,
\item[2.] $\langle x, \ensuremath{\mathcal{L}}\xspace x \rangle = \langle \Pi x, \ensuremath{\mathcal{L}}\xspace \Pi x \rangle$.
\item[3.] For all real numbers $a$ and $b$,
$\ensuremath{\mathcal{L}}\xspace(a x_1 + b x) = b \ensuremath{\mathcal{L}}\xspace(x)$.
\end{compactitem} \end{lemma}
\begin{proof}
For the first statement, observe that since the diffusion process
is defined on a closed system, the total measure given by $\sum_{u \in V} \varphi_u$ does not change.
Therefore, $0 = \langle \ensuremath{\mathbf{1}}, \frac{d \varphi}{d t} \rangle = \langle \Wh \ensuremath{\mathbf{1}}, \frac{d x}{d t} \rangle$,
which implies that $\ensuremath{\mathcal{L}}\xspace x = - \frac{d x}{d t} \perp x_1$.
For the second statement,
observe that from Lemma~\ref{lemma:ray_disc},
we have
$
\langle x, \ensuremath{\mathcal{L}}\xspace x \rangle
=\sum_{e\in E}\w{e}\{
\beta^e_0\max_{s,i\in e}(\frac{x_s}{\sqrt{w_s}}-\frac{x_i}{\sqrt{w_i}})^2
+\sum_{j\in e}\beta^e_{j}[
(\max_{s\in e}\frac{x_s}{\sqrt{w_s}}-\frac{x_j}{\sqrt{w_j}})^2
+(\frac{x_j}{\sqrt{w_j}}-\min_{i\in e}\frac{x_i}{\sqrt{w_i}})^2
]
\}
=\langle (x + \alpha x_1), \ensuremath{\mathcal{L}}\xspace (x + \alpha x_1) \rangle,
$
where the last equality holds for all real numbers $\alpha$.
Observe that $\Pi x = x + \alpha x_1$, for some suitable real $\alpha$.
For the third statement, it is more convenient
to consider transformation into the density space $f = \Wmh x$.
It suffices to show that $\ensuremath{\mathsf{L}\xspace}_w (a \ensuremath{\mathbf{1}} + b f) = b \ensuremath{\mathsf{L}\xspace}_w(f)$.
Observe that in the diffusion process, only pairwise difference in densities among vertices matters.
Hence, we immediately have $\ensuremath{\mathsf{L}\xspace}_w (a \ensuremath{\mathbf{1}} + b f) = \ensuremath{\mathsf{L}\xspace}_w (b f)$.
For $b \geq 0$, observe that all the rates are scaled by the same factor $b$.
Hence, we have $\ensuremath{\mathsf{L}\xspace}_w(b f) = b \ensuremath{\mathsf{L}\xspace}_w(f)$.
Finally, if we reverse the sign of every coordinate of $f$,
then the roles of $S_e(f)$ and $I_e(f)$ are switched. Moreover,
the direction of every component of the measure flow is reversed with the same magnitude.
Hence, $\ensuremath{\mathsf{L}\xspace}_w(-f) = - \ensuremath{\mathsf{L}\xspace}_w(f)$, and the result follows.
$\Box$ \end{proof}
\begin{proofof}{Theorem~\ref{th:hyper_lap}} This follows the same argument as in~\cite{chan2018jacm}.
Suppose $\ensuremath{\mathcal{L}}\xspace$ is the normalized Laplacian
induced by the diffusion process in Lemma~\ref{lemma:define_lap}.
Let $\gamma_2 := \min_{\ensuremath{\mathbf{0}} \neq x \perp \Wh \ensuremath{\mathbf{1}}} \ensuremath{\mathcal{R}}\xspace(x)$
be attained by some minimizer $x_2$.
We use the isomorphism between the three spaces:
$\Wmh \varphi = x = \Wh f$.
The third statement of Lemma~\ref{lemma:deriv}
can be formulated in terms of the normalized space,
which states that $\frac{d \ensuremath{\mathcal{R}}\xspace(x)}{d t} \leq 0$,
where equality holds \emph{iff} $\ensuremath{\mathcal{L}}\xspace x \in \operatorname{span}(x)$.
We claim that $\frac{d \ensuremath{\mathcal{R}}\xspace(x_2)}{d t} = 0$.
Otherwise, suppose $\frac{d \ensuremath{\mathcal{R}}\xspace(x_2)}{d t} < 0$.
From Lemma~\ref{lemma:lap_proj},
we have $\frac{dx}{dt} = - \ensuremath{\mathcal{L}}\xspace x \perp \Wh \ensuremath{\mathbf{1}}$.
Hence, it follows that at this moment, the current normalized
vector is at position $x_2$, and is moving
towards the direction given by
$x' := \frac{d x}{dt}|_{x=x_2}$ such that
$x' \perp \Wh \ensuremath{\mathbf{1}}$, and $\frac{d \ensuremath{\mathcal{R}}\xspace(x)}{d t}|_{x=x_2} < 0$.
Therefore, for sufficiently small $\epsilon > 0$,
it follows that $x_2' := x_2 + \epsilon x'$ is a non-zero vector
such that $x_2' \perp \Wh \ensuremath{\mathbf{1}}$
and $\ensuremath{\mathcal{R}}\xspace(x_2') < \ensuremath{\mathcal{R}}\xspace(x_2) = \gamma_2$, contradicting the definition of $x_2$.
Hence, it follows that $\frac{d \ensuremath{\mathcal{R}}\xspace(x_2)}{d t} = 0$,
which implies that $\ensuremath{\mathcal{L}}\xspace x_2 \in \operatorname{span}(x_2)$.
Since $\gamma_2 = \ensuremath{\mathcal{R}}\xspace(x_2) = \frac{\langle x_2, \ensuremath{\mathcal{L}}\xspace x_2 \rangle}{\langle x_2, x_2 \rangle}$,
it follows that $\ensuremath{\mathcal{L}}\xspace x_2 = \gamma_2 x_2$, as required. \end{proofof}
{
}
\appendix
\section{Rayleigh Quotient Coincides with Discrepancy Ratio} \label{sec:ray_disc}
To prove Lemma~\ref{lemma:ray_disc}, we first re-interpret the diffusion rules in Definition~\ref{defn:rules} by considering the interaction between every pair of nodes. Observe that the rules sometimes say that some measure is flow from one subset of vertices to another subset. Hence, at the moment, the exact pairwise interactions are not specified. In fact, we know that in general, the pairwise interactions are not uniquely determined. Fig.~\ref{fig:diffusion_framework} captures this non-deterministic nature of the pairwise interactions.
\begin{figure}
\caption{Constraints on Pairwise Interactions}
\label{fig:diffusion_framework}
\end{figure}
\begin{proofof}{Lemma~\ref{lemma:ray_disc}} It suffices to show that $$ \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w = \sum_{e\in E}\w{e}\left\{\beta^e_0\max_{s,i\in e}\left(\f{s}-\f{i}\right)^2+\sum_{j\in e}\beta^e_{j}\left[\left(\max_{s\in e}\f{s}-\f{j}\right)^2 +\left(\f{j}-\min_{i\in e}\f{i}\right)^2\right]\right\}. $$
Recall that $\varphi = \ensuremath{\mathsf{W}}\xspace f$, and $\ensuremath{\mathsf{L}\xspace}_w = \ensuremath{\mathsf{I}_n\xspace} - \ensuremath{\mathsf{W}^{-1}}\xspace A_f$, where $A_f$ satisfies the constaints in Fig.~\ref{fig:diffusion_framework}.
Hence, it follows that \begin{align} \begin{split} \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w&=f^\ensuremath{\mathsf{T}}\xspace (\ensuremath{\mathsf{W}}\xspace - A_f) f = \sum_{uv \in {V \choose 2}} a_{uv} (f_u - f_v)^2 =\sum_{uv \in {V \choose 2}} \sum_{e \in E} \sum_{j \in [e]} a^{(e,j)}_{uv} (f_u - f_v)^2 \\ &=\sum_{uv \in {V \choose 2}} \sum_{e \in E} \left[a^{(e,0)}_{uv} (f_u - f_v)^2+\sum_{j \in e} a^{(e,j)}_{uv} (f_u - f_v)^2\right] \end{split} \nonumber \end{align}
For the first term, we have \begin{align} \begin{split} &\sum_{uv \in {V \choose 2}} \sum_{e \in E} a^{(e,0)}_{uv} (f_u - f_v)^2\\ =&\sum_{uv \in {V \choose 2}} \sum_{e \in E:\{uv,vu\}\cap S_e \times I_e\not= \emptyset} a^{(e,0)}_{uv} (f_u - f_v)^2 \\ =&\sum_{e \in E} \sum_{si \in {e \choose 2}:\{si,is\}\cap S_e \times I_e\not= \emptyset} a^{(e,0)}_{si} (f_s - f_i)^2 \\ =&\sum_{e \in E} \sum_{si \in S_e \times I_e} a^{(e,0)}_{si} (f_s - f_i)^2 \\ =&\sum_{e \in E} w_e\beta^e_0 \max_{s,i\in e}\left(f_{s}-f_{i}\right)^2 \end{split} \nonumber \end{align}
For the second term, we have \begin{align} \begin{split} &\sum_{uv \in {V \choose 2}} \sum_{e \in E} \sum_{j \in e} a^{(e,j)}_{uv} (f_u - f_v)^2\\ =&\sum_{uv \in {V \choose 2}} \sum_{e \in E} \left[ \sum_{j\in e: \{uv,vu\}\cap S_e \times j\not= \emptyset} a^{(e,j)}_{uv} (f_u - f_v)^2 +\sum_{j\in e: \{uv,vu\}\cap j \times I_e\not= \emptyset} a^{(e,j)}_{uv} (f_u - f_v)^2 \right] \\ =&\sum_{e \in E} \sum_{si \in {e \choose 2}} \left[ \sum_{j\in e: \{si,is\}\cap S_e \times j\not= \emptyset} a^{(e,j)}_{si} (f_s - f_i)^2 +\sum_{j\in e: \{si,is\}\cap j \times I_e\not= \emptyset} a^{(e,j)}_{si} (f_s - f_i)^2 \right] \\ =&\sum_{e \in E}\sum_{j\in e}\left[ \sum_{s\in S_e}a_{sj}^{(e,j)} (f_{s}-f_{j})^2 +\sum_{i\in I_e}a_{ji}^{(e,j)} (f_{j}-f_{i})^2\right] \\ =&\sum_{e \in E}\sum_{j\in e}\left[ w_e\beta^e_{j} \left(\max_{s\in e}f_{s}-f_{j}\right)^2 +w_e\beta^e_{j} \left(f_{j}-\min_{i\in e}f_{i}\right)^2 \right] \\ =&\sum_{e \in E}w_e\sum_{j\in e} \beta^e_{j}\left[\left(\max_{s\in e}f_{s}-f_{j}\right)^2 + \left(f_{j}-\min_{i\in e}f_{i}\right)^2 \right] \end{split} \nonumber \end{align}
Thus, we conclude \begin{align} \begin{split} \langle f, \ensuremath{\mathsf{L}\xspace}_w f \rangle_w &=\sum_{uv \in {V \choose 2}} \sum_{e \in E} \left[a^{(e,0)}_{uv} (f_u - f_v)^2+\sum_{j \in e} a^{(e,j)}_{uv} (f_u - f_v)^2\right] \\ &=\sum_{e\in E}\w{e}\left\{\beta^e_0\max_{s,i\in e}\left(\f{s}-\f{i}\right)^2 +\sum_{j\in e}\beta^e_{j}\left[\left(\max_{s\in e}\f{s}-\f{j}\right)^2 +(\f{j}-\min_{i\in e}\f{i})^2\right]\right\} \end{split} \nonumber \end{align} as required. \end{proofof}
\section{Improved Upper Bound for Hypergraph Cheeger Inequality} \label{sec:up_bound_cheeger}
Recall that we denote $\ensuremath{\mathsf{Q}\xspace}^0(f):=\sum_{e\in E}\w{e} \max_{s,i\in e}\left(\f{s}-\f{i}\right)^2$ for the special case in (\refeq{eq:quad}) when
$\beta_0^e=1$ for all $e\in E$, and $\gamma^0_2 := \min_{\ensuremath{\mathbf{0}} \neq f \perp_w \ensuremath{\mathbf{1}}} \frac{\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}$.
\begin{theorem}[Upper Bound for Hypergraph Cheeger Inequalities]
\label{thm:hyper-cheeger}
Given an edge-weighted hypergraph $H$, we have:
\[ \phi_H \leq
\min_{\ensuremath{\mathbf{0}} \not=f \perp_w \ensuremath{\mathbf{1}}} \sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}
=\sqrt{2\gamma^0_2},\]
where $\phi_H$ is the hypergraph conductance defined in~(\ref{eq:hyper_exp}). \end{theorem}
\begin{proposition}
\label{prop:find_hyper_cut}
Given an edge-weighted hypergraph $H = (V,E,w)$
and a non-zero vector $f \in \mathbb{R}^{V}$ such that $f \perp_w \ensuremath{\mathbf{1}}$,
there exists a set $S\subseteq {\sf supp}(f)$ such that
$$\phi(S) \leq
\sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}.$$ \end{proposition}
\begin{proof}
Without loss of generality,
suppose $-1\leq f_u \leq 1$ for all $u\in V$
since we can scale $f$ if not.
Let $t$ be a random variable that
is uniformly distributed in $(0,1]$.
Define $S_t:=\{u\in V: f_u^2\geq t\}$.
Then $S_t\subseteq {\sf supp}(f)$ by definition.
We consider the expected value of $w(S_t)$ and $w(\partial S_t)$.
\begin{align}
\begin{split}
\mathbb{E}[w(S_t)]=\sum_{u\in V}w_u\Pr[u\in S_t]
=\sum_{u\in V}w_u\Pr[f_u^2\geq t]
=\sum_{u\in V}w_uf_u^2.
\end{split}
\nonumber
\end{align}
\begin{align}
\begin{split}
&\mathbb{E}\left[w(\partial S_t)\right]
=\sum_{e\in E}w_e\Pr\left[e\in \partial S_t\right]
=\sum_{e\in E}w_e\Pr\left[
\min_{i\in e}f_i^2<t\leq\max_{s\in e}f_s^2
\right]
\\
=&\sum_{e\in E}w_e\left(\max_{s\in e}f_s^2-\min_{i\in e}f_i^2\right)
=\sum_{e\in E}w_e\max_{s,i\in e}\left(f_s^2-f_i^2\right)
=\sum_{e\in E}w_e\max_{s,i\in e}\left(\Abs{f_s}^2-\Abs{f_i}^2\right)
\\
=&\sum_{e\in E}w_e\max_{s,i\in e}
\left(\Abs{f_s}-\Abs{f_i}\right)\left(\Abs{f_s}+\Abs{f_i}\right)
\leq\sum_{e\in E}w_e\max_{s,i\in e}\left(\Abs{f_s}-\Abs{f_i}\right)
\max_{u,v\in e}\left(\Abs{f_u}+\Abs{f_v}\right)
\\
\leq&\sum_{e\in E}w_e\max_{s,i\in e}\left(f_s-f_i\right) \cdot
\max_{u,v\in e}\left(\Abs{f_u}+\Abs{f_v}\right)
\\
&\mbox{(By Cauchy--Schwarz inequality)}
\\
\leq&
\sqrt{\sum_{e\in E}w_e
\max_{s,i\in e}\left(f_s-f_i\right)^2} \cdot
\sqrt{\sum_{e\in E}w_e
\max_{u,v\in e}\left(\Abs{f_u}+\Abs{f_v}\right)^2}
\\
\leq&\sqrt{\ensuremath{\mathsf{Q}\xspace}^0(f)} \cdot \sqrt{\sum_{e\in E}w_e
\max_{u,v\in e}2\left(f_u^2+f_v^2\right)}
\leq\sqrt{\ensuremath{\mathsf{Q}\xspace}^0(f)} \cdot \sqrt{\sum_{e\in E}w_e \cdot
2\sum_{u\in e}f_u^2}
\\
=&\sqrt{\ensuremath{\mathsf{Q}\xspace}^0(f)} \cdot \sqrt{2\sum_{u\in V}w_uf_u^2}
=\sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}
\cdot \sum_{u\in V}w_uf_u^2.
\end{split}
\nonumber
\end{align}
Thus,
$\frac{\mathbb{E}\left[w(\partial S_t)\right]}{\mathbb{E}\left[w(S_t)\right]}
\leq \sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}$,
which means
$\mathbb{E}\left[w(\partial S_t)
-\sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}\cdot w(S_t)\right]
\leq 0$.
Therefore, there exists a $t$ such that
the induced $S_t$ satisfies
$\phi(S_t)=
\frac{w(\partial S_t)}{w(S_t)}
\leq \sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}$.
$\Box$ \end{proof}
For any $g \in \mathbb{R}^{V}$, define $g_+,g_- \in \mathbb{R}^{V}$ such that $g_+(v)=\max\{0,g(v)\}$ and $g_-(v)=\min\{0,g(v)\}$ for $v\in V$.
\begin{proposition}
\label{prop:find_hyper_general_cut}
Given an edge-weighted hypergraph $H = (V,E,w)$ and a non-zero vector $f \in \mathbb{R}^{V}$ such that $f \perp_w \ensuremath{\mathbf{1}}$,
let $g=f-c\ensuremath{\mathbf{1}}$ where $c$ is a constant such that
$w({\sf supp}(g_+))\leq \frac{w(V)}{2}$
and $w({\sf supp}(g_-))\leq \frac{w(V)}{2}$.
Then we have the following:
\begin{compactitem}
\item[1.] $\|g\|_w^2\geq \|f\|_w^2$,
\item[2.] $\ensuremath{\mathsf{Q}\xspace}^0(g) = \ensuremath{\mathsf{Q}\xspace}^0(f)$.
\item[3.] $\min\left\{
\frac{\ensuremath{\mathsf{Q}\xspace}^0(g_+)}{\|g_+\|_w^2},
\frac{\ensuremath{\mathsf{Q}\xspace}^0(g_-)}{\|g_-\|_w^2}
\right\}
\leq \frac{\ensuremath{\mathsf{Q}\xspace}^0(g)}{\|g\|_w^2}
\leq \frac{\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}$.
\end{compactitem} \end{proposition}
\begin{proof}
For the first statement, let
$h(c)=\|g\|_w^2=\sum_{u\in V}w_ug(u)^2=\sum_{u\in V}w_u(f_u-c)^2$.
Then we have $h'(c)=\sum_{u \in V}(-2w_uf_u+2cw_u)=\sum_{u \in V}2cw_u$,
since $f \perp_w \ensuremath{\mathbf{1}}$.
We also have $h''(c)=\sum_{u \in V}2w_u>0$.
Thus, $h(c)$ is minimized when $h'(c)=0$, i.e. $c=0$.
Then $\|g\|_w^2=h(c)\geq h(0)
=\sum_{u\in V}w_uf_u^2=\|f\|_w^2$.
For the second statement, we have
$\ensuremath{\mathsf{Q}\xspace}^0(g)
=\sum_{e\in E}w_e\max_{u,v\in e}[g(u)-g(v)]^2
=\sum_{e\in E}w_e\max_{u,v\in e}[f(u)-f(v)]^2
=\ensuremath{\mathsf{Q}\xspace}^0(f).$
For the third statement, notice that
$\|g\|_w^2=\sum_{u\in V}w_ug(u)^2
=\sum_{u\in V}w_ug_+(u)^2+\sum_{u\in V}w_ug_-(u)^2
=\|g_+\|_w^2+\|g_-\|_w^2
$.
We claim that $\ensuremath{\mathsf{Q}\xspace}^0(g)\geq \ensuremath{\mathsf{Q}\xspace}^0(g_+)+\ensuremath{\mathsf{Q}\xspace}^0(g_-)$ since
for all $e\in E$,
$\max_{u,v\in e}[g(u)-g(v)]^2
\geq \max_{u,v\in e}[g_+(u)-g_+(v)]^2
+\max_{u,v\in e}[g_-(u)-g_-(v)]^2$
\noindent by considering the following two cases:
\begin{compactitem}
\item[1.] $g(u)$ and $g(u)$ have the same sign.
Then it must be either $\forall u\in e, g(u)\geq 0$
or $\forall u\in e, g(u)\leq 0$.
Thus, one of $\max_{u,v\in e}[g_+(u)-g_+(v)]^2$
and $\max_{u,v\in e}[g_-(u)-g_-(v)]^2$ equals zero.
Then $\max_{u,v\in e}[g(u)-g(v)]^2
= \max_{u,v\in e}[g_+(u)-g_+(v)]^2
+\max_{u,v\in e}[g_-(u)-g_-(v)]^2$.
\item[2.] $g(u)$ and $g(u)$ have the opposite signs.
With out loss of generality, we assume $g(u)>g(v)$.
Then we have $g_+(u)>0$, $g_-(v)<0$
and $g_+(v)=g_-(u)=0$.
Thus,
\begin{align}
\begin{split}
\max_{u,v\in e}[g(u)-g(v)]^2
&=\max_{u,v\in e}[g(u)^2-2g(u)g(v)+g(v)^2]
\\
&\geq \max_{u\in e:g(u)\geq 0}g(u)^2
+\max_{v\in e:g(v)\leq 0}g(v)^2
\\
&=\max_{u\in e}g_+(u)^2
+\max_{v\in e}g_-(v)^2
\\
&=\max_{u,v\in e}[g_+(u)-g_+(v)]^2
+\max_{u,v\in e}[g_-(u)-g_-(v)]^2.
\end{split}
\nonumber
\end{align}
\end{compactitem} Then by the first and second statements, we have
$$\frac{\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}
\geq \frac{\ensuremath{\mathsf{Q}\xspace}^0(g)}{\|g\|_w^2} \geq \frac{\ensuremath{\mathsf{Q}\xspace}^0(g_+)+\ensuremath{\mathsf{Q}\xspace}^0(g_-)}{
\|g_+\|_w^2+\|g_-\|_w^2} \geq \min\left\{
\frac{\ensuremath{\mathsf{Q}\xspace}^0(g_+)}{\|g_+\|_w^2},
\frac{\ensuremath{\mathsf{Q}\xspace}^0(g_-)}{\|g_-\|_w^2} \right\}. $$
$\Box$ \end{proof}
Now we can prove Theorem~\ref{thm:hyper-cheeger}.
\begin{proofof}{Theorem~\ref{thm:hyper-cheeger}}~
By Proposition~\ref{prop:find_hyper_cut},
there exist $S_{t+}\subseteq {\sf supp}(g_+)$
and $S_{t-}\subseteq {\sf supp}(g_-)$
such that for any non-zero vector
$f\in \mathbb{R}^V$ satisfying $f \perp_w \ensuremath{\mathbf{1}}$:
\begin{align}
\begin{split}
\min\left\{\phi(S_{t+}),\phi(S_{t-})\right\}
\leq\min\left\{
\sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(g_+)}{\|g_+\|_w^2}},
\sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(g_-)}{\|g_-\|_w^2}}
\right\}
\leq \sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}
\end{split}
\nonumber
\end{align}
where the last inequality follows from the third statement of Proposition~\ref{prop:find_hyper_general_cut}.
Moreover, $w(S_{t+})\leq\frac{w(V)}{2}$ and $w(S_{t-})\leq\frac{w(V)}{2}$
by Proposition~\ref{prop:find_hyper_general_cut}.
Thus, for any non-zero vector
$f\in \mathbb{R}^V$ satisfying $f \perp_w \ensuremath{\mathbf{1}}$:
$$
\phi_H
\leq \min\left\{\phi(S_{t+}),\phi(S_{t-})\right\}
\leq \sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}.
$$
Then we have $\phi_H \leq
\min_{\ensuremath{\mathbf{0}}\not=f \perp_w \ensuremath{\mathbf{1}}}
\sqrt{\frac{2\ensuremath{\mathsf{Q}\xspace}^0(f)}{\|f\|_w^2}}=\sqrt{2\gamma^0_2}$. \end{proofof}
\end{document} | arXiv |
\begin{document}
\title{Algebras simple with respect to a Taft algebra action}
\author{A.\,S.~Gordienko} \address{Vrije Universiteit Brussel, Belgium} \email{[email protected]}
\keywords{Associative algebra, polynomial identity, skew-derivation, Taft algebra, $H$-module algebra, codimension.}
\begin{abstract} Algebras simple with respect to an action of a Taft algebra $H_{m^2}(\zeta)$ deliver an interesting example of $H$-module algebras that are $H$-simple but not necessarily semisimple. We describe finite dimensional $H_{m^2}(\zeta)$-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial $H_{m^2}(\zeta)$-identities. In particular, we show that the Hopf PI-exponent of an $H_{m^2}(\zeta)$-simple algebra $A$ over an algebraically closed field of characteristic $0$ equals $\dim A$. The groups of automorphisms preserving the structure of an $H_{m^2}(\zeta)$-module algebra are studied as well. \end{abstract}
\subjclass[2010]{Primary 16W22; Secondary 16R10, 16R50, 16T05, 16W25.}
\thanks{Supported by Fonds Wetenschappelijk Onderzoek~--- Vlaanderen Pegasus Marie Curie post doctoral fellowship (Belgium) and RFBR grant 13-01-00234a (Russia).}
\maketitle
The notion of an $H$-(co)module algebra is a natural generalization of the notion of a graded algebra, an algebra with an action of a group by automorphisms, and an algebra with an action of a Lie algebra by derivations.
In particular, if $H_{m^2}(\zeta)$ is the $m^2$-dimensional Taft algebra, an $H_{m^2}(\zeta)$-module algebra is an algebra endowed both with an action of the cyclic group of order $m$ and with a skew-derivation satisfying certain conditions. The Taft algebra $H_4(-1)$ is called Sweedler's algebra.
The theory of gradings on matrix algebras and simple Lie algebras is a well developed area~\cite{ BahtKochMont, BahturinZaicevSeghalSimpleGraded}. Quaternion $H_4(-1)$-extensions and related crossed products were considered in~\cite{DoiTakeuchi}. In~\cite{ASGordienko11}, the author classified all finite dimensional $H_4(-1)$-simple algebras. Here we classify finite dimensional $H_{m^2}(\zeta)$-simple algebras over an algebraically closed field (Sections~\ref{SectionTaftSimpleSemisimple}--\ref{SectionTaftSimpleNonSemisimple}).
Amitsur's conjecture on asymptotic behaviour of codimensions of ordinary polynomial identities was proved by A.~Giambruno and M.\,V.~Zaicev~\cite[Theorem~6.5.2]{ZaiGia} in 1999.
Suppose an algebra is endowed with a grading, an action of a group $G$ by automorphisms and anti-automorphisms, an action of a Lie algebra by derivations or a structure of an $H$-module algebra for some Hopf algebra $H$. Then it is natural to consider, respectively, graded, $G$-, differential or $H$-identities~\cite{BahtGiaZai, BahtZaiGradedExp, BahturinLinchenko, BereleHopf, Kharchenko}.
The analog of Amitsur's conjecture for polynomial $H$-identities was proved under wide conditions by the author in~\cite{ASGordienko8, ASGordienko9}. However, in those results the $H$-invariance of the Jacobson radical was required. Until now the algebras simple with respect to an action of $H_4(-1)$ were the only example where the analog of Amitsur's conjecture was proved for an $H$-simple non-semisimple algebra~\cite{ASGordienko11}.
In this article we prove the analog of Amitsur's conjecture for all finite dimensional $H_{m^2}(\zeta)$-simple algebras not necessarily semisimple (Section~\ref{SectionTaftSimpleAmitsur}) assuming that the base field is algebraically closed and of characteristic $0$.
\section{Introduction}
An algebra $A$ over a field $F$ is an \textit{$H$-module algebra} for some Hopf algebra $H$ if $A$ is endowed with a homomorphism $H \to \End_F(A)$ such that $h(ab)=(h_{(1)}a)(h_{(2)}b)$ for all $h \in H$, $a,b \in A$. Here we use Sweedler's notation $\Delta h = h_{(1)} \otimes h_{(2)}$ where $\Delta$ is the comultiplication in $H$. We refer the reader to~\cite{Danara, Montgomery, Sweedler}
for an account
of Hopf algebras and algebras with Hopf algebra actions.
Let $A$ be an $H$-module algebra for some Hopf algebra $H$ over a field $F$. We say that $A$ is \textit{$H$-simple} if $A^2\ne 0$ and $A$ has no non-trivial two-sided $H$-invariant ideals.
Let $m \geqslant 2$ be an integer and let $\zeta$ be a primitive $m$th root of unity in a field $F$. (Such root exists in $F$ only if $\ch F \nmid m$.) Consider the algebra $H_{m^2}(\zeta)$ with unity generated by elements $c$ and $v$ satisfying the relations $c^m=1$, $v^m=0$, $vc=\zeta cv$. Note that $(c^i v^k)_{0 \leqslant i, k \leqslant m-1}$ is a basis of $H_{m^2}(\zeta)$. We introduce on $H_{m^2}(\zeta)$ a structure of a coalgebra by
$\Delta(c)=c\otimes c$, $\Delta(v) = c\otimes v + v\otimes 1$, $\varepsilon(c)=1$, $\varepsilon(v)=0$. Then $H_{m^2}(\zeta)$ is a Hopf algebra with the antipode $S$ where $S(c)=c^{-1}$ and $S(v)=-c^{-1}v$. The algebra $H_{m^2}(\zeta)$ is called a \textit{Taft algebra}.
\begin{remark} Note that if $A$ is an $H_{m^2}(\zeta)$-module algebra, then the group $\langle c \rangle \cong \mathbb Z_m$ is acting on $A$ by automorphisms. Every algebra $A$ with a $\mathbb Z_m$-action by automorphisms is a $\mathbb Z_m$-graded algebra: $$A^{(i)} = \lbrace a \in A \mid ca = \zeta^i a\rbrace,$$ $A^{(i)}A^{(k)}\subseteq A^{(i+k)}$. Conversely, if $A = \bigoplus_{i=0}^{m-1} A^{(i)}$ is a $\mathbb Z_m$-graded algebra, then $\mathbb Z_m$ is acting on $A$ by automorphisms: $c a^{(i)} = \zeta^i a^{(i)}$ for all $a^{(i)} \in A^{(i)}$. Moreover, the notions of $\mathbb Z_m$-simple and simple $\mathbb Z_m$-graded algebras are equivalent. \end{remark}
\begin{remark} \cite[Theorems~5 and~6]{BahturinZaicevSeghalGroupGrAssoc} imply that every $\mathbb Z_m$-grading on $M_n(F)$, where $F$ is an algebraically closed field, is, up to a conjugation, \textit{elementary}, i.e. there exist $g_1, g_2, \ldots, g_n \in \mathbb Z_m$ such that each matrix unit $e_{ij}$ belongs to $A^{(g_i^{-1} g_j)}$. Rearranging rows and columns, we may assume that every $\mathbb Z_m$-action on $M_n(F)$ is defined by $c a = Q^{-1} a Q$ for some matrix $$Q=\diag\{\underbrace{1, \ldots, 1}_{k_0}, \underbrace{\zeta, \ldots, \zeta}_{k_1}, \ldots, \underbrace{\zeta^{m-1}, \ldots, \zeta^{m-1}}_{k_{m-1}}\}.$$ \end{remark}
\section{Semisimple $H_{m^2}(\zeta)$-simple algebras}\label{SectionTaftSimpleSemisimple}
In this section we treat the case when an $H_{m^2}(\zeta)$-simple algebra $A$ is semisimple.
\begin{theorem}\label{TheoremTaftSimpleSemisimple} Let $A$ be a semisimple $H_{m^2}(\zeta)$-simple algebra over an algebraically closed field~$F$. Then
$$A \cong \underbrace{M_k(F) \oplus M_k(F) \oplus \dots \oplus M_k(F)}_t\qquad \text{(direct sum of ideals)}$$ for some $k,t\in\mathbb N$, $t \mid m$, and there exist $P \in M_k(F)$ and $Q \in \GL_k(F)$ where $Q^{\frac{m}{t}} = E_k$, $E_k$ is the identity matrix $k\times k$, $Q P Q^{-1}=\zeta^{-t} P$, $P^m = \alpha E_k$ for some $\alpha \in F$, such that \begin{equation}\label{EqSSTaftSimple1} c\, (a_1, a_2, \ldots, a_t) = (Q a_t Q^{-1}, a_1, \ldots, a_{t-1}), \end{equation}\begin{equation}\label{EqSSTaftSimple2} v\,(a_1, a_2, \ldots, a_t)=(Pa_1 - (Q a_t Q^{-1}) P, \zeta (P a_2 - a_1 P), \ldots, \zeta^{t-1}(Pa_t - a_{t-1} P))\end{equation} for all $a_1, a_2, \ldots, a_t \in M_k(F)$. \end{theorem}
\begin{remark} Diagonalizing $Q$, we may assume that $$Q = \diag\lbrace\underbrace{1,\ldots,1}_{k_1}, \underbrace{\zeta^t,\ldots,\zeta^t}_{k_2}, \dots, \underbrace{\zeta^{t\left(\frac{m}{t}-1\right)},\ldots,\zeta^{t\left(\frac{m}{t}-1\right)}}_{k_{\frac{m}{t}}}\rbrace$$ for some $k_1,\ldots,k_{\frac{m}{t}}\in \mathbb Z_+$, $k_1+\ldots+k_{\frac{m}{t}}=k$. Now $QPQ^{-1} = \zeta^{-1} P$ imply that $P=(P_{ij})$ is a block matrix where $P_{ij}$ is an matrix $k_{i-1} \times k_{j-1}$ and $P_{ij} = 0$ for all $j \ne i+1$ and $(i,j)\ne \left(\frac{m}{t}, 1\right)$. \end{remark}
We begin with three auxiliary lemmas. In the first two, we prove all the assertions of Theorem~\ref{TheoremTaftSimpleSemisimple} except $P^m=\alpha E_k$. In Lemma~\ref{LemmaTaftSimpleMatrix} we treat the case when $A$ isomorphic to a full matrix algebra.
\begin{lemma}\label{LemmaTaftSimpleMatrix} Let $A$ be an $H_{m^2}(\zeta)$-module algebra over an algebraically closed field $F$, isomorphic as an algebra to $M_k(F)$ for some $k\in \mathbb N$. Then there exist matrices $P\in M_k(F)$, $Q\in \GL_k(F)$, $Q^m=E_k$ such that $Q P Q^{-1}=\zeta^{-1} P$ and $A$ is isomorphic as an $H_{m^2}(\zeta)$-module algebra to $M_k(F)$ with the following $H_{m^2}(\zeta)$-action: $ca=QaQ^{-1}$ and $va=Pa-(QaQ^{-1})P$ for all $a\in M_k(F)$. \end{lemma} \begin{proof} All automorphisms of full matrix algebras are inner. Hence $ca=QaQ^{-1}$ for some $Q\in \GL_k(F)$. Since $c^m = 1$, the matrix $Q^m$ is scalar. Multiplying $Q$ by the $m$th root of the corresponding scalar, we may assume that $Q^m = E_k$.
Recall that $v$ is acting on $A$ by a skew-derivation. We claim\footnote{This result is a ``folklore'' one. I am grateful to V.\,K.~Kharchenko who informed me of a simple proof of it.} that this skew-derivation is \textit{inner}, i.e. there exists a matrix $P \in A$ such that $va = Pa-(ca)P$ for all $a\in A$. Indeed, $$Q^{-1}(v(ab)) = Q^{-1}((ca)(vb)+(va)b) = Q^{-1}((QaQ^{-1})(vb)+(va)b)=a(Q^{-1}(vb))+(Q^{-1}(va))b$$ for all $a,b \in A$. Hence $Q^{-1}(v(\cdot))$ is a derivation and $Q^{-1}(va)=P_0a-aP_0$ for all $a\in A$ for some $P_0 \in A$.
Thus $$va = QP_0a- Q a P_0= QP_0 a - Q aQ^{-1}QP_0= Pa-(QaQ^{-1})P
\text{ for all }a\in A$$ where $P=QP_0$, i.e. $v$ acts as an inner skew-derivation.
Note that $vc=\zeta cv$ implies $c^{-1}v=\zeta v c^{-1}$, $$Q^{-1}(Pa-(QaQ^{-1})P)Q = \zeta P(Q^{-1}aQ)-aP,$$ $$Q^{-1}PaQ-aQ^{-1}PQ = \zeta PQ^{-1}aQ-\zeta aP,$$ $$Q^{-1}Pa-aQ^{-1}P = \zeta PQ^{-1}a-\zeta aPQ^{-1},$$ $$Q^{-1}Pa-\zeta PQ^{-1}a = aQ^{-1}P-\zeta aPQ^{-1},$$ $$(Q^{-1}P-\zeta PQ^{-1})a = a(Q^{-1}P-\zeta PQ^{-1}) \text{ for all } a\in A.$$ Hence $Q^{-1}P-\zeta PQ^{-1} = \alpha E_k$ for some $\alpha \in F$. Now we replace $P$ with $(P-\frac{\alpha}{1-\zeta} Q)$. Then $v$ is the same but $Q^{-1}P-\zeta PQ^{-1} = 0$ and $Q PQ^{-1} = \zeta^{-1} P$. \end{proof}
Here we treat the general case.
\begin{lemma}\label{LemmaTaftSimpleSemisimpleFirst} Let $A$ be a semisimple $H_{m^2}(\zeta)$-simple algebra over an algebraically closed field $F$. Then
$A \cong \underbrace{M_k(F) \oplus M_k(F) \oplus \dots \oplus M_k(F)}_t$ (direct sum of ideals) for some $k, t \in\mathbb N$, $t \mid m$, and there exist $P \in M_k(F)$ and $Q \in \GL_k(F)$, $Q^{\frac{m}{t}} = E_k$, $Q P Q^{-1}=\zeta^{-t} P$, such that~(\ref{EqSSTaftSimple1}) and~(\ref{EqSSTaftSimple2}) hold for all $a_1, a_2, \ldots, a_t \in M_k(F)$. \end{lemma} \begin{proof} If $A$ is semisimple, then $A$ is the direct sum of $\mathbb Z_m$-simple subalgebras. Let $B$ be one of such subalgebras. Then $vb= v(1_B b)=(c1_B)(vb)+(v1_B)b \in B$ for all $b\in B$. Hence $B$ is an $H_{m^2}(\zeta)$-submodule, $A=B$, and $A$ is a $\mathbb Z_m$-simple algebra. Therefore, $A \cong \underbrace{M_k(F) \oplus M_k(F) \oplus \dots \oplus M_k(F)}_t$ (direct sum of ideals) for some $k,t \in\mathbb N$, $t \mid m$, and $c$ maps the $i$th component to the $(i+1)$th.
In the case $t=1$, the assertion is proved in Lemma~\ref{LemmaTaftSimpleMatrix}. Consider the case $t\geqslant 2$. Note that $c^t$ maps each component onto itself. Since every automorphism of the matrix algebra is inner, there exist $Q$ such that $c^t(a, 0, \ldots, 0) = (QaQ^{-1}, 0, \ldots, 0)$ for any $a \in M_k(F)$. Now $c^m = \id_A$ implies that $Q^{\frac{m}{t}}$ is a scalar matrix and we may assume that $Q^{\frac{m}{t}} = E_k$ since the field $F$ is algebraically closed and we can multiply $Q$ by the $m$th root of the corresponding scalar. Therefore, we may assume that~(\ref{EqSSTaftSimple1}) holds.
Let $\pi_i \colon A \to M_k(F)$ be the natural projections on the $i$th component. Consider
$\rho_{ij} \in \End_F(M_k(F))$, $1 \leqslant i,j\leqslant t$, defined by $\rho_{ij}(a):= \pi_i(v\,(\underbrace{0, \ldots, 0}_{j-1}, a,0, \ldots, 0))$ for $a \in M_k(F)$. Then \begin{equation*}\begin{split}\rho_{ij}(ab)=\pi_i(v\,(0,\ldots, 0, ab,0, \ldots, 0))=\\ \pi_i(v((0,\ldots, 0, a,0, \ldots, 0)(0,\ldots, 0, b,0, \ldots, 0)))=\\ \pi_i((c(0,\ldots, 0, a,0, \ldots, 0))v(0,\ldots, 0, b,0, \ldots, 0))+ \\ \pi_i((v(0,\ldots, 0, a,0, \ldots, 0))(0,\ldots, 0, b,0, \ldots, 0))= \\ \delta_{ij}\,\rho_{ii}(a)b+\delta_{j, i-1}\,a\rho_{i,i-1}(b)+\delta_{i1}\delta_{jt}\,QaQ^{-1}\rho_{1t}(b) \end{split}\end{equation*} for all $a,b \in M_k(F)$ where $\delta_{ij}$ is the Kronecker delta.
Let $\rho_{ii}(E_k)=P_i$, $\rho_{i, i-1}(E_k)=Q_i$, $\rho_{1t}(E_k)=Q_1$ where $P_i, Q_i \in M_k(F)$. Then \begin{equation*}\begin{split}v(a_1, \ldots, a_m)=(\pi_1(v\,(a_1,\ldots, a_t)), \ldots, \pi_1(v\,(a_1,\ldots, a_t)))= \\ (P_1 a_1 + (Qa_tQ^{-1}) Q_1, P_2 a_2 + a_1 Q_2, \ldots, P_t a_t + a_{t-1} Q_t). \end{split}\end{equation*} Now we notice that \begin{equation*}\begin{split}0=v((\underbrace{0, \ldots, 0}_{i-1}, E_k,0, \ldots, 0)(\underbrace{0, \ldots, 0}_i, E_k,0, \ldots, 0))=\\ (c(\underbrace{0, \ldots, 0}_{i-1}, E_k,0, \ldots, 0))(v(\underbrace{0, \ldots, 0}_i, E_k,0, \ldots, 0))+\\ (v(\underbrace{0, \ldots, 0}_{i-1}, E_k,0, \ldots, 0))(\underbrace{0, \ldots, 0}_i, E_k,0, \ldots, 0)=\\ (\underbrace{0, \ldots, 0}_i, P_{i+1}+Q_{i+1}, 0, \ldots, 0).\end{split}\end{equation*} Thus $Q_{i+1}=-P_{i+1}$.
Note that \begin{equation*}\begin{split}(-\zeta QP_tQ^{-1}, 0, \ldots, 0, \zeta P_{t-1})=\zeta cv(0, \ldots, 0, E_k, 0)=vc(0, \ldots, 0, E_k, 0)=\\ v(0, \ldots, 0, E_k)=(-P_1,0, \ldots, 0, P_t),\end{split}\end{equation*} \begin{equation*}\begin{split}(\zeta QP_tQ^{-1}, -\zeta P_1,0, \ldots, 0)=\zeta cv(0, \ldots, 0, E_k)=vc(0, \ldots, 0, E_k)=\\ v(E_k,0, \ldots, 0)=(P_1, -P_2,0, \ldots, 0),\end{split}\end{equation*} and $P_1 = \zeta Q P_t Q^{-1}$, $P_2 = \zeta P_1$, $P_t = \zeta P_{t-1}$.
Moreover, if $t > 2$, \begin{equation*}\begin{split}(\underbrace{0, \ldots, 0}_i, \zeta P_i, -\zeta P_{i+1},0, \ldots, 0)=\zeta cv(\underbrace{0, \ldots, 0}_{i-1}, E_k,0, \ldots, 0)=\\ vc(\underbrace{0, \ldots, 0}_{i-1}, E_k,0, \ldots, 0)=v(\underbrace{0, \ldots, 0}_i, E_k,0, \ldots, 0)=(\underbrace{0, \ldots, 0}_i, P_{i+1}, -P_{i+2},0, \ldots, 0)\end{split}\end{equation*} for $1\leqslant i \leqslant t-2$. Therefore, $P_{i+1}=\zeta P_i$ for all $1\leqslant i \leqslant t-1$. Let $P:= P_1$. Then $P_i = \zeta^{i-1} P$, $\zeta^t QP Q^{-1} = P$, (\ref{EqSSTaftSimple2}) holds, and the lemma is proved. \end{proof}
Recall the definition of \textit{quantum binomial coefficients}: $$\binom{n}{k}_\zeta := \frac{n!_\zeta}{(n-k)!_\zeta\ k!_\zeta}$$ where $n!_\zeta := n_\zeta (n-1)_\zeta \cdot \dots \cdot 1_\zeta$ and $n_\zeta := 1 + \zeta + \zeta^2 + \dots + \zeta^{n-1}$.
\begin{lemma}\label{LemmaTaftSimpleSemisimpleFormula} Let $v$ be the operator defined on $M_k(F)^t$ by~(\ref{EqSSTaftSimple2}) where $QPQ^{-1}=\zeta^{-t} P$. Then $$v^\ell (a_1, a_2, \ldots, a_t) =(b_1, b_2, \ldots, b_t)$$ where \begin{equation}\label{EqSSTaftSimple3} b_k = \zeta^{\ell(k-1)} \sum_{j=0}^\ell (-1)^j \zeta^{-\frac{j(j-1)}{2}} \binom{\ell}{j}_{\zeta^{-1}} P^{\ell-j} a_{k-j} P^j \end{equation} and $a_{-j} := Q a_{t-j} Q^{-1}$, $j \geqslant 0$, $a_i \in M_k(F)$, $1\leqslant \ell \leqslant m$. \end{lemma} \begin{proof} We prove the assertion by induction on $\ell$. The base $\ell=1$ is evident. Suppose~(\ref{EqSSTaftSimple3}) holds for $\ell$. Then $$v^{\ell+1} (a_1, a_2, \ldots, a_t) =(\tilde b_1, \tilde b_2, \ldots, \tilde b_t)$$ where $\tilde b_k = \zeta^{k-1}(P b_k - b_{k-1} P)$, $1\leqslant k \leqslant t$, and $b_0 := Q b_t Q^{-1}$. Then \begin{equation*}\begin{split} \tilde b_k = \zeta^{k-1}\left( \zeta^{\ell(k-1)} \sum_{j=0}^\ell (-1)^j \zeta^{-\frac{j(j-1)}{2}} \binom{\ell}{j}_{\zeta^{-1}} P^{\ell-j+1} a_{k-j} P^j -\right. \\ \left. \zeta^{\ell(k-2)} \sum_{j=0}^\ell (-1)^j \zeta^{-\frac{j(j-1)}{2}} \binom{\ell}{j}_{\zeta^{-1}} P^{\ell-j} a_{k-j-1} P^{j+1} \right)= \\ \zeta^{k-1}\left( \zeta^{\ell(k-1)} \sum_{j=0}^\ell (-1)^j \zeta^{-\frac{j(j-1)}{2}} \binom{\ell}{j}_{\zeta^{-1}} P^{\ell-j+1} a_{k-j} P^j -\right. \\ \left. \zeta^{\ell(k-2)} \sum_{j=1}^{\ell+1} (-1)^{j-1} \zeta^{-\frac{(j-2)(j-1)}{2}} \binom{\ell}{j-1}_{\zeta^{-1}} P^{\ell-j+1} a_{k-j} P^j \right)= \\ \zeta^{(\ell+1)(k-1)}\left( \sum_{j=0}^\ell (-1)^j \zeta^{-\frac{j(j-1)}{2}} \binom{\ell}{j}_{\zeta^{-1}} P^{\ell-j+1} a_{k-j} P^j +\right. \\ \left. \sum_{j=1}^{\ell+1} (-1)^j \zeta^{-\frac{j(j-1)}{2}} \zeta^{j-\ell-1} \binom{\ell}{j-1}_{\zeta^{-1}} P^{\ell-j+1} a_{k-j} P^j \right)=\\ \zeta^{(\ell+1)(k-1)} \sum_{j=0}^{\ell+1} (-1)^j \zeta^{-\frac{j(j-1)}{2}} \binom{\ell+1}{j}_{\zeta^{-1}} P^{\ell-j+1} a_{k-j} P^j.\end{split}\end{equation*} Therefore, (\ref{EqSSTaftSimple3}) holds for every $1\leqslant \ell \leqslant m$. \end{proof} \begin{proof}[Proof of Theorem~\ref{TheoremTaftSimpleSemisimple}] Recall that $v^m=0$ and $\binom{m}{j}_{\zeta^{-1}}=0$ for $1\leqslant j \leqslant m-1$. Thus Lemmas~\ref{LemmaTaftSimpleSemisimpleFirst} and \ref{LemmaTaftSimpleSemisimpleFormula} imply \begin{equation*}\begin{split}v^m(a_1, \ldots, a_t)=(P^m a_1 - a_{1-m}P^m, P^m a_2 - a_{2-m}P^m, \ldots, P^m a_t - a_{t-m}P^m)=\\ ([P^m, a_1], [P^m, a_2], \ldots, [P^m, a_t]) = 0\end{split}\end{equation*} for all $a_i \in M_k(F)$ since $Q^{\frac{m}{t}}=E_k$. Therefore, $P^m = \alpha E_k$ for some $\alpha\in F$, and we get the theorem. \end{proof}
\begin{remark} Conversely, for every $k, t \in\mathbb N$, $t \mid m$, and matrices $P \in M_k(F)$ and $Q \in \GL_k(F)$ such that $Q^{\frac{m}{t}} = E_k$, $Q P Q^{-1}=\zeta^{-t} P$, $P^m = \alpha E_k$ for some $\alpha \in F$, we can define the structure of an $H_{m^2}(\zeta)$-simple algebra on $A \cong \underbrace{M_k(F) \oplus M_k(F) \oplus \dots \oplus M_k(F)}_t$ (direct sum of ideals) by~(\ref{EqSSTaftSimple1}) and~(\ref{EqSSTaftSimple2}), and this algebra $A$ is even $\mathbb Z_m$-simple. \end{remark}
\begin{theorem}\label{TheoremTaftSimpleSSIso} Let $A \cong \underbrace{M_k(F) \oplus M_k(F) \oplus \dots \oplus M_k(F)}_t$ (direct sum of ideals) be a semisimple $H_{m^2}(\zeta)$-simple algebra over a field $F$ defined by matrices $P_1 \in M_k(F)$ and $Q_1 \in \GL_k(F)$
by~(\ref{EqSSTaftSimple1}) and~(\ref{EqSSTaftSimple2}), and let $A_2$ be another such algebra defined by matrices $P_2 \in M_k(F)$ and $Q_2 \in \GL_k(F)$. Then $A_1 \cong A_2$ as algebras and $H_{m^2}(\zeta)$-modules if and only if $P_2 =\zeta^r\, T P_1 T^{-1}$ and $Q_2 = \beta T Q_1 T^{-1}$ for some $r\in\mathbb Z$, $\beta \in F$, and $T \in \GL_k(F)$. \end{theorem} \begin{proof} Note that in each of $A_1$ and $A_2$ there exist exactly $t$ simple ideals isomorphic to $M_k(F)$. Moreover, each isomorphism of $M_k(F)$ is inner. Therefore, if $\varphi \colon A_1 \to A_2$ is an isomorphism of algebras and $H_{m^2}(\zeta)$-modules, then there exist matrices $T_i \in \GL_k(F)$ and a number $0 \leqslant r < t$ such that \begin{equation*}\begin{split}\varphi(a_1, \ldots, a_m)=(T_{r+1} a_{r+1} T_{r+1}^{-1}, T_{r+2} a_{r+2} T_{r+2}^{-1}, \ldots, T_t a_t T_t^{-1},\\ T_1 a_1 T_1^{-1}, T_2 a_2 T_2^{-1}, \ldots, T_r a_r T_r^{-1})\end{split}\end{equation*} for all $a_i \in M_k(F)$. (Here we use the fact that $\varphi$ must commute with $c$.) Using $c\varphi = \varphi c$ once again, we get \begin{equation*}\begin{split}c\varphi(a_1, \ldots, a_m)=(Q_2 T_r a_r T_r^{-1} Q_2^{-1}, T_{r+1} a_{r+1} T_{r+1}^{-1}, T_{r+2} a_{r+2} T_{r+2}^{-1}, \ldots, T_t a_t T_t^{-1},\\ T_1 a_1 T_1^{-1}, T_2 a_2 T_2^{-1}, \ldots, T_{r-1} a_{r-1} T_{r-1}^{-1})=\\ \varphi c(a_1, \ldots, a_m)=(T_{r+1} a_r T_{r+1}^{-1}, T_{r+2} a_{r+1} T_{r+2}^{-1}, \ldots, T_t a_{t-1} T_t^{-1},\\ T_1 Q_1 a_t Q_1^{-1} T_1^{-1}, T_2 a_1 T_2^{-1}, \ldots, T_r a_{r-1} T_r^{-1})\end{split}\end{equation*} for all $a_i \in M_k(F)$. Therefore, $T_i$ is proportional to $T_{i+1}$ for $1 \leqslant i \leqslant r-1$, $r+1\leqslant i \leqslant t-1$. In addition, $Q_2 T_r$ is proportional to $T_{r+1}$, and $T_t$ is proportional to $T_1 Q_1$. Multiplying $T_i$ by scalars, we may assume that $T_1 = \ldots = T_r$, $T_{r+1}=\ldots=T_t = T_1 Q_1$. Let $T := T_{r+1}$. Then \begin{equation*}\begin{split}\varphi(a_1, \ldots, a_m)=(T a_{r+1} T^{-1}, T a_{r+2} T^{-1}, \ldots, T a_t T^{-1},\\ TQ_1^{-1} a_1 Q_1 T^{-1}, T Q_1^{-1} a_2 Q_1 T^{-1}, \ldots, TQ_1^{-1} a_r Q_1T^{-1} )\end{split}\end{equation*} and $Q_2 = \beta T Q_1 T^{-1}$ for some $\beta \in F$.
Using $v\varphi = \varphi v$, we get \begin{equation*}\begin{split}(P_2 T a_{r+1} T^{-1} - T a_r T^{-1} P_2, \zeta (P_2 T a_{r+2} T^{-1} - T a_{r+1} T^{-1} P_2), \ldots,\\ \zeta^{t-r-1}(P_2 T a_t T^{-1} - T a_{t-1} T^{-1} P_2),\\ \zeta^{t-r}(P_2 T Q_1^{-1} a_1 Q_1 T^{-1}- T a_t T^{-1} P_2), \zeta^{t-r+1}(P_2 T Q_1^{-1} a_2 Q_1 T^{-1} - T Q_1^{-1} a_1 Q_1 T^{-1} P_2), \ldots,\\ \zeta^{t-1}(P_2 T Q_1^{-1} a_r Q_1 T^{-1}- T Q_1^{-1} a_{r-1} Q_1 T^{-1}P_2 ))=\\ (P_2 T a_{r+1} T^{-1} - Q_2 T Q_1^{-1} a_r Q_1 T^{-1} Q_2^{-1} P_2, \zeta (P_2 T a_{r+2} T^{-1} - T a_{r+1} T^{-1} P_2), \ldots,\\ \zeta^{t-r-1}(P_2 T a_t T^{-1} - T a_{t-1} T^{-1} P_2),\\ \zeta^{t-r}(P_2 T Q_1^{-1} a_1 Q_1 T^{-1}- T a_t T^{-1} P_2), \zeta^{t-r+1}(P_2 T Q_1^{-1} a_2 Q_1 T^{-1} - T Q_1^{-1} a_1 Q_1 T^{-1} P_2), \ldots,\\ \zeta^{t-1}(P_2 T Q_1^{-1} a_r Q_1 T^{-1}- T Q_1^{-1} a_{r-1} Q_1 T^{-1}P_2 ))=\\ v(T a_{r+1} T^{-1}, T a_{r+2} T^{-1}, \ldots, T a_t T^{-1},\\ TQ_1^{-1} a_1 Q_1 T^{-1}, T Q_1^{-1} a_2 Q_1 T^{-1}, \ldots, TQ_1^{-1} a_r Q_1T^{-1})=v\varphi(a_1, \ldots, a_m)=\\ \varphi v(a_1, \ldots, a_m) =\varphi (P_1 a_1 - (Q_1 a_t Q_1^{-1}) P_1, \zeta (P_1 a_2 - a_1 P_1), \ldots, \zeta^{t-1}(P_1 a_t - a_{t-1} P_1))=\\ (\zeta^r T (P_1 a_{r+1} - a_r P_1) T^{-1}, \zeta^{r+1} T (P_1 a_{r+2} - a_{r+1} P_1) T^{-1}, \ldots, \zeta^{t-1} T (P_1 a_t - a_{t-1} P_1) T^{-1}, \\ TQ_1^{-1}(P_1 a_1 - (Q_1 a_t Q_1^{-1}) P_1)Q_1 T^{-1}, \zeta TQ_1^{-1}(P_1 a_2 - a_1 P_1)Q_1 T^{-1}, \ldots,\\ \zeta^{r-1}T Q_1^{-1}(P_1 a_r - a_{r-1} P_1)Q_1 T^{-1})=\\ (\zeta^r (T P_1 a_{r+1} T^{-1} - T a_r P_1 T^{-1}), \zeta^{r+1} (T P_1 a_{r+2} T^{-1} - T a_{r+1} P_1 T^{-1}), \ldots,\\ \zeta^{t-1} (T P_1 a_t T^{-1} - T a_{t-1} P_1 T^{-1}), \\ TQ_1^{-1} P_1 a_1 Q_1 T^{-1} - T a_t (Q_1^{-1} P_1 Q_1) T^{-1}, \zeta (TQ_1^{-1} P_1 a_2 Q_1 T^{-1} - T Q_1^{-1} a_1 P_1 Q_1 T^{-1}), \ldots,\\ \zeta^{r-1}(T Q_1^{-1}P_1 a_r Q_1 T^{-1} - T Q_1^{-1} a_{r-1} P_1 Q_1 T^{-1}))\end{split}\end{equation*} for all $a_i \in M_k(F)$. Hence $$P_2 = \zeta^r\, T P_1 T^{-1} = \zeta^{r-t}\, T Q_1^{-1} P_1 Q_1 T^{-1} $$ if $r > 0$, and $P_2 = T P_1 T^{-1}$ if $r=0$. Taking $Q_1 P_1 Q_1^{-1} = \zeta^{-t} P_1$ into account, we reduce both conditions to $P_2 =\zeta^r\, T P_1 T^{-1}$.
The converse assertion is proved explicitly. If $P_2 =\zeta^r\, T P_1 T^{-1}$ for some $r \in \mathbb Z$, we can always make $0 \leqslant r < t$ conjugating $P_1$ by $Q_1$. \end{proof}
\begin{remark} Therefore every automorphism of a semisimple $H_{m^2}(\zeta)$-simple algebra $A$ that corresponds to a number $t \in \mathbb N$, and matrices $P \in M_k(F)$, $Q \in \GL_k(F)$, can be identified with a pair $(\bar T, r)$, $0\leqslant r < t$ where $T \in \GL_k(F)$, $QTQ^{-1}T^{-1} = \beta E_k$ for some $\beta \in F$, $P =\zeta^r\, T P T^{-1}$. (Here by $\bar T$ we denote the class of a matrix $T \in \GL_k(F)$ in $\PGL_k(F)$.) If we transfer the multiplication from the automorphism group to the set of such pairs, we get $$(\overline W,s)(\overline T,r)=\left\lbrace\begin{array}{llcc} (\overline{WT}, & r+s) & \text{ if }& r+s < t, \\ (\overline{WTQ^{-1}}, & r+s-t) & \text{ if }& r+s \geqslant t. \end{array}\right.$$ Therefore, the automorphism group of $A$ is an extension of a subgroup of $\mathbb Z_m$ by a subgroup of $\PGL_k(F)$. \end{remark}
\begin{remark} The case $m=2$ is worked out in detail in~\cite{ASGordienko11}. Below we list several examples that are consequences of Theorems~\ref{TheoremTaftSimpleSemisimple} and~\ref{TheoremTaftSimpleSSIso}. \end{remark}
\begin{example}
In the case of $m=2$ and $A\cong M_2(F)$ we have the following variants:
\begin{enumerate}
\item $A = A^{(0)}= M_2(F)$, $A^{(1)}= 0$, $ca=a$, $va=0$ for all $a\in A$;
\item $A = A^{(0)} \oplus A^{(1)}$ where $$A^{(0)}=\left\lbrace\left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta
\end{array}\right) \mathbin{\biggl|} \alpha,\beta \in F\right\rbrace$$
and $$A^{(1)}=\left\lbrace\left(\begin{array}{cc} 0 & \alpha \\ \beta & 0
\end{array}\right) \mathbin{\biggl|} \alpha,\beta \in F\right\rbrace,$$
$ca=(-1)^{i}a$, $va=0$ for $a\in A^{(i)}$;
\item $A = A^{(0)} \oplus A^{(1)}$ where $$A^{(0)}=\left\lbrace\left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta
\end{array}\right) \mathbin{\biggl|} \alpha,\beta \in F\right\rbrace$$
and $$A^{(1)}=\left\lbrace\left(\begin{array}{cc} 0 & \alpha \\ \beta & 0
\end{array}\right) \mathbin{\biggl|} \alpha,\beta \in F\right\rbrace,$$
$ca=(-1)^{i}a$, $va = Pa-(ca)P$ for $a\in A^{(i)}$ where $P=\left(\begin{array}{cc} 0 & 1 \\ \gamma & 0
\end{array}\right)$ and $\gamma \in F$ is a fixed number.
\end{enumerate}
\end{example}
\begin{example} Every semisimple $H_4(-1)$-simple algebra $A$ over an algebraically closed field $F$, $\ch F \ne 2$, that is not simple as an ordinary algebra, is isomorphic to $M_k(F) \oplus M_k(F)$ (direct sum of ideals) for some $k \geqslant 1$ where $$ c\, (a, b) = (b,a),\qquad v\,(a,b)=(Pa-bP,aP-Pb)$$ for all $a,b \in M_k(F)$ and \begin{enumerate} \item either $P=(\underbrace{\alpha,\alpha,\ldots, \alpha}_{k_1}, \underbrace{-\alpha,-\alpha,\ldots, -\alpha}_{k_2})$ for some $\alpha \in F$ and $k_1 \geqslant k_2$, $k_1+k_2=k$, \item or $P$ is a block diagonal matrix with several blocks $\left(\begin{smallmatrix}
0 & 1 \\
0 & 0\\
\end{smallmatrix}\right)$ on the main diagonal (the rest cells are filled with zeros)
\end{enumerate} and these algebras are not isomorphic for different $P$. \end{example}
\section{Non-semisimple algebras}\label{SectionTaftSimpleNonSemisimple}
First we construct an example of an $H_{m^2}(\zeta)$-simple algebra and then we prove that every non-semisimple $H_{m^2}(\zeta)$-simple algebra is isomorphic to one of the algebras below.
\begin{theorem}\label{TheoremTaftSimpleNonSemiSimplePresent} Let $B$ be a simple $\mathbb Z_m$-graded algebra over a field $F$. Suppose $F$ contains some primitive $m$th root of unity $\zeta$. Define $\mathbb Z_m$-graded vector spaces $W_i$, $1\leqslant i \leqslant m-1$, $W_0 := B$, with linear isomorphisms $\varphi \colon W_{i-1} \to W_i$ (we denote the isomorphisms by the same letter), $1 \leqslant i \leqslant m-1$, such that $\varphi(W_{i-1}^{(\ell)})=W_i^{(\ell+1)}$. Let $\varphi(W_{m-1})=0$. Consider $H_{m^2}(\zeta)$-module $A=\bigoplus_{i=0}^{m-1} W_i$ (direct sum of subspaces) where $v\varphi(a)=a$ for all $a \in W_i$, $0\leqslant i \leqslant m-2$, $vB=0$, and $c a^{(i)}=\zeta^i a^{(i)}$, $a^{(i)} \in A^{(i)}$, $A^{(i)} := \bigoplus_{i=0}^{m-1} W_i^{(i)}$ (direct sum of subspaces).
Define the multiplication on $A$ by $$\varphi^k(a)\varphi^\ell(b)=\binom{k+\ell}{k}_\zeta\ \varphi^{k+\ell}((c^\ell a)b) \text{ for all }a, b\in B \text{ and } 0 \leqslant k,\ell < m.$$ Then $A$ is an $H_{m^2}(\zeta)$-simple algebra. \end{theorem} \begin{proof} We check explicitly that the formulas indeed define on $A$ a structure of an $H_{m^2}(\zeta)$-module algebra.
Suppose that $I$ is an $H_{m^2}(\zeta)$-invariant ideal of $A$. Then $v^m I = 0$. Let $t \in \mathbb Z_+$ such that $v^t I \ne 0$, $v^{t+1} I = 0$. Then $0 \ne v^t I \subseteq I \cap \ker v$. However, $\ker v = B$ is a simple graded algebra. Thus $\ker v \subseteq I$. Since $1_A \in I$, we get $I = A$. Therefore, $A$ is an $H_{m^2}(\zeta)$-simple algebra. \end{proof}
Now we prove that we indeed have obtained all non-semisimple $H_{m^2}(\zeta)$-simple algebras.
\begin{theorem}\label{TheoremTaftSimpleNonSemiSimpleClassify} Suppose $A$ is a finite dimensional $H_{m^2}(\zeta)$-simple algebra over a perfect field $F$ and $J:=J(A)\ne 0$. Then $A$ is isomorphic to an algebra from Theorem~\ref{TheoremTaftSimpleNonSemiSimplePresent}. \end{theorem} \begin{corollary} Let $A$ be a finite dimensional $H_{m^2}(\zeta)$-simple algebra over $F$ where $F$ is a field of characteristic $0$, an algebraically closed field, or a finite field. Suppose $J:=J(A)\ne 0$. Then $A$ is isomorphic to an algebra from Theorem~\ref{TheoremTaftSimpleNonSemiSimplePresent}. \end{corollary}
In order to prove Theorem~\ref{TheoremTaftSimpleNonSemiSimpleClassify}, we need several auxiliary lemmas.
Let $M_1$ and $M_2$ be two $(A,A)$-graded bimodules for a $\mathbb Z_m$-graded algebra $A$. We say that a linear isomorphism $\varphi \colon M_1 \to M_2$ is a \textit{$c$-isomorphism} of $M_1$ and $M_2$ if there exists $r\in\mathbb Z$ such that $c\varphi(b) = \zeta^{-r} \varphi(cb)$, $\varphi(ab)=(c^r a)\varphi(b)$, $\varphi(ba)=\varphi(b)a$ for all $b\in M_1$, $a\in A$.
\begin{lemma}\label{LemmaTaftSimpleNonSemiSimpleClassifySumDirect} Suppose $A$ is a finite dimensional $H_{m^2}(\zeta)$-simple algebra over a field $F$ and $J:=J(A)\ne 0$. Let $J^\ell = 0$, $J^{\ell-1} \ne 0$. Choose a minimal $\mathbb Z_m$-graded $A$-ideal $\tilde J \subseteq J^{\ell-1}$. Then for any $k$, $J_k := \sum_{i=0}^{i=k} v^i \tilde J$ is a graded ideal of $A$ and $A = \bigoplus_{i=0}^t v^i \tilde J$ (direct sum of graded subspaces) for some $1 \leqslant t \leqslant m-1$. Moreover, $J_k/J_{k-1}$, $0 \leqslant k \leqslant t$, are irreducible graded $(A,A)$-bimodules $c$-isomorphic to each other. (Here $J_{-1} := 0$.) \end{lemma} \begin{proof} Since for any $a \in \tilde J$, $b\in A$, the element $(v^k a) b$ can be presented as a linear combination of elements $v^i((c^{k-i} a)(v^{k-i}b))$, each $J_k := \sum_{i=0}^{i=k} v^i \tilde J$ is a graded ideal of $A$.
Recall that $v^m =0$. Thus $J_m$ is an $H_{m^2}(\zeta)$-invariant ideal of $A$. Hence $A=J_m$.
Let $\varphi_k \colon J_k/J_{k-1} \to J_{k+1}/J_k$ where $0 \leqslant k \leqslant m-1$, be the map defined by $\varphi_k (a + J_{k-1}) = va + J_k$. Then $c\varphi_k (\bar b) = \zeta^{-1}\varphi_k (c\bar b)$, $$\varphi_k (a \bar b) = v(ab)+J_k = (ca)(vb)+(va)b+J_k = (ca)(vb)+J_k =(ca)\varphi_k (\bar b),$$ $$\varphi_k (\bar b a) = v(ba)+J_k = (cb)(va)+(vb)a+J_k = (vb)a+J_k =\varphi_k (\bar b) a$$ for all $a\in A$, $b \in J_k$. Note that $\tilde J = J_0/J_{-1}$ is an irreducible graded $(A,A)$-bimodule. Therefore, $J_{k+1}/J_k$ is an irreducible graded $(A,A)$-bimodule or zero for any $0 \leqslant k \leqslant m-1$. Thus if $A = J_t$, $A \ne J_{t-1}$, then $\dim J_t = (t+1)\dim \tilde J$ and $A = \bigoplus_{i=0}^t v^i \tilde J$ (direct sum of graded subspaces). \end{proof}
\begin{lemma}\label{LemmaTaftSimpleNonSemiSimpleClassifyUnity} Suppose $A$ is a finite dimensional $H_{m^2}(\zeta)$-simple algebra over a perfect field $F$ where $J(A)\ne 0$. Then $A$ has unity, $A/J(A)$ is a simple $\mathbb Z_m$-graded algebra, and $J_{t-1} = J(A)$. (The ideal $J_{t-1}$ was defined in Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifySumDirect}.) \end{lemma} \begin{proof} Note that $J(A)$ annihilates all irreducible $(A,A)$-bimodules. Thus
$J_k/J_{k-1}$
are irreducible $(A/J(A),A/J(A))$-bimodules. By~\cite{Taft}, there exists a maximal $\mathbb Z_m$-graded semisimple subalgebra $B\subseteq A$ such that $A = B \oplus J(A)$ (direct sum of $\mathbb Z_m$-graded subspaces), $B \cong A/J(A)$. Note that $J_k/J_{k-1}$ are irreducible $(B,B)$-bimodules.
Let $e$ be the unity of $B$. Then $$A = eAe \oplus (\id_A-e)A e \oplus e A(\id_A-e) \oplus (\id_A-e)A(\id_A-e)\text{ (direct sum of graded subspaces)}$$
where $\id_A$ is the identity map.
Note that $eAe$ is a completely reducible graded left $B \otimes B^{\mathrm{op}}$-module,
$(\id_A-e)A e$ is a completely reducible graded right $B$-module,
$e A(\id_A-e)$ is a completely reducible graded left $B$-module,
and $(\id_A-e)A(\id_A-e)$ is a graded subspace with zero $B$-action.
Thus $A$ is a sum of irreducible graded $(B,B)$-bimodules or bimodules with zero $B$-action. Since by Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifySumDirect}, the algebra $A$ has a series of graded $(B,B)$-subbimodules with $c$-isomorphic irreducible factors, the only possibility is that $A=eAe$, $A/J(A)$ is a simple graded algebra.
Therefore, $J(A)$ is the unique maximal graded ideal. Note that all $J_k/J_{k-1}\cong A/J(A)$ and, in particular, $A/J_{t-1}=J_t/J_{t-1} \cong A/J(A)$ (as vector spaces). Hence $\dim J_t = \dim J(A)$ and $J_{t-1} = J(A)$. \end{proof}
\begin{lemma}\label{LemmaTaftSimpleNonSemiSimpleClassifyFormula} Suppose $A$ is a finite dimensional $H_{m^2}(\zeta)$-simple algebra over a perfect field $F$ where $J(A)\ne 0$. Define the linear map $\varphi \colon A \to A$ by $\varphi(v^k a) = v^{k-1} a$ for all $a \in\tilde J$, $1 \leqslant k \leqslant t$, $\varphi(\tilde J) = 0$. (See Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifySumDirect}.) Then \begin{equation}\label{EqQuantumBinomPhi} \varphi^k(a)\varphi^\ell(b)=\binom{k+\ell}{k}_\zeta\ \varphi^{k+\ell}((c^\ell a)b) \text{ for all }a, b\in \ker v \text{ and } 0 \leqslant k,\ell < m. \end{equation} \end{lemma} \begin{proof} Note that $\varphi(va)=a$ for all $a\in J_{t-1}$. Thus the properties of $v$ imply $c\varphi(a) = \zeta \varphi(ca)$, $\varphi(ba)=(c^{-1}b)\varphi(a)$, $\varphi(ab)=\varphi(a)b$ for all $a\in v J_{t-1}$, $b\in \ker v$, and therefore for all $a\in A$, $b\in \ker v$ since $A = v J_{t-1} \oplus \tilde J$ (direct sum of graded subspaces) and $\varphi(\tilde J)=0$. This proves~(\ref{EqQuantumBinomPhi}) for $k=0$ or $\ell=0$.
Recall that, by Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifyUnity}, $\im \varphi = J_{t-1}=J(A)$. Hence $(\im \varphi) \tilde J= \tilde J(\im \varphi) = 0$. Moreover $v\varphi(a)-a \in \tilde J$ for all $a\in A$. Now the case of arbitrary $k$ and $\ell$ is done by induction: \begin{equation*}\begin{split} \varphi^k(a)\varphi^\ell(b)= \varphi(v(\varphi^k(a)\varphi^\ell(b))) =\varphi((c\varphi^k(a))\varphi^{\ell-1}(b)+\varphi^{k-1}(a)\varphi^\ell(b))=\\ \varphi\left((\zeta^k\varphi^k(ca)\varphi^{\ell-1}(b)+\varphi^{k-1}(a)\varphi^\ell(b)\right)=\\ \varphi\left(\zeta^k\binom{k+\ell-1}{k}_\zeta\ \varphi^{k+\ell-1}((c^\ell a)b)+\binom{k+\ell-1}{k-1}_\zeta\ \varphi^{k+\ell-1}((c^\ell a)b) \right)=\\ \left(\zeta^k \binom{k+\ell-1}{k}_\zeta + \binom{k+\ell-1}{k-1}_\zeta\right) \varphi^{k+\ell}((c^\ell a)b) =\\ \binom{k+\ell}{k}_\zeta\ \varphi^{k+\ell}((c^\ell a)b) \end{split}\end{equation*} since \begin{equation*}\begin{split} \zeta^k \binom{k+\ell-1}{k}_\zeta + \binom{k+\ell-1}{k-1}_\zeta = \frac{\zeta^k (k+\ell-1)!_\zeta}{k!_\zeta (\ell-1)!_\zeta}+ \frac{(k+\ell-1)!_\zeta}{(k-1)!_\zeta \ell!_\zeta}=\\(\zeta^k \ell_\zeta + k_\zeta)\frac{(k+\ell-1)!_\zeta}{k!_\zeta \ell!_\zeta}=(k+\ell)_\zeta \frac{(k+\ell-1)!_\zeta}{k!_\zeta \ell!_\zeta} = \frac{(k+\ell)!_\zeta}{k!_\zeta \ell!_\zeta}=\binom{k+\ell}{k}_\zeta.\end{split}\end{equation*} \end{proof}
\begin{proof}[Proof of Theorem~\ref{TheoremTaftSimpleNonSemiSimpleClassify}.] By Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifyUnity}, there exists unity $1_A \in A$. Note that $1_A \notin J_{t-1}$ (see the definition in Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifySumDirect}), since $J_{t-1}$ is an ideal. Hence $\varphi^t(1_A)\ne 0$. (See the definition of the map $\varphi$ in Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifyFormula}.) Since $v\varphi(a)-a \in \tilde J$ for all $a\in A$, we have $v\varphi^t(1_A)=\varphi^{t-1}(1_A)+j_1$ and $v\varphi(1_A)=1_A+j_2$ for some $j_1, j_2 \in \tilde J$. Note that $\varphi^t(1_A) \varphi(1_A)=\binom{t+1}{t}_\zeta \varphi^{t+1}(1_A) = 0$. However \begin{equation*}\begin{split}0=v(\varphi^t (1_A)\varphi(1_A)) =(v\varphi^t(1_A))\varphi(1_A)+(c\varphi^t(1_A))v\varphi(1_A) =\\ (\varphi^{t-1}(1_A)+j_1)\varphi(1_A)+\zeta^t\varphi^t(1_A)(1_A+j_2) =\varphi^{t-1}(1_A)\varphi(1_A)+\zeta^t\varphi^t(1_A)1_A =\\ \left(\binom t{t-1}_\zeta+\zeta^t\right)\varphi^t(1_A) =(t+1)_\zeta\ \varphi^t(1_A)\end{split}\end{equation*} since by Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifyUnity},
$(\im \varphi)\tilde J = J_{t-1}\tilde J=J(A)\tilde J = 0$. Hence $(t+1)_\zeta = 0$ and $t=m-1$. By Lemma~\ref{LemmaTaftSimpleNonSemiSimpleClassifyUnity}, $J(A)=J_{m-2}$. Thus $\ker v = v^{m-1}\tilde J \cong A/J(A)$. Now~(\ref{EqQuantumBinomPhi}) implies the theorem. \end{proof}
\begin{remark} Since the maximal semisimple subalgebra $\ker v$ is uniquely determined, any two such $H_{m^2}(\zeta)$-simple algebras $A$ are isomorphic as $H_{m^2}(\zeta)$-module algebras if and only if their subalgebras $\ker v$ are isomorphic as $\mathbb Z_m$-algebras. Moreover, all automorphisms of $A$ as an $H_{m^2}(\zeta)$-module algebra are induced by the automorphisms of $\ker v$ as a $\mathbb Z_m$-algebra. Indeed, let $\psi \colon A \to A$ be an automorphism of $A$ as an $H_{m^2}(\zeta)$-module algebra. Since $\tilde J = J(A)^{m-1}$, $\psi(\tilde J) = \tilde J$ and $$v^{m-1}\psi(\varphi^{m-1}(a))=\psi(a) \text{ for all }a\in \ker v$$ implies $$\psi(\varphi^{m-1}(a))=\varphi^{m-1}(\psi(a)).$$ Applying $v^{m-k-1}$, we get $\psi(\varphi^k(a))=\varphi^k(\psi(a))$ for all $a\in\ker v$ and $0 \leqslant k < m$. \end{remark}
\section{Growth of polynomial $H_{m^2}(\zeta)$-identities}\label{SectionTaftSimpleAmitsur}
Here we apply the results of Section~\ref{SectionTaftSimpleNonSemisimple} to polynomial $H_{m^2}(\zeta)$-identities.
First we introduce the notion of the free associative $H$-module algebra. Here we follow~\cite{BahturinLinchenko}. Let $F \langle X \rangle$ be the free associative algebra without $1$
on the set $X := \lbrace x_1, x_2, x_3, \ldots \rbrace$.
Then $F \langle X \rangle = \bigoplus_{n=1}^\infty F \langle X \rangle^{(n)}$
where $F \langle X \rangle^{(n)}$ is the linear span of all monomials of total degree $n$.
Let $H$ be a Hopf algebra over $F$. Consider the algebra $$F \langle X | H\rangle
:= \bigoplus_{n=1}^\infty H^{{}\otimes n} \otimes F \langle X \rangle^{(n)}$$
with the multiplication $(u_1 \otimes w_1)(u_2 \otimes w_2):=(u_1 \otimes u_2) \otimes w_1w_2$
for all $u_1 \in H^{{}\otimes j}$, $u_2 \in H^{{}\otimes k}$,
$w_1 \in F \langle X \rangle^{(j)}$, $w_2 \in F \langle X \rangle^{(k)}$. We use the notation $$x^{h_1}_{i_1} x^{h_2}_{i_2}\ldots x^{h_n}_{i_n} := (h_1 \otimes h_2 \otimes \ldots \otimes h_n) \otimes x_{i_1} x_{i_2}\ldots x_{i_n}.$$ Here $h_1 \otimes h_2 \otimes \ldots \otimes h_n \in H^{{}\otimes n}$, $x_{i_1} x_{i_2}\ldots x_{i_n} \in F \langle X \rangle^{(n)}$.
Note that if $(\gamma_\beta)_{\beta \in \Lambda}$ is a basis in $H$,
then $F\langle X | H \rangle$ is isomorphic to the free associative algebra over $F$ with free formal generators $x_i^{\gamma_\beta}$, $\beta \in \Lambda$, $i \in \mathbb N$.
We refer to the elements
of $F\langle X | H \rangle$ as \textit{associative $H$-polynomials}.
In addition, $F \langle X | H \rangle$ becomes an $H$-module algebra with the $H$-action defined by $h(x^{h_1}_{i_1} x^{h_2}_{i_2}\ldots x^{h_n}_{i_n})=x^{h_{(1)}{h_1}}_{i_1} x^{h_{(2)}{h_2}}_{i_2}\ldots x^{h_{(n)}{h_n}}_{i_n}$ for $h\in H$.
Let $A$ be an associative $H$-module algebra. Any map $\psi \colon X \to A$ has a unique homomorphic extension $\bar\psi
\colon F \langle X | H \rangle \to A$ such that $\bar\psi(h w)=h\psi(w)$
for all $w \in F \langle X | H \rangle$ and $h \in H$.
An $H$-polynomial
$f \in F\langle X | H \rangle$
is an \textit{$H$-identity} of $A$ if $\varphi(f)=0$
for all homomorphisms $\varphi \colon F \langle X | H \rangle \to A$ of algebras and $H$-modules.
In other words, $f(x_1, x_2, \ldots, x_n)$
is an $H$-identity of $A$ if and only if $f(a_1, a_2, \ldots, a_n)=0$ for any $a_i \in A$.
In this case we write $f \equiv 0$. The set $\Id^{H}(A)$ of all $H$-identities of $A$ is an $H$-invariant ideal of $F\langle X | H \rangle$.
We denote by $P^H_n$ the space of all multilinear $H$-polynomials in $x_1, \ldots, x_n$, $n\in\mathbb N$, i.e. $$P^{H}_n = \langle x^{h_1}_{\sigma(1)} x^{h_2}_{\sigma(2)}\ldots x^{h_n}_{\sigma(n)}
\mid h_i \in H, \sigma\in S_n \rangle_F \subset F \langle X | H \rangle.$$ Then the number $c^H_n(A):=\dim\left(\frac{P^H_n}{P^H_n \cap \Id^H(A)}\right)$ is called the $n$th \textit{codimension of polynomial $H$-identities} or the $n$th \textit{$H$-codimension} of $A$.
The analog of Amitsur's conjecture for $H$-codimensions can be formulated as follows.
\begin{conjecture} There exists
$\PIexp^H(A):=\lim\limits_{n\to\infty}
\sqrt[n]{c^H_n(A)} \in \mathbb Z_+$. \end{conjecture}
In the theorem below we consider the case $H=H_{m^2}(\zeta)$.
\begin{theorem}\label{TheoremTaftSimpleAmitsur} Let $A$ be a finite dimensional $H_{m^2}(\zeta)$-simple algebra over an algebraically closed field $F$ of characteristic $0$. Then there exist constants $C > 0$, $r\in \mathbb R$ such that $$C n^{r} (\dim A)^n \leqslant c^{H_{m^2}(\zeta)}_n(A) \leqslant (\dim A)^{n+1}\text{ for all }n \in \mathbb N.$$ \end{theorem} \begin{corollary} The analog of Amitsur's conjecture holds
for such codimensions. In particular, $\PIexp^H(A)=\dim A$. \end{corollary}
In order to prove Theorem~\ref{TheoremTaftSimpleAmitsur}, we need one lemma.
Let $k\ell \leqslant n$ where $k,n \in \mathbb N$ are some numbers.
Denote by $Q^H_{\ell,k,n} \subseteq P^H_n$ the subspace spanned by all $H$-polynomials that are alternating in $k$ disjoint subsets of variables $\{x^i_1, \ldots, x^i_\ell \} \subseteq \lbrace x_1, x_2, \ldots, x_n\rbrace$, $1 \leqslant i \leqslant k$.
\begin{lemma}\label{LemmaTaftNSSAltHPolynomial} Let $A$ be an $H_{m^2}(\zeta)$-simple non-semisimple associative algebra over an algebraically closed field $F$ of characteristic $0$, $\dim A=\ell m$. Then there exists a number $n_0 \in \mathbb N$ such that for every $n\geqslant n_0$ there exist disjoint subsets $X_1$, \ldots, $X_k \subseteq \lbrace x_1, \ldots, x_n \rbrace$, $k = \left[\frac{n-n_0}{2\ell m}\right]$,
$|X_1| = \ldots = |X_{k}|=\ell m$ and a polynomial $f \in P^{H_{m^2}(\zeta)}_n \backslash \Id^{H_{m^2}(\zeta)}(A)$ alternating in the variables of each set $X_j$. \end{lemma} \begin{proof} By Theorem~\ref{TheoremTaftSimpleNonSemiSimpleClassify}, $A=\bigoplus_{i=0}^{m-1} v^i \tilde J$ (direct sum of subspaces) where $\tilde J^2=0$ and $v^{m-1}\tilde J$ is a $\mathbb Z_m$-simple subalgebra.
Fix the basis $a_1, \ldots, a_\ell;\ va_1, \ldots, va_\ell; \ldots; v^{m-1}a_1, \ldots, v^{m-1}a_\ell$ in $A$ where $a_1, \ldots, a_\ell$ is a basis in $\tilde J$.
Since $v^{m-1}\tilde J$ is a $\mathbb Z_m$-simple subalgebra, by~\cite[Theorem~7]{ASGordienko3}, there exist $T \in \mathbb Z_+$ and $\bar z_1, \ldots, \bar z_T \in v^{m-1}\tilde J$ such that for any $k \in \mathbb N$ there exists $$f_0=f_0(x_1^1, \ldots, x_\ell^1; \ldots; x^{2k}_1, \ldots, x^{2km}_\ell;\ z_1, \ldots, z_T;\ z) \in Q^{F\mathbb Z_m}_{\ell, 2km, 2k\ell m+T+1}$$ such that for any $\bar z \in v^{m-1} \tilde J$ we have $$f_0(v^{m-1} a_1, \ldots, v^{m-1} a_\ell; \ldots; v^{m-1} a_1, \ldots, v^{m-1} a_\ell; \bar z_1, \ldots, \bar z_T; \bar z) = \bar z.$$
Take $n_0=T+1$, $k=\left[\frac{n-n_0}{2\ell m}\right]$, and consider \begin{equation*}\begin{split}f(x_1^1, \ldots, x_{\ell m}^1; \ldots; x^{2k}_1, \ldots, x^{2k}_{\ell m};\ z_1, \ldots, z_T; \ z;\ y_1, \ldots, y_{n-2k\ell m-T-1})=\\ \Alt_1 \Alt_2 \ldots \Alt_{2k} f_0(x_1^1, \ldots, x_{\ell}^1;\ \left(x_{\ell+1}^1\right)^v, \ldots, \left(x_{2\ell}^1\right)^v; \ \left(x_{2\ell+1}^1\right)^{v^2}, \ldots, \left(x_{3\ell}^1\right)^{v^2};
\ldots;\\ \left(x_{\ell(m-1)+1}^1\right)^{v^{m-1}}, \ldots, \left(x_{\ell m}^1\right)^{v^{m-1}}; \ldots; \\ x_1^{2k}, \ldots, x_{\ell}^{2k};\ \left(x_{\ell+1}^{2k}\right)^v, \ldots, \left(x_{2\ell}^{2k}\right)^v; \ \left(x_{2\ell+1}^{2k}\right)^{v^2}, \ldots, \left(x_{3\ell}^{2k}\right)^{v^2};
\ldots;\\ \left(x_{\ell(m-1)+1}^{2k}\right)^{v^{m-1}}, \ldots, \left(x_{\ell m}^{2k}\right)^{v^{m-1}};\ z_1, \ldots, z_T;\ z)\ y_1 y_2 \ldots y_{n-2k\ell m-T-1}\in P^{H_{m^2}(\zeta)}_n\end{split}\end{equation*} where $\Alt_i$ is the operator of alternation on the set $X_i:=\lbrace x_1^i, \ldots, x_{\ell m}^i\rbrace$.
Now we notice that \begin{equation*}\begin{split} f(v^{m-1}a_1, \ldots, v^{m-1}a_{\ell},\ \ldots\ , va_1, \ldots, va_{\ell},\ a_1, \ldots, a_{\ell};\ \ldots\ ; \\ v^{m-1} a_1, \ldots, v^{m-1} a_{\ell},\ \ldots\ , va_1, \ldots, va_{\ell},\ a_1, \ldots, a_{\ell};\ \bar z_1, \ldots, \bar z_T; \ 1_A, \ldots, 1_A) =(\ell!)^{2k m} 1_A\end{split}\end{equation*} since $v^m=0$. The lemma is proved. \end{proof}
\begin{proof}[Proof of Theorem~\ref{TheoremTaftSimpleAmitsur}.] If $A$ is semisimple, then Theorem~\ref{TheoremTaftSimpleAmitsur} follows from~\cite[Theorem~5]{ASGordienko3}. If $A$ is not semisimple, we repeat verbatim the proof of~\cite[Lemma~11 and Theorem~5]{ASGordienko3} using Lemma~\ref{LemmaTaftNSSAltHPolynomial} instead of~\cite[Lemma~10]{ASGordienko3} and~\cite[Lemma~4]{ASGordienko3} instead of~\cite[Theorem~6]{ASGordienko3}. \end{proof}
\end{document} | arXiv |
Work, energy and power
5.5 Power (ESCMJ)
Now that we understand the relationship between work and energy, we are ready to look at a quantity related the rate of energy transfer. For example, a mother pushing a trolley full of groceries can take \(\text{30}\) \(\text{s}\) or \(\text{60}\) \(\text{s}\) to push the trolley down an aisle. She does the same amount of work, but takes a different length of time. We use the idea of power to describe the rate at which work is done.
The unit watt is named after Scottish inventor and engineer James Watt (19 January 1736 - 19 August 1819) whose improvements to the steam engine were fundamental to the Industrial Revolution. A key feature of it was that it brought the engine out of the remote coal fields into factories.
Power is defined as the rate at which work is done or the rate at which energy is transfered to or from a system. The mathematical definition for power is:
\(P=\frac{W}{t}\)
Power is easily derived from the definition of work. We know that:
\(W=F \Delta x \cos \theta\)
Power is defined as the rate at which work is done. Therefore,
\begin{align*} P & = \frac{W}{t} \\ & = \frac{F \Delta x \cos \theta}{t} \\ & \text{in the case where }F \text{ and }\Delta x \text{ are in the same direction} \\ & = \frac{F \Delta x}{t} \\ & = F \frac{ \Delta x}{t} \\ & = F v \end{align*}
In the case where the force and the velocity are in opposite directions the power will be negative.
The unit of power is watt (symbol W).
Historically, the horsepower (symbol hp) was the unit used to describe the power delivered by a machine. One horsepower is equivalent to approximately \(\text{750}\) \(\text{W}\). The horsepower was derived by James Watt to give an indication of the power of his steam engine in terms of the power of a horse, which was what most people used to for example, turn a mill wheel.
Worked example 11: Power calculation 1
Calculate the power required for a force of \(\text{10}\) \(\text{N}\) applied to move a \(\text{10}\) \(\text{kg}\) box at a speed of \(\text{1}\) \(\text{m·s$^{-1}$}\) over a frictionless surface.
Determine what is given and what is required.
We are given the force, \(F=10~\text{N}\).
We are given the speed, \(v=1~\text{m}·{\text{s}}^{-1}\).
We are required to calculate the power required.
Draw a force diagram
Determine how to approach the problem
From the force diagram, we see that the weight of the box is acting at right angles to the direction of motion. The weight does not contribute to the work done and does not contribute to the power calculation. We can therefore calculate power from: \(P=F·v\).
Calculate the power required
\begin{align*} P& = F·v\\ & = \left(10 \text{N}\right)\left(1 \text{m}·{\text{s}}^{-1}\right)\\ & = 10 \text{W} \end{align*}
Write the final answer
\(\text{10}\) \(\text{W}\) of power are required for a force of \(\text{10}\) \(\text{N}\) to move a \(\text{10}\) \(\text{kg}\) box at a speed of \(\text{1}\) \(\text{m·s$^{-1}$}\) over a frictionless surface.
Machines are designed and built to do work on objects. All machines usually have a power rating. The power rating indicates the rate at which that machine can do work upon other objects.
A car engine is an example of a machine which is given a power rating. The power rating relates to how rapidly the car can accelerate. Suppose that a 50 kW engine could accelerate the car from \(\text{0}\) \(\text{km·hr$^{-1}$}\) to \(\text{60}\) \(\text{km·hr$^{-1}$}\) in \(\text{16}\) \(\text{s}\). Then a car with four times the power rating (i.e. \(\text{200}\) \(\text{kW}\)) could do the same amount of work in a quarter of the time. That is, a \(\text{200}\) \(\text{kW}\) engine could accelerate the same car from \(\text{0}\) \(\text{km·hr$^{-1}$}\) to \(\text{60}\) \(\text{km·hr$^{-1}$}\) in \(\text{4}\) \(\text{s}\).
A forklift lifts a crate of mass \(\text{100}\) \(\text{kg}\) at a constant velocity to a height of \(\text{8}\) \(\text{m}\) over a time of \(\text{4}\) \(\text{s}\). The forklift then holds the crate in place for \(\text{20}\) \(\text{s}\). Calculate how much power the forklift exerts in lifting the crate? How much power does the forklift exert in holding the crate in place?
Determine what is given and what is required
mass of crate: m=\(\text{100}\) \(\text{kg}\)
height that crate is raised: h=\(\text{8}\) \(\text{m}\)
time taken to raise crate: \({t}_{r}=4~\text{s}\)
time that crate is held in place: \({t}_{s}=20~\text{s}\)
We are required to calculate the power exerted.
We can use:
\(P=Fv=F\frac{\Delta x}{\Delta t}\)
to calculate power. The force required to raise the crate is equal to the weight of the crate.
Calculate the power required to raise the crate
\begin{align*} P& = F\frac{\Delta x}{\Delta t}\\ & = m·g\frac{\Delta x}{\Delta t}\\ & = \left(100 \text{kg}\right)\left(9,8 \text{m}·{\text{s}}^{-2}\right)\frac{8 \text{m}}{4 \text{s}}\\ & = 1960 \text{W} \end{align*}
Calculate the power required to hold the crate in place
While the crate is being held in place, there is no displacement. This means there is no work done on the crate and therefore there is no power exerted.
\(\text{1 960}\) \(\text{W}\) of power is exerted to raise the crate and no power is exerted to hold the crate in place.
Worked example 13: Stair climb
What is the power output for a \(\text{60,0}\) \(\text{kg}\) woman who runs up a \(\text{3,00}\) \(\text{ m}\) high flight of stairs in \(\text{3,50}\) \(\text{s}\), starting from rest but having a final speed of \(\text{2,00}\) \(\text{m·s$^{-1}$}\)? Her power output depends on how fast she does this.
Analyse the question
The work going into mechanical energy is \(W= E_k + E_p\). At the bottom of the stairs, we take both \(E_k\) and the potential energy due to gravity, \(E_{p,g}\), as initially zero; thus, \(W=E_{k,f}+ E_{p,g}=\frac{1}{2}mv_f^2+mgh\), where \(h\) is the vertical height of the stairs. Because all terms are given, we can calculate \(W\) and then divide it by time to get power.
Calculate power
Substituting the expression for \(W\) into the definition of power given in the previous equation, \(P=\frac{W}{t}\) yields:
\begin{align*} P &=\frac{W}{t}\\ &=\frac{\frac{1}{2}mv_f^2+mgh}{t} \\ &=\frac{\frac{1}{2}(\text{60,0})(\text{2,00})^2+(\text{60,0})(\text{9,80})(\text{3,00})}{\text{3,50}} \\ & = \frac{120 +1764}{\text{3,50}} \\ & = \text{538,3}\text{ W} \end{align*}
Quote the final answer
The power generated is \(\text{538,0}\) \(\text{W}\).
The woman does \(\text{1 764}\) \(\text{J}\) of work to move up the stairs compared with only \(\text{120}\) \(\text{J}\) to increase her kinetic energy; thus, most of her power output is required for climbing rather than accelerating.
Worked example 14: A borehole
What is the power required to pump water from a borehole that has a depth \(h=\text{15,0}\text{ m}\) at a rate of \(\text{20,0}\) \(\text{l·s$^{-1}$}\)?
We know that we will have to do work on the water to overcome gravity to raise it a certain height. If we ignore any inefficiencies we can calculate the work, and power, required to raise the mass of water the appropriate height.
We know how much water is required in a single second. We can first determine the mass of water: \(\text{20,0}\text{ l} \times\frac{\text{1}\text{ kg}}{\text{1}\text{ l}} = \text{20,0}\text{ kg}\).
The water will also have non-zero kinetic energy when it gets to the surface because it needs to be flowing. The pump needs to move \(\text{20,0}\) \(\text{kg}\) from the depth of the borehole every second, we know the depth so we know the speed that the water needs to be moving is \(v=\frac{h}{t}=\frac{\text{15,0}}{1}=\text{15,0}\text{ m·s$^{-1}$}\).
Work done to raise the water
We can use
\begin{align*} W_{\text{non-conservative}} & = \Delta E_k + \Delta E_p \\ & = E_{k,f} - E_{k,i} + E_{p,f} - E_{p,i} \\ & = \frac{1}{2}mv^2 -(0) + mgh - 0 \\ & = \frac{1}{2}(20)(15)^2 + (20)(\text{9,8})(15) \\ & = \text{2,25} \times \text{10}^{\text{3}} + \text{2,94} \times \text{10}^{\text{3}} \\ & = \text{5,19} \times \text{10}^{\text{3}}\text{ J} \end{align*}
\begin{align*} P &= \frac{W}{t} \\ &= \frac{\text{5,19} \times \text{10}^{\text{3}}}{1} \\ & = \text{5,19} \times \text{10}^{\text{3}}\text{ W} \end{align*}
The minimum power required from the pump is \(\text{5,19} \times \text{10}^{\text{3}}\) \(\text{W}\).
Simple measurements of human power
You can perform various physical activities, for example lifting measured weights or climbing a flight of stairs to estimate your output power, using a stop watch. Note: the human body is not very efficient in these activities, so your actual power will be much greater than estimated here.
Success in Maths and Science unlocks opportunities
Sign up to get a head start on bursary and career opportunities. Use Siyavula Practice to get the best marks possible.
Sign up to unlock your future
[IEB 2005/11 HG] Which of the following is equivalent to the SI unit of power:
\(\text{V·A}\)
\(\text{V·A$^{-1}$}\)
\(\text{kg·m·s$^{-1}$}\)
Two students, Bill and Bob, are in the weight lifting room of their local gym. Bill lifts the \(\text{50}\) \(\text{kg}\) barbell over his head 10 times in one minute while Bob lifts the \(\text{50}\) \(\text{kg}\) barbell over his head 10 times in \(\text{10}\) \(\text{seconds}\). Who does the most work? Who delivers the most power? Explain your answers.
The displacement is 0 and so there is no work done. Since power is the rate at which work is done, the power is also 0.
Jack and Jill ran up the hill. Jack is twice as massive as Jill; yet Jill ascended the same distance in half the time. Who did the most work? Who delivered the most power? Explain your answers.
With comparative problems we always write the values for one of the objects/people in terms of the values for the other. So:
\begin{align*} m_{Jack}&= 2\times m_{Jill} \\ \frac{1}{2}t_{Jack}&= t_{Jill} \\ t_{Jack}&= 2\times t_{Jill} \end{align*}
The work done was to increase the gravitational potential energy of the system. If the hill has a height of \(h\) then:
\begin{align*} W_{Jill}&= m_{Jill}gh \\ W_{Jack}&= m_{Jack}gh \\ W_{Jack}&= (2\times m_{Jill})gh \\ W_{Jack}&= 2\times (m_{Jill}gh \\ W_{Jack}&= 2\times W_{Jill} \end{align*}
The power comparison is similar:
\begin{align*} P_{Jill}&= \frac{W_{Jill}}{t_{Jill}} \\ P_{Jack}&= \frac{W_{Jack}}{t_{Jack}} \\ P_{Jack}&= \frac{2\times W_{Jill}}{2\times t_{Jill}} \\ P_{Jack}&= \frac{ W_{Jill}}{t_{Jill}} \\ P_{Jack}&= P_{Jill} \end{align*} Jack did twice as much work as Jill but the same power.
When doing a chin-up, a physics student lifts her \(\text{40}\) \(\text{kg}\) body a distance of \(\text{0,25}\) \(\text{m}\) in \(\text{2}\) \(\text{s}\). What is the power delivered by the student's biceps?
The work done by the biceps (a non-conservative force) increased the net mechanical energy of the system. The kinetic energy is zero at the top and bottom so we are only interested in the gravitational potential energy.
\begin{align*} W_{\text{non-conservative}} & = \Delta E_k + \Delta E_p \\ & = E_{p,f} - E_{p,i} \\ & = mgh(h_f-h_i) \\ & = (40)(9,8)(0,25)\\ & = \text{98}\text{ J} \end{align*} \begin{align*} P&=\frac{W}{t} \\ &= \frac{(98)}{2} \\ &= \text{49}\text{ W} \end{align*} 49 W
The unit of power that is used on a monthly electricity account is kilowatt-hours (symbol \(\text{kWh}\)). This is a unit of energy delivered by the flow of \(\text{1}\) \(\text{kW}\) of electricity for \(\text{1}\) \(\text{hour}\). Show how many joules of energy you get when you buy \(\text{1}\) \(\text{kWh}\) of electricity.
\(1 \text{kWh} = 1000 \text{Wh} = 1000 \text{Wh} \times \frac{3600}{\text{h}}=\text{3 600 000}\text{ J}\)
An escalator is used to move 20 passengers every minute from the first floor of a shopping mall to the second. The second floor is located 5-meters above the first floor. The average passenger's mass is \(\text{70}\) \(\text{kg}\). Determine the power requirement of the escalator in order to move this number of passengers in this amount of time.
The work done will be to move the total mass of the 20 passengers vertically 5 m. We are not given the mass of the elevator itself so we assume we can ignore it. The total mass will be: \(m_{total}=(20)(70)=\text{1 400}\text{ kg}\)
\begin{align*} W&=mgh \\ &= (1400)(9,8)(5) \\ &= \text{68 600}\text{ J} \end{align*} \begin{align*} P&=\frac{W}{t} \\ &=\frac{68600}{60} \\ &= \text{1 143,33}\text{ W} \end{align*} \(\text{1 143,33}\) \(\text{W}\) | CommonCrawl |
\begin{document}
\def\mathcal {A}{\mathcal {A}} \def\mathcal {B}{\mathcal {B}} \def{\mathcal C}{\mathcal {C}} \def\mathbb P{\mathbb P} \def\mathbb N{\mathbb N} \def\mathbb H{\mathbb H} \def\mathbb V{\mathbb V} \def\mathcal {H}{\mathcal {H}} \def\mathcal {K}{\mathcal {K}} \def\mathcal {M}{\mathcal {M}} \def\mathbb {R}{\mathbb {R}} \def\mathbb {T}{\mathbb {T}} \def\mathcal {R}{\mathcal {R}} \def\mathcal {U}{\mathcal {U}} \defD(\mathcal E){D(\mathcal E)} \def\mathcal {F}{\mathcal {F}} \def\mathcal {E}{\mathcal {E}} \def\mathbb {E}{\mathbb {E}} \def\1{\,{\makebox[0pt][c]{\normalfont 1}
\makebox[2.5pt][c]{\raisebox{3.5pt}{\tiny {$\|$}}}
\makebox[-2.5pt][c]{\raisebox{1.7pt}{\tiny {$\|$}}} \makebox[2.5pt][c]{} }} \def\1 {\1 } \def{[0,1]}{[0,1]} \def\varepsilon{\varepsilon} \def\partial{\partial}
\def\1 {\1 }
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[]{Corollary}
\newtheorem{defn}[]{Definition} \newtheorem{bem}[]{Remark} \newtheorem{bemn}[bem]{Remarks} \newtheorem{bsp}[]{Example} \newtheorem{bspe}{Examples}
\newcommand\iitem{\stepcounter{idesc}\item[(\roman{idesc})]} \newenvironment{idesc}{\setcounter{idesc}{0} \begin{description}} {\end{description}}
\def{d \ol{\normalfont v}}{{d \ol{\normalfont v}}}
\def\mathop{\Rightarrow}\limits^d{\mathop{\Rightarrow}\limits^d} \def{\mathcal P}{{\mathcal P}} \def{\mathbb Z}{{\mathbb Z}} \defCM[0,1]^2{CM[0,1]^2} \def{[0,1]}{{[0,1]}} \def{L^2}{{L^2}} \def\mathcal P _{ac}{\mathcal P _{ac}}
\def\langle \langle{\langle \langle} \def\rangle \rangle{\rangle \rangle} \def\mathrm{d}{\mathrm{d}}
\def\mathbb P^\Beta{\mathbb P^\Beta}
\def\mathcal G{\mathcal G}
\def{\mathbb Q^\beta}{{\mathbb Q^\beta}}
\def{
$\Box$}{{
$\Box$}}
\def{\mathcal C}{{\mathcal C}}
\def{\mathcal D}{{\mathcal D}}
\newcommand{\mathrm{comp}\,}{\mathrm{comp}\,} \newcommand{{\mbox{Leb}}}{{\mbox{Leb}}} \newcommand{{\mathfrak{C}}}{{\mathfrak{C}}} \newcommand{{\mathfrak{S}}}{{\mathfrak{S}}} \newcommand{{\mathfrak{Z}}}{{\mathfrak{Z}}} \def{\mbox{$\int$}}{{\mbox{$\int$}}}
\newcommand{\mbox{\rm Ent}}{\mbox{\rm Ent}}
\newcommand{\mbox{\rm gaps}}{\mbox{\rm gaps}}
\newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition}
\newcommand{{\mathbb{Q}}}{{\mathbb{Q}}} \newcommand{{\mathbb{P}}}{{\mathbb{P}}}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\author{Alexei Kulik \footnote{Institute of Mathematics, NAS of Ukraine, 3, Tereshchenkivska str., 01601 Kyiv, Ukraine \sf{[email protected]}}, Michael Scheutzow \footnote{Institut f\"ur Mathematik, MA 7-5, Technische Universit\"at Berlin, 10623 Berlin, Germany \sf{[email protected]}}}
\title{Generalized couplings and convergence of transition probabilities\\ }
\maketitle ~\\
\begin{abstract} We provide sufficient conditions for the uniqueness of an invariant measure of a Markov process as well as for the weak convergence of transition probabilities to the invariant measure. Our conditions are formulated in terms of generalized couplings. We apply our results to several SPDEs for which unique ergodicity has been proven in a recent paper by Glatt-Holtz, Mattingly, and Richards and show that under essentially the same assumptions the weak convergence of transition probabilities actually holds true. \end{abstract}
\section{Introduction}
In this article, we provide sufficient conditions in terms of (generalized) couplings for the uniqueness of an invariant measure and weak convergence to the invariant measure for a Markov chain taking values in a Polish space $E$. Such criteria have already been established for the uniqueness of an invariant measure in \cite{BM05} and \cite{HMS11} but -- to the best of our knowledge -- not for the weak convergence (or asymptotic stability) of transition probabilities. In \cite{KPS10} and \cite{BKS}, uniqueness and asymptotic stability were shown for so-called e-processes (which we explain below) and a similar approach was used in \cite{HMS11} to prove asymptotic stability. Our aim is to present a unified approach to both uniqueness and asymptotic stability in terms of generalized (asymptotic) couplings. Here, a probability measure $\xi$ on a product space is called a {\em generalized} coupling of $\mu$ and $\nu$ if the marginals of $\xi$ are not necessarily equal to $\mu$ and $\nu$ but only absolutely continuous. We point out that we do not assume the e-property to hold (which indeed does not hold in all cases of interest -- see e.g. Example \ref{ex54} -- and even if it does it is often cumbersome to verify). Our uniqueness statement, Theorem \ref{unique}, is a slight generalization of \cite[Theorem 1.1]{HMS11}. We will show in Example \ref{comparison} that our conditions are indeed strictly weaker than those in \cite{HMS11}. At the same time the proof is quite short and elementary.
We then proceed to formulate sufficient conditions for the convergence of transition probabilities assuming existence of an invariant measure $\mu$. Our main results are Theorems \ref{conv1} and \ref{conv2}, the former one providing a sufficient condition for weak convergence of transition probabilities for $\mu-$almost all initial conditions and the latter one for weak convergence of the transiton probabilities starting from a given point $x \in E$. In both theorems, the conditions are formulated in terms of generalized couplings. In Theorem \ref{conv1} convergence is in fact in probability, i.e. the measure $\mu$ of the set of initial conditions for which the distance of the transition probability to the invariant measure $\mu$ after $n$ steps is larger than $\varepsilon$ converges to 0 for every $\varepsilon>0$. It seems to be an open question if convergence even holds true in an almost sure sense. We point out that our proof of Theorem \ref{conv1} requires the chain to be {\em Feller}, while Theorems \ref{unique} and \ref{conv2} do not rely on this property. However, the Feller property is required to hold true with respect to some metric $d$ on $E$ which in general may differ from the original one, hence this assumption is quite flexible and non-restrictive.
The main results are proved in Sections \ref{s3} and \ref{s4}. In Section \ref{s5} several examples are given which illustrate the conditions imposed in the main results and clarify the relations of these results with some other available in the field.
To illustrate the usefulness of our results, in Section \ref{sSDDE} we give two groups of their applications. First, in Section \ref{s31}, we consider the same example as in \cite{HMS11}, namely a stochastic delay equation which has the space of continuous functions on $[-1,0]$ as its natural state space. The solution is a Feller process and it is not hard to find a generalized coupling which satisfies the conditions of Theorem \ref{unique}. This coupling is actually a simplified version of the one used in the proof of \cite[Theorem 3.1]{HMS11}; namely, our Theorem \ref{unique}, unlike \cite[Theorem 1.1]{HMS11}, does not require the equivalence of the marginal distributions, thus the ``localization in time'' part of the construction of the generalized coupling can be omitted. Remarkably, such a simplified construction appears well applicable in Theorems \ref{conv1} and \ref{conv2} as well, so that without any additional work we get asymptotic stability for free (unlike in \cite{HMS11}).
This method to improve a result from unique ergodicity to asymptotic stability looks quite generic, and in our second group of applications essentially the same method leads to several new statements. We reconsider the results from the recent paper \cite{GMR15}, where the generalized coupling technique (or {\em asymptotic coupling}, in their terminology) is used to prove uniqueness of an ergodic measure for several types of non-linear SPDEs. Each of the five SPDE models considered therein is analytically quite involved, and \cite{GMR15} perfectly illustrates the flexibility of the generalized coupling approach, which appears to be well applicable in complicated models. In Section \ref{s32} we show that just minor modifications in the construction of the generalized couplings from \cite{GMR15} make them applicable in our Theorems \ref{conv1} and \ref{conv2}, as well, providing asymptotic stability (almost) for free and thus illustrating the power of our approach.
\section{Main results}\label{mainresults} \subsection{Basic definitions and notation}\label{basic} Let $(E,\rho)$ be a Polish (i.e.~separable, complete metric) space with Borel $\sigma$-algebra $\mathcal {E}$, and let $X=\{X_n, n\in {\mathbb{Z}}_+\}$ be a Markov chain with state space $(E, \mathcal{E})$, where ${\mathbb{Z}}_+:=\{0,1,\cdots\}$. Transition probabilities and $n$-step transition probabilities for $X$ are denoted respectively by $P(x,\mathrm{d} y)$ and $P_n(x,\mathrm{d} y)$. Let $E^\infty:=E^{{\mathbb{Z}}_+}$. The law of the sequence $\{X_n\}$ in $(E^\infty, \mathcal{E}^{\otimes \infty})$ with initial distribution $\mathrm{Law}\, (X_0)=\mu$ is denoted by $\mathbb{P}_\mu$, the respective expectation is denoted by $\mathbb{E}_\mu$; in case $\mu=\delta_x$ we write simply $\mathbb{P}_x, \mathbb{E}_x$.
Recall that an invariant probability measure for $X$ is a probability measure $\mu$ on $(E, \mathcal{E})$ such that \begin{equation}\label{invar} \mu(\mathrm{d} y)=\int_EP(x,\mathrm{d} y)\mu(\mathrm{d} x). \end{equation} Equivalently, a probability measure $\mu$ is invariant if the sequence $\{X_n, n\in \mathbb{Z}_+\}$ is strictly stationary under $\mathbb{P}_\mu$. An invariant probability measure $\mu$ for $X$ is {\em ergodic}, if the left shift on the space $(E^\infty, \mathcal{E}^{\otimes \infty})$ is ergodic with respect to $\mathbb{P}_\mu$. Recall that a strictly stationary sequence $\zeta_n, n\in \mathbb{Z}_+$ is called \emph{mixing} if for any bounded measurable functions $f,g:E\to \mathbb {R}$ \begin{equation}\label{mixing} \mathbb {E} f(\zeta_0)g(\zeta_{n})\to \mathbb {E} f(\zeta_0)\mathbb {E} g(\zeta_0), \quad n\to \infty. \end{equation}
For a measurable space $(S, \mathcal{S})$, we denote the set of all probability measures on $(S, \mathcal{S})$ by $\mathcal{P}(S)$.
For given $\mu, \nu\in \mathcal{P}(S)$, define $$ C(\mu,\nu)=\Big\{\xi\in \mathcal{P}(S\times S):\pi_1(\xi)=\mu,\, \pi_2(\xi)=\nu\Big\}, $$ where $\pi_i(\xi)$ denotes the $i$-th marginal distribution of $\xi, i=1,2$. Any $\xi\in C(\mu,\nu)$ is called a \emph{coupling} for $\mu,\nu.$ We also introduce the following two extensions of the notion of a coupling. Recall that $\mu \ll \nu$ means that $\mu$ is absolutely continuous with respect to $\nu$ and $\mu \sim \nu$ means that $\mu$ and $\nu$ are equivalent, i.e. mutually absolutely continuous. Define $$ \widetilde C(\mu,\nu)=\Big\{\xi\in \mathcal{P}(S\times S):\pi_1(\xi)\sim \mu, \pi_2(\xi)\sim\nu\Big\}, $$ $$ \widehat C(\mu,\nu)=\Big\{\xi\in \mathcal{P}(S\times S):\pi_1(\xi)\ll\mu, \pi_2(\xi)\ll\nu\Big\}, $$ and call any probability measure from one of the classes $\widetilde C(\mu,\nu), \widehat C(\mu,\nu)$ a \emph{generalized coupling} for $\mu,\nu.$
In what follows, we use this notation mainly in the following two frameworks: (a) $S=E$, (b) $S=E^\infty$; that is, the probability measures are considered
either on
the initial state space or on the trajectories space. To distinguish the notation, we denote probability measures by $\mu,\nu, \dots$ and
$\mathbb P,{\mathbb{Q}}, \dots$
respectively in the first and the second cases.
For given $p>1$ and $R \ge 1$, denote by $\widehat C^R_p(\mathbb P,{\mathbb{Q}})$ the set of generalized couplings $\xi\in \widehat C(\mathbb P,{\mathbb{Q}})$ such that $$ \left(\int_{E^\infty}\left({\mathrm{d} \pi_1(\xi)\over \mathrm{d}\mathbb P}\right)^p\,\mathrm{d}\mathbb P\right)^{1/p}\leq R, \quad \left(\int_{E^\infty}\left({\mathrm{d}\pi_2(\xi)\over \mathrm{d}{\mathbb{Q}}}\right)^p\,\mathrm{d}{\mathbb{Q}}\right)^{1/p}\leq R. $$
Let $(S, \rho)$ be a metric space and $h:S\times S\to [0,1]$ be a \emph{distance-like function}; that is, $h$ is symmetric, lower semicontinuous, and $h(x,y)=0\Leftrightarrow x=y$. The associated \emph{minimal} (or \emph{coupling}) distance on $\mathcal{P}(S)$ (denoted by the same letter $h$) is defined by $$ h(\mu,\nu)=\inf_{\eta\in C(\mu,\nu)}\int_{S\times S} h(x,y)\, \eta(\mathrm{d} x,\mathrm{d} y), \quad \mu,\nu\in \mathcal{P}(S). $$ When $\rho\leq 1$ and $h(x,y)=\rho(x,y),$ the above definition coincides with the definition of the Kantorovich-Rubinshtein metric (also commonly called 1-Wasserstein metric) on $\mathcal{P}(S)$, and it is well known that $\mathcal{P}(S)$ with this metric is a Polish space; cf. \cite{Dudley}, Chapter 11.
Without loss of generality we assume furthermore that the metric $\rho$ on $E$ satisfies $\rho\leq 1$ (otherwise we introduce an equivalent metric $\rho\wedge 1$). We define the metric $\rho^{(\infty)}$ on $E^\infty$ by $$ \rho^{(\infty)}(x,y)=\sum_{n=0}^\infty{1\over 2^{n+1}}d(x_n,y_n), \quad x=(x_n)_{n\geq 0}, \, y=(y_n)_{n\geq 0}\in E^\infty, $$ and the metric $\rho^{(\infty,\infty)}$ on $E^\infty\times E^\infty$ by $$ \rho^{(\infty,\infty)}\Big((x,x'), (y, y')\Big)=\rho^{(\infty)}(x,y)+\rho^{(\infty)}(x', y'). $$ We consider $\mathcal{P}(E)$, $\mathcal{P}(E^\infty)$, and $\mathcal{P}(E^\infty\times E^\infty)$ as Polish spaces w.r.t.~the corresponding Kantorovich-Rubinshtein metrics $\rho,\rho^{(\infty)}$ and $\rho^{(\infty,\infty)}$. The metric $\rho$ on $\mathcal{P}(E)$ induces weak convergence which we will denote by $\Rightarrow$.
Recall the following facts about the structure of the set of invariant probability measures; e.g. \cite{DZ96}, Section 3.2 or \cite[Theorem 5.7]{H06}: \begin{itemize}
\item The set $\mathcal{I}_X$ of the invariant probability measures for $X$ is a convex compact set in $\mathcal{P}(E)$.
\item Each two different ergodic invariant probability measures are mutually singular.
\item Every extreme point of the set $\mathcal{I}_X$ is an ergodic invariant probability measure,
and each invariant probability measure $\mu$ has a representation of the form
\begin{equation}\label{decomp} \mu=\int_{\mathcal{P}(E)} \nu \, \kappa (\mathrm{d}\nu),
\end{equation} where $\kappa$ is a probability measure on the space $\mathcal{P}(E)$ which is concentrated on the extreme points of the set $\mathcal{I}_X$. \end{itemize}
Together with the initial metric $\rho$ on $E$, we will consider another metric $d$, and assume that it is bounded and continuous with respect to the metric $\rho$. The metric $d$ is not assumed to be complete. All measurability and continuity statements refer to $\rho$ rather than $d$ unless we explicitly say something different. Considering two metrics, one to deal with measurability issues and one to prove convergence, is motivated by applications to SPDE models, see Section \ref{s32} below. In many cases of interest however one can avoid such complications and choose $d$ and $\rho$ the same.
When $d\not=\rho$, we denote by $\overline{E}^d$ the completion of $E$ with respect to $d$, and regard $E$ as a subset in $\overline{E}^d$. Note that $(\overline{E}^d,d)$ is a Polish space (where we denote the extended metric again by $d$). We also assume that for any $y\in E$ there exist a sequence of $d$-continuous functions $\rho_n^y:\overline{E}^d\to [0, \infty)$ such that for $x\in \overline{E}^d$ \begin{equation}\label{approx} \rho_n^y(x)\to \left\{
\begin{array}{ll}
\rho(x,y), & x\in E \\
\infty, & \hbox{otherwise}
\end{array}
\right.,\quad n\to \infty; \end{equation}
cf. \cite{GMR15}, Appendix A. This ensures that the image in $\overline{E}^d$ of any open ball in $E$ is a $d$-Borel subset. Because $(E,d)$ is separable, this implies that $E$ is a $d$-Borel subset in $\overline{E}^d$ and guarantees that the \emph{trace $\sigma$-algebra}
$$
\{A\cap E, A\in \mathcal{B}(\overline{E}^d)\}
$$
on $E$ coincides with $\mathcal {E}$, hence allowing us to identify $\mathcal{P}(E)$ with the set of measures from $\mathcal{P}(\overline{E}^d)$ which provide a full measure for $E$.
We will use a separate notation $\mathop{\Rightarrow}\limits^d$ for the weak convergence in $\mathcal{P}(E)$ with respect to $d$. We will call the Markov chain {\em $d$-Feller} if for each bounded and $d$-continuous function $f:E \to \mathbb{R}$, the map $x \mapsto \int f(y)P(x,\mathrm{d} y)$ is $d$-continuous.
Finally, recall that $X$ is called an \emph{e-chain} with respect to the metric $d$, if its transition probability function is \emph{$d$-equicontinuous}: for any $x\in E, \varepsilon>0$ there exists $\delta>0$ such that $$ d(P_n(x,\cdot), P_n(y,\cdot))\leq \varepsilon, \quad n\geq 0, \quad d(x,y)<\delta. $$ We note that our definition is equivalent to that in \cite[Definition 2.1]{KPS10} by the Kantorovich-Rubinshtein duality theorem, see \cite{Dudley}, Chapter 11.
\subsection{Main theorems}
Our first main result is aimed at uniqueness of an invariant probability measure. It is a slight generalization of \cite[Theorem 1.1]{HMS11}.
\begin{theorem}\label{unique} Let $\mu_1$ and $\mu_2$ be ergodic invariant probability measures. Assume that for some set $M\in \mathcal {E}\otimes \mathcal {E}$ with $\mu_1 \otimes \mu_2 (M)>0$ for each $(x,y)\in M$ there exists $\alpha_{x,y}>0$ such that for each $\varepsilon>0$ there exists $\xi_{x,y}^\varepsilon \in \widehat C(\mathbb P_x,\mathbb P_y)$ which satisfies \begin{equation}\label{limsup} \limsup_{n \to \infty} \frac 1n \sum_{i=0}^{n-1}\xi_{x,y}^\varepsilon \big( d(X_i,Y_{i})\le \varepsilon\big) \ge \alpha_{x,y}. \end{equation}\ Then $\mu_1=\mu_2$. \end{theorem}
Theorem \ref{unique} combined with the representation \eqref{decomp} immediately implies the following statement.
\begin{corollary}\label{corounique} Let $M \in \mathcal {E}$ be such that $\mu(M)>0$ for every invariant probability measure $\mu$ and for each $x,y\in M$ there exists some $\alpha_{x,y}>0$ such that for each $\varepsilon>0$ there exists $\xi_{x,y}^\varepsilon \in \widehat C(\mathbb P_x,\mathbb P_y)$ which satisfies \begin{equation}\label{limsupnew} \limsup_{n \to \infty} \frac 1n \sum_{i=0}^{n-1}\xi_{x,y}^\varepsilon \big( d(X_i,Y_{i})\le \varepsilon\big) \ge \alpha_{x,y}. \end{equation}
Then there exists at most one invariant probability measure, and if there exists one this measure is ergodic. \end{corollary}
The second main result provides the weak convergence of transition probabilities to an invariant probability measure $\mu$ in a somewhat unusual form of the ``weak convergence in probability''.
\begin{theorem}\label{conv1} I. Assume that $X$ is $d$-Feller. Let $\mu$ be an invariant probability measure, and assume that for some $M \in \mathcal {E}\otimes \mathcal {E}$ with $(\mu\otimes \mu)(M)=1$ the following condition holds true for each $(x,y)\in M$: \begin{itemize} \item[\rm{(i)}] $$\lim_{\varepsilon \downarrow 0+}\liminf_{n \to \infty} \sup_{\xi \in C(\mathbb P_x,\mathbb P_y)} \xi\big( d(X_n,Y_n) \le \varepsilon \big) >0. $$ \end{itemize} Then \begin{equation}\label{weak_in_prob} \mu\left(x:d\Big(P_n(x,\cdot),\mu\Big)>\varepsilon\right)\to 0, \quad n\to \infty, \quad \varepsilon>0; \end{equation} that is, $P_n(\cdot,\cdot),\, n\geq 0$, considered as a sequence of $\mathcal{P}(E)$-valued random elements on $(E, \mathcal{E},\mu)$, $d$-converges in probability to the random element identically equal to $\mu$.
II. The following condition is equivalent to (i):\ \begin{itemize} \item[\rm{(ii)}] For some $p >1$, $R \ge 1$, $$ \lim_{\varepsilon \downarrow 0+}\liminf_{n \to \infty} \sup_{\xi \in \widehat C^R_p(\mathbb P_x,\mathbb P_y)} \xi\big( d(X_n,Y_n) \le \varepsilon \big) >0. $$ \end{itemize} Further, the following condition implies (ii) (and (i)): \begin{itemize} \item[\rm{(iii)}] $$\sup_{\xi \in \widehat C(\mathbb P_x,\mathbb P_y)}\lim_{\varepsilon \downarrow 0+}\liminf_{n \to \infty} \xi \big( d(X_n,Y_n)\le \varepsilon\big) >0.$$ \end{itemize} \end{theorem}
Combining the two statements of the theorem, we directly get the following corollary, formulated in terms similar to those used in Theorem \ref{unique}.
\begin{corollary}\label{coroconverge} Let $X$ be $d$-Feller, $\mu$ be an ergodic invariant probability measure and $M \in \mathcal {E} \otimes \mathcal {E}$ be such that $\mu \otimes \mu(M)=1$, and assume that for each $(x,y) \in M$ there exist $\xi_{x,y}\in \widehat C(\mathbb P_x,\mathbb P_y)$ and $\alpha_{x,y}>0$ such that $$ \liminf_{n \to \infty} \xi_{x,y} \big( d(X_n,Y_n)\le \varepsilon\big) \geq\alpha_{x,y} $$ for every $\varepsilon>0$. Then \eqref{weak_in_prob} holds true. \end{corollary}
In the following theorem a stronger (and more typical) type of convergence is obtained at the cost of making a stronger assumption: the e-chain property (which is essentially the uniform Feller property) instead of the usual Feller one. \begin{theorem}\label{conv11} Let $X$ be an e-chain w.r.t. $d$, and one of the assumptions (i) -- (iii) of Theorem \ref{conv1} hold true.
Then $P_n(x,\cdot)\mathop{\Rightarrow}\limits^d \mu$ for $\mu$-a.a. $x\in E$. \end{theorem}
A proper benchmark for Theorem \ref{conv11} is the \emph{modified Doob theorem}, given in \cite[Theorem 2]{KS15}. Let, for a while, $d(x,y)=1_{x\not=y}$ be the discrete metric (which is however is not included into our setting since it is not continuous). Then by the \emph{Coupling Lemma} (e.g. \cite[Lemma 1]{KS15}) the corresponding probability distance equals $1/2$ of the total variation distance. In \cite[Theorem 2]{KS15} it is assumed that for $\mu\otimes\mu$-a.a. $(x,y)$ there exists $n=n_{x,y}$ such that $P_n(x, \cdot)\not\perp P_n(y, \cdot)$, which by the Coupling Lemma is equivalent to existence of a coupling $\xi_{x,y}$ and positive $\alpha_{x,y}$ such that $$ \xi_{x,y} \big( d(X_n,Y_n)=0\big)\geq \alpha_{x,y}. $$ One can extend the coupling $\xi_{x,y}$ in such a way that $X_N=Y_N, N\geq n$, hence the above assumption actually coincides with the one from Corollary \ref{coroconverge}. That is, Theorem \ref{conv11} is a direct analogue of the modified Doob theorem, which operates with weak convergence of the transition probabilities instead of total variation convergence.
Note that the discrete metric $d$ is \emph{non-expanding}: since the discrete metric $d$ takes values $0,1$ only, $$ d(P_n(x,\cdot), P_n(y,\cdot))\leq d(x,y), \quad x,y\in E, \quad n\geq 1. $$ This property has the same meaning as the e-chain property, which in Theorem \ref{conv11} is imposed as an additional assumption because general
metric $d$ may fail to be non-expanding. The ergodicity under the e-chain (actually, the e-process) property was systematically studied in \cite{BKS}, \cite{KPS10}, see also \cite[Theorem 3.7]{HMS11}, where the e-chain property was used essentially without naming it explicitly. We remark that the e-chain property, although being quite typical for ergodic processes, \emph{does not follow} from the fact that the transition probabilities converge to the (unique) invariant probability distribution: see Example \ref{ex54} below, which in particular shows that Proposition 6.4.2 in \cite{MeTwee} is incorrect. In that concern, the clearly seen advantage of Theorem \ref{conv1} is that there we avoid the quite non-elementary (and sometimes not easy to verify) e-chain assumption. We remark that both of the proofs of Theorem \ref{conv1} and Theorem \ref{conv11} exploit the typical ``coupling'' idea. Namely, we make one ``coupling attempt'' with the probability of success being close to the presumably maximal possible one, and then we show that if the latter probability is $<1$, another ``coupling attempt'' will increase the overall probability of success significantly. In that strategy of the proof, a kind of the ``non-expansion'' property is crucial in order to preserve the positive result of the first ``coupling attempt''. We note that, in the proof of Theorem \ref{conv1}, only the basic $d$-Feller property is used to provide such ``non-expansion''.
Theorem \ref{conv1} provides the following important corollary.
\begin{corollary}\label{coromixing} Under the conditions of Theorem \ref{conv1}, the (stationary) chain $X$ is mixing w.r.t. $\mathbb P_\mu$. \end{corollary}
This corollary gives a good prerequisite for our third main result, which provides a sufficient condition for the transition probabilities $P_n(x,.)$ of a given $x \in E$ to converge to the invariant measure $\mu$.
\begin{theorem}\label{conv2} Let $\mu$ be an invariant probability measure and $X$ be mixing w.r.t. $\mathbb P_\mu$. Fix $x \in E$ and assume that there exists a set $M \in \mathcal {E}$ such that $\mu(M)>0$ and for every $y \in M$ there exists some $\xi_{x,y} \in \widehat C(\mathbb P_x,\mathbb P_y)$ such that $\pi_1(\xi_{x,y})\sim \mathbb P_x$ and \begin{equation}\label{prob1} \lim_{n \to \infty} \xi_{x,y} \big( d(X_n,Y_n) \le \varepsilon\big) =1 \end{equation} for every $\varepsilon>0$. Then $P_n(x,\cdot)\mathop{\Rightarrow}\limits^d \mu$. \end{theorem}
Combining Corollary \ref{coroconverge}, Corollary \ref{coromixing}, and Theorem \ref{conv2} we easily derive the following corollary, which provides weak convergence of $P_n(x, \cdot)$ for any starting point $x$ in terms of generalized couplings.
\begin{corollary}\label{coroconv} Let $X$ be $d$-Feller, and assume that for any $(x,y) \in E\times E$ there exists some $\xi_{x,y} \in \widehat C(\mathbb P_x,\mathbb P_y)$ such that $\pi_1(\xi_{x,y})\sim \mathbb P_x$ and \eqref{prob1} holds true for every $\varepsilon>0$.
Then there exists at most one invariant probability measure, and if such a measure $\mu$ exists then $P_n(x,\cdot)\mathop{\Rightarrow}\limits^d \mu$ for any $x\in E$. \end{corollary}
\begin{remark}\label{rem29} The assumption of Corollary \ref{coroconv} is well designed to be easily applied in various particular settings; we illustrate this in Section \ref{sSDDE} below considering Markov chains generated by stochastic functional delay equations (SFDEs) and stochastic partial differential equations (SPDEs). On the other hand, this assumption is quite precise and can not be essentially weakened. For instance, the required statement may fail if one assumes only \eqref{prob1} for some $\xi_{x,y} \in \widehat C(\mathbb P_x,\mathbb P_y)$ without the additional condition $\pi_1(\xi_{x,y})\sim \mathbb P_x$, see Example \ref{ex55} below. \end{remark}
\begin{remark} The \emph{existence} of an invariant probability measure is a much easier topic, studied in great detail in the literature, and we do not address it here, referring e.g. to \cite{DZ96}. \end{remark}
\begin{remark}\label{rem_cont} All the main statements, formulated above in the discrete-time case, have straightforward analogues in the
continuous-time case. Namely, if it is assumed that the process $X_t, t\in [0,\infty)$ for any $x$ has a c\`adl\`ag modification with $X_0=x$, we can repeat the arguments literally, with the space $E^\infty$ changed to $\mathbb{D}([0,\infty), E)$ and $\mathbb P_x, x\in E$ being the respective laws of $X$ in $\mathbb{D}([0,\infty), E)$. We remark that in some specific but important cases it may happen that the Markov process $X_t, t\in [0,\infty)$ is stochastically continuous, but fails to have a c\`adl\`ag modification. For such an example in the framework of L\'evy driven SPDEs we refer to \cite{6Authors}. In this case the proofs of Theorem \ref{conv1}, Theorem \ref{conv11} and Theorem \ref{conv2} also can be adapted, and the statements of these theorems and respective Corollaries \ref{coroconverge}, \ref{coromixing}, and \ref{coroconv} hold true. The technical difficulty which arise here is that now we do not have a ``good'' space of trajectories, hence the statements of Proposition \ref{jan_cor} and Proposition \ref{prop} on the measurable choice can not be applied directly. These statements can be modified properly, but in order not to overburden the exposition we will not go into further details. \end{remark}
\section{Proofs of Theorem \ref{unique} and Theorem \ref{conv2}}\label{s3} In this section we provide proofs of two of our main results, which are comparatively simple and are mainly based on the Ergodic theorem.
\begin{proof}[Proof of Theorem \ref{unique}]
If $\mu_1\not=\mu_2$ then $\mu_1\perp\mu_2$. The fact that $E$ is Polish implies that for any probability measure $\mu$ and each set $A \in \mathcal {E}$, we have $\mu(A)=\sup \mu(K)$, where the supremum is taken over all compact subsets of $E$ which are contained in $A$ (this is sometimes called {\em Ulam's theorem} or {\em inner regularity} of $\mu$; e.g. \cite{Billingsley}, Theorems 1.1 and 1.4). Therefore, for every $m\geq 1$ there exist compact sets $K^m_{1,2}$ such that $K^m_1\cap K^m_2=\emptyset$ and $\mu_i(K_i^m)>1-1/m$. Since $d$ is continuous and $K^m_{1,2}$ are compact and disjoint, the $d$-distance between $K^m_1$ and $K^m_{2}$ is positive. Then there exists a $d$-Lipschitz function $f^m:E \to [0,1]$ such that $f^m|_{K_1^m}\equiv 0$, $f^m|_{K_2^m}\equiv 1$.
Choose $(x,y) \in M$, and let $\alpha_{x,y}$ be as in the statement of the theorem. We can and will assume in addition that $x,y$ are chosen such that \begin{equation}\label{add} \frac 1n \sum_{i=0}^{n-1}f^m (X_i)\to \int f^m \, \mathrm{d}\mu_1\quad \hbox{a.s. w.r.t. }\mathbb{P}_x, \quad \frac 1n\sum_{i=0}^{n-1}f^m (Y_{i})\to \int f^m\, \mathrm{d}\mu_2\quad \hbox{a.s. w.r.t. }\mathbb{P}_y \end{equation} for every $m\geq 1$. Take $m_0>2/\alpha_{x,y}$ and fix $\varepsilon >0$ such that $$ \varepsilon <\frac{\alpha_{x,y}-2/m_0}{\alpha_{x,y} \mathrm{Lip}(f^{m_0})}. $$ Let $\xi_{x,y}^\varepsilon \in \widehat C(\mathbb P_x,\mathbb P_y)$ be as in the statement of the theorem, then $$ \liminf_{n\to \infty}\mathbb {E}^{\xi_{x,y}^\varepsilon}\left(\frac 1n \sum_{i=0}^{n-1}(f^{m_0} (Y_{i})-f^{m_0} (X_i))\right)\leq (1-\alpha_{x,y})+\varepsilon \mathrm{Lip}(f^{m_0}) \alpha_{x,y}<1-2/{m_0}. $$
Because the distribution of $\{X_i\}$ (resp. $\{Y_i\}$) w.r.t. $\xi_{x,y}^\varepsilon$ is absolutely continuous w.r.t. $\mathbb{P}_x$ (resp. $\mathbb{P}_y$) and $f^{m_0}$ is bounded, it follows from (\ref{add}) that $$ \mathbb {E}^{\xi_{x,y}^\varepsilon}\left(\frac 1n\sum_{i=1}^n(f^{m_0} (Y_{i})-f^{m_0} (X_i))\right)\to \int f^{m_0} \, \mathrm{d}\mu_2 -\int f^{m_0} \, \mathrm{d}\mu_1\geq 1-2/{m_0},\quad n\to \infty, $$ which is a contradiction. Therefore $\mu_1=\mu_2$ follows. \end{proof} In the proof Theorem \ref{conv2} we will use the following proposition, whose proof is postponed to Appendix B. \begin{proposition}\label{jan_cor} Under the assumptions of Theorem \ref{conv2}, there exists a measurable mapping $$ M\ni y\mapsto \xi_{x,y} \in \mathcal{P}(E^\infty\times E^\infty) $$ such that for $\mu$-a.a. $y\in M$ one has $\xi_{x,y} \in \widehat C(\mathbb P_x,\mathbb P_y)$, $\pi_1(\xi_{x,y})\sim \mathbb P_x$, and \eqref{prob1} holds true. \end{proposition}
\begin{proof}[Proof of Theorem \ref{conv2}] Define the measure $\xi\in \mathcal{P}(E^\infty\times E^\infty)$ as follows: $$ \xi(A)={1\over \mu(M)}\int_{M}\xi_{x,y}(A)\mu(dy), \quad A\in\mathcal{E}^{\otimes\infty}\otimes \mathcal{E}^{\otimes\infty}, $$ where $\{\xi_{x,y}, y\in M\}$ is defined in Proposition \ref{jan_cor}. Because $\pi_1(\xi_{x,y})\sim \mathbb P_x, \pi_2(\xi_{x,y})\ll \mathbb P_\mu$ for $\mu$-a.a. $y\in M$, we have that
$\xi\in \widehat C(\mathbb P_x,\mathbb P_\mu)$ and $\pi_1(\xi)\sim \mathbb P_x$. In addition, we have
\begin{equation}\label{prob11} \lim_{n \to \infty} \xi \big( d(X_n,Y_n) \le \varepsilon\big) =1 \end{equation} for every $\varepsilon>0$.
Denote $$ \mu_n(C)=\xi(X_n\in C), \quad \nu_n(C)=\xi(Y_n\in C), \quad C\in\mathcal{E}. $$ Observe first that the family $\{\nu_n, n\geq 1\}\subset \mathcal{P}(E)$ is tight: recall that $\pi_2(\xi)\ll \mathbb P_\mu$ and thus for every $\varepsilon>0$ there exists $\delta>0$ such that for any $B\in \mathcal{E}^{\otimes \infty}$ with $\mathbb P_\mu(B)\leq \delta$ one has $\pi_2(\xi)\big(B\big)\leq \varepsilon$. Therefore $$ \nu_n(K)=\xi\big(Y_n\in K\big)\leq \varepsilon, \quad n\geq 1 $$ if a compact set $K\subset E$ is chosen such that $\mu(K)\geq 1-\delta.$
Since the embedding $(E,\rho)$ into $(\overline{E}^d,d)$ is continuous, this yields that $\{\nu_n, n\geq 1\}\subset \mathcal{P}(\overline{E}^d)$ is tight, as well. Then using using (\ref{prob11}), we deduce that $\{\nu_n, n\geq 1\}\subset \mathcal{P}(\overline{E}^d)$ is tight, as well. Because $\mathbb P_x\ll \pi_1(\xi)$, this finally yields the tightness of $\{P_n(x,.), n\geq 1\}\subset \mathcal{P}(\overline{E}^d)$.
The metric space $(\overline{E}^d,d)$ is complete by the construction and is separable since $(E,\rho)$ is separable. Hence if we assume that
that $P_n(x,.), n\geq 1$ does not weakly converge to $\mu$ w.r.t. $d$, then there exists some probability measure $\nu \neq \mu$ on $\overline{E}^d$ and a subsequence $P_{m_k}(x,.) \mathop{\Rightarrow}\limits^d\nu$. Fix a bounded $d$-Lipschitz continuous function $f:\overline{E}^d\to \mathbb {R}$ such that $\bar f:=\int_E f\,\mathrm{d} \mu \neq \int_{\overline{E}^d} f\,\mathrm{d} \nu$ and put $$ U_n=\frac 1n \sum_{k=1}^n f(X_{m_k}). $$ Recall that the chain $X$ is stationary and mixing w.r.t. $\mathbb P_{\mu}$, hence $$
\mathrm{Cov}_\mu(f(X_m), f(X_n)):=\mathbb {E}_\mu \Big( f(X_m)f(X_n)\Big) -(\bar f)^2\to 0, \quad |n-m|\to \infty. $$ Then $$ \mathbb {E}_{\mu}(U_n-\bar f)^2=\frac 1{n^2}\sum_{i=1}^n\sum_{j=1}^n \mathrm{Cov}_\mu(f(X_{m_i}), f(X_{m_j}))\to 0, \quad n\to \infty,
$$ and furthermore there exists a sequence $\{n_r\}$ such that $U_{n_r}\to \bar f, r\to \infty$ a.s. w.r.t. $\mathbb P_\mu$. Because $\pi_2(\xi)\ll \mathbb P_\mu$, we have finally that $U_{n_r}\to \bar f, r\to \infty$ a.s. with respect to $\pi_2(\xi)$.
Recall that $f$ is Lipschitz continuous, then by (\ref{prob11}) the sequence $$
\Delta_m:=|f(X_m)-f(Y_m)| \leq \mathrm{Lip}(f)d(X_m,Y_m), \quad m\geq 1 $$ converges to $0$ in $\xi$-probability. Because $f$ is bounded, this convergence holds true also in the mean sense, which then implies $$ \frac 1n \sum_{k=1}^n f(X_{m_k})-\frac 1n \sum_{k=1}^n f(Y_{m_k})\to 0, \quad n\to \infty $$ in the mean sense w.r.t. $\xi$. Since $\{Y_n\}$ has the law $\pi_2(\xi)$ w.r.t. $\xi$ and $U_{n_r}\to \bar f, r\to \infty$ a.s. w.r.t. $\pi_2(\xi)$, we have then $$ \Big( \frac 1{n_r} \sum_{k=1}^{n_r} f(X_{m_k}), \quad \frac 1{n_r} \sum_{k=1}^{n_r}f(Y_{m_k})\Big)\to (\bar f,\bar f), \quad r\to \infty $$ in $\xi$-probability,
hence $U_{n_r}\to \bar f, r\to \infty$ in probability with respect to $\pi_1(\xi)$. Because $\mathbb P_x\ll \pi_1(\xi)$, we deduce finally that
$U_{n_r}\to \bar f, r\to \infty$ in probability with respect to $\mathbb P_x$. Since $f$ is bounded, the sequence $U_{n_r}, r\geq 1$ is bounded by the same constant, and then
we have
$$
\mathbb {E}_x U_{n_r}\to \bar f, \quad r\to \infty.
$$ On the other hand, by the assumption $P_{m_k}(x,.) \mathop{\Rightarrow}\limits^d \nu$, we have $\int_{\overline{E}^d} f(z)P_{m_k}(x,\mathrm{d} z)\to \int_{\overline{E}^d} f\mathrm{d}\nu$ and therefore $$\mathbb {E}_x\frac 1n \sum_{k=1}^n f(X_{m_k})=\frac 1n \sum_{k=1}^n\int_E f(z)P_{m_k}(x,\mathrm{d} z)=\frac 1n \sum_{k=1}^n\int_{\overline{E}^d} f(z)P_{m_k}(x,\mathrm{d} z)\to \int_E f \,\mathrm{d} \nu \neq \bar f,\quad n\to \infty,$$ which is a contradiction finishing the proof of the theorem. \end{proof}
\section{Proofs of Theorem \ref{conv1}, Theorem \ref{conv11}, and Corollary \ref{coromixing}}\label{s4} In this section we prove Theorem \ref{conv1}, which is the most complicated of our main results. We also show how (parts of) the proof can be modified in order to obtain Theorem \ref{conv11}, and prove Corollary \ref{coromixing}.
Before proceeding with these proofs, we formulate several auxiliary results.
\begin{proposition}\label{newlemma} I. Let $p >1$ and $R>0$ be fixed. Then for each $\alpha>0$ there exists some $\alpha'>0$ such that the following holds true: for every $\mathbb P,{\mathbb{Q}}\in \mathcal{P}(E^\infty)$ and every $\xi\in \widehat C^R_p(\mathbb P,{\mathbb{Q}})$ there exists some $\zeta \in C(\mathbb P,{\mathbb{Q}})$ such that for each $A \in \mathcal {E}^{\otimes\infty} \otimes \mathcal {E}^{\otimes\infty}$ satisfying $\xi(A)\ge \alpha$ we have $\zeta(A)\ge \alpha'$.
II. For each $\alpha>0$ there exists some $\alpha'>0$ and $R\geq 1$ such that the following holds true: for every $p\geq 1$, every $\mathbb P,{\mathbb{Q}}\in \mathcal{P}(E^\infty)$ and every $\xi\in \widehat C(\mathbb P,{\mathbb{Q}})$ there exists some $\zeta \in \widehat C^R_p(\mathbb P,{\mathbb{Q}})$ such that for each $A \in \mathcal {E}^{\otimes\infty} \otimes \mathcal {E}^{\otimes\infty}$ satisfying $\xi(A)\ge \alpha$ we have $\zeta(A)\ge \alpha'$. \end{proposition}
The proof of this proposition is given in Appendix A.
\begin{proposition}\label{prop} Let $S_1, S_2$ be Polish spaces and $Q:S_1\to \mathcal{P}(S_2)$ be a continuous mapping. Let $h:S_2\times S_2\to [0,1]$ be a distance-like function.
Then there exists a measurable mapping $\eta:S_1\times S_1\ni (x,y)\mapsto \eta_{x,y}\in \mathcal{P}(S_2\times S_2)$ such that $$ \eta_{x,y}\in C(Q(x), Q(y)), \quad \int_{S_2\times S_2}h(x', y')\eta_{x,y}(\mathrm{d} x', \mathrm{d} y')=h(Q(x), Q(y)), \quad x,y\in S_1. $$ \end{proposition}
The proof of this proposition is given in Appendix C.
\begin{corollary}\label{measur_coup} For a given $d$-Feller chain $X$ and given $n\in \mathbb{N}, \varepsilon>0$, denote $$ \gamma_{x,y}^{n,\varepsilon}:=\sup_{\xi \in C(\mathbb P_x,\mathbb P_y)}\xi(d(X_n,Y_n)\le \varepsilon), \quad (x,y)\in E\times E. $$ The following statements hold.
I. The function $\gamma_{\cdot}^{n,\varepsilon}:E\times E\to [0,1]$ is $\mathcal {E} \otimes \mathcal {E}-\mathcal{B}([0,1])$ measurable.
II. There exists a measurable function $$ \xi^{n,\varepsilon}:E\times E\ni(x,y)\mapsto \xi^{n,\varepsilon}_{x,y}\in \mathcal{P}(E^\infty\times E^\infty) $$
such that for every $(x,y)\in E\times E$ the following properties hold: \begin{itemize}
\item[(i)] $\xi^{n,\varepsilon}_{x,y}\in C(\mathbb P_x,\mathbb P_y)$;
\item[(ii)]
$$
\xi^{n,\varepsilon}_{x,y}(d(X_n,Y_n)\le \varepsilon)=\gamma_{x,y}^{n,\varepsilon}.
$$
\end{itemize} \end{corollary}
\begin{proof} Consider the Polish metric space $(\overline{E}^d, d)$. Consider also the space $(\overline{E}^d)^\infty$ with the metric $d^{(\infty)}$ introduced in the same way with the metric $\rho^{(\infty)}$, and the space $\mathcal{P}((\overline{E}^d)^\infty)$ with the Kantorovich-Rubinshtein distance $d^{(\infty)}$; see Section \ref{basic}. By \eqref{approx} we have that the image of $E^\infty$ under the natural embedding is a measurable subset in $(\overline{E}^d)^\infty$, and $\mathcal{P}(E^\infty)$ can be identified as the (measurable) set of those measures from $\mathcal{P}((\overline{E}^d)^\infty)$ which provide a full measure for $E^\infty$. The same remarks are valid for the spaces $(\overline{E}^d)^\infty\times (\overline{E}^d)^\infty$ and $\mathcal{P}((\overline{E}^d)^\infty\times (\overline{E}^d)^\infty)$ which are defined analogously.
Observe that, because the chain $X$ is $d$-Feller, the mapping $$ E\ni x\mapsto \mathbb P_x\in \mathcal{P}((\overline{E}^d)^\infty) $$ is continuous. Hence we can apply Proposition \ref{prop} with $S_1=E, S_2=(\overline{E}^d)^\infty$, $Q(x)=\mathbb P_x, x\in E$, and $$ h(x,y)=\1_{d(x_n, y_n)> \varepsilon} =1-\1_{d(x_n, y_n)\leq \varepsilon}, \quad x=(x_k)_{k\geq 1}, \, y=(y_k)_{k\geq 1}\in (\overline{E}^d)^\infty. $$ Then there exists a measurable function $$ \xi^{n,\varepsilon}:E\times E\ni(x,y)\mapsto \xi^{n,\varepsilon}_{x,y}\in \mathcal{P}((\overline{E}^d)^\infty\times (\overline{E}^d)^\infty) $$ which satisfies properties (i), (ii) in statement II of the corollary. In addition, each $\mathbb P_x, x\in E$ assigns full measure to $E^\infty$, hence by the property (i) each measure $\xi^{n,\varepsilon}_{x,y}, (x,y)\in E\times E$ assigns full measure to $E^\infty\times E^\infty$. Therefore $\xi^{n,\varepsilon}$ can be considered as a measurable mapping taking values in $\mathcal{P}(E^\infty\times E^\infty)$, which completes the proof of statement II.
Statement I follows immediately, because the mapping $$ (x,y)\mapsto \xi^{n,\varepsilon}_{x,y}(d(X_n,Y_n)\le \varepsilon) $$ is measurable.
\end{proof}
\begin{remark} Proposition \ref{prop} and Corollary \ref{measur_coup} give a natural extension of the ``Coupling Lemma for transition probabilities'' (Lemma 1 in \cite{KS15}). This lemma provides a probability kernel which in a point-wise sense minimizes the particular distance-like function $h(x,y)=1_{x\not=y}$, while Proposition \ref{prop} provides such a kernel for an arbitrary distance-like function. The proof of Lemma 1 in \cite{KS15} exploits an explicit construction of a maximal coupling based on the splitting representation of a probability law, and it can not be extended to our current setting. We use instead the general measurable selection theorem which dates back to Kuratovskii and Ryll-Nardzevski theorem combined with some measurability criteria for set-valued maps explained in \cite{Stroock_Varad}, Chapter 12.1. We mention that our proof is similar to that of Lemma 4.13 in \cite{HMS11}, which also provides a probability kernel which is maximal w.r.t. $h$, but in our setting we avoid using an additional assumption on $h$ to be continuous. \end{remark}
\begin{remark}\label{r45} We mention for future reference that the kernel $\xi^{n, \varepsilon}$ can be modified such that it possesses the following additional property, which is a direct analogue of property (ii) of the maximal coupling kernel constructed in \cite[Lemma 1]{KS15}: \begin{itemize}
\item[(iii)] the measure $\xi^{n,\varepsilon}_{x,y}$ conditioned by $\{d(X_n,Y_n)> \varepsilon\}$ is absolutely continuous w.r.t.
$\mathbb P_x\otimes \mathbb P_y$ with the respective Radon-Nikodym density being bounded from above by $(1-\gamma_{x,y}^{n,\varepsilon})^{-1}$. \end{itemize} Namely, denote by $\eta^{n,\varepsilon}_{x,y}$ and $\zeta^{n,\varepsilon}_{x,y}$ the initial measure $\xi^{n,\varepsilon}_{x,y}$ conditioned by the event $\{d(X_n,Y_n)\leq \varepsilon\}$ and by its complement, respectively. Then it is easy to see that the modified function $$ \gamma^{n,\varepsilon}_{x,y}\eta^{n,\varepsilon}_{x,y}+(1-\gamma^{n,\varepsilon}_{x,y})\pi_1(\zeta^{n,\varepsilon}_{x,y})\otimes \pi_1(\zeta^{n,\varepsilon}_{x,y}) $$ satisfies (i) -- (iii); see the proof of Theorem \ref{conv1} below for a more detailed discussion of this construction in a slightly different setting. \end{remark}
\begin{proposition}\label{singular} If $\nu_1$ and $\nu_2$ are singular probability measures on a Polish space $(E,\rho)$, then $$ \lim_{\varepsilon \downarrow 0} \sup_{\zeta \in C(\nu_1,\nu_2)}\zeta \big( d(X,Y)\le \varepsilon\big) =0. $$ \end{proposition}
The proof of this proposition is given in Appendix A.
\begin{corollary}\label{ergomu} Under the conditions of Theorem \ref{conv1}, the measure $\mu$ is ergodic. \end{corollary} \begin{proof} Consider the ergodic decomposition \eqref{decomp} of $\mu$. Then $$ 1=(\mu\otimes\mu)(M)=\int_{E\times E}(\nu_1\otimes \nu_2)(M)\kappa(\mathrm{d} \nu_1)\kappa(\mathrm{d} \nu_2). $$ If $\mu$ is not ergodic then $\kappa$ is non-degenerate and there exist two mutually singular invariant probability measures $\nu_1, \nu_2$ such that \begin{equation}\label{ass} (\nu_1\otimes \nu_2)(M)=1. \end{equation} Define for a given $n\geq 1, \varepsilon>0$ the measure $\eta^{n,\varepsilon}\in \mathcal{P}(E^\infty\times E^\infty)$ by $$ \eta^{n, \varepsilon}=\int_{E\times E} \xi_{x,y}^{n, \varepsilon}\,\nu_1(\mathrm{d} x)\nu_2(\mathrm{d} y), $$ where $\xi_{x,y}^{n, \varepsilon}$ is defined as in Corollary \ref{measur_coup}, and denote by $\zeta^{n, \varepsilon}$ the law of $(X_n, Y_n)$ under $\eta^{n, \varepsilon}$. Because $\pi_1(\eta^{n, \varepsilon})=\mathbb P_{\nu_1}, \pi_2(\eta^{n, \varepsilon})=\mathbb P_{\nu_2}$, and $\nu_1, \nu_2$ are invariant, we have $$ \zeta^{n, \varepsilon}\in C(\nu_1,\nu_2) $$ for any $n\geq 1, \varepsilon>0$.
On the other hand, $$ \zeta^{n, \varepsilon}( d(X,Y)\le \varepsilon)= \int_{E\times E}\gamma_{x,y}^{n, \varepsilon}\,\nu_1(\mathrm{d} x)\nu_2(\mathrm{d} y) $$ and therefore $$ \sup_{\zeta \in C(\nu_1,\nu_2)}\zeta \big( d(X,Y)\le \varepsilon\big)\geq \int_{E\times E}\gamma_{x,y}^{n, \varepsilon}\,\nu_1(\mathrm{d} x)\nu_2(\mathrm{d} y). $$ Denote $$ \gamma_{x,y}^\varepsilon= \liminf_{n \to \infty} \gamma_{x,y}^{n,\varepsilon}, \quad \gamma_{x,y}= \lim_{\varepsilon\to 0+} \gamma_{x,y}^{\varepsilon}, $$ then by the Fatou lemma and the monotone convergence theorem $$ \lim_{\varepsilon \downarrow 0} \sup_{\zeta \in C(\nu_1,\nu_2)}\zeta \big( d(X,Y)\le \varepsilon\big)\geq \int_{E\times E}\gamma_{x,y}\,\nu_1(\mathrm{d} x)\nu_2(\mathrm{d} y). $$ By condition (i) of Theorem \ref{conv1}, we have $\gamma_{x,y}>0$ for any $(x,y)\in M$, hence the above inequality combined with \eqref{ass} contradicts Proposition \ref{singular}. \end{proof}
\begin{corollary}\label{ergomu2} Under the conditions of Theorem \ref{conv1}, the measure $\mu\otimes \mu$ is ergodic for the product chain. \end{corollary} \begin{proof} Denote by the same symbol $d$ the metric on the product space $E\times E$ $$ d\big((x,u), (y,v)\big)=d(x,y)\wedge d(u,v), $$ and by $\mathbb P_{(x,u)}$ the distribution of the product chain with the initial value $(x,u)$. For any $x,y,u,v\in E$ and $n\geq 1, \varepsilon>0$ consider the probability measure on $E^\infty\times E^\infty\times E^\infty\times E^\infty$ $$ \xi^{n,\varepsilon}_{x,y,u,v}=\xi^{n,\varepsilon}_{x,y}\otimes \xi^{n,\varepsilon}_{u,v}, $$ where $\xi^{n,\varepsilon}_{x,y}$ is defined in Corollary \ref{measur_coup}. Then the projections $\pi_{1,3}$ and $\pi_{2,4}$ of this measure on the coordinates 1,2 and 2,4, respectively equal $\mathbb P_{(x,u)}$ and $\mathbb P_{(y,v)}$. On the other hand, $$ \lim_{\varepsilon \downarrow 0+}\liminf_{n \to \infty} \xi^{n,\varepsilon}_{x,y,u,v} \Big( d\big((X_n,U_n),(Y_n,V_n)\big) \le \varepsilon \Big)= \lim_{\varepsilon \downarrow 0+}\liminf_{n \to \infty} \gamma^{n,\varepsilon}_{x,y}\gamma^{n,\varepsilon}_{u,v}\geq \gamma_{x,y}\gamma_{u,v}>0 $$ for any $$ (x,y,u,v)\in M':=M\times M. $$
Thus the above inequality yields the following analogue of
condition (i) of Theorem \ref{conv1} for the product chain: for any $(x,y,u,v)\in M'$, \begin{equation}\label{prod}\lim_{\varepsilon \downarrow 0+}\liminf_{n \to \infty} \sup_{\pi_{1,3}(\xi)=\mathbb P_{(x,u)},\pi_{2,4}(\xi)=\mathbb P_{(y,v)}} \xi\Big( d\big((X_n,U_n),(Y_n,V_n)\big) \le \varepsilon \Big) >0. \end{equation}
Clearly, $(\mu\otimes \mu\otimes \mu\otimes \mu)(M')=1$, hence the required statement follows by the previous corollary. \end{proof}
Now we proceed with the proof of Theorem \ref{conv1}. The second statement of the theorem follows easily: since $$ C(\mathbb P,{\mathbb{Q}}) \subset \widehat C_p^R(\mathbb P,{\mathbb{Q}}) $$
for each $p>1$ and $R \ge 1$, condition (i) immediately implies (ii). The inverse implication follows from the first statement in Proposition \ref{newlemma}
while the second statement in this proposition shows that (iii) implies (ii).
\begin{proof}[Proof of Theorem \ref{conv1}, statement I] Our aim is to prove that for every $\varepsilon>0$ \begin{equation}\label{weak_in_prob2} \Gamma^{n,\varepsilon}:=\int_{E\times E}\,\gamma_{x,y}^{n, \varepsilon}\mu(\mathrm{d} x)\mu(\mathrm{d} y)\to 1, \quad n\to \infty. \end{equation} This yields the required statement. Indeed, for given $\varepsilon>0$ and $n \ge 1$, consider a random element $\eta$ with law $\mu$ and a sequence $Z_k=(X_k,Y_k), k\geq 0$ with $Z_0=(x,\eta)$ and the conditional law of $Z$ under $\sigma(Z_0)$ equal $\xi^{n,\varepsilon}_{x,\eta}$; the measurable mapping $\xi^{n,\varepsilon}$ is introduced in Corollary \ref{measur_coup}. By property (i) of this mapping, the law of $Z$ belongs to $C(\mathbb P_x, \mathbb P_\mu)$. We have assumed $d\leq 1$, hence $$ d\Big(P_n(x,\cdot),\mu\Big)\leq \varepsilon+\xi^{n,\varepsilon}_{x,\eta} \Big(d(X_n, Y_n)> \varepsilon\Big). $$ By property (ii) of the mapping $\xi^{n,\varepsilon}$, we have $$ \xi^{n,\varepsilon}_{x,\eta}\Big(d(X_n, Y_n)\leq \varepsilon\Big)=\int_{E}\,\gamma_{x,y}^{n, \varepsilon}\mu(\mathrm{d} y). $$ Consequently, it follows from \eqref{weak_in_prob2} that $$ \liminf_{n\to \infty}\int_{E}d\Big(P_n(x,\cdot),\mu\Big)\, \mu(\mathrm{d} x)\leq \liminf_{n\to \infty}\left(\varepsilon+(1-\Gamma^{n,\varepsilon})\right)=\varepsilon $$ for any $\varepsilon>0$, which then yields \eqref{weak_in_prob}.\\
Now we proceed with the proof of \eqref{weak_in_prob2}. Take an independent coupling $Z_k=(X_k,Y_k),\,k\in {\mathbb Z}_+$ with $\mathrm{Law}\,(X_0)=\mathrm{Law}\,(Y_0)=\mu$ on a probability space $(\Omega, {\mathcal{F}}, \mathbb P)$ and observe that for every fixed $\varepsilon, n, k$ \begin{equation}\label{submart}
\gamma_{Z_k}^{n+1,\varepsilon}\geq \mathbb {E}[ \gamma_{Z_{k+1}}^{n,\varepsilon}|\mathcal{F}^Z_k]. \end{equation} Indeed, the expression on the left hand side means that one fixes the position of $Z$ at the time instant $k$ and optimizes the probability for the coordinates of $Z$ to stay $\varepsilon$-close at the time instant $n+k+1$, while the expression at the right hand side means that one makes an independent step first, and then optimizes the same the probability; the optimal probability in the second case is smaller because due to the more restricted set of possible couplings.
Using Fatou's lemma and inequality \eqref{submart} we obtain: $$
\mathbb {E}[\gamma_{Z_{k+1}}^{\varepsilon}|\mathcal{F}^Z_k]=
\mathbb {E}[\liminf_n\ \gamma_{Z_{k+1}}^{n,\varepsilon}|\mathcal{F}^Z_k]\leq \liminf_n \mathbb {E}[\gamma_{Z_{k+1}}^{n,\varepsilon}|\mathcal{F}^Z_k]\leq \liminf_n \gamma_{Z_k}^{n+1,\varepsilon}=
\gamma_{Z_k}^{\varepsilon}, $$ where $\gamma^\varepsilon, \gamma$ are defined as in the proof of Corollary \ref{ergomu}. Hence $\gamma_{Z_n}^{\varepsilon}, n\geq 1$ is a non-negative super-martingale for every $\varepsilon>0$, and so is $\gamma_{Z_n}, n\geq 1$. Therefore, the $\mathbb P^Z$-a.s. limits $$
\gamma_{Z_n}\to \gamma, \quad \gamma_{Z_n}^{\varepsilon}\to \gamma^{\varepsilon}, \quad n\to \infty
$$ exist. On the other hand, since $Z$ is stationary, the sequences $\gamma_{Z_n}^{\varepsilon}, n\geq 1$ and $\gamma_{Z_n}, n\geq 1$ are stationary as well, and thus each $\gamma_{Z_n}$ (resp. $\gamma_{Z_n}^{\varepsilon}$) has the same law as $\gamma$ (resp. $\gamma^{\varepsilon}$). By Corollary \ref{ergomu2},
the process $Z$ is ergodic and therefore Birkhoff's ergodic theorem implies $$ \gamma=\lim_{n\to \infty}{1\over n}\sum_{k=1}^n\gamma_{Z_k}, \quad \gamma^{\varepsilon}=\lim_{n\to \infty}{1\over n}\sum_{k=1}^n\gamma_{Z_k}^{\varepsilon} $$ are almost surely constant. We can therefore assume that $\gamma$ and $\gamma^{\varepsilon}$ are deterministic. It follows that $$ \gamma^\varepsilon_{x,y}=\gamma^\varepsilon, \quad \gamma_{x,y}=\gamma $$ for $\mu\otimes\mu$-a.a. $(x,y)\in E\times E$. Observe that $\gamma^\varepsilon\geq \gamma$, and by assumption (i) of the theorem we have $\gamma>0$.
The same reasoning as the one we have used to prove \eqref{submart} shows that $$ \Gamma^{n+1,\varepsilon}\geq \Gamma^{n, \varepsilon}, $$ and clearly $\varepsilon \mapsto \Gamma^{n, \varepsilon}$ is non-decreasing. Hence there exist the limits $$ \Gamma^\varepsilon=\lim_{n\to \infty}\Gamma^{n,\varepsilon}, \quad \Gamma=\lim_{\varepsilon\to 0}\Gamma^\varepsilon. $$ To show \eqref{weak_in_prob2}, we just need to show that $\Gamma=1$. Observe that $\Gamma^{n,\varepsilon}$ equals the maximal probability of the event $\{d(X_n, Y_n)\leq \varepsilon\}$ over all couplings $Z=(X,Y)$ such that $\mathrm{Law}(Z_0)=\mu\otimes\mu$ and the conditional distributions of $X,Y$ conditioned by $\sigma(Z_0)$ equal $\mathbb P_{X_0}, \mathbb P_{Y_0}$, respectively.
Assuming $\Gamma<1$, we will construct for any fixed $\varepsilon>0$ and any $n$ large enough a coupling $Z=(X,Y)$ having the same properties as above such that \begin{equation}\label{terminal} \mathbb P(d(X_n, Y_n)\leq \varepsilon)\geq \Gamma+{(1-\Gamma)\gamma\over 2}. \end{equation} This yields the contradictory inequality $$ \Gamma=\lim_{\varepsilon\to 0}\lim_{n\to \infty}\Gamma^{n, \varepsilon}\geq \Gamma+{(1-\Gamma)\gamma\over 2}>\Gamma, $$ and hence $\Gamma<1$ is impossible. Note however that our proof does not show that $\gamma=1$.
Fix $\gamma'\in (\gamma/2, \gamma)$ and $\Gamma'\in (0, \Gamma)$ close enough to $\Gamma$, so that $$ \Gamma'+(1-\Gamma')\gamma'> \Gamma+{(1-\Gamma)\gamma\over 2}. $$ Then choose $\delta>0$ small enough, so that \begin{equation}\label{delta} (1-\delta)(\Gamma'-2\delta)+\gamma'(1-\Gamma')-\delta> \Gamma+{(1-\Gamma)\gamma\over 2}. \end{equation}
Let us proceed with a preliminary analysis which will give us several auxiliary objects we will use in the construction below. First,
since $\gamma_{x,y}^\varepsilon=\gamma^\varepsilon\geq \gamma$ for $\mu\otimes\mu$-a.a. $(x,y)\in E\times E$, by the definition of $\gamma_{x,y}^\varepsilon$ we have that the time moment $$ T_{x,y}^{\varepsilon}=\min\{T: \gamma_{x,y}^{n,\varepsilon}>\gamma', \quad n\geq T \} $$ is finite for $\mu\otimes\mu$-a.a. $(x,y)\in E\times E$. Fix some $N$ such that $$ (\mu\otimes\mu)\Big((x,y):T_{x,y}^\varepsilon>N\Big)<\delta, $$ and denote $O_N^\varepsilon=\{(x,y):T_{x,y}^\varepsilon\leq N\}$.
Next, recall that $X$ is assumed to be $d$-Feller, and $d$ is continuous. Then for given $\varepsilon>0$, ащк $N\geq 1$ chosen above, and ащк any compact set $K\subset E$ $$ \gamma_{x,y}^{N, \varepsilon}\to 1 $$ when $d(x,y)\to 0, (x,y)\in K\times K$. Fix a compact set $K\subset E$ such that $\mu(K)>1-\delta,$ and choose $\varepsilon_1>0$ such that $$ \gamma_{x,y}^{N, \varepsilon}>1-\delta, \quad d(x,y)\leq \varepsilon_1, \quad (x,y)\in K\times K. $$
Finally, we observe that by the definition of $\Gamma^{\varepsilon_1}\geq \Gamma>\Gamma'$, there exists $N_0\in \mathbb N$ such that $\Gamma^{\varepsilon_1, N_0}\geq\Gamma'$. Hence for arbitrary $n\geq N_0$ there exists a coupling $Z^{n,\varepsilon_1}=(X^{n,\varepsilon_1},Y^{n,\varepsilon_1})$ such that $\mathrm{Law}(Z_0^{n,\varepsilon_1})=\mu\otimes\mu$, the conditional distributions of $X^{n,\varepsilon_1},Y^{n,\varepsilon_1}$ conditioned by $\sigma(Z_0^{n,\varepsilon_1})$ equal $\mathbb P_{X_0^{n,\varepsilon_1}}, \mathbb P_{Y_0^{n,\varepsilon_1}}$ respectively, and \begin{equation}\label{ineq} \mathbb P\Big(d(X_{n}^{n,\varepsilon_1}, Y_{n}^{n,\varepsilon_1})\leq \varepsilon_1\Big)\geq \Gamma'. \end{equation} We modify this coupling by the same construction we have mentioned in Remark \ref{r45}. Namely, denote the law of $Z$ by $\mathbb P^Z$ and consider the set $$ C=\Big\{d(X_{n}^{n,\varepsilon_1}, Y_{n}^{n,\varepsilon_1})\leq \varepsilon_1\Big\}. $$ Then $$
\mathbb P^Z=\Gamma' \mathbb P^Z(\cdot|C)+(1-\Gamma'){\mathbb{Q}}^{Z, \Gamma', C}, $$ where $$
{\mathbb{Q}}^{Z, \Gamma', C}=(1-\Gamma')^{-1}(\mathbb P^Z-\Gamma' \mathbb P^Z(\cdot|C)) $$ is a probability measure on $E^\infty\times E^\infty$. Recall that the projections $\pi_1, \pi_2$ of $\mathbb P^Z$ equal $\mathbb P_\mu$, hence the projections of ${\mathbb{Q}}^{Z, \Gamma', C}$ are absolutely continuous w.r.t. $\mathbb P_\mu$ with their Radon-Nikodym derivatives $\leq (1-\Gamma')^{-1}$. Taking instead of $\mathbb P^Z$ the measure $$
\Gamma' \mathbb P^Z(\cdot|C)+(1-\Gamma')\pi_1({\mathbb{Q}}^{Z, \Gamma', C})\otimes \pi_2({\mathbb{Q}}^{Z, \Gamma', C}), $$ we obtain a new coupling such that \eqref{ineq} for this coupling still holds true, but in addition the distribution conditioned by the complement to the set $C$ is absolutely continuous w.r.t. $\mathbb P_\mu\otimes\mathbb P_\mu$ with the Radon-Nikodym density bounded by $(1-\Gamma')^{-1}$. With a slight abuse of notation which however does not cause misunderstanding, we denote this modified coupling by the same symbol $Z^{n, \varepsilon_1}$.
Now for an arbitrary $n\geq N_0+N$ we construct the required coupling $Z$ such that \eqref{terminal} holds true. We define $Z$ as follows: \begin{itemize}
\item the law of $Z_k, k\leq n-N$ is the same as the law of $Z_k^{n-N,\varepsilon_1}, k\leq n-N$ (recall that $n-N\geq N_0$ hence $Z^{n-N,\varepsilon_1}$ is well defined);
\item the conditional law of $Z_{l+n-N}, l\geq 0$ w.r.t. $\sigma(Z_k, k\leq n-N)$ equals $\xi^{N, \varepsilon}_{X_{n-N},Y_{n-N}}$, where $\xi^{n,\varepsilon}$ is the function
constructed in Lemma \ref{measur_coup}. \end{itemize}
To estimate the probability of the event $A=\{d(X_{n},Y_{n})\leq \varepsilon\}$, denote $$ B=\{d(X_{n-N},Y_{n-N})\leq \varepsilon_1\}, \quad C=\{Z_{n-N}\in K\times K\}, \quad D=\{Z_{n-N}\in O_{N}^{\varepsilon}\}. $$ Observe that, when conditioned by $B\cap C$, the event $A$ has probability $\geq 1-\delta$ because the components $X,Y$ start $\varepsilon_1$-close from the compact set $K$ and hence by the choice of $N$ they stay $\varepsilon$-close after the time $N$ with probability $\geq 1-\delta$.
On the other hand, when conditioned by $\overline B\cap D$, event $A$ has probability $\geq \gamma'$ by the definitions of the set $O^\varepsilon_N$ and the event $D$, and according to our construction of the coupling $Z$. Therefore $$ \mathbb P(A)\geq (1-\delta)\mathbb P(B\cap C)+\gamma'\mathbb P(\overline B\cap D). $$ Recall that each of the components $X,Y$ has law $\mathbb P_\mu$, hence $$ \mathbb P(B\cap C)\geq \mathbb P(B)-\mathbb P(X_{n-N}\not\in K)-\mathbb P(Y_{n-N}\not\in K)\geq \mathbb P(B)-2\delta. $$ Next, the law of $Z_{n-N}$ conditioned by $\overline B$ is absolutely continuous w.r.t. $\mu\otimes \mu$ with Radon-Nikodym density $\leq (1-\Gamma')^{-1}$. Hence $$
\mathbb P(\overline D|\overline B)\leq (1-\Gamma')^{-1}(\mu\otimes \mu)((E\times E)\setminus O_{N}^{\varepsilon})\leq (1-\Gamma')^{-1}\delta $$ and therefore $$
\mathbb P(\overline B\cap D)=\mathbb P(D|\overline B)(1-\mathbb P(B))\geq \Big(1- (1-\Gamma')^{-1}\delta\Big)(1-\mathbb P(B)). $$ Recall that $\mathbb P(B)=\Gamma'$, so we finally obtain $$ \mathbb P(A)\geq (1-\delta)(\Gamma'-2\delta)+\gamma'(1-\Gamma')-\delta. $$ By \eqref{delta}, this yields \eqref{terminal} and completes the proof. \end{proof}
\begin{proof}[Proof of Theorem \ref{conv11}] Like in the previous proof, it is sufficient to show that for any $\varepsilon>0$ the constant $\gamma^\varepsilon$ constructed above equals 1. Fix $x_0$ in the (topological) support of $\mu$ and observe that by the e-chain property and the triangle inequality for the metric $d$ on $\mathcal{P}(E)$, for any $\kappa>0$ there exists $r>0$ such that \begin{equation}\label{21} d(P_n(x,\cdot), P_n(y,\cdot))\leq \kappa, \quad n\geq 0, \quad x,y\in B(x_0, r), \end{equation} where $B_d(x_0, r)$ is the open ball in $E$ w.r.t. $d$ with center $x_0$ and radius $r$. Note that for any $\varepsilon>0$ and any coupling $\xi\in C(\mathbb P_x, \mathbb P_y)$, \begin{equation}\label{22} \mathbb {E}^\xi d(X_n, Y_n)\geq \varepsilon\xi(d(X_n, Y_n)> \varepsilon)\geq \varepsilon(1-\gamma_{x,y}^{n,\varepsilon}), \end{equation} hence $$ \gamma_{x,y}^{n,\varepsilon}\geq 1-{1\over \varepsilon}\mathbb {E}^\xi d(X_n, Y_n) $$
Since $$ d\Big(P_n(x,\cdot), P_n(y,\cdot)\Big)=\min_{\xi\in C(\mathbb P_x, \mathbb P_y)}\mathbb {E}^\xi d(X_n, Y_n), $$
combining \eqref{21} and \eqref{22} we get $$ \gamma_{x,y}^{n,\varepsilon}\geq 1-{\kappa\over \varepsilon}, \quad n\geq 0, \quad x,y\in B(x_0, r). $$ Because $B(x_0, r)\times B(x_0, r)$ has positive measure $\mu\otimes \mu$ for any $r>0$, and $$ \liminf_{n\to \infty}\gamma^{n, \varepsilon}_{x,y}=\gamma^\varepsilon $$ for $\mu\otimes \mu$-a.a. $(x,y)$, the above inequality yields $$ \gamma^\varepsilon\geq 1-{\kappa\over \varepsilon} $$ for any $\kappa>0$; that is, $\gamma^\varepsilon=1$. \end{proof}
\begin{proof}[Proof of Corollary \ref{coromixing}] Let $g:E \to {\mathbb{R}}$ be Lipschitz continuous w.r.t. $d$. Then $$
\Big|\mathbb {E}_\mu[g(X_n)|X_j, j\leq 0]-\mathbb {E}_\mu g(X_0)\Big|\leq \mathrm{Lip}(g) d( P_n(X_0,\cdot), \mu). $$ Since $\mathrm{Law}\,(X_0)=\mu$, \eqref{mixing} follows from \eqref{weak_in_prob} by the dominated convergence theorem (recall that we assume $d\leq 1$).
For an arbitrary bounded $g$, the usual approximation arguments can be applied since the time shift is an isometry on $L_2(E^\infty, \mathbb P_\mu)$ and the class of $d$-Lipschitz continuous functions is dense in $L_2(E, \mu)$. \end{proof}
\section{Examples}\label{s5}
In this section we give several examples which illustrate the conditions imposed in our main results and clarify the relations of these results with some other available in the field.
\begin{example}\label{examplelimsup} This example shows that assumption \eqref{limsup} in Theorem \ref{unique} cannot be replaced by the assumption \begin{equation}\label{limsupmod} \limsup_{n \to \infty} \xi_{x,y} \big( d(X_n,Y_n)\le \varepsilon\big) =1, \end{equation}
even if the chain is Feller and generalized couplings are replaced by couplings. Consider the torus $E=[0,1)$ equipped with the Euclidean metric $d(x,y)=|y-x|\wedge (1-|y-x|)$ and consider the deterministic map $x \mapsto 2x$ mod 1, $\mu_1=\delta_0$, $\mu_2=\lambda$, where $\lambda$ is the Lebesgue measure on $E$. Both $\delta_0$ and $\lambda$ are invariant and ergodic and for $\lambda$-almost all $y \in E$ there exists a (deterministic) sequence along which the transition probabilities starting from $y$ converge to $\delta_0$ weakly. \end{example}
\begin{example}\label{ex52} This example shows that the assumptions of Theorem \ref{unique} or Corollary \ref{corounique} do not guarantee weak convergence of transition probabilities. Take $E=\{0,1\}$ with transition probabilities $p_{0,1}=p_{1,0}=1$. The assumptions hold with $M=\{(0,0)\}$ respectively $M=\{0\}$ and $\alpha=1$ and there exists a unique invariant measure $\mu$ but the transition probabilities do not converge to $\mu$. Note however that under the assumptions of Theorem \ref{unique} or Corollary \ref{corounique}, for any ergodic invariant measure $\mu$ the (time-){\em averaged} transition probabilities converge to $\mu$ for $\mu$-almost all initial conditions $y \in E$ by the ergodic theorem. \end{example}
\begin{example}\label{ex53} This example shows that Theorem \ref{conv2} fails if \eqref{prob1} is replaced by a corresponding averaged limit.
Consider a deterministic dynamics on an unbounded countable subset $E=\{0,a_1,a_2,...\}$ of $[0,\infty)$ which maps 0 to 0 and $a_1 \to a_2 \to ...$. Choosing the
sequence such that it has only 0 as an accumulation point we can ensure that the chain is Feller. Clearly, $\delta_0$ is an invariant measure. On the other hand,
if the average of the first $n$ members of the sequence converges to 0 then for every $x=a_j\in E$ and $y=0$ the (deterministic) coupling
$X_n=a_{j+n}, Y_n=0, n\geq 0$ satisfies the averaged analogue of \eqref{prob1}. However, if $a_n\not \to 0$, we have $P_n(x,\cdot) \not\Rightarrow \delta_0$
for each $x\not=0$. \end{example}
\begin{example}\label{ex54} This example shows that an ergodic Feller chain is not necessarily an e-chain.
Consider a deterministic dynamics on the unbounded countable set $E=\{0\}\cup\{2^{-k}, k\geq 0\}$ which maps 0 to 0, 1 to 0, and $2^{-k}$ to $2^{-k+1}$, $k\geq 1$. Clearly, the chain is Feller and for every $x\in E$, $P_n(x,\cdot)$ converge as $n\to \infty$ to the unique invariant measure $\mu=\delta_0$ -- even in the total
variation distance. However, for any two points $x,y\in E\setminus\{0,1\}$ with, say, $x>y$ there exists $n\geq 1$ such that
$2^nx=1, 2^ny\in (0,1/2],$ and therefore
$$
\sup_n d(P_n(x,\cdot), P_n(y,\cdot))\geq {1\over 2}.
$$
On the other hand, for any $\delta>0$ there exist $k,m$ large enough so that $d(2^{-k}, 2^{-m})<\delta$. That is, this chain is not an e-chain. Note that this example is also not {\em asymptotically strong Feller} (see \cite{HM06} for the definition of this concept). The example does however satisfy the assumptions of all theorems in Section \ref{mainresults}. \end{example}
\begin{example}\label{ex55} This example shows that Theorem \ref{conv2} may fail if the assumption $\pi_1(\xi_{x,y})\sim \mathbb P_x$ is omitted, and \eqref{prob1} holds true just for $\xi_{x,y}\in \widehat C(\mathbb P_x, \mathbb P_y)$. Consider $E=\{0,1,2,...\}$ with transition probabilities $p_{0,0}=1$ and $p_{i,i-1}=1/3$ and $p_{i,i+1}=2/3$ for $i=1,2,...$. Clearly, $\mu=\delta_0$ is the unique invariant measure, transition probabilities $P_n(x,.)$ do not converge to $\mu$ for $x \neq 0$ and for each $x \in \mathbb N$ there exists some $\xi\in \widehat C(\mathbb P_x,\mathbb P_0)$ such that $X_n \to 0$ almost surely under $\xi$. \end{example}
\begin{example}\label{comparison} This final example clarifies the relation between condition \eqref{prob1} and the condition \begin{equation}\label{conv_as}
\xi_{x,y}(\lim_{n\to \infty}d(X_n, Y_n)=0)>0, \end{equation} which was used in \cite[Theorem 3.1]{HMS11}. Namely, we show that the ``convergence in probability'' type assumption \eqref{prob1} is strictly weaker than the ``convergence with positive probability'' one \eqref{conv_as}. Since this difference may not be too crucial, in order not to overburden the exposition we just outline the construction and omit detailed proofs.
Let $E=[0,1) \times \{-1,1\}$ be equipped with the metric $d((u,i),(v,j))=\widetilde d(u,v) +|j-i|$, where $\widetilde d$ denotes the Euclidean metric on the torus $T=[0,1)$ and let $r \in (0,1)\backslash {\mathbb{Q}}$. Define a Markov operator $P$ on $E$ as follows: for any $x=(u, i)\in E$, \begin{eqnarray*} P\big((u,i),\big\{ (u+r \mbox{ mod } 1,i)\big\} \big)&=&1/2\\ P\big( (u,i),\{(u,-i)\} \big)&=&1/2. \end{eqnarray*} It is clear that $P$ is Feller, and there exists at least one invariant probability measure, namely $\mu=\lambda \otimes \big( \frac 12 \delta_0 + \frac 12 \delta_1\big)$, where $\lambda$ denotes Lebesgue measure on $T$.
In the following we distinguish between {\em components} and {\em coordinates}, the former referring to the first or second element of a pair $(X,Y)\in E^\infty\times E^\infty$, and the latter referring to the first or second element of a point $x=(u,i) \in E$. For any generalized coupling $\xi_{x,y}\in \widehat{C}(\mathbb P_x, \mathbb P_y)$ the distance of the two components remains constant as long as their second coordinates are the same, and if the second coordinates differ the distance is at least two. Therefore the only way the distance of the two components can converge to zero is that they coincide eventually. Hence for any $x=(u,i), y=(v,j)$ such that $u-v$ is {\em not} an integer multiple of $r$ (mod 1), it is clear that there is no $\xi \in \widehat C(\mathbb P_x,\mathbb P_y)$ for which the distance between the two components converges to zero with positive probability. That is, there are no sets $M_1,M_2 \in \mathcal {E}$ of positive $\mu$-measure such that for each $(x,y) \in M_1 \times M_2$ there exists some $\xi_{x,y} \in \widehat C(\mathbb P_x,\mathbb P_y)$ satisfying \eqref{conv_as}.
On the other hand, for any $x,y\in E$ we can find a coupling $\xi_{x,y} \in C(\mathbb P_x,\mathbb P_y)$ such that \eqref{prob1} holds (and hence $P_n(x,\cdot)\Rightarrow\mu$ for any $x$). Fix $x,y\in E$ and define $\xi_{x,y} \in C(\mathbb P_x,\mathbb P_y)$ as a Markov chain $\{(X_n, Y_n), n\geq 0\}$ defined as follows with the function $p(z),z \in [0,1)$ yet to be determined: \begin{itemize}
\item if the second coordinates of $X_{n}, Y_{n}$ differ, then with probability $1/2$, $X_{n+1}$ changes the second coordinate and $Y_{n+1}$ doesn't and the same
holds for $X$ and $Y$ interchanged, so in both cases the second coordinates of $X_{n+1}, Y_{n+1}$ coincide;
\item if the second coordinates of $X_{n}, Y_{n}$ coincide, and the difference (mod 1) between the first coordinates of $X_n, Y_n$ is $z \in [0,1)$, then
the second coordinates of $X_{n+1}$ and $Y_{n+1}$ either change or stay the same simultaneously, with the probability of each of these two possibilities $1/2(1-p(z))$, and the probabilities that the second coordinate of $X_{n+1}$ (resp. $Y_{n+1}$) changes while $Y_{n+1}$ (resp. $Y_{n+1}$) doesn't, are equal $1/2p(z)$. \end{itemize} By construction, if at some moment the second coordinates differ, they become equal immediately afterwards. Consider the sequence $\{Z_n\}$ of differences of the first coordinates of $\{(X_n, Y_n)\}$. If $Z_n=z$ and the second coordinates of $X_n$ and $Y_n$ coincide, then $Z$ will keep taking the value $z$ for a geometric number of steps with expected value $1/p(z)$, then the second coordinates of $X,Y$ will be different for one time unit after which they become the same again and $Z$ takes the values $z,\,z+2r$, and $z-2r$ with probabilities $1/2,\, 1/4$, and $1/4$ respectively. It is not hard to see (and easy to believe) that if the continuous function $z \mapsto p(z)$ is chosen such that $p(0)=0$, $p(z)>0$ for $z \neq 0$ and $p(z)$ approaches 0 as $z \to 0$ sufficiently fast, then both $Z_n$ and the indicator $1_{X_n\neq Y_n}$ will converge to 0 in probability since $Z_n$ is very likely to take a value close to 0 when $n$ is large.
\end{example}
\section{Applications: SFDEs and SPDEs}\label{sSDDE}
In this section we illustrate our main results applying them to stochastic functional differential equations (SFDEs) and stochastic partial differential equations (SPDEs).
\subsection{Stochastic delay equations}\label{s31} Denote $C:=C([-1,0],\mathbb {R}^m)$, and for a function or a process $X$ defined on $[-1,t]$ write $X_t (s) := X(t + s), s \in [-1, 0]$. Consider the SFDE \begin{align}\label{SFDE} \mathrm{d} X(t)&=F(X_t)\,\mathrm{d} t + G(X_t)\,\mathrm{d} W(t),\\ X_0&=f \in C, \end{align} where $F:C \to \mathbb {R}^m$ and $G:C \to \mathbb {R}^{m\times m}$ satisfy a global Lipschitz condition with respect to the supremum norm and $W$ is a standard Wiener process in $\mathbb {R}^m$. Assume the non-degeneracy condition \begin{equation}\label{non-deg}
\sup_{f \in C}\big | G^{-1}(f))\big |<\infty, \end{equation} where $G^{-1}(f)$ denotes the generalized (or Moore-Penrose) inverse matrix of $G(f)$, $f\in C$.
This model was well studied in \cite{HMS11}, where it was proved that the $C$-valued solution process $X_t$, $t \ge 0$ is uniquely defined, is a Feller process, and has at most one invariant probability measure $\mu$ in which case all transition probabilities converge to $\mu$ weakly (for ease of exposition we have imposed slightly stronger assumptions on $F$ and $G$ compared to \cite{HMS11}).
Here we use this model to benchmark our results. Namely, we will show that these results can be applied yielding the same conclusions, but in a considerably easier and more straightforward way.
Like in \cite{HMS11}, we fix a pair of initial conditions $f$ and $g$ in $C$, and consider the pair of equations $$ \begin{aligned}
\mathrm{d} X(t)&=F(X_t)\,\mathrm{d} t +G(X_t)\,\mathrm{d} W(t),&X_0=f,\\ \mathrm{d} Y(t)&=F(Y_t)\,\mathrm{d} t + \lambda(X(t)-Y(t))\,\mathrm{d} t + G(Y_t)\,\mathrm{d} W(t),\quad &Y_0=g. \end{aligned} $$ It is shown in \cite{HMS11}, Section 3 that if $\lambda >0$ is sufficiently large (when compared with the Lipschitz constants for $f,g$), then with probability 1 $$
|X(t)-Y(t)|\to 0, \quad t\to \infty $$ exponentially fast, and thus \begin{equation}\label{delta_2}
\int_0^\infty \big |X(t)-Y(t)\big |^2 \,\mathrm{d} t <\infty \end{equation} (the proofs are not very long and are based on basic stochastic calculus arguments). Observe that the equation for $Y$ can be re-written to the form \begin{equation}\label{SDDE} \mathrm{d} Y(t)=F(Y_t)\,\mathrm{d} t + \lambda(X(t)-Y(t))\,\mathrm{d} t + G(Y_t)\,\mathrm{d} \widehat W(t),\quad Y_0=g \end{equation} with $$ \widehat W(t)=W(t)+\int_0^t\beta_s\,\mathrm{d} s, \quad \beta_t:=\lambda(X(t)-Y(t))G^{-1}(Y_t). $$ Combining \eqref{delta_2} with \eqref{non-deg}, we see that $$ \int_0^\infty\beta_t^2\, dt<\infty $$ with probability 1. Then by the Girsanov theorem the law of $\widehat W$ on $C([0, \infty), \mathbb {R}^m)$ is absolutely continuous w.r.t. the law of the Wiener process $W$; cf. \cite{LipSher}, Theorem 7.4. Because $Y$ is the strong solution to \eqref{SDDE}, this yields immediately that the law of of $Y(t), t\in [-1, \infty)$ is absolutely continuous with respect to the law of the solution to \eqref{SFDE} with initial condition $g$. On the other hand, $X$ is just the solution to \eqref{SFDE} with initial condition $f$, hence the joint law $\xi$ of the pair $X,Y$ is a generalized coupling from the class $\widehat{C}(\mathbb P_f, \mathbb P_g)$ which satisfies the additional condition $\pi_1(\xi)=\mathbb P_f$. Applying the continuous-time version of Corollary \ref{coroconv}, we directly obtain weak convergence of all transition probabilities to the unique invariant probability measure (in the case it exists).
We note that the simple construction explained above can not be applied directly within the approach developed in \cite{HMS11}. Theorem 3.1 in \cite{HMS11}, which provides uniqueness of the invariance measure, exploits a generalized coupling which belongs to the class $\widetilde{C}(\mathbb P_f, \mathbb P_g)$. It is difficult to guarantee the \emph{equivalence} of the law of $Y$ to $\mathbb P_g$ using just the Girsanov theorem; this is the reason why in the proof of uniqueness in \cite{HMS11} a more sophisticated construction of the generalized coupling is used which involves localization in time. The proof of Theorem 3.7 in \cite{HMS11}, which states the weak convergence of transition probabilities to the invariant measure, contains an extra analysis which actually shows that $X$ is an e-process. None of these additional considerations are required in our approach. This is a clearly seen advantage, which makes it possible to extend the uniqueness results to asymptotic stability (almost) for free. Below we show that such a possibility is quite generic and is available as well in SPDE setting.
\subsection{SPDEs}\label{s32}
In this section we show that in each of the five SPDE models studied in \cite{GMR15} only a minor modification of the constriction of a generalized coupling allows us to apply
Corollary \ref{coroconv} and thus to obtain the asymptotic stability of the model rather than just unique ergodicity. Such a drastic improvement becomes possible thanks to
Theorem \ref{conv1} and Theorem \ref{conv2}, and illustrates the usefulness of these results. To simplify the cross-references, within this section we mainly adopt the notation
from \cite{GMR15} even if it does not correspond to the notation introduced in Section \ref{basic}. The methodology will be similar for all the five models, hence we explain most
details for the first one and then just sketch the argument for the other four. Throughout this section we denote by $H^r$ the Sobolev classes $H^r_2({\mathcal D})$ with a domain ${\mathcal D}$
which varies from
model to model. The $L_2$-norm and the $H^1$-norm are denoted $|\cdot|$ and $\|\cdot\|$ respectively, for all other norms are indicated explicitly.
We also denote by $\lambda_n, n\geq 1$ the increasingly enumerated eigenvalues of an operator $A$, which will be specified in each model separately, and by
$P_N$ the projector onto the span of the
respective first $N$ eigenvectors.
\subsubsection{2D Navier-Stokes on a domain}
Consider the 2D stochastic Navier-Stokes equation on ${\mathcal D}\subset \mathbb {R}^2$ \begin{equation}\label{2dNS1} \mathrm{d} \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} \,\mathrm{d} t = (\nu\Delta \mathbf{u} +\nabla \pi + \mathbf{f}) \,\mathrm{d} t +\sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \nabla \cdot \mathbf{u} = 0, \end{equation} with the unknown velocity field $\mathbf{u} = (u_1, u_2)$ and the unknown pressure $\pi$. The bounded domain ${\mathcal D}$ is assumed to have smooth $\partial{\mathcal D}$, and the no-slip (Dirichlet) boundary condition on $\mathbf{u}$ is imposed: \begin{equation}\label{2dNS2}
\mathbf{u}|_{\partial{\mathcal D}} = 0. \end{equation} The deterministic vector fields $\mathbf{f}, \sigma_1, \dots, \sigma_m\in L_2({\mathcal D})^2$ and independent standard Brownian motions $W_1, . . . ,W_m$ are fixed.
Denote by $V$ the subspace of $H^1({\mathcal D})^2$, which contains $\mathbf{u}$ such that $\nabla \cdot \mathbf{u} = 0$ and $\mathbf{u}\cdot \mathbf{n} = 0$ (where $\mathbf{n}$ denotes the outward normal for $\partial{\mathcal D}$). Denote by $H$ the completion of $V$ w.r.t. the $L_2({\mathcal D})^2$-norm, by $P_H$ the projector in $L_2({\mathcal D})^2$ on $H$, and by $A=-P_H\Delta$ the \emph{Stokes operator}.
It is known that for any $\mathbf{u}_0\in H$ the system \eqref{2dNS1}, \eqref{2dNS2} with the initial data $\mathbf{u}_0\in H$ admits a unique (strong) solution with values in $H$, and this solution depends continuously on $\mathbf{u}_0\in H$. That is, \eqref{2dNS1}, \eqref{2dNS2} defines a Feller Markov process valued in $H$; we refer for details to \cite{GMR15}, Section 3.1.1.
Now we explain the generalized coupling construction for this system. Fix arbitrary $\mathbf{u}_0,\widetilde{\mathbf{u}}_0\in H$ and consider $\mathbf{u}=\mathbf{u}(\cdot , \mathbf{u}_0)$ solving \eqref{2dNS1}, \eqref{2dNS2} with initial data $\mathbf{u}_0$, and $\widetilde{\mathbf{u}}$ solving $$ \mathrm{d}\widetilde{\mathbf{u}} +\widetilde{\mathbf{u}} \cdot \nabla\widetilde{\mathbf{u}} \,\mathrm{d} t = (\nu\Delta\widetilde{\mathbf{u}} + \lambda P_N(\mathbf{u} - ˜\widetilde{\mathbf{u}}) + \nabla \varpi + \mathbf{f}) \,\mathrm{d} t +\sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \nabla \cdot\widetilde{\mathbf{u}} = 0, $$ $$
\widetilde{\mathbf{u}}|_{\partial{\mathcal D}} = 0 $$ with initial data $\widetilde{\mathbf{u}}_0$; here $\lambda$ and the number $N$
are yet to be chosen; recall that $P_N$ is the projector which is defined in the terms of the operator $A$.
This construction is based on the ``stochastic control'' argument, similar to the one developed in \cite{HMS11}, Section 3; see also
\cite{GMR15}, Section 2.4.
The similar coupling construction in \cite{GMR15}, Section 3.1.2 involves an additional localization term $1_{\tau_K>t}$, and the corresponding
generalized coupling is
defined as the conditional law of the pair $({\mathbf{u}}, \widetilde{\mathbf{u}})$ on the set $\{\tau_K=\infty\}$. This gives a generalized
coupling from the class $\widehat C(\mathbb P_{\mathbf{u}}, \mathbb P_{\widetilde{\mathbf{u}}_0})$. Because of the conditioning, the law of the first component have
no reason to be equivalent to $\mathbb P_{\mathbf{u}}$. The latter condition is however crucial for our Theorem \ref{conv2}; see Remark \ref{rem29} and Example \ref{ex55}.
We resolve this difficulty in a similar way we did in Section \ref{s31}. Namely, we remove the localization term and consider the law of the pair
$({\mathbf{u}}, \widetilde{\mathbf{u}})$ as the required generalized coupling. This leads only to minor modifications in the respective calculus, as we
explain below, but it allows to apply our main results in order to derive asymptotic stability.
The difference $\mathbf{v}:=\mathbf{u}-\widetilde{\mathbf{u}}$ satisfies \begin{equation}\label{eq} \mathrm{d} \mathbf{v}-\nu \Delta \mathbf{v}\, \mathrm{d} t+1_{\tau>t}\lambda P_N\mathbf{v}\, \mathrm{d} t=-\nabla \pi+\nabla\varpi
+\widetilde{\mathbf{u}}\cdot \nabla \mathbf{u} +\mathbf{u}\cdot \nabla\widetilde{\mathbf{u}}, \quad \nabla \cdot \mathbf{v}=0, \quad \mathbf{v}|_{\partial{\mathcal D}} = 0. \end{equation} Like in \cite{GMR15}, Section 3.1.2, multiplying \eqref{eq} by $\mathbf{v}$, integrating over ${\mathcal D}$, and using that $\mathbf{u},\widetilde{\mathbf{u}},$ and $\mathbf{v}$ are all divergence free and satisfy the Dirichlet boundary condition, one gets $$ \begin{aligned}
\frac{1}{2}\mathrm{d} |\mathbf{v}|^2 + \nu\|\mathbf{v}\|^2\, \mathrm{d} t+ \lambda |P_N \mathbf{v}|^2 \mathrm{d} t
&\leq \left|\int_{{\mathcal D}} \mathbf{v}\cdot \nabla \mathbf{u}\cdot \mathbf{v}\, \mathrm{d} x\right|\, \mathrm{d} t
\\&\leq C_{\mathcal D} |\mathbf{v}|\|\mathbf{v}\|\|\mathbf{u}\|\, \mathrm{d} t\leq \left(\frac{\nu}{2}\|\mathbf{v}\|^2+
\frac{C_{\mathcal D}}{2\nu}|\mathbf{v}|^2\|\mathbf{u}\|^2\right)\, \mathrm{d} t \end{aligned} $$ with a universal constant $C_{\mathcal D}$ which involves the quantities from Sobolev embedding. By the Poincar\'e inequalities \cite{GMR15} (3.3), for the particular choice $\lambda =\nu \lambda_N/2$ we get $$
\lambda |P_N \mathbf{v}|^2+\frac{\nu}{2}\|\mathbf{v}\|^2\geq \lambda|\mathbf{v}|^2, \quad \lambda\leq \nu \lambda_N/2. $$ Taking $\lambda=\nu \lambda_N/2$ we obtain $$
\mathrm{d} |\mathbf{v}|^2\leq \Big(-\nu \lambda_N1_{\tau>t}+\frac{C_{\mathcal D}}{\nu}\|\mathbf{u}\|^2\Big)|\mathbf{v}|^2\, \mathrm{d} t, $$ and finally by Gronwall's lemma \begin{equation}\label{2Dexp}
|\mathbf{v}(t)|^2\leq |\mathbf{u}_0-\widetilde{\mathbf{u}}_0|^2\exp\left(-\nu\lambda_N t+\frac{C_{\mathcal D}}{\nu}\int_0^t\|\mathbf{u}(s)\|^2\, \mathrm{d} s\right), \quad t\geq 0. \end{equation}
One has with probability 1 \begin{equation}\label{liminf}
\limsup_{t\to \infty}\frac{1}{t}\int_0^t \|\mathbf{u}(s)\|^2\, \mathrm{d} s\leq \frac{|A^{-1/2}\mathbf{f}|^2}{2\nu^2}+\frac{|\sigma|^2}{\nu}, \end{equation}
where $|\sigma|^2:= \sum_{k=1}^m |\sigma_k|^2$; this follows from the energy estimate \cite{GMR15} (3.5). Hence $N$ satisfies \begin{equation}\label{sup}
\lambda_N>C_{\mathcal D}\left(\frac{|A^{-1/2}\mathbf{f}|^2}{2\nu^4}+\frac{|\sigma|^2}{\nu^3}\right), \end{equation} with probability 1 the right hand side term in \eqref{2Dexp} tends to 0 exponentially fast.
On the other hand, consider $\sigma$ as a linear operator $\mathbb {R}^m\to H$ and assume that, for the given $N$, \begin{equation}\label{range} H_N:=P_NH\subset\mathrm{Range}\, (\sigma)=\mathrm{Span}\, (\sigma_k, k=1, \dots, m). \end{equation} Then the corresponding pseudo-inverse operator $\sigma^{-1}: H_N\to \mathbb {R}^m$ is well defined and bounded. Then the principal equation for $\widetilde{\mathbf{u}}$ can be written in the form $$ \mathrm{d}\widetilde{\mathbf{u}} +\widetilde{\mathbf{u}} \cdot \nabla\widetilde{\mathbf{u}} \,\mathrm{d} t = (\nu\Delta\widetilde{\mathbf{u}} + \nabla \varpi + \mathbf{f}) \,\mathrm{d} t +\sum_{k=1}^m \sigma_k \mathrm{d} \widetilde W_k $$ with $$ \widetilde W(t)=W(t)+\int_0^t\beta_s\,\mathrm{d} s, \quad \beta_t:=\lambda \sigma^{-1}P_N\mathbf{v}(t) $$
Since $\sigma^{-1}$ is bounded and $|v(t)|$ tends to $0$ exponentially fast, we have \begin{equation}\label{beta}
\mathbb P\left(\int_0^t\|\beta_s\|^2_{\mathbb {R}^m}\,\mathrm{d} s<\infty\right)=1, \end{equation}
and the law of $\widehat W$ is absolutely continuous w.r.t. the law of $W$. Thus the law of $\widetilde{\mathbf{u}}$ is absolutely continuous w.r.t. the law of the solution
to \eqref{2dNS1}, \eqref{2dNS2} with initial data $\widetilde{\mathbf{u}}_0$. Note that the law of the first component w.r.t. this coupling just equals $\mathbb P_{\mathbf{u}_0}$
and the distance between the components tend to $0$ exponentially fast as $t\to \infty$. Hence the law of the pair $(\mathbf{u}(\cdot),\widetilde{\mathbf{u}}(\cdot))$ can
be used as the coupling required in Corollary \ref{coroconv} with $E=H,\rho=d=|\cdot-\cdot|\wedge 1$. We conclude that in the framework of Proposition 3.1 \cite{GMR15}, which
states unique ergodicity for \eqref{2dNS1}, \eqref{2dNS2}, the following stabilization property actually holds true: \begin{center} \emph{for any $\mathbf{u}\in H$, the transition probabilities $P_t(\mathbf{u}, \cdot)\in \mathcal{P}(H)$ weakly converge as $t\to \infty$ to the unique invariant measure.} \end{center}
\subsubsection{2D Hydrostatic Navier-Stokes Equations}
Next, following \cite{GMR15} Section 3.2, we consider a stochastic version of the 2D Hydrostatic Navier-Stokes equation \begin{equation}\label{HNS1}\begin{aligned}
\mathrm{d} u&+ (u\partial_x u+w\partial_zu+\partial_xp-\nu\Delta u)\, \mathrm{d} t =\sum_{k=1}^m \sigma_k\mathrm{d} W_k,\\ \partial_zp &= 0,\\ \partial_xu &+\partial_z w = 0 \end{aligned} \end{equation} for an unknown velocity field $(u,w)$ and pressure $p$ evolving on the domain ${\mathcal D} = (0,L)\times(−h, 0)$. The boundary $\partial D$ is decomposed into its vertical sides $\Gamma_v = [0,L] \times \{0,−h\}$ and lateral sides $\Gamma_l = \{0,L\} \times [−h, 0]$, where the boundary conditions are imposed: \begin{equation}\label{HNSboundary}
u = 0 \quad \hbox{on $\Gamma_l$}, \quad \partial_zu = w = 0 \quad \hbox{on $\Gamma_v$}. \end{equation} Denote $$
H=\left\{f\in L_2({\mathcal D}):\int_{-h}^0u\, \mathrm{d} z\equiv 0\right\}, \quad V=\left\{u\in H^1({\mathcal D}):\int_{-h}^0u\, \mathrm{d} z\equiv 0, \, u|_{\Gamma_l}=0\right\}. $$ Denote also by $P_H$ the projector in $L_2({\mathcal D})$ on $H$, and put $A=-P_H\Delta$.
It is known (see \cite{GMR15}, Section 3.2.1) that under a proper condition on the family $\{\sigma_k\}$ for a given $u_0\in V$ the system (\ref{HNS1}), \eqref{HNSboundary}
has a unique strong solution, which in addition depends continuously on $u_0\in V$. Thus the system (\ref{HNS1}), \eqref{HNSboundary} defines a Feller Markov process in $E=V$.
Now we explain the generalized coupling construction for (\ref{HNS1}), \eqref{HNSboundary}. For fixed $u_0, \widetilde u_0\in V$, consider the solution $u$ to (\ref{HNS1}), \eqref{HNSboundary} with the initial data $u_0$ and the solution $\widetilde u$ to a similar system with the first equation changed to $$ \mathrm{d} \widetilde u+ (\widetilde u\partial_x \widetilde u+w\partial_z \widetilde u+\partial_x \widetilde p-\nu\Delta \widetilde u+\lambda P_N (u-\widetilde u))\, \mathrm{d} t =\sum_{k=1}^m \sigma_k\mathrm{d} W_k $$ with $\lambda=\nu\lambda_N/2$. One has \begin{equation}\label{Hlip}
|v(t)|\leq \exp\left(-2\lambda t+C\int_0^t\Big(\|u(s)\|^2+|\partial_z u(s)|\|\partial_z u(s)\|\Big)\, \mathrm{d} s\right)|v(0)|, \quad t\geq 0 \end{equation}
with a constant $C$ depending only on $\nu$ and ${\mathcal D}$; see (3.22), \cite{GMR15}. Next, there exists $C_1$ depending only on $\nu, {\mathcal D}$, and $|\sigma|^2+|\partial_z\sigma|^2$ such that \begin{equation}\label{Hliminf}
\limsup_{t\to \infty}\frac{1}{t}\int_0^t \Big(\|u(s)\|^2+\|\partial_z u(s)\|^2\Big)\, \mathrm{d} s\leq C_1 \end{equation}
with probability 1; this follows from the energy estimates (3.16), (3.18) \cite{GMR15}. If $N$ is large enough, so that $2\lambda=\nu\lambda_N>C C_1$, the above inequalities yield that the $H$-norm $|v(t)|$ tends to zero as $t\to \infty$ exponentially fast. If in addition for such $N$ \eqref{range} holds true, then one can interpret $\widetilde u$ as the solution to (\ref{HNS1}), \eqref{HNSboundary} with the initial data $\widetilde u_0$ and $W$ changed to $$ \widetilde W(t)=W(t)+\int_0^t\beta_s\,\mathrm{d} s, \quad \beta_t:=\lambda \sigma^{-1}P_N {v}(t). $$
Since the pseudo-inverse operator $\sigma^{-1}:H_N\to \mathbb {R}^m$ is bounded and $|v(t)|$ decays exponentially fast, we have (\ref{beta}). Hence the law of $\widetilde W$ is absolutely continuous w.r.t. the law of $W$ and therefore the law of $\widetilde u$ in $C([0, \infty), V)$ is absolutely continuous w.r.t. $\mathbb P_{\widetilde u_0}$. Recall that the Markov process which corresponds to (\ref{HNS1}), \eqref{HNSboundary} is well defined and is Feller on $V$. However, it is an easy observation that this process is $H$-Feller, as well. Namely, inequality \eqref{Hlip} actually holds true for any $\lambda\leq \nu\lambda_N/2$, and taking $\lambda=0$ we easily deduce the $H$-continuity of the semigroup.
We take $E=V$, $\rho=\|\cdot-\cdot\|\wedge 1, d=|\cdot-\cdot|\wedge 1$; note that condition \eqref{approx} holds true with $\rho_n^y(x)=|P_n(y-x)|, n\geq 1, y\in E.$ In this setting, we apply continuous time version of Corollary \ref{coroconv} with the generalized coupling $\xi$
defined as the joint law of processes $u(\cdot), \widetilde u(\cdot)$ defined above. We conclude that in the framework of Proposition 3.2 \cite{GMR15}, which states unique
ergodicity for (\ref{HNS1}), \eqref{HNSboundary}, in addition the following (weak) $L_2$-stabilization property holds: \begin{center} \emph{for any $u\in V$, the transition probabilities $P_t(u, \cdot)\in \mathcal{P}(V)$ weakly converge in the $L_2$-topology as $t\to \infty$ to the unique invariant measure $\mu\in\mathcal{P}(V)$.} \end{center}
\subsubsection{The fractionally dissipative Euler model}
Next, following \cite{GMR15} Section 3.3, we consider the fractionally dissipative Euler model, described by the system \begin{equation}\label{dEu} \mathrm{d} \xi+\left(\Lambda^\gamma\xi+\mathbf{u}\cdot \nabla\xi\right)\, \mathrm{d} t=\sum_{k=1}^m\sigma_k \mathrm{d} W_k, \quad \mathbf{u}=\mathcal {K}*\xi \end{equation} for an unknown vorticity field $\xi$ (this is the notation borrowed from \cite{GMR15}, which is not to be mixed with the notation for a coupling we used previously). Here $\Lambda^\gamma = (-\Delta)^\gamma$ is the fractional Laplacian with $\gamma\in (0, 2]$, $\mathcal {K}$ is the Biot-Savart kernel, so that $\nabla^\perp \cdot \mathbf{u}= \xi$ and $\nabla \mathbf{u} = 0$, and \eqref{dEu} is posed on the periodic box $\mathbb {T}^2 = [−\pi, \pi]^2$. In the velocity formulation, \eqref{dEu} has the form \begin{equation}\label{dEu2} \mathrm{d} \mathbf{u} + \Big(\Lambda^\gamma \mathbf{u} + \mathbf{u}\cdot \nabla \mathbf{u}+ \nabla \pi\Big) \,\mathrm{d} t = \sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \nabla \cdot \mathbf{u} = 0, \end{equation} where the unknowns are the velocity field $\mathbf{u}$ and the pressure $\pi$. It is known that for a fixed $r>2$ for any given $\mathbf{u}_0\in H^r$ there exists a unique strong solution to \eqref{dEu2} taking values in $H^r$ and this solution depends on the initial data $\mathbf{u}_0\in H^r$ continuously. That is, \eqref{dEu2} defines a Feller Markov process valued in $H^r$; see \cite{GMR15}, Section 3.3.1.
For fixed $\mathbf{u}_0, \widetilde{\mathbf{u}}_0\in H^r$, consider the function $\mathbf{u}(\cdot)$ solution to \eqref{dEu2} with the initial data $\mathbf{u}_0$ and the function $\widetilde{\mathbf{u}}(\cdot)$ solving $$ \mathrm{d}\widetilde{\mathbf{u}} + \Big(\Lambda^\gamma\widetilde{\mathbf{u}}-\lambda P_N(\mathbf{u}-\widetilde{\mathbf{u}}) +\widetilde{\mathbf{u}}\cdot \nabla \widetilde{\mathbf{u}}+ \nabla \widetilde{\pi}\Big) \,\mathrm{d} t = \sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \nabla \cdot \widetilde{\mathbf{u}} = 0 $$ with the initial data $\widetilde{\mathbf{u}}_0$ and $P_N$ which now denotes the projector which corresponds to the eigenfunctions of $A=\Lambda^\gamma$. This is actually the generalized coupling construction from \cite{GMR15}, Section 3.3.2, where in the additional control term we remove the localization term $1_{\tau_K>t}$. Denote $\mathbf{v}=\mathbf{u}-\widetilde{\mathbf{u}} $, then for $\lambda\leq \lambda_N/2$ $$
|\mathbf{v}(t)|^2\leq \exp\left(-2\lambda t+C\int_0^t\|\xi(s)\|^2_{L_p}\, \mathrm{d} s\right)|\mathbf{v}(0)|^2, \quad t\geq 0 $$ with properly chosen $p>1$ and universal $C$; see \cite{GMR15}, (3.28). On the other hand, there exists a universal $C_1$ such that \begin{equation}\label{dEuliminf}
\limsup_{t\to \infty}\frac{1}{t}\int_0^t \|\xi(s)\|^2_{L_p}\, \mathrm{d} s\leq C_1\|\sigma\|^2_{L_p} \end{equation} with probability 1, where $$
\|\sigma\|_{L_p}=\left(\int_{\mathbb {T}^2}\left(\sum_{k=1}^m\sigma_k^2\right)^{p/2}\, \mathrm{d} x\right)^{1/p}. $$ This follows from the energy estimate (3.31) \cite{GMR15}.
Now we can repeat literally the argument from the previous subsection. Taking in the above calculation $\lambda=0$, we see that the Markov process is $H$-Feller with $H=L_2(\mathbb {T}^2)$. Taking $\lambda=\lambda_N/2$, we get that if, for some $N$, $\lambda_N>C C_1$ and \eqref{range} holds, then the law of the pair $(\mathbf{u}(\cdot),\widetilde{\mathbf{u}}(\cdot))$ can be used as the coupling required in Corollary \ref{coroconv} with
$E=H^r,\rho=\|\cdot-\cdot\|_{H^r}\wedge 1, d=|\cdot-\cdot|\wedge 1$ (again, condition \eqref{approx} is easy to verify). We conclude that in the framework of Proposition 3.3 \cite{GMR15}, which states unique ergodicity for \eqref{dEu2}, in addition the following (weak) $L_2$-stabilization property holds: \begin{center} \emph{for any $u\in H^r$, transition probabilities $P_t(u, \cdot)\in \mathcal{P}(H^r)$ weakly converge in the $L_2$-topology as $t\to \infty$ to the unique invariant measure $\mu\in\mathcal{P}(H^{r})$.} \end{center}
\subsubsection{The damped stochastically forced Euler-Voigt model}
Next, we consider an inviscid ``Voigt-type'' regularization of a damped stochastic Euler equation: \begin{equation}\label{EV} \mathrm{d} \mathbf{u} + \Big(\gamma \mathbf{u} + \mathbf{u}_\alpha\cdot \nabla \mathbf{u}_\alpha+ \nabla p\Big) \,\mathrm{d} t = \sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \nabla \cdot \mathbf{u} = 0, \end{equation} with for some $\gamma>0$ and the unknown vector field $\mathbf{u}$, where the non-linear terms are subject to an $\alpha$-degree regularization $$ (-\Delta)^{\alpha/2}\mathbf{u}_\alpha=\Lambda^\alpha\mathbf{u}_\alpha=\mathbf{u}. $$ The absence of a parabolic regularization mechanism brings specific difficulties to the analysis of the model, we refer to \cite{GMR15}, Sections 3.4.1 -- 3.4.3 for details. Surprisingly, the construction of the generalized coupling which leads to the stability of the model does not bring substantial novelties and can be provided within the same lines we discussed previously. To shorten the exposition, we consider only the case of a 2D model evolving on the periodic box $\mathbb {T}^2$, and assume $\alpha>2/3$. In this case, Proposition 3.4, \cite{GMR15} shows that for any $\mathbf{u}_0\in H^{1-\alpha/2}$ there exists unique strong solution to \eqref{EV} with the initial data, and the corresponding semigroup is Feller w.r.t. $H^{-\alpha/2}$ norm.
Next, for fixed $\mathbf{u}_0, \widetilde{\mathbf{u}}_0\in H^{1-\alpha/2}$, consider the function $\mathbf{u}(\cdot)$ solution to \eqref{EV} with the initial data $\mathbf{u}_0$ and the function $\widetilde{\mathbf{u}}(\cdot)$ solving $$ \mathrm{d} \widetilde{\mathbf{u}} + \Big(\gamma \mathbf{u} -\lambda P_N(\mathbf{u}-\widetilde{\mathbf{u}}) + \widetilde{\mathbf{u}}_\alpha\cdot \nabla \widetilde{\mathbf{u}}_\alpha+ \nabla \widetilde {p}\Big) \,\mathrm{d} t = \sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \nabla \cdot \mathbf{u} = 0 $$ with the initial data $\widetilde{\mathbf{u}}_0$; now $P_N$ denotes the projector on the span of $N$ first elements in the sinusoidal basis. Since $\alpha>2/3$, there exists $\delta>0$ such that $H^{\alpha/2-\delta}\subset L^3$. For such $\delta$, inequalities (3.45), (3.46) \cite{GMR15} provide the following bound for $\mathbf{v}=\mathbf{u}-\widetilde{\mathbf{u}} $: $$
{1\over 2}\mathrm{d} \|\mathbf{v}\|^2_{H^{-\alpha/2}}+\Big(\gamma-C\bigg(\lambda^{-1}+N^{-\delta}\Big)\Big(1+\|\xi\|^2_{H^{-\alpha/2}}\Big)\bigg)\|\mathbf{v}\|^2_{H^{-\alpha/2}}\, \mathrm{d} t\leq 0, $$ where $\xi=\mathrm{curl}\, \mathbf{v}$ and constant $C$ depends only on $\delta, \alpha$. On the other hand, with probability 1 $$
\limsup_{t\to \infty}\frac{\gamma}{t}\int_0^t \|\xi(s)\|^2_{H^{-\alpha/2}}\, ds\leq \|\varsigma\|^2_{H^{-\alpha/2}} $$
with $\varsigma= \mathrm{curl}\, \sigma$; see \cite{GMR15}, Section 3.4.1. Hence, if $\lambda, N$ are taken large enough, $\|\mathbf{v}(t)\|^2_{H^{-\alpha/2}}$ tends to 0 as $t\to \infty$ exponentially fast. Repeating literally the same arguments as before we obtain that, if $H_N\subset \mathrm{Range}(\sigma)$, the law of the pair $(\mathbf{u}(\cdot),\widetilde{\mathbf{u}}(\cdot))$ can be used as the coupling required in Corollary \ref{coroconv} with
$E=H^{1-\alpha/2}, \rho=\|\cdot-\cdot\|_{H^{1-\alpha/2}}\wedge 1, d=\|\cdot-\cdot\|_{H^{-\alpha/2}}\wedge 1$ (again, condition \eqref{approx} is easy to verify). We conclude that in the framework of Proposition 3.4 \cite{GMR15}, which states unique ergodicity for \eqref{EV}, in addition the following (weak) $H^{-\alpha/2}$-stabilization property holds: \begin{center} \emph{for any $u\in H^{1-\alpha/2}$, the transition probabilities $P_t(u, \cdot)\in \mathcal{P}(H^{1-\alpha/2})$ weakly converge in the $H^{-\alpha/2}$-topology as $t\to \infty$ to the unique invariant measure $\mu\in\mathcal{P}(H^{1-\alpha/2})$.} \end{center}
\subsubsection{The damped nonlinear wave equation}
Finally, we consider the damped Sine-Gordon equation which is written as the system of stochastic partial differential equations \begin{equation}\label{SG} \mathrm{d} v + \Big(\alpha v - \Delta u +\beta\sin(u)\Big)\, \mathrm{d} t = \sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \mathrm{d} u=v\, \mathrm{d} t, \end{equation}
where the unknown $u$ evolves on a bounded domain ${\mathcal D}\subset \mathbb {R}^n$ with smooth boundary, and satisfies the Dirichlet boundary condition $u|_{\partial{\mathcal D}}\equiv0$. The parameter $\alpha$ is strictly positive and $\beta$ is a real number.
It is known (see \cite{GMR15}, Section 3.5.1) that for any initial data $U_0=(u_0, v_0)\in X:=H^1_0({\mathcal D})\times L_2({\mathcal D})$ there exists a unique strong solution to \eqref{SG}, and moreover \eqref{SG} defines a Feller Markov process in $X$. In this final example the generalized coupling construction proposed in \cite{GMR15}, Section 3.5.2 is already well adapted for our purposes. Within this construction, they put $$ \mathrm{d} \widetilde{v} + \Big(\alpha \widetilde{v} - \Delta \widetilde{u} +\beta\sin(\widetilde{u})-\beta 1_{\tau_K>t}P_N(\sin({u})-\sin(\widetilde{u}))\Big)\, \mathrm{d} t = \sum_{k=1}^m \sigma_k \mathrm{d} W_k,\quad \mathrm{d} u=v\, \mathrm{d} t, $$ with the initial data $(\widetilde{u}_0,\widetilde{v}_0)$ and $$
\tau_K=\inf\left\{t:\int_0^t|u(t)-\widetilde u(s)|^2\, ds\geq K\right\}. $$ They prove that for $N,K$ sufficiently large $\tau_K=\infty$ a.s., and the difference $w=u-\widetilde u$ satisfies $$
\|w(t)\|^2+|\partial_t w(t)|^2\to 0, \quad t\to \infty $$ exponentially fast. This means that, if $H_N\subset \mathrm{Range}(\sigma)$, the joint law of the solutions $U=(u,v), \widetilde U=(\widetilde{u},\widetilde{v})$
can be used as the coupling required in Corollary \ref{coroconv} with $E=X$, $\rho=d=\|\cdot-\cdot\|_{X}\wedge 1$. We conclude that in the framework of Proposition 3.5 \cite{GMR15}, which states unique ergodicity for \eqref{SG}, in addition the following stabilization property holds: \begin{center} \emph{for any $(u,v)\in X=H^1_0({\mathcal D})\times L_2({\mathcal D})$, transition probabilities $P_t\big((u,v), \cdot\big)\in \mathcal{P}(X)$ weakly converge as $t\to \infty$ to the unique invariant measure.} \end{center}
\appendix
\section{Proofs of Propositions \ref{newlemma} and \ref{singular}}
\begin{proof}[Proof of Proposition \ref{newlemma}]
\emph{I.} Take an arbitrary $\xi\in \widehat C^R_p(\mathbb P,{\mathbb{Q}})$, and consider the sets $$ B_\gamma^1=\left\{x: {\mathrm{d} \pi_1(\xi)\over \mathrm{d}\mathbb P}(x)\leq \gamma^{-1}\right\},\quad B_\gamma^2=\left\{x: {\mathrm{d} \pi_2(\xi)\over \mathrm{d}{\mathbb{Q}}}(x)\leq \gamma^{-1}\right\},\quad C_\gamma=B_\gamma^1\times B_\gamma^2, \quad \gamma\in (0,1). $$ Define the sub-probability measure $\eta_\gamma$ on $(E^\infty\times E^\infty, \mathcal{E}^{\otimes \infty}\otimes \mathcal{E}^{\otimes \infty})$ by $$ \eta_\gamma(A)=\gamma \xi(A\cap C_\gamma). $$ Then the ``marginal distributions'' $\pi_i(\eta_\gamma), i=1,2$ (which now are sub-probability measures, as well) satisfy $$ \pi_1(\eta_\gamma)\leq \mathbb P, \quad \pi_2(\eta_\gamma)\leq {\mathbb{Q}}. $$ Denote $$ \beta_\gamma=\eta_\gamma(E^\infty\times E^\infty)=\gamma\xi(C_\gamma)\leq \gamma<1, $$ then each of the measures $\mathbb P-\pi_1(\eta_\gamma), {\mathbb{Q}}-\pi_2(\eta_\gamma)$ has total mass $1-\beta_\gamma.$ We put \begin{equation}\label{split} \zeta_\gamma=\eta_\gamma+(1-\beta_\gamma)^{-1}\big(\mathbb P-\pi_1(\eta_\gamma)\big)\otimes \big({\mathbb{Q}}-\pi_2(\eta_\gamma)\big), \end{equation} which by construction belongs to $C(\mathbb P,{\mathbb{Q}})$. Let us show that $\gamma$ can be chosen small enough, so that $\zeta=\zeta_\gamma$ possesses the required property.
Let $\alpha >0$. For $A \in \mathcal {E} \otimes \mathcal {E}$ satisfying $\xi(A)\ge \alpha$, we have $$ \zeta_\gamma(A)\geq \eta_\gamma(A)\geq \gamma\Big(\alpha- \xi\big((E^\infty\times E^\infty)\setminus C_\gamma\big)\Big). $$ Next, $$ \xi\big((E^\infty\times E^\infty)\setminus C_\gamma\big)\leq \xi\big((E^\infty\setminus B_\gamma^1)\times E^\infty\big)+
\xi\big(E^\infty\times (E^\infty\setminus B_\gamma^2)\big)=\pi_1(\xi)\big(E^\infty\setminus B_\gamma^1\big)+\pi_2(\xi)\big(E^\infty\setminus B_\gamma^2\big) $$ and by the definition of the sets $B_\gamma^i, i=1,2$ $$ \pi_1(\xi)\big(E^\infty\setminus B_\gamma^1\big)=\int_{E^\infty\setminus B_\gamma^1} {\mathrm{d} \pi_1(\xi)\over \mathrm{d}\mathbb P}\,\mathrm{d}\mathbb P\leq \gamma^{p-1} \int_{E^\infty}\left({\mathrm{d} \pi_1(\xi)\over \mathrm{d}\mathbb P}\right)^p\,\mathrm{d}\mathbb P\leq \gamma^{p-1}R^p, $$ $$ \pi_2(\xi)\big(E^\infty\setminus B_\gamma^2\big)=\int_{E^\infty\setminus B_\gamma^2} {\mathrm{d} \pi_2(\xi)\over \mathrm{d}{\mathbb{Q}}}\,\mathrm{d}{\mathbb{Q}}\leq \gamma^{p-1} \int_{E^\infty}\left({\mathrm{d} \pi_2(\xi)\over \mathrm{d}{\mathbb{Q}}}\right)^p\,\mathrm{d}{\mathbb{Q}}\leq \gamma^{p-1}R^p. $$ Hence, if $\gamma$ is taken small enough for $4\gamma^{p-1}R\leq \alpha$, for every $A$ with $\xi(A)\geq \alpha$ we have for $\zeta=\zeta_\gamma$ $$ \zeta(A)\geq {\gamma\alpha\over 2}=:\alpha', $$ which completes the proof of statement I.
\emph{II.} We fix $\xi\in \widehat C(\mathbb P,{\mathbb{Q}})$ and modify slightly the construction from the previous part of the proof. Let $B_\gamma^i, i=1,2, C_\gamma$ be as above, then we define $$ \widetilde\eta_\gamma(A)=\xi(A\cap C_\gamma). $$ We fix $\gamma\in (0,1)$ small enough, so that $$ \xi\big((E^\infty\times E^\infty)\setminus C_\gamma\big)\leq \alpha/2, $$ where $\alpha$ is as in the statement of the lemma.
We have $\widetilde\eta_\gamma=\gamma^{-1}\eta_\gamma$, and thus the total mass of the measure $\widetilde\eta_\gamma$ equals $\gamma^{-1}\beta_\gamma=\xi(C_\gamma)\leq 1$. In addition, $$ \pi_1(\widetilde\eta_\gamma)\leq \gamma^{-1}\mathbb P, \quad \widetilde\pi_2(\eta_\gamma)\leq \gamma^{-1}{\mathbb{Q}}, $$ and the total mass for each of the measures $\gamma^{-1}\mathbb P-\pi_1(\widetilde\eta_\gamma), \gamma^{-1}{\mathbb{Q}}-\pi_2(\tilde\eta_\gamma)$ equals $\gamma^{-1}(1-\beta_\gamma)\geq \gamma^{-1}-1$. Then $$ \widetilde\zeta_\gamma:=\widetilde\eta_\gamma+(1-\gamma^{-1}\beta_\gamma)\Big(\gamma^{-1}(1-\beta_\gamma)\Big)^{-2}\Big(\gamma^{-1}\mathbb P-\pi_1(\widetilde \eta_\gamma)\Big)\otimes \Big(\gamma^{-1}{\mathbb{Q}}-\pi_2(\widetilde\eta_\gamma)\Big) $$ is a probability measure with $$ \widetilde\zeta_\gamma(A)\geq \xi(A)-\xi\big((E^\infty\times E^\infty)\setminus C_\gamma\big)\geq \xi(A)-{\alpha\over 2}; $$ that is, $\widetilde\zeta_\gamma(A)\geq \alpha':=\alpha/2$ as soon as $\xi(A)\geq \alpha$. In addition, the marginal distributions of $\widetilde\zeta_\gamma$ equal $$ (1-\beta_\gamma)^{-1}\Big((1-\gamma^{-1}\beta_\gamma)\mathbb P+(1-\gamma)\pi_1(\widetilde\eta_\gamma)\Big),\quad (1-\beta_\gamma)^{-1}\Big((1-\gamma^{-1}\beta_\gamma){\mathbb{Q}}+ (1-\gamma)\pi_2(\widetilde\eta_\gamma)\Big), $$ and their Radon-Nikodym densities w.r.t. $\mathbb P,{\mathbb{Q}}$ respectively are bounded by $$ R:=(1-\beta_\gamma)^{-1}\Big((1-\gamma^{-1}\beta_\gamma)+(1-\gamma)\gamma^{-1}\Big)=\gamma^{-1}, $$ hence $\widetilde \zeta_\gamma\in \widehat C_p^R(\mathbb P,{\mathbb{Q}})$ for every $p\geq 1$. \end{proof}
\begin{proof}[Proof of Proposition \ref{singular}]
There exist two increasing sequences $K_1^n,\, K_2^n, n\geq 1$ of compact subsets of $E$ such that $K_1^n\cap K_2^n=\emptyset$, $\nu_1(K_1^n)\ge 1-1/n$, and $\nu_2(K_2^n)\ge 1-1/n$ for $n\geq 1$. Let $$ \delta_n=d(K_1^n,K_2^n), \quad n\geq 1. $$ Clearly $\delta_n, n\geq 1$ is non-increasing and $\delta_n>0$ for all $n\geq 1$ since $d:E \times E \to [0,\infty)$ is continuous with respect to $\rho \otimes \rho$. On the other hand, for any $\xi \in C(\nu_1,\nu_2)$ we have $$ \xi \big( d(X,Y)< \delta_n\big)\le {2\over n}, $$ proving the proposition. \end{proof} \section{Jankov's lemma and the proof of Proposition \ref{jan_cor}}
Recall that a measurable space $(\mathbb{X}, \mathcal{X})$ is called \emph{(standard) Borel} if it is measurably isomorphic to a Polish space equipped with its
Borel $\sigma$-algebra. For any
Borel space $(\mathbb{X}, \mathcal{X})$ and any set $A\in \mathcal{X}$, this set endowed with its {trace $\sigma$-algebra}
$$
\mathcal{X}_A:=\{A\cap B, B\in \mathcal{X}\}
$$ is a Borel measurable space, see \cite[Corollary 13.4]{K95}.
Our proof of Proposition \ref{jan_cor} is based on the following lemma.
\begin{lemma} \emph{(Jankov's lemma, \cite[Appendix 3 \S 1]{Dynkin_Yushk})}. Let $(\mathbb{X}, \mathcal{X})$, $(\mathbb{Y}, \mathcal{Y})$ be Borel measurable spaces and let $f:\mathbb{Y}\to \mathbb{X}$ be a measurable mapping with $f(\mathbb{Y})=\mathbb{X}$.
Then for any probability measure $\nu$ on $(\mathbb{X}, \mathcal{X})$ there exists a measurable function $\phi: \mathbb{X}\to \mathbb{Y}$ such that $f(\phi(x))=x$ for $\nu$-a.a. $x\in \mathbb{X}$. \end{lemma}
In the framework of Proposition \ref{jan_cor}, we put $\mathbb{X}=M, \mathcal{X}=\mathcal {E}_M$ (the trace $\sigma$-algebra), then $(\mathbb{X}, \mathcal{X})$ is a Borel space. We define $\nu$ as the measure $\mu$ conditioned by $M$.
Before proceeding with the construction, we mention several simple facts we will use. First, let $\mathbb{S}$ be a Polish space and
$\mathcal{P}(\mathbb{S})$ be endowed by the corresponding Kantorovich-Rubinshtein metric. Then the subset $\Delta\subset \mathcal{P}(\mathbb{S})$ consisting of all
$\delta$-measures (that is, measures concentrated in one point) is closed, and $\mathbb{S}$ and $\Delta$ are isomorphic.
Second,
the mapping $\theta$ from $\mathcal{P}(E^\infty\times E^\infty)$ to $\mathcal{P}(E\times E)$ which maps the law of $\{(X_n, Y_n), n\geq 0\}$ to the law of $(X_0, Y_0)$ is
(Lipschitz) continuous. Hence, the subset
$$
\Xi:=\{\xi\in \mathcal{P}(E^\infty\times E^\infty):\theta(\xi)\hbox{ is a $\delta$-measure}\}
$$
is closed. In addition, the mapping $\varrho:\Xi\to E\times E$ which transforms $\xi\in \Xi$ to the (unique) point $(x,y)\in E$ such that $\theta(\xi)=\delta_{(x,y)}$, is
continuous. Then $\Xi$ endowed with the trace $\sigma$-algebra is a Borel space and $\varrho$ is a measurable mapping on this space with $\varrho(\Xi)=E\times E$.
Denote by $\varrho_{1,2}$ the (measurable) mappings $\Xi\to E$ such that $\varrho(\xi)=(\varrho_1(\xi), \varrho_2(\xi)), \xi\in \Xi.$
Now we can proceed with the construction which deduces Proposition \ref{jan_cor} from Jankov's lemma. We fix $x\in E$, put
$$
\mathbb{Y}=\{\xi\in \Xi: \varrho_1(\xi)=x, \varrho_2(\xi)\in M, \pi_1(\xi)\sim \mathbb P_x, \pi_2(\xi)\ll \mathbb P_{\varrho_2(\xi)}\},
$$
and $f(\xi)=\varrho_2(\xi), \xi\in \mathbb{Y}$. Clearly, $f(\mathbb{Y})=M$ and $f$ is a restriction on $\mathbb{Y}$ of a measurable mapping $\Xi\to E$ (the projection
of $\varrho$ on the second coordinate). Hence in order to be able to apply Jankov's lemma we need only to show that $\mathbb{Y}$ is a measurable subset of $\Xi$.
Because $\varrho_{1,2}$ are measurable and $\{x\}, M\in \mathcal {E}$, the sets
$$
\{\xi\in \Xi: \varrho_1(\xi)=x\}, \quad \{\xi\in \Xi: \varrho_2(\xi)\in M\} $$ are measurable.
Next, recall that for any two probability measures $\mathbb P, {\mathbb{Q}}$ on $(E^\infty, \mathcal{E}^{\otimes \infty})$ one has $\mathbb P\ll {\mathbb{Q}}$ if, an only if, for every $\varepsilon>0$ there exists $\delta>0$ such that
$$
\mathbb P(A)\leq \varepsilon\quad \hbox{for any $A\in \mathcal{E}^{\otimes \infty}$ such that}\quad {\mathbb{Q}}(A)\leq \delta.
$$
Because $E^\infty$ is a Polish space, there exists a
countable algebra
$\mathcal{A}$ which generates $\mathcal{E}^{\otimes \infty}$, and then for any $\gamma>0, A\in \mathcal{E}^{\otimes \infty}$ there exists $A_\gamma\in \mathcal{A}$
such that
$$
\mathbb P(A\triangle A_\gamma)<\gamma, \quad {\mathbb{Q}}(A\triangle A_\gamma)<\gamma.
$$
Then in the above characterization of the absolute continuity the class $\mathcal{E}^{\otimes \infty}$ can be replaced by $\mathcal{A}$. Hence
$$ \{\xi\in \Xi: \pi_2(\xi)\ll \mathbb P_{\varrho_2(\xi)}\}=\bigcap_{m=1}^\infty\bigcup_{k=1}^\infty\bigcap_{A\in \mathcal{A}}B_{m,k}(A), $$ where $$ B_{m,k}(A)=\left\{\xi: \pi_2(\xi)(A)\leq m^{-1}, \mathbb P_{\varrho_2(\xi)}(A)\leq k^{-1}\right\}\bigcup \left\{\xi: \mathbb P_{\varrho_2(\xi)}(A)> k^{-1}\right\}. $$ Since the mappings $\varrho_2:\Xi\to E, \pi_2:\Xi\to \mathcal{P}(E^\infty)$, $E\ni v\mapsto \mathbb P_v\in \mathcal{P}(E^\infty)$, and $$ \mathcal{P}(E^\infty)\ni \mathbb P\mapsto \mathbb P(A)\in \mathbb{R}, \quad A\in \mathcal{A} $$ are measurable, each of the sets $B_{m,k}(A)$ is measurable. Therefore the set $$ \{\xi\in \Xi: \pi_2(\xi)\ll \mathbb P_{\varrho_2(\xi)}\} $$ is measurable, as well. Finally, a similar and simpler argument shows that the set $$ \{\xi\in \Xi: \pi_1(\xi)\sim \mathbb P_{x}\} $$ is measurable (we omit the explicit expression for this set here).
Summarizing, we have that $\mathbb{Y}$ is a measurable subset of $\Xi$ and therefore, being endowed with the trace $\sigma$-algebra, is a Borel space. We finish the proof of Proposition \ref{jan_cor} by applying Jankov's lemma to the Borel spaces $\mathbb{X}$, $\mathbb{Y}$, the mapping $f$, and the measure $\nu$ specified above.
\section{Kuratovskii and Ryll-Nardzevski's theorem and the proof of Proposition \ref{prop}} Our proof of Proposition \ref{prop} is based on measurability and measurable selection results discussed in \cite{Stroock_Varad}, Chapter 12.1. Let us survey the required results briefly.
Let $\mathbb{X}$ be a Polish space with complete metric $\rho$. Denote by $\mathrm{comp}\,(\mathbb{X})$ the space of all non-empty compact subsets of $\mathbb{X}$, endowed with the Hausdorff metric.
\begin{theorem}\label{tKR} (\cite[Theorem 12.1.10]{Stroock_Varad} Let $(E, \mathcal {E})$ be a measurable space and $\Phi:E \to \mathrm{comp}\,(\mathbb{X})$ be a measurable map. Then there exists a measurable map $\phi: E\to \mathbb{X}$ such that $\phi(q)\in \Phi(q), q\in E$. \end{theorem} The above theorem is a weaker version of the Kuratovskii and Ryll-Nardzevski's theorem on measurable selection for a set-valued mapping which takes values in the space of closed subsets of $\mathbb{X}$; e.g. \cite{W80}.
In the set-up of Proposition \ref{prop}, for $\mu,\nu\in \mathcal{P}(S_2)$, we denote by $C_{\mathrm{opt}}(\mu, \nu)$ the subset of $C(\mu, \nu)$ consisting of all couplings which minimize the distance-like function $h$; that is, $$ \eta\in C_{\mathrm{opt}}(\mu, \nu)\quad \Leftrightarrow\quad \eta\in C(\mu, \nu), \quad \int_{S_2\times S_2}h(u,v)\eta(\mathrm{d} u, \mathrm{d} v)=h(\mu, \nu). $$ We prove the following simple facts. \begin{lemma} \begin{enumerate} For any $\mu,\nu\in \mathcal{P}(S_2)$:
\item the set $C_{\mathrm{opt}}(\mu, \nu)$ is non-empty;
\item the sets $C(\mu, \nu)$, $C_{\mathrm{opt}}(\mu, \nu)$ are compact.
\end{enumerate}
\end{lemma} \begin{proof} Since $\pi_1, \pi_2: \mathcal{P}(S_2\times S_2)\to \mathcal{P}(S_2)$ are continuous, any weak limit point of a sequence from $C(\mu, \nu)$ belongs to $C(\mu, \nu)$. Because the marginal distributions of all $\eta\in C(\mu, \nu)$ are the same, the set $C(\mu, \nu)$ is tight, which by the Prokhorov theorem completes the proof of compactness of $C(\mu, \nu)$.
Next, the mapping $$ \mathcal{P}(S_2\times S_2)\ni \eta\mapsto I_h(\eta):=\int_{S_2\times S_2}h(u,v)\eta(\mathrm{d} u, \mathrm{d} v) \in [0,1] $$ is lower semicontinuous. To see that, consider a sequence $\eta_n\Rightarrow \eta$ and use the Skorokhod ``common probability space principle'': there exist random elements $X_n, n\geq 1, X$ with $\mathrm{Law}\,(X_n)=\eta_n, \mathrm{Law}\,(X)=\eta$ such that $X_n\to X$ a.s. (see \cite[Theorem 11.7.2]{Dudley}. Since $h$ is bounded and lower semicontinuous, we have $$ \mathbb {E} h(X)\leq \mathbb {E}\liminf_nh(X_n)\leq \liminf_n \mathbb {E} h(X_n), $$ which proves the required semicontinuity of $I_h$. By this semicontinuity (a) the function $I_h$ attains its minimum on the compact set $C(\mu, \nu)$, i.e. $C_{\mathrm{opt}}(\mu, \nu)$ is non-empty; (b) the set $C_{\mathrm{opt}}(\mu, \nu)$ is closed, and since it is a subset of the compact set $C(\mu, \nu)$, it is compact.
\end{proof}
To prove Proposition \ref{prop}, we apply Theorem \ref{tKR} in the following setting: $E=S_1\times S_1$, $\mathbb{X}=\mathcal{P}(S_2\times S_2)$, and $$ \Phi\big((x,y)\big)=C_{\mathrm{opt}}(Q(x), Q(y)), \quad (x,y)\in E. $$ We represent $\Phi$ as a composition of $\Psi$ and $\Upsilon$, where $$ \Psi\big((x,y)\big)=C(Q(x), Q(y)), \quad (x,y)\in E $$ and $$ \Upsilon(K)=\left\{\eta\in \mathbb{X}: I_h(\eta)=\min_{\zeta\in K}I_h(\zeta)\right\}\in \mathrm{comp}\,(\mathbb{X}), \quad K\in \mathrm{comp}\,(\mathbb{X}). $$ Clearly, the minimization of $I_h$ is equivalent to maximization of $1-I_h$, and $1-I_h$ is upper semicontinuouus. Hence the mapping $\Upsilon:\mathrm{comp}\,(\mathbb{X})\to \mathrm{comp}\,(\mathbb{X})$ is measurable by \cite{Stroock_Varad}, Lemma 12.1.7. On the other hand, for any sequence $(x_n,y_n)\to (x,y)$ and $\eta_n\in \Psi((x_n, y_n))$ we have that the marginal distributions of $\eta_n$ weakly converge to $Q(x), Q(y)$ respectively. Then by the Prokhorov theorem there exist a weakly convergent subsequence $\eta_{n_k}$, and in addition the weak limit has the marginal distributions $Q(x), Q(y)$, that is, belongs to $\Psi((x,y))$. Then the mapping $\Psi:E\to \mathrm{comp}\,(\mathbb{X})$ is measurable by \cite[Lemma 12.1.8]{Stroock_Varad}. Hence $\Phi$ is measurable, as well, and we obtain the statement of Proposition \ref{prop} as a straightforward corollary of Theorem \ref{tKR}. \qed
\end{document} | arXiv |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$
Let $P_n$ represent the probability that the bug is at its starting vertex after $n$ moves. If the bug is on its starting vertex after $n$ moves, then it must be not on its starting vertex after $n-1$ moves. At this point it has $\frac{1}{2}$ chance of reaching the starting vertex in the next move. Thus $P_n=\frac{1}{2}(1-P_{n-1})$. $P_0=1$, so now we can build it up:
$P_1=0$, $P_2=\frac{1}{2}$, $P_3=\frac{1}{4}$, $P_4=\frac{3}{8}$, $P_5=\frac{5}{16}$, $P_6=\frac{11}{32}$, $P_7=\frac{21}{64}$, $P_8=\frac{43}{128}$, $P_9=\frac{85}{256}$, $P_{10}=\frac{171}{512}$,
Thus the answer is $171+512=\boxed{683}$ | Math Dataset |
\begin{document}
\begin{titlingpage}
\title{ Solitons of Curve Shortening Flow and Vortex Filament Equation\\ \Large UROP+ Final Paper, Summer 2017}
\begin{abstract}
In this paper we explore the nature of self-similar solutions of the Curve Shortening Flow and the Vortex Filament Equation, also known as the Binormal Flow. We explore some of their fundamental conservation properties and describe the behavior of their self-similar solutions. For Curve Shortening Flow we mainly expose the results of Huisken, Grayson, and Halldorsson concerning the equation's basic properties and self-similar solutions in the plane. For the Vortex Filament Equation we present the results by Banica and Vega, Arms and Hama, and Hasimoto. We also derive the evolution equations of the normal, binormal and tangent vectors in the Frenet frame for the vortex filament as well as those of curvature and torsion. We give a proof that circles are the only planar translating self-similar solutions and also derive a system of ordinary differential equations that govern the behavior for rotating self-similar solutions.
\end{abstract}
\end{titlingpage}
\tableofcontents
\section{Preliminaries}
\subsection{Sobolev Spaces}
We begin with an understanding of Sobolev spaces which are the natural extension of smooth functions that the solutions of PDEs will live in. These spaces serve will also become important in the formulation of dilating self-similar solutions of the Vortex Filament Equation.
To explain Sobolev Spaces it is first necessary for us to introduce the notion of the weak derivative of a function.
Let $ (\Omega, \Sigma, \mu) $ be a measurable space, $f$ be a function on $\Omega$. If $ f\in L^{p}(\Omega)$ $ (1\leq p<\infty) $ then \[ \lVert f\rVert_{p = }(\int_{\Omega}\lvert f\rvert^{p}d\mu)^{\frac{1}{p}}<\infty\]. If $ p =\infty $ then the norm is instead the essential supremum, defined as follows:
\begin{definition}
The essential supremum of a measurable function $ f:\Omega\rightarrow\mathbb{R} $ with measure $ \mu $ is the smallest number $ \alpha $ such that the set \[ \{x\big||f(x)|>\alpha\} \] has measure zero. If no such $ \alpha $ exists then it is taken to be infinity.
\end{definition}
We shall denote this norm as $\lVert f \rVert_{\infty} =\text{ess sup}_{\Omega}\lvert f \rvert $.
To say $ f $ is in $ C^{\infty}_{c}(\Omega) $ means that f is indefinitely differentiable and compactly supported.
\begin{definition}[Weak Derivative]
Let $ \Omega\subset \mathbb{R}^{n} $ be an open set. a function $ f\in L^{1}_{loc}(\Omega) $ is said to be weakly differentiable to the $i$th variable if there exists $ g_{i}\in L^{1}_{loc}(\Omega) $ such that \[ \int_{\Omega}f\partial_{i}\phi dx = -\int_{\Omega}g_{i}\phi dx \] for all $ \phi\in C^{\infty}_{c}(\Omega) $. We then call $ g_{i} $ the weak $ i $th partial derivative of $ f $, denoted as usual $ \partial_{i}f $.
\end{definition}
It is clear from the definition that the weak derivative coincides with the common pointwise derivative of a continuously differentiable function. However, a function that is not pointwise differentiable almost everywhere can still have a weak derivative. For higher order derivatives the following definition holds.
\begin{definition}
Let $ \alpha = (\alpha_{1},...,\alpha_{n})\in \mathbb{N}^{n} $ be a multi-index and $ \lvert \alpha\rvert = \alpha_{1}+...+\alpha_{n}=k $. A function $ f\in L^{1}_{loc}(\Omega) $ has a weak derivative of order $ k $, denoted by $ D^{\alpha}f $, if \[ \int_{\Omega} (D^{\alpha}f)\phi dx = (-1)^{\lvert\alpha\rvert}\int_{\Omega}f(D^{\alpha}\phi) dx \] for all $ \phi\in C^{\infty}_{c}(\Omega) $.
\end{definition}
We now present some definitions and basic properties of Sobolev spaces without proof. We direct the reader who seeks proofs of theorems as well as a more thorough understanding of Sobolev spaces to chapter 5 of Evans' work on PDEs ([1]) from where the notation and theorems are taken.
\begin{definition}[Sobolev Space]
We denote with \[ W^{k,p}(\Omega) \] as the space of all locally summable functions $ f:\Omega\rightarrow\mathbf{R} $ such that for each multi-index $ \alpha $ with $ \lvert\alpha\rvert\leq k $, $ D^{\alpha}f $ exists in the weak sense and belongs to $ L^{p}(\Omega) $. If p = 2 it is common to use the notation\[ H^{k}(\Omega) = W^{k,2}(\Omega) \text{ } (k =0,1,...) \]
\end{definition}
\begin{definition}[Norm of Sobolev Spaces]
Let $ f\in W^{k,p}(\Omega) $, we define the norm to be \[ \lVert f \rVert_{W^{k,p}}:=\begin{cases}
(\Sigma_{\lvert\alpha\rvert\leq k}\int_{\Omega}\lvert D^{\alpha}f\rvert^{p}dx)^{1/p} &(1\leq p <\infty)\\
\Sigma_{\lvert\alpha\rvert\leq k} \text{ess sup}_{\Omega}\lvert D^{\alpha}f\rvert &(p = \infty)
\end{cases}
\]
\end{definition}
\begin{definition}[Convergence in Sobolev Spaces]
(i) Let $ f,\{f_{m}\}^{\infty}_{m = 1} \in W^{k,p}(\Omega) $. We say that $ f_{m} $ converges to $ f $ in $ W^{k,p}(\Omega) $, written \[f_{m}\rightarrow f \mbox{ in } W^{k,p}(\Omega),\] given that \[\lim_{m\rightarrow\infty} \rVert f_{m}-f\lVert_{W^{k,p}(\Omega)} = 0.\]
(ii) We denote by \[ W^{k,p}_{0}(\Omega) \] as the closure of $ C^{\infty}_{c}(\Omega) $ in $ W^{k,p}(\Omega) $. That is to say that $ f\in W^{k,p}_{0}(\Omega) $ if there exists a sequence $ f_{m}\in C^{\infty}_{c}(\Omega) $ such that $ f_{m}\rightarrow f $ in $ W^{k,p}(\Omega) $.
\end{definition}
The Fourier transform of a function is defined as follows:
\begin{definition}
Let $ f:\mathbb{R}^{n}\rightarrow\mathbb{C} $ be a measurable function. Then the corresponding Fourier transform is defined as \[ \hat{f}(\zeta) = \int_{-\infty}^{\infty}f(x)e^{-2\pi i x\cdot\zeta}d^{n}x \].
\end{definition}
\begin{theorem}[$ H^{k} $ by Fourier Transform]
Let $ f\in L^{2}(\mathbf{R}^{n}) $, then $ f\in H^{k}(\mathbf{R}^{n}) $ if and only if \[(1+\lvert y \rvert^{k})\hat{f}\in L^{2}(\mathbf{R}^{n}) \].
In addition there exists a positive constant C such that \[\frac{1}{C}\lVert f\rVert_{H^{k}(\mathbf{R}^{n})}\leq \lVert (1+\lvert y \rvert^{k})\hat{f}\rVert_{L^{2}(\mathbb{R}^{n})}\leq C\lVert f \rVert_{H^{k}(\mathbf{R}^{n})} \] for each $ f\in H^{k}(\mathbf{R}^{n}) $.
\end{theorem}
\begin{definition}[Non-integer Sobolev Spaces]
Assume $0 < s <\infty$ and $f\in L^{2}(\mathbf{R}^{n}).$ Then $ f\in H^{s}(\mathbf{R}^{n}) $ if $ (1+\lvert y \rvert^{s})\hat{f}\in L^{2}(\mathbf{R}^{n}) $. For non-integer $ s $ the norm becomes \[\lVert f \rVert_{H^{s}(\mathbf{R}^{n})} := \lVert(1+\lvert y \rvert^{s})\hat{f}\rVert_{L^{2}(\mathbf{R}^{n})}. \]
\end{definition}
To finish this section we present a small discussion of negative order Sobolev spaces which will be relevant when discussing dilating solutions of the Vortex Filament Equation.
\begin{definition}
Denote as $ H^{-1}(\Omega) $ as the dual to $ H^{1}(\Omega) $. The norm of this space will be defined as follows \[ \lVert f \rVert_{H^{-1}(\Omega)} = sup\{\langle f, u \rangle|u\in H^{1}_{0}(\Omega), \lVert u\rVert_{H^{1}_{0}(\Omega)}\leq 1 \}. \]
\end{definition}
\begin{theorem}[Characterization of the Dual Space]
Assume $ f \in H^{-1}(U) $. Then there exist functions $ f^{0},f^{1},...,f^{1} $ in $ L^{2}(\Omega) $ such that \[ \langle f,\nu\rangle = \int_{\Omega}f^{0}\nu+\sum_{i =1}^{n}f^{i}\nu_{x_{i}}dx \mbox{ \text{(}$ \nu\in H^{1}_{0}(\Omega) $\text{)}} \]
Furthermore, \[ \lVert f \rvert_{H^{-1}(\Omega)} = \mbox{inf}\{ (\int_{\Omega}\sum_{i = 0}^{n}|f^{i}|^{2}dx)^{1/2}| \mbox{ $ f $ satisfies (1) for $ f^{0},...,f^{n}\in L^{2}(\Omega) $} \} \]
\end{theorem}
For spaces $ H^{-k} $ it coincides with the sense of the previous definition in that they will be the dual of the Sobolev space $ H^{k}_{0} $. For the rest of this paper the reader may assume that all derivatives are to be taken in the point-wise sense unless otherwise required in the ambient space.
\subsection{Self Similar Solutions and Geometric PDEs}
In this section we shall discuss the nature of self-similar solutions as well as provide some background on Geometric PDEs, which are the main focus of this paper.
The term self-similar is often used to describe solutions to partial differential equations that demonstrate a particular invariance towards scaling, or in a sense `look the same' at all times. Their importance comes from the fact that they can be used to observe the behavior of a PDE at a singularity by `blowing it up'. In a sense the scaling invariance allows one to create a blow up sequence and rescale time in order to get a clearer picture of the behavior of that PDE at the singularity.
This technique will become clearer in our discussion of Grayson's Theorem when it is used to see the behavior of a point that is `blown up' backwards in time. Here we shall give a basic example of a self-similar solution to the 2-dimensional heat equation, however, for Geometric PDEs, self-similar solutions can be taken to not only be invariant in rescaling but also in translations and rotations. These invariances then give dilating, translating, and rotating self-similar solutions. It is important to note that one can have self-similar solutions that express more than one of these properties.
\begin{theorem}[Self-similar solution of the heat equation]
Consider the heat equation with the following initial value problem \[u_{t}-ku_{xx}= 0\] \[ t>0, -\infty<x<\infty \]
Then \[ u(x,t) = \frac{1}{\sqrt{4kt}}e^{-x^{2}/(4kt)}\] is a self-similar solution of this equation, i.e. it satisfies $ u_{\alpha}(x,t) = \alpha u(\alpha x, \alpha^{2}t) $ for any $ \alpha>0 $.
\end{theorem}
\begin{proof}
Before beginning we must set the following condition on the solution:
\begin{align}
&I(t):=\int_{-\infty}^{\infty}u(x,t)dx<\infty,\\
&\lim_{x\rightarrow\pm\infty}u_{x}(x,t) = 0
\end{align}
To find a self-similar solution we first begin by finding a solution $ u_{\alpha}(x,t) $ of the form $ u_{\alpha}(x,t) = \alpha u(\alpha x, \alpha^{2}t) $, such that $ u(x,t) = u_{\alpha}(x,t) $ for all $ \alpha, t>0, x $. The previous condition satisfies the invariance of a self-similar solution, and this method of rescaling variables is called parabolic rescaling.
If we then take $ \alpha = t^{-1/2} $ to get rid of the time variable, we have that \[ u(x,t) = \frac{1}{\sqrt{t}}u(\frac{x}{\sqrt{t}},1) = \frac{1}{\sqrt{t}}\phi(\frac{x}{\sqrt{t}}) \].
Then we get the following equations by taking the derivatives:\begin{align} u_{t} &= -\frac{1}{2}t^{-3/2}[\phi(\frac{x}{\sqrt{t}})+\frac{x}{\sqrt{t}}\phi'(\frac{x}{\sqrt{t}})]\\
u_{x}&=\frac{1}{t}\phi'(\frac{x}{\sqrt{t}})\\
u_{xx} &=t^{-3/2}\phi''(\frac{x}{\sqrt{t}})
\end{align}
We then make the substitution $ \zeta = \frac{x}{\sqrt{t}} $ and plug into our heat equation to obtain the following ODE:\[ -\frac{1}{2}[\phi(\zeta)+\zeta\phi'(\zeta)]-k\phi''(\zeta) = 0 \].
Finally, solving this ODE gives \[ \phi(\zeta) = Ce^{-\frac{\zeta^{2}}{4k}} \]
so that $ u(x,t) = \frac{1}{\sqrt{4\pi kt}}e^{-\frac{\zeta^{2}}{4k}} $ where $ C = \frac{1}{\sqrt{4k\pi}} $ in order to normalize $ I(t) $.\qed
\end{proof}
The method used in the previous proof of parabolically rescaling the variable is used when attempting to find dilating solutions of the vortex filament equation.
In the description of geometric properties of manifolds there are often situations that arise which are modeled with a system of PDEs, this then allows us to use the tools of PDE theory to be able to investigate the geometric, analytic, and topological properties of the objects these equations describe. Geometric PDEs have a wide range of applications, from aiding in the solution of previously open problems such as the Poincar\'e conjecture and the differentiable sphere theorem, to applications in image and sound processing.
In this paper we mainly concern ourselves with Geometric Flows. These are systems of equations that describe the deformation of metrics on Riemannian manifolds driven by the geometric quantities such as curvature, volumes, etc. In this paper we present a discussion on two quasi-parabolic geometric flows: Curve Shortening Flow and the Vortex Filament Equation.
\section{Curve Shortening Flow}
Curve Shortening Flow (CSF) is a geometric quasi-parabolic equation for curves that serves as an analogous flow to the Vortex Filament Equation. This geometric flow equation has been extensively studied and we have derived many of its more important properties such as monotonicity formulas, and maximum principle estimates that combine with the blowup analysis we mentioned earlier while discussing self-similar solutions to give Huisken's ([3]) proof of Grayson's Theorem.
\begin{definition}[Curve Shortening Flow]
A family of embedded curves $ \{\Gamma_t\subset \mathbf{R}^2 \}_{t \in I }$ moves by curve shortening flow if the normal velocity at each point is given by the curvature vector:
\begin{equation}
\partial_t p = \vec{\kappa}(p)
\end{equation}
for all $ p \in \Gamma_t $ and all $t \in I $. Here, $I$ is an interval, $\partial_t p $ is the normal velocity at $p$, and $\vec{\kappa}(p)$ is the curvature vector at $p$.
\end{definition}
We now provide an important example of an explicit solution which also happens to be the only self-similar translating solution of curves evolving under CSF:
\begin{example}[Grim Reaper Curve]
Take $ y(x,t) =t-\log \cos(x) $ then under CSF this curve moves upwards without changing its shape. Any curve similar, by either scaling translation or rotation, to the grim reaper is also translated in the direction of the axis of symmetry without changing shape or orientation, satisfying what is to be expected of a self-similar solution. Interestingly it is the only curve with this property ([13]).
\end{example}
\subsection{Properties of Curve Shortening Flow}
We will focus on the evolution of closed embedded curves. For this section we mainly present the theorems of Haslhofer's notes on CSF ([2]) and his discussion on blow-up methods for singularities. The main part of this section will be the presentation by Haslhofer of Huisken's proof of Grayson's Theorem ([6]) from his lecture notes ([2]), for which we shall present all the necessary machinery without proof. Although there are various other proofs of Grayson's Theorem, some more geometric than others and requiring less machinery, the purpose of this one is to demonstrate the utility of self-similar solutions and why we seek to discover them.
CSF can also be rewritten by taking
\begin{equation}
\gamma = \gamma(\cdot , t) : S^1 \times I \rightarrow \mathbb{R}^2
\end{equation}
with $\Gamma_t = \gamma (S^1, t)$. Setting $p = \gamma (x,t)$, the equation transforms into
\begin{equation}
\partial_t\gamma(x,t) = \kappa(x,t)N(x,t).
\end{equation}
\begin{remark}
The evolution can also be written in the form
\begin{equation}
\partial_t\gamma = \partial_s^2\gamma,
\end{equation}
where s denotes arc length if one were to transform from the Frenet frame of reference to the Cartesian plane.
\end{remark}
First we present some facts of CSF and derive some of its basic properties such as evolution for the length functional.
\begin{theorem}[Evolution of Arclength]
Define $ L(t) := \int ds $ to be the arclength functional. A curve $ \gamma $ evolving under CSF has decreasing arclength which obeys the following evolution equation:\begin{equation}
\partial_{t}L = -\int\kappa^{2}ds.
\end{equation}
\end{theorem}
\begin{proof}
From calculus we have that $ L(t) = \int ds = \int_{S^{1}}\langle\gamma_{x},\gamma_{x} \rangle^{1/2}dx $ where $ x\in S^{1} $.
We take the derivative with respect to time and proceed to differentiate under the integral to obtain\begin{align} \partial_{t}L &= \int_{S^{1}}\langle\partial_{xt}\gamma,T\rangle du\\
&=\int_{S^{1}}\langle\partial_{tx}\gamma,T\rangle du\\
&=\int_{S^{1}}\langle \partial_{x}(\kappa N),T\rangle dx
\end{align}
were $ T $ and $ N $ are the tangent and normal vectors, respectively. From the Frenet equations we have that $ \partial_{x}T = s'(x)\kappa N $ and $ \partial_{x}N=-s'(x) \kappa T $. So substituting this gives that the only component not zero by orthogonality is $ -\kappa^{2} $, turning (13) into \[ \partial_{t}L =-\int_{S^{1}}\kappa^{2}s'(x)dx = -\int\kappa^{2}ds \] as desired.
\end{proof}
\begin{remark}
This could also be easily derived from the first variation formula of arc-length, which says that if a curve moves with normal velocity $ v $ the length of the curve changes by $ -\int \kappa v $. This theorem also shows why it is considered that curves under CSF decrease their length most efficiently.
\end{remark}
\begin{proposition}[Evolution of Curvature]
Suppose that $ \gamma $ is a curve that satisfies CSF, then its curvature satisfies the evolution equation \begin{equation}
\kappa_{t} = \kappa_{ss}+\kappa^{3}.
\end{equation}
\end{proposition}
\begin{proof}
We work here with a parametrization such that $ |\partial_{x}\gamma| =1 $ and $ \langle\partial^{2}_{x}\gamma,N\rangle =0$ at the point $(x,t) $. By definition $ \kappa = |\partial_{x}\gamma|^{-2}\langle\partial^{2}_{x}\gamma,N\rangle $ compute \begin{equation}
\kappa_{t} = \partial_{t}\langle\partial_{xx}\gamma,N\rangle-2\langle T,\partial_{xt}\gamma,N\rangle\\
=\langle\partial_{tx}\gamma,N\rangle-2\kappa\langle T,\partial_{tx}\gamma\rangle
\end{equation}
since $ N_{t} $ moves in the tangent direction and $ \partial_{xx}\gamma =\kappa N$. Using the orthogonality relations between tangent and normal vectors and plugging in (8) we obtain\begin{align}
\partial_{t}\kappa &= \partial_{xx}\kappa+\kappa\langle\partial_{xx}N,N\rangle-2\kappa^{2}\langle T,\partial_{x}N\rangle\\
&= \partial_{xx}\kappa-\kappa\langle\partial_{x}N,\partial_{x}N\rangle +2\kappa^{3}\\
&= \kappa_{ss}+\kappa^{3}
\end{align}
where we used that $ \partial_{x}N = -\kappa T $.
\end{proof}
We now present the following corollary without proof:
\begin{corollary}[Conservation of Convexity]
Convexity is preserved under curve shortening flow, i.e. if $ \kappa>0 $ at $ t = 0 $ then $ \kappa>0 $ for all $ t\in[0,\infty) $.
\end{corollary}
We now present an important formula without proof derived by Huisken ([4]) that is useful when trying to look at blow-up analysis on singularities of CSF as described in Section 1.2 since this equation is invariant under parabolic scaling.
\begin{definition}[Heat Kernel]
Let $ X_{0} = (x_{0},t_{0}) $ and denote by $ \rho_{X_{0}}(x,t) $ the backwards Heat Kernel. We then define it as\begin{equation}
\rho_{X_{0}}(x,t) = (4\pi(t_{0}-t))^{-1/2}e^{-\frac{|x-x_{0}|^{2}}{4(t_{0}-t)}}
\end{equation}
for $ t<t_{0} $.
\end{definition}
\begin{theorem}[Huisken's Monotonicity Formula]
Let $ \{\Gamma_{t}\} $ be a family of curves that move by CSF, then
\begin{equation}
\frac{d}{dt}\int_{\Gamma_{t}} \rho_{X_{0}}ds =-\int_{\Gamma_{t}}\Big|\kappa+\frac{\langle\gamma,N\rangle}{2(t_{0}-t)} \Big|^{2}\rho_{X_{0}}ds\qquad(t<t_{0`})
\end{equation}
\end{theorem}
Although the goal of the following theorem is to prove existence and uniqueness for solutions of CSF its importance is in the maximal existence time curvature.
\begin{theorem}[Existence and Uniqueness]
Let $ \gamma_{0}:S^{1}\rightarrow\mathbb{R}^{2} $ be an embedded curve. Then there exists a unique smooth solution $ \gamma:S^{1}\times[0,T)\rightarrow\mathbb{R}^{2} $ of curve shortening flow defined on a maximal interval $ [0,T) $. The maximal existence time is characterized by \begin{equation}
\sup_{S^{1}\times[0,T)}|\kappa| = \infty
\end{equation}
\end{theorem}
\begin{theorem}[Local Regularity Theorem $ \text{([7],[8])} $]
Let $ X = (x,t) $ be a point in space-time and $ P_{r}(X)=B_{r}(x)\times(t-r^{2},t] $ for the parabolic ball with center $ X $ and radius $ r $. There exist universal constants $ \epsilon >0 $ and $ C<\infty $ with the following property. If $\{\Gamma_{t}\subset\mathbb{R}^{2}\}_{t\in(t_{0}-2r^{2},t]} $ is a curve shortening flow with \begin{equation}
\sup_{\bar{X_{0}}\in P_{r}(X_{0})}\Theta\big(\{\Gamma _{t}\}, \bar{X_{0}},r \big) :=\int_{\Gamma_{t_{0}}-r^{2}}\rho_{(x_{0},t_{0})}ds <1+\epsilon,
\end{equation}
then \begin{equation}
\sup_{P_{r/2}(X_{0})}|\kappa|\leq\frac{C}{r}
\end{equation}
\end{theorem}
The importance of Theorem 2.1.7 shall be when we create blowup sequences that are close to a circle as we approach the singularity, and won't allow for it to be a straight line. This is a fact we shall use to show that the curve must become a circle in the proof of Grayson's Theorem.
\begin{theorem}[Hamilton's Harnack Inequality$ \text{([9])} $]
If $ \{ \Gamma_{t}\subset\mathbf{R}^{2} \}_{t\in[0,T)} $ is a convex solution of CSF then $\frac{\kappa_{t}}{\kappa}-\frac{\kappa_{s}^{2}}{\kappa^{2}}+\frac{1}{2t}\geq0 $
\end{theorem}
We now present Huisken's distance comparison principle between the extrinsic and intrinsic distances ([5]). It shows that embededness is preserved by CSF.
\begin{theorem}[Huisken's Distance Comparison Principle $ \text{([5])} $]
If a family of closed embedded curves $ X $ in the plane evolves by CSF, then the following equation \begin{equation}
R(t) :=\sup_{x\neq y} \frac{L(t)}{\pi d(x,y,t)}sin\frac{\pi l(x,y,t)}{L(t)},
\end{equation}
where $ L(t) $ is the total length of the curve, $ l(x,y,t) $ is the intrinsic distance between $ X(x,t) $ and $ X(y,t) $, and $ d(x,y,t) = |X(x,t)-X(y,t)| $, is non-increasing in time.
\end{theorem}
The important part to take away from this theorem is that the intrinsic and extrinsic distance equation is bounded by $ R(0)<\infty $. Particularly this implies that the grim reaper solutions cannot arise as a blow-ip limit of CSF closed embedded curves.
We finish this section with Huisken's proof of Grayson's Theorem ([6]) via the method of singularity blow-up which we present in full to demonstrate the utility of self-similar solutions. First we make the following definition of a blow-up point for CSF.
\begin{definition}[Blow-up Point]
We say that $ x_{0}\in\mathbb{R}^{2} $ is a blowup point if there are sequences $ t_{i}\rightarrow T $, $ p_{i}\in \Gamma_{t} $ such that $ |\kappa|(p_{i})\rightarrow\infty $ and $ p_{i}\rightarrow x_{0}$, i.e it achieves the curvature of the maximal existence point as defined in Theorem 2.1.6.
\end{definition}
\begin{theorem}[Grayson's Theorem $ \text{([6])} $]\label{Grayson}
If $ \Gamma\subset\mathbb{R}^{2} $ is a closed embedded curve, then the curve shortening flow $ \{\Gamma_{t}\}_{t\in[0,T)} $ with $ \Gamma_{0}=\Gamma $ exists until $ T = \frac{A_{\Gamma}}{2\pi} $ and converges for $ t\rightarrow T $ to a round point, i.e. there exists a unique point $ x_{0}\in\mathbb{R}^{2} $ such that the rescaled flows \begin{equation}
\Gamma^{\lambda}_{t}:=\lambda\cdot\big( \Gamma_{T+\lambda^{-2}t}-x_{0}\big)
\end{equation}
converge for $ \lambda\rightarrow\infty $ to the round shrinking circle $ \{\partial B_{\sqrt{-2t}} \}_{t\in(-\infty,0)} $
\end{theorem}
To prove this theorem the following lemma is needed\begin{lemma}
Along CSF we have that \begin{equation}
\frac{d}{dt}\int_{\Gamma_{t}}|\kappa|ds = -2\sum_{x:\kappa(x,t) = 0}|\kappa_{s}|(x,t)
\end{equation}
\end{lemma}
\begin{proof}
Since solutions of CSF are analytic there are only a finite number of inflection points, giving \[ \frac{d}{dt}\Big( \int_{\{\kappa\geq0\}}\kappa ds-\int_{\{\kappa\leq0\}}\kappa ds \Big) = \int_{\{\kappa\geq0\}}\kappa_{ss} ds-\int_{\{\kappa\leq0\}}\kappa_{ss} ds \] and integrating by parts gives the results.
\end{proof}
\begin{proof}
(Proof of Theorem \ref{Grayson}) Let $ T<\infty $ be the maximal existence time of CSF starting at $ \Gamma $. Suppose towards a contradiction \begin{equation}
\limsup_{t\rightarrow T}\Big( (T-t)\max_{\Gamma_{t}}\kappa^{2} \Big) = \infty
\end{equation}
i.e. we have a type II blow-up. For any integer $ k\geq1/T $ we let $ t_{k}\in [0,T-\frac{1}{k}], x_{k}\in S^{1}$ be such that \begin{equation}
\kappa^{2}(x_{k},t_{k})(T-1/k-t_{k}) = \max_{t\leq T-1/k,x\in S^{1}}\kappa^{2}(x,t)(T-1/k-t).
\end{equation}
We also set \begin{equation*}
\lambda_{k}=\kappa(x_{k},t_{k}),\quad t^{(0)}_{k} = -\lambda^{2}_{k}t_{k}, \quad t^{(1)}_{k} = \lambda^{2}_{k}(T-1/k-t_{k}).
\end{equation*}
We can then say, thanks to (27), that for any $ M\leq \infty $ there exist $ \bar{t}< T $ and $ \bar{x}\in S^{1} $ such that $ \kappa^{2}(\bar{x},\bar{t})(T-\bar{t})
>2M $. For $ k $ large enough we have \begin{equation}
\bar{t}<T-1/k,\quad\kappa^{2}(\bar{x},\bar{t})(T-\bar{t}-1/k)>M.
\end{equation}
Then it follows that \begin{equation}
t^{(1)}_{k} = \kappa^{2}(x_{k},t_{k})(T-1/k-t_{k})\geq\kappa^{2}(\bar{x},\bar{t})(T-1/k-\bar{t})>M.
\end{equation}
So then since $ t_{k}^{(1)} $ is increasing and $ M $ is arbitrary, this implies $ t^{(1)}_{k}\rightarrow\infty $, so then $ \lambda_{k}\rightarrow\infty,t_{k}\rightarrow T $ and $ t^{(0)}_{k}\rightarrow-\infty $.
Then consider the sequence of the rescaled flow \begin{equation}
\Gamma_{t}^{k} = \lambda_{k}\cdot\Big( \Gamma_{t_{k}+\lambda_{k}^{-2}t}-x_{k} \Big),\qquad t\in[t_{k}^{(0)}, t_{k}^{(1)}).
\end{equation}
and we find that by construction, $ \Gamma^{k}_{t} $ has $ \kappa_{k}=1 $ at $ t =0 $ at the origin. Then by our choice $ (x_{k},t_{k}) $ implies \begin{equation}
\kappa^{2}_{k}(x,t)\leq\frac{T-1/k-t_{k}}{T-1/k-t_{k}-\lambda^{2}_{k}t} = \frac{t^{(1)}_{k}}{t^{(1)}_{k}-t},\quad t\in[t_{k}^{(0)},t^{(1)}_{k}).
\end{equation}
Then we have that after passing to a subsequence, we get the smooth limit $ \{\Gamma_{t}^{\infty} \}t\in(-\infty,\infty) $. Then we have by our previous formulation that the limit $ \kappa = 1 $ at the time 0 at the origin, and $ \kappa^{2}\leq1 $ at every point for all time. Then by the previous Lemma the limit satisfies \begin{equation}
\int_{-\infty}^{\infty}\sum_{x:\kappa(x,t) = 0}|\kappa_{s}|(x,t)dt = 0
\end{equation}
i.e. if $ \kappa = 0 $ then $ \kappa_{s} = 0 $ as well. Then, using the evolution equations and analyticity this implies that $ \{\Gamma_{t}^{\infty}\}_{t\in(-\infty,\infty)} $ is a straight line, a contradiction.
This then gives that $ \kappa>0 $, and by equality in the case of Hamilton's Harnack Inequality, and the fact that a translating soliton for CSF must be a grim reaper curve, this contradicts the bound for the ratio between intrinsic and extrinsic distance.
So then we have shown the a type I blow-up rate \begin{equation}
\limsup_{t\rightarrow T}\big( (T-t)\max_{\Gamma_{t}}\kappa^{2} \big)<\infty
\end{equation}
To finish the discussion we simply need to prove the following claim:\begin{claim}
Define the parabolic scaling of a family of curves that satisfy CSF $\{\Gamma^{\lambda}_{t}\}_{t\in[-\lambda^{2}T,0)} $ where $ \Gamma^{\lambda}_{t}:=\lambda\cdot(\Gamma_{T+\lambda^{-2}t}-x_{0}) $. Then, for the limit as $ \lambda\rightarrow\infty $, these converge smoothly to a family of round shrinking circles $ \{ \partial B_{\sqrt{-2t}}(0) \}_{t\in(-\infty,0)}. $
\end{claim}
To finish the proof we rescale the blowup rate and try to show that the limit intersects families of round shrinking circles.
\begin{equation}
\max_{\Gamma^{\lambda}_{t}}|\kappa|\leq\frac{C}{\sqrt{-t}},\qquad t\in[-\lambda^{2}T,0).
\end{equation}
We have in our previous argument found a subsequence of $ \lambda_{k} $ such that $ \{ \Gamma^{\lambda_{k_{i}}}_{t} \} $ converges smoothly to a limit. By construction, the limit is an ancient solution of CSF. Then we can use the definition of blow-up points and comparing with round shrinking circles we can see that $ \Gamma^{\lambda}_{-1}\cap B_{2}(x_{0}) \neq\emptyset$ for $ \lambda $ large enough. So then the limit is not empty. Then by Huisken's Monotonicity Formula for all $ t_{1}<t_{2}<0 $ it gives that \begin{equation}
\int_{t_{1}}^{t_{2}}\int_{\Gamma_{t}^{\lambda}}\Big|\kappa+\frac{\langle\gamma,N\rangle}{2(t_{0}-t)} \Big|^{2}\rho dsdt =-\Big[\int_{\Gamma_{t}^{\lambda}} \rho_{X_{0}}ds\Big]^{T-t_{2}/\lambda^{2}}_{T-t_{1}/\lambda^{2}}\rightarrow 0
\end{equation}
as $ \lambda\rightarrow 0 $. So then the limit is self-similarly shrinking and completely determined by its slice at $ t=-1 $ satisfying \begin{equation}
\kappa+\frac{\langle \gamma,N\rangle}{2} = 0.
\end{equation}
Then by the local regularity theorem it cannot be a straight line, so then $ \Gamma_{-1} $ must be a circle of radius $\sqrt{2} $ which completes the theorem.
\end{proof}
\subsection{Self-Similar Solutions of Curve Shortening Flow}
In this section we summarize the work of Halldorsson ([10]) who gave the classification of all self-similar embedded curves that evolve under CSF in the plane. We also direct the reader to the work of Altschuler et al ([11]) who classified the solitons of CSF in $ \mathbb{R}^{n} $ through the creation of a group acting on curves evolving under CSF that produced self-similar solutions that turned the PDE into an ordinary differential equation.
We return now to the work of Halldorsson and follow his steps to derive the family of ODE's that provide self-similar solutions and state his theorems in full. In [10] self-similar solutions were classified under the following classifications:\begin{itemize}
\item Translating Curves: Only the Grim Reaper curve
\item Expanding Curves: A one dimensional family of curves. Each is properly embedded and asymptotic to the boundary of a cone.
\item Shrinking Curves: A one-dimensional family of curves. Each is contained in an annulus and consists of identical excursions between both boundaries.
\item Rotating Curves: A one-dimensional family of curves. Each is properly embedded and spirals out to infinity.
\item Rotating and Expanding: A two dimensional family of curves, properly embedded that spiral out to infinity.
\item Rotating and Shrinking Curves: A two-dimensional family of curves, with an end asymptotic to a circle and the other either similarly asymptotic or spiraling out to infinity.
\end{itemize}
Consider instead of CSF acting on curves mapping $\mathbb{R} $ to $ \mathbb{R}^{2}$ that they were mapped instead to the $ \mathbb{C} $, the change of space allows one to simplify the action of rotations on the curve. Now, following the steps of Halldorsson let $ X:\mathbb{R}\times I\rightarrow\mathbb{C} $ be a curve evolving under CSF, and being a self-similar solution it is of the form \begin{equation}
\hat{X}(x,t) = g(t)e^{if(t)}X(x)+H(t)
\end{equation}
where $ I $ is an interval 0, and $ f,g,H $ are all differentiable functions such that $ f(0) = 0,g(0) =1 $ and $ H(0) =0 $ so that $ \hat{X}(x,0) = X(x) $.
In $ \mathbb{C} $ we define the normal vector $ N(x,t) =iT(x,t) $. Then plugging in (38) into the definition of CSF gives\begin{equation}
g^{2}(t)f'(t)\langle X(x),T(x)\rangle +g(t)g'(t)\langle X,N\rangle+g(t)\langle e^{-if(t)}H'(t),N\rangle = \kappa(x).
\end{equation}
Since this equation must hold for all $ (x,t)\in\mathbb{R}\times I $. For $ t =0 $ the curve must then satisfy the following ODE:\begin{equation}
A\langle X,T\rangle+B\langle X,N\rangle+\langle C,N\rangle=\kappa(x)
\end{equation}
where $ A =f'(0),\quad B =g'(0),\quad\text{and}\quad C=H'(0) $.
Looking at solutions where the dilation term vanishes gives \begin{equation}
A\langle X,T\rangle+B\langle X,N\rangle =\kappa.
\end{equation}
We now look to prove the following theorem:
\begin{theorem}[Existence of Embedded Self-Similar Curves $ \text{[10]} $]
For each value of A and B there exists an immersed curve $ X $ satisfying (41).
\end{theorem}
To ensure that (40) exists for all time choose $ g^{2}f' =A $ and $ g(t)g'(t)=B $ for all $ t\in I $. We choose $ f $ and $ g $ to be \[ f(t )=\begin{cases}
\frac{A}{2B}\log(2Bt+1)\quad&\text{if}\quad B\neq0,\\
At\quad&\text{if}\quad B=0
\end{cases}\] and \begin{equation}
g(t) =\sqrt{2Bt+1}.
\end{equation}
Thanks to these equations we see that $ X $ rotates around the origin, with exception of if $ A=0 $ of course, and dilates outwards for $ B>0 $ and inwards if $ B<0 $. Including the rotation term $ C $ only causes the curve to screw-dilate around the point $ \frac{-C}{B+iA} $. For curves that only translate we have that it can only give a Grim Reaper Curve as was shown in [13].
Now consider a parametrization by arclength and use the Frenet equations to obtain the following relations\begin{align}
\frac{d}{ds}\langle X,T\rangle &= 1+\kappa\langle X,N\rangle,\\
\frac{d}{ds}\langle X,N\rangle &=-\kappa\langle X,N\rangle.
\end{align}
Then define the equations\begin{align}
x &= A\langle X,T\rangle+B\langle X,N\rangle\\
y &= -N\langle X,T\rangle+A\langle X,N\rangle\\
x+iy &= (A-iB)(\langle X,T\rangle+i\langle X,N\rangle),
\end{align}
that satisfy \begin{align}
x' = \kappa y+A,\\
y' = -\kappa x-B.
\end{align}
and from these equations we can rewrite $ X $ to be \begin{equation}
X =e^{i\theta(s)}\frac{x+iy}{A-iB}
\end{equation}
where $ \theta(s) = \int_{0}^{s}\kappa(z)dz+\theta_{0} $ and $ T(0) = e^{i\theta_{0}} $. Since we now look for curves that satisfy $ x =\kappa $, we substitute this into (48) and (49), as well as the definition of $ \theta $ to obtain \begin{align}
x'&=xy+A\\
y'&=-x^{2}-B
\end{align}
with initial conditions $ x_{0} $ and $ y_{0} $.
We then see that by these equations \[ X' = e^{i\theta}, \] so that then $ T = e^{i\theta} $ and $ X $ is parametrized by arclength, so then $ \kappa = \theta' =x $. To finish the proof we simply run through the following calculation\begin{align}
A\langle X,T\rangle+B\langle X,N\rangle &=\langle X,(A+iB)Y\rangle\\
&=Re(X(A-iB)e^{-i\theta})\\
&=x\\
&=\kappa
\end{align}
finishing the theorem.
All the possible values of $ A,B $ create then a 2 parameter family of ODEs that govern the creation of self-similar curves for CSF, and are classified as in the beginning of this section. The differing possible values for $ A $ and $ B $ are what splits CSF into the following 4 cases:\begin{itemize}
\item $ A\neq0 $ and $ B\geq 0$ gives rotating expanding curves.
\item $ A\neq0 $ and $ B< 0$ gives rotating shrinking curves.
\item $ A=0 $ and $ B< 0$ gives only shrinking curves.
\item $ A=0 $ and $ B> 0$ gives only expanding curves.
\end{itemize}
Halldorson then goes on a case by case basis of solving the ODEs and finding their significant properties and deriving the following theorems which we present without proof and direct the reader to [10] in order to not only see their proof but also the accompanying graphics.
\begin{theorem}[$ A\neq0 $ and $ B\geq 0$]
The curves are properly embedded, have one point closest to the
origin and consist of two arms coming out from this point which strictly go away from the origin to infinity. Each arm has infinite total curvature and spirals infinitely many circles around the origin. The curvature goes to 0 along each arm, and
their limiting growing direction is $ B + iA $ times the location.
The curves form a one-dimensional family parametrized by their distance to the origin, which can take on any value in $[ 0, \infty) $.
If $ B = 0 $, then under the CSF the curves rotate forever with constant angula speed A.
If $ B > 0 $, then under the CSF the curves rotate and expand forever with angular function $ \frac{A}{2B}\log(2Bt + 1) $ and scaling function $
\sqrt{2Bt + 1}. $
\end{theorem}
\begin{theorem}[$ A\neq0 $ and $ B< 0$]
In this case there are two types of curves.
1) Curves such that the limiting behavior when going along the curve in each direction is wrapping around the circle of radius $\frac{1}{\sqrt{-B}} $, clockwise if $ A < 0 $ and
counterclockwise if $ A > 0 $. These curves form a one-dimensional family.
2) Curves such that the curvature never changes sign and the two ends behave very differently. One end wraps around the circle with radius $ \frac{1}{\sqrt{-B}} $
in its limiting
behavior, clockwise if $ A < 0 $ and counterclockwise if $ A > 0 $. The other end spirals infinitely many circles around the origin out to infinity and has infinite total curvature. The curvature goes to 0 along it, and its limiting growing direction is
$ -B - iA $ times the location. There is at least one curve of this type, and we call it the comet spiral.
These curves rotate and shrink with angular function $\frac{A}{2B}\log(2Bt+1) $ and scaling
function $ \sqrt{2Bt + 1} $ under the CSF. A singularity forms at time $ t = -\frac{1}{2B} $. The curves of type 1 are bounded, so they disappear into the origin.
\end{theorem}
\begin{theorem}[$ A=0 $ and $ B< 0$]
Each of the curves is contained in an annulus around the origin
and consists of a series of identical excursions between the two boundaries
of the annulus. The curvature is an increasing function of the radius and never changes sign. The inner and outer radii of the annulus, $ r_{min} $ and $ r_{max} $, satisfy
$r_{min}\exp(Br_{min}^{2}/2)=r_{max}\exp(Br_{max}^{2}/2) $ and take on every value in $ (0, \frac{1}{\sqrt{-B}}] $ and $ [\frac{1}{\sqrt{-B}},\infty) $,
respectively.
The curves form a one-dimensional family parametrized by $ r_{min} $ and are divided
into two sets:
1) Closed curves, i.e., immersed $ \mathbb{S}^{1} $ (Abresch-Langer curves ([12])). In addition to the circle, there is a curve with rotation number p which touches each boundary of the annulus $ q $ times for each pair of mutually prime positive integers $ p, q $ such that $ \frac{1}{2}<\frac{p}{q} $.
2) Curves whose image is dense in the annulus.
Under the CSF these curves shrink with scaling function $g(t) =\sqrt{2Bt + 1} $ until
they disappear into the origin at time $ t = -\frac{1}{2B}$.
\end{theorem}
\begin{theorem}[$ A=0 $ and $ B \geq 0 $]
Each of the curves is convex, properly embedded and asymptotic to
the boundary of a cone with vertex at the origin. It is the graph of an even function.
The curves form a one-dimensional family parametrized by their distance to the
origin, which can take on any value in $ [0, \infty) $.
Under the CSF these curves expand forever as governed by the scaling function $ g(t) = \sqrt{2Bt + 1} $.
\end{theorem}
\section{The Vortex Filament Equation}
The Vortex Filament Equation (VFE) \[ \partial_{t}\gamma = \partial_{s}\gamma\times\partial_{ss}\gamma, \] where $ s $ is the arc-length, arises in the consideration of vortices with infinitesimal thickness of size and whose effects at infinity can be ignored. It has applications in the description on the shape of vortices and the interaction of vortex lines and has applications in aerodynamics as well as high energy quantum fluids. Here we present a derivation of the equation by the Local Induction Principle as given by Arms and Hama ([14]).
Hama and Arms first consider the Biot-Savart law:\begin{equation}
d\mathbf{q}_{ij} = -\frac{k}{4\pi}\mathbf{r}^{-3}_{ij}\frac{\partial \mathbf{r}_{ij}}{\partial s_{i}}\times \mathbf{r}_{ij}ds_{j}.
\end{equation}
Here $ k $ is a scalar and is the strength of the vortex, $ d\mathbf{q}_{ij} $ is the induced velocity at the point $ r_{i} $ by the vortex segment $ ds_{j} $ at the point $ \mathbf{r}_{i} $, and $ \mathbf{r}_{ij} $ is the vector distance between the points $ \mathbf{r}_{i} $ and $ \mathbf{r}_{j} $.
Begin by expanding through a Taylor Series the vector \[ \mathbf{r}_{ij}(\zeta,t) = \mathbf{r}_{i}(s_{i},t) -\mathbf{r}_{j}(s_{i}+\zeta,t). \] Assuming that $ \zeta $ is small then the expression becomes:\begin{equation}
\mathbf{r}_{ij}(\zeta) = \mathbf{a}_{1}\zeta+\mathbf{a}_{2}\zeta^{2}+...
\end{equation}
where the substitutions \begin{equation}
\mathbf{a}_{1} = \partial_{\zeta}\mathbf{r}_{ij},
\quad\mathbf{a}_{2}= \partial^{2}_{\zeta}\mathbf{r}_{ij},\text{... at }\zeta = 0
\end{equation}
are made.
This gives that $ \partial\mathbf{r}_{ij}/\partial s_{i} = \partial\mathbf{r}_{ij}/\partial\zeta = \mathbf{a}_{1}+2\mathbf{a}_{2}\zeta+... $ and \begin{align}
-\partial\mathbf{r}_{ij}/\partial s_{i}\times\mathbf{r}_{ij} &= (\mathbf{a}_{1}\zeta+\mathbf{a}_{2}\zeta^{2}+...)\times(\mathbf{a}_{1}+2\mathbf{a}_{2}\zeta+...)\\
&=(\mathbf{a}_{1}\times\mathbf{a}_{2})\zeta^{2}+O(\zeta^{3})\\
&= (\mathbf{a}_{1}\times\mathbf{a}_{2})|\zeta|^{2}.
\end{align}
Similarly to find the value of the distance vector, \begin{align}
|\mathbf{r}_{ij}|^{2} &= |(\mathbf{a}_{1}\zeta+\mathbf{a}_{2}\zeta^{2}+...)^{2}|\\
&= |\mathbf{a}_{1}|^{2}|\zeta|^{2}+2\mathbf{a}_{1}\cdot\mathbf{a}_{2}\zeta^{3}+...\\
r_{ij} &= |\mathbf{a}_{1}||\zeta|\Big(1+2\frac{\mathbf{a}_{1}\cdot\mathbf{a}_{2}}{|\mathbf{a}_{1}|^{2}}\zeta+... \Big)
\end{align}
so that we can then take the exponent and expand the binomial to obtain \[ r_{ij}^{-3} = |\mathbf{a}_{1}|^{-3}|\zeta|^{-3}\Big(1-3\frac{\mathbf{a}_{1}\cdot\mathbf{a}_{2}}{|\mathbf{a}_{1}|^{2}}\zeta+... \Big). \]
Finally, this gives the expression \begin{equation}
\mathbf{q}_{ij} = \frac{k}{4\pi}\int\Big[ \frac{\mathbf{a}_{1}\times\mathbf{a}_{2}}{|\mathbf{a}_{1}|^{3}}\frac{1}{|\zeta|}+O(1)\Big]d\zeta.
\end{equation}
Arms and Hama then consider integration over the limit $ \epsilon\leq|\zeta|<1 $, one obtains that \[\mathbf{q}_{ij} =\frac{k}{2\pi} \frac{\mathbf{a}_{1}\times\mathbf{a}_{2}}{|\mathbf{a}_{1}|^{3}}\frac{1}{|\epsilon|}+O(1)\Big] \] and by re-substituting $ \mathbf{a}_{n} = \frac{1}{n!}\partial^{n}\mathbf{r}_{ij}/\partial\zeta^{n} $ they obtain:
\begin{equation}
\frac{4\pi}{k}\mathbf{q}_{ij} = \frac{(\partial\mathbf{r}/\partial s)_{i}\times(\partial^{2}\mathbf{r}/\partial s^{2})_{i}}{|(\partial\mathbf{r}/\partial s)_{i}|^{3}}\log(\frac{1}{\epsilon})+O(1)
\end{equation}
Then considering the infinitesimal limit for $ \epsilon<<1 $ and ignoring the terms $ O(1) $, whch is equivalent to ignoring long distance effects, we may write the previous expression as \begin{equation}
\frac{\partial\mathbf{r}}{\partial t} = \frac{(\partial\mathbf{r}/\partial s)\times(\partial^{2}\mathbf{r}/\partial s^{2})}{|(\partial\mathbf{r}/\partial s)|^{3}}
\end{equation}
The high nonlinearity of this equation makes for explicit solutions to be hard to find. We then look for self-similar solutions to understand the behavior of vortices under this equation.
\subsection{Properties of Vortex Filament Equation}
We present in this section some of the basic properties of VFE and conclude it with a proof that the only self-similar translating solutions that lie in a plane for all time are circles moving in the binormal direction.
\begin{lemma}[Arc-Length Commutator]
Let $\gamma$ be a curve evolving by the Vortex Filament Equation. Then it satisfies the following commutator relation, where $x$ is arclength:
\begin{equation}
\Large{\Big[\frac{\partial}{\partial t}, \frac{\partial}{\partial s}\Big]}
\normalsize :=
\Large \frac{\partial^{2}}{\partial t \partial s}-\frac{\partial^{2}}{\partial s \partial t} \normalsize =0.
\end{equation}
In other words,
\begin{align}
\gamma_{ts}=\gamma_{st}
\end{align}
\end{lemma}
\begin{proof}
Using $\frac{\partial}{\partial s}=\lvert\frac{\partial\gamma}{\partial u}\rvert^{-1}\frac{\partial}{\partial u}$, where $u$ is an arbitrary parameter, we compute
\begin{align}
\gamma_{st} &= (\lvert\gamma_{u}\rvert^{-1}\gamma_{u})_{t}\\
&=-\lvert\gamma_{u}\rvert^{-3}\langle\gamma_{tu},\gamma_{u}\rangle\gamma_{u}+\lvert\gamma_{u}\rvert^{-1}\gamma_{tu}\\
&=-\lvert\gamma_{u}\rvert^{-3}\langle\kappa B_{u},\gamma_{u}\rangle\gamma_{u}+\gamma_{ts}\\
&=\kappa\langle B_{s},T\rangle\gamma_{s}+\gamma_{ts}\\
&=\gamma_{ts} ,
\end{align}
because $B_{s}=-\tau N$.$\qed$
\end{proof}
\begin{corollary}[Evolution of the Normal Vector, Curvature and Torsion]
Let $\gamma$ be a curve evolving under Binormal Flow, $N$ be its corresponding normal vector and $ F(s,t) = \tau^{2}-\frac{\kappa_{ss}}{\kappa} $. Then the evolution of the normal vector satisfies $\partial_{t}N = \tau\kappa T-F(s,t)B$, the evolution for curvature satisfies $ \kappa_{t} = -(2\kappa_{s}\tau+\tau_{s}\kappa)$, and the evolution equation for torsion satisfies $\partial_{t}\tau = -\kappa\kappa_{s}+F_{s}(s,t)$.
\end{corollary}
\begin{proof}
We have that $\lVert N\rVert^{2} = 1$ so then $\partial_{t}(\lVert N\rVert^{2}) = 2\langle N_{t},N\rangle = 0$. We can therefore separate $\partial_{t}N$ into tangential and binormal components. We now define $F(x,t) = \langle N,B_{t}\rangle$ and calculate
\begin{align}
\frac{d}{dt}\langle N,T\rangle &= \langle N_{t},T\rangle+\langle N,T_{t}\rangle=0\\
\langle N_{t},T\rangle &=-\langle N,T_{t}\rangle=\tau\kappa\\
\langle N_{t},B\rangle &=-\langle N,B_{t}\rangle = -F(s,t).
\end{align}
Since by the definition of VFE, $T_{t}=\partial_{s}(\kappa B) = \kappa_{s}B-\tau\kappa N$. Using that $\langle N,T\rangle = \langle N,B\rangle = 0$, we compute
\begin{align}
N_{t} &= \Big[\frac{\partial_{ss}\gamma}{\kappa}\Big]_{t}\\
&=-\frac{\kappa_{t}}{\kappa^{2}}\gamma_{ss}+\frac{1}{\kappa}\gamma_{sst}.
\end{align}
We now calculate $ \gamma_{sst} $ as follows:\begin{align}
\partial_{s}\gamma &= \frac{1}{|\gamma_{x}|}\gamma_{x}\\
\partial_{ss}\gamma &= \frac{1}{|\gamma_{x}|}\partial_{x}(|\gamma_{x}|^{-1}\gamma_{x})\\
&= \frac{\gamma_{xx}}{|\gamma_{x}|^{2}}\\
\partial_{sst}\gamma &=\partial_{t}(\frac{\gamma_{xx}}{|\gamma_{x}|^{2}})\\
&= -2|\gamma_{x}|^{-3}\langle\gamma_{xt},\gamma_{x}\rangle\gamma_{xx}+ \frac{\gamma_{xxt}}{|\gamma_{x}|^{2}}\\
&= \frac{\gamma_{txx}}{|\gamma_{x}|^{2}} = (\kappa B)_{ss} = \kappa_{ss}B+2\kappa_{s}B_{s}+\kappa B_{ss}\\
&= \kappa_{ss}B-2\kappa_{s}\tau N+\kappa(-\tau_{s}N+\kappa\tau T-\tau^{2}B).
\end{align}
Combining our results we get that \begin{align}
N_{t} &= -\frac{\kappa_{t}}{\kappa}N+\frac{\kappa_{ss}}{\kappa}B-2\frac{\kappa_{s}\tau}{\kappa}N-\tau_{s}N+\kappa\tau T-\tau^{2}B.\\
&= -\Big(\frac{\kappa_{t}}{\kappa}+2\frac{\kappa_{s}\tau}{\kappa}+\tau_{s}\Big)N+\tau\kappa T+\Big(\frac{\kappa_{ss}}{\kappa}-\tau^{2}\Big)B.
\end{align}
From this we can conclude that $ \kappa_{t} = -(2\kappa_{s}\tau+\tau_{s}\kappa) $ since $ \langle N_{t},N\rangle = 0$, and that $ F(s,t) = \tau^{2}-\frac{\kappa_{ss}}{\kappa} $.
Our final expression for the evolution of the normal vector is then \begin{equation}
N_{t} = \tau\kappa T+\Big(\frac{\kappa_{ss}}{\kappa}-\tau^{2}\Big)B.
\end{equation}
To compute the evolution equation for torsion we first compute the evolution for the Binormal Vector:
\begin{align}
B &= T\times N\\
B_{t}&=T_{t}\times N+T\times N_{t}\\
&=-\kappa_{s}T+F(s,t)N = -\kappa_{s}T+\Big(\tau^{2}-\frac{\kappa_{ss}}{\kappa}\Big)N.
\end{align}
Continuing, we calculate
\begin{align}
\partial_{t}(\tau N) &=\partial_{t}(\partial_{s}B)\\
&= \partial_{s}(\partial_{t}B)\\
&= \partial_{s}\Big[-\kappa_{s}T+F(s,t)N\Big]\\
&= -\kappa_{ss}T-\kappa_{s}\kappa N+F_{s}(s,t)N-\kappa F(s,t) T+\tau F(s,t) B\\
&= \Big[-\kappa_{ss}-\kappa F(s,t)\Big]T - \Big[\kappa_{s}\kappa+F_{s}(s,t)\Big]N+\tau F(s,t)B.
\end{align}
Thus since $\partial_{t}(\tau N) = \partial_{t}\tau N+\tau N_{t}$ we have that \[ \partial_{t}\tau= -\kappa\kappa_{s}+2\tau\tau_{s}-\frac{\kappa_{sss}}{\kappa}+\frac{\kappa_{ss}\kappa_{s}}{\kappa^{2}}. \]
\end{proof}
\begin{lemma}[Planar Translating Self-similar Solutions]
The only solutions of VFE that lie in a translating plane are the line and the circle, with the latter traveling in the binormal direction.
\end{lemma}
\begin{proof}
Take $\gamma = (f(s,t),g(s,t),h(s,t))$ to be a space curve in $\mathbb{R}^{3}$ parametrized by arclength and having torsion 0.
Then $\partial_{t}\gamma = (\partial_{t}f(s,t),\partial_{t}g(s,t),\partial_{t}h(s,t))$, $\partial_{s}\gamma = (\partial_{s}f(s,t),\partial_{s}g(s,t),\partial_{s}h(s,t))$, and $\partial_{ss}\gamma = ( \partial_{ss}f(s,t),\partial_{ss}g(s,t),\partial_{ss}h(s,t))$.
Using the definition of VFE equation we calculate that
\begin{align}
\partial_{t}\gamma&= \kappa B\\
&= \partial_{s}\gamma\times\partial_{ss}\gamma \\
&= ( g_{s}h_{ss}-g_{ss}h_{s},f_{ss}h_{s}-f_{s}h_{ss}, f_{s}g_{ss}-f_{ss}g_{s})
\end{align}
Since curvature is invariant under rotation and the solutions of the curve are invariant under translations, it suffices to solve for equations lying in the x-y plane of our chosen coordinate system.
Also the Frenet Equations under the initial conditions become
\begin{align}
\partial_{s}T &= \kappa N\\
\partial_{s}N &= -\kappa T\\
\partial_{s}B &= 0
\end{align}
From (104) we can conclude that $ B(s,t) =B(0,t) $. Similarly $\kappa(x,t) = (f_{s}g_{ss}-g_{ss}f_{s})(s,t)$ since the curve lies on the x-y plane, and by Corollary 3.1.2 we have that $ \kappa_{t} = 0 $. Hence \[ \kappa(s,t) = \kappa(s,0) = (f_{s}g_{ss}-g_{ss}f_{s})(s,0). \]
Then, since for all $ t $ we have that \[ \partial _{t}h(s,t) = (f_{s}g_{ss}-g_{s}f_{ss})(s,t)=\kappa(s,t), \] the previous argument gives that $ \partial_{t}h(s,t) = \kappa(s,0) $. Integrating this last expression gives us that \[ h(s,t) = \kappa(s,0)t. \]
So then since $\lVert B(t)\rVert = 1$ we have that $\lVert\partial_{t}\gamma(s,t)\rVert = \lvert\kappa^{2}\rvert$, and this can be rewritten as $f_{t}^{2}+g_{t}^{2}+h_{t}^{2} =f_{t}^{2}+g_{t}^{2}+\kappa^{2}= \kappa^{2}$ and this gives that $f_{t}^{2} = g_{t}^{2} = 0$.
This implies that $f(s,t) = w(s)$ and $g(s,t) = z(s)$ completely determine the solutions to the binormal flow equation with $\tau = 0$ so long as $w(x)$ and $z(s)$ satisfy
\begin{align}
w'(s)^{2}+z'(s)^{2}+\kappa_{s}^{2}t^{2} &= 1
\end{align}
for all $s$ and $t$, where $\kappa = w'z''-z'w''$, due to the parametrization by arc length. Since this must hold for all $t \in[0,\infty)$ it forces $\kappa_{s} = 0$
Putting this all together we have that the equations $w(s)$ and $z(s)$ determine a solution to the binormal equation with $\tau = 0$ so long as they solve the system of differential equations:
\begin{align}
w'(s)^{2}+z'(s)^{2} &= 1\\
w'z''-z'w'' &= C
\end{align}
where $C$ is a constant.
The only solutions of these equations depend on the value assigned to $C$. If $C$ is 0, then it follows that $w(s) = as$ and $z(s) = \sqrt{1-a^{2}}s$, and if $C\neq0$ then the solutions are circles with a radius of $1/C$ since $\tau = 0$ and curvature is constant.$\qed$
\end{proof}
\subsection{The Hasimoto Transform and Hasimoto's Explicit Solution}
We now present the results obtained by Hasimoto on his paper on VFE and his explicit solution for a traveling wave soliton ([15]). The most striking part of his paper is the Hasimoto Transform which manages to turn VFE into a form that satisfies a Nonlinear Cubic Schroedinger Partial Differential Equation (NLCSE). This then allows one to find solutions to VFE by finding solutions to the fully integrable NLCSE. It was through this process that Hasimoto found one of the first explicit soliton solutions to VFE. We now present the results from his paper.
\begin{theorem}[Hasimoto Transform $ \text{[15]} $]
Let $ \Gamma:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}^{3} $ be a differentiable curve parametrized by arclength of curvature $ \kappa $ and torsion $ \tau $ that satisfies the Vortex Filament Equation. If we take the transformation $ \psi = \kappa \exp\Big(i\int_{0}^{s}\tau ds\Big) $ then $ \psi $ will satisfy the following NLCSE:\[ \frac{1}{i}\partial_{t}\psi = \partial_{ss}\psi+\frac{1}{2}(|\psi|^{2}+A)\psi. \]
\end{theorem}
\begin{proof}
The Frenet system of coordinates gives the following equations \begin{align}
\Gamma_{s} &= T\\
T_{s} &= \kappa N\\
N_{s} &= \tau B -\kappa T\\
B_{s} &= -\tau N.
\end{align}
Combining the last two equations in the following form gives \begin{equation}
(N+iB)_{s}=-i\tau(N+iB)-\kappa T
\end{equation}
which leads to the introduction of the new variables $ \mathfrak{N} $ and $ \psi $ defined as \begin{align}
\mathfrak{N} &= (N+iB)\exp\Big(i\int_{0}^{s}\tau ds\Big)\\
\psi &= \kappa\exp\Big(i\int_{0}^{s}\tau ds\Big).
\end{align}
From the Frenet equations we obtain the following expressions, where the last one follows from the defintion of VFE and use of (66) and (67): \begin{align}
\mathfrak{N}_{s} &= -\psi T\\
T_{s}&= \text{Re}\big[\psi\mathfrak{N}\big] = \frac{1}{2}\Big(\bar{\psi}\mathfrak{N}+\psi\bar{\mathfrak{N}}\Big)\\
T_{t} &= \text{Re}\big[i\psi'\bar{\mathfrak{N}} \big] = \frac{1}{2}i\big(\psi'\bar{\mathfrak{N}}-\bar{\psi}'\mathfrak{N}\big).
\end{align}
The following relations shows that this creates an orthogonal system of equations \begin{equation*}
T\cdot T = 1,\qquad\mathfrak{N}\cdot\bar{\mathfrak{N}} = 2,\qquad\mathfrak{N}\cdot\mathfrak{N} = 0,\qquad \mathfrak{N}\cdot T = 0,\quad \text{etc.}
\end{equation*}
Now we derive the evolution equation for $ \mathfrak{N} $ by expressing it in this new orthogonal system as\begin{equation}
\mathfrak{N}_{t} = \alpha\mathfrak{N}+\beta\bar{\mathfrak{N}}+\gamma T.
\end{equation}
We now determine the values of the coefficients as follows \begin{align}
\alpha+\bar{\alpha} &=\frac{1}{2}\big(\bar{\mathfrak{N}}_{t}\cdot\mathfrak{N}+\mathfrak{N}\cdot\bar{\mathfrak{N}}_{t}\big)= \frac{1}{2}\partial_{t}(\bar{\mathfrak{N}}\cdot\mathfrak{N}) = 0,\qquad \alpha = iR\\
\beta &= \frac{1}{2}\mathfrak{N}\cdot\bar{\mathfrak{N}} =\frac{1}{4}\partial_{t}\big(\mathfrak{N}\cdot\mathfrak{N}\big) = 0\\
\gamma&=-\mathfrak{N}\cdot T_{t} = -i\psi
\end{align}
where $ R $ is a real function. This then simplifies to \begin{equation}
\mathfrak{N}_{t} = i\big(R\mathfrak{N}-\psi T\big).
\end{equation}
We now take the time derivative of (68) and the arclength derivative of (75) obtaining \begin{align}
\mathfrak{N}_{st} &=-\psi T-\psi T_{t}=-\psi T-\frac{1}{2}i\psi\big(\psi_{s}\bar{\mathfrak{N}}-\bar{\psi}_{s}\mathfrak{N}\big)\\
\mathfrak{N}_{ts} &= i\big[R_{s}\mathfrak{N}-R\psi T-\psi_{ss}T-\frac{1}{2}\psi_{s}\big(\bar{\psi}\mathfrak{N}+\psi\bar{\mathfrak{N}}\big)\big].
\end{align}
We can now equate the coefficients of $ T $ and $ i\mathfrak{N} $ giving us \begin{align}
-\psi = -i\big(\psi_{ss}+R\psi\big)\\
\frac{1}{2}|\psi|^{2} = R_{s}-\frac{1}{2}\psi_{s}\bar{\psi}.
\end{align}
Solving this last equation for $ R $ gives that $ R = \frac{1}{2}\big(|\psi|^{2}+A\big) $ which reduces the first equation down to \begin{equation}
\frac{1}{i}\partial_{t}\psi = \partial_{ss}\psi+\frac{1}{2}(|\psi|^{2}+A)\psi.
\end{equation}
\end{proof}
\begin{theorem}[Hasimoto's Traveling Wave $ \text{[15]} $]
Consider a soliton solution to NLCSE defined by Theorem 3.2.1 such that $ \kappa = 0 $ as $ s\rightarrow\infty $. This then gives as a solution a traveling wave with a kink that becomes a line at infinity.
\end{theorem}
\begin{proof}
First we introduce the new variable $ \zeta = s-ct $ where $ c $ can be taken to be the 'speed' of the translation. This variable is introduced in order to be able to work with a soliton of the Schroedinger equation derived in the previous theorem. Using this variable then gives \begin{equation}
\psi = \kappa(\zeta)\exp\Big[i\int_{0}^{s}\tau(\zeta)ds\Big].
\end{equation}
We plug this variable into the Schroedinger equation yielding the following real and imaginary parts, respectively, \begin{align}
\-c\kappa[\tau(\zeta)(-ct)] &= \kappa''-\kappa\tau^{2}+\frac{1}{2}(\kappa^{2}+A)\kappa\\
c\kappa' &= 2\kappa'\tau+\kappa\tau'
\end{align}
and integrating the last equation gives \begin{equation}
(c-2\tau)\kappa^{3} = 0
\end{equation}
where we have used our curvature limit to determine the constant of integration. By (129) we then have \[
\tau =\tau_{0}= \frac{1}{2}c = \text{constant}
\]
assuming that curvature is not identically zero. Using this we can integrate (130)
\[ \kappa = 2\nu \sech(\nu\zeta) \]
so long as $ A $ is a constant determined by $ A = 2(\tau_{0}^{2}-\nu^{2}) $. Now that the torsion and curvature are determined we can get the actual shape of the filament by substituting these into our original Frenet frame of reference and solving the following equation for the binormal vector:\begin{align}
\tau_{0}(T'-\kappa N) = \big[(1/\kappa)\big(B''+\tau_{0}^{2}B\big)\big]'+\kappa B' &= 0\\
\frac{d^{3}}{d\eta^{3}}B+\tanh(\eta)\frac{d^{2}}{d\eta^{2}}B+\big(S^{2}+\sech^{2}(\eta)\big)\frac{d}{d\eta}B+S^{2}\tanh(\eta)B &= 0
\end{align}
where we have made the substitutions $ \eta = \nu\zeta $ and $ S = \frac{\tau_{0}}{\nu} $.
A solution to this equation can then be obtained if we note that \[ \mathbf{C} = \frac{dB}{d\eta}+\tanh(\eta)B \] is a solution of the equation \[ d^{2}\mathbf{C}/d\eta^{2}+\big(S^{2}+2\sech^{2}(\eta)\big)\mathbf{C} = 0 \] which has as solutions \begin{equation}
\quad(1-S^{2}\mp2iS\tanh(\eta))e^{\pm iS\eta}.
\end{equation}
This finally gives us the binormal vector\begin{equation}
B = \sech(\eta),\quad(1-S^{2}\mp2iS\tanh(\eta))e^{\pm iS\eta}.
\end{equation}
which we can then substitute into our equations of the Frenet Frame and using the assumption, without loss of generality, that the filament is parallel to the x-axis at infinity: \begin{align}
T_{x}\rightarrow1\quad\text{as}\quad\eta\rightarrow\infty\\
N_{y}+iN_{z} = -i\big(B_{y}+iB_{z}\big) = e^{i(\tau_{0}\zeta+\sigma(t))}
\end{align}
Here $ \sigma(t) $ is a real function of $ t $ and the subscripts denote the component of the vector in the x, y, or z axis.
Finally straightforward calculation gives the final expressions
\begin{align}
\begin{cases}
\Gamma:& x = s-\frac{2\mu}{\nu}\tanh(\eta), \quad y+iz = re^{i\Theta},\\
T:& T_{x} = 1-2\mu\sech^{2}(\eta),\quad T_{y}+iT_{z} =-\nu r(\tanh(\eta)-iS)e^{i\Theta}\\
N:& N_{x} = 2\mu\sech^{2}(\eta)\sinh(\eta),\quad N_{y}+iN_{z} = -[1-2\mu(\tanh(\eta)-iS)\tanh(\eta)]e^{i\Theta}\\
B:& B_{x} = 2\mu S\sech(\eta), B_{y}+iB_{z} = i\mu(1-S^{2}-2iS\tanh(\eta))e^{i\Theta}
\end{cases}
\end{align}
where \begin{align}
\mu &= \frac{1}{1+T^2} = \frac{\nu^{2}}{\nu^{2}+\tau_{0}^{2}},\quad r =\frac{2\mu}{\nu}\sech(\eta)\\
\eta &= \nu\zeta = \nu(s-2\tau_{0}t), \quad\Theta = S\eta+\nu^{2}(1+S^{2})t = \tau_{0}s+(\nu^{2}-\tau_{0}^{2})t.
\end{align}
This then provides the traveling wave soliton solution that was first described by Hasimoto.
\end{proof}
\begin{remark}
Another way one can arrive to the Hasimoto Transform and solution is by assuming the y and z components to be dependent on the x component. Using Taylor expansions of the norm of the curve where second order terms are thrown out to simplify the system of equations obtained, one can then substitute $ \Psi \equiv -(y+iz) $ to find that this $ \Psi $ satisfies the NLCSE. If one then looks for a soliton solution of this equation that is slowly varying one finds Hasimoto's traveling wave solution.
\end{remark}
\subsection{Self Similar Dilating Solutions of the Vortex Filament Equation}
In this section we present a summary the results of Banica and Vega on the discovery of self-similar dilating solutions of curves with a corner ([16],). They developed a set of solutions that only just fail to exist in $ H^{3/2} $ and develop a corner in finite time. We present their general results as well as present a rough sketch of their proof, but we suggest further reading the works of Banica and Vega to illustrate the depth of the problem in finding such solutions and some of the applications of VFE to scattering theory for Schroedinger equations.
First it is necessary to find a way to translate back from the tangent and normal vectors a filament that satisfies VFE where curvature is allowed to become 0. Banica and Vega overcome this difficulty by creating another frame of reference $ (T, e_{1}, e _{2}) $ governed by \[
\begin{bmatrix}
T\\
e_{1}\\
e_{2}
\end{bmatrix}_{x} = \begin{bmatrix}
0 &\alpha &\beta\\
-\alpha &0 &0\\
-\beta &0 &0
\end{bmatrix}\begin{bmatrix}
T\\
e_{1}\\
e_{2}
\end{bmatrix}.
\]
This is in turn a reformulation of the Hasimoto Transform since from this coordinate frame change we can define \[
\psi(t,x) = \alpha(x,t)+i\beta(x,t)
\] and find that this solves the NLCSE with $ A(t) = \alpha^{2}(t,0)+\beta^{2}(t,0) $. Then we can define $ N = e_{1}+ie_{2}$ and find that this is in turn equivalent to the $ \mathfrak{N} $ from section 3.2, showing that the Hasimoto transform is one way that one could obtain this new frame of reference.
These vectors can then be combined to yield the evolution equations \begin{equation*}
T_{x} = \text{Re}(\bar{\psi}N),\quad N_{x} = -\psi T, \quad T_{t} =\text{Im}(\bar{\psi}N), \quad N_{t} = -i\psi_{x}T+i(|\psi|^{2}-A(t))N
\end{equation*}
which can then be used to construct these vectors with the aid of imposing $ (T,N) = (e_{0},e_{1}+ie_{2}) $. Banica and Vega then define \begin{equation}
\gamma(t,x) := P+\int_{t}^{t_{0}}(T\wedge T_{xx})(\tau, x_{0})d\tau+\int_{x}^{x_{0}}T(t,s)ds
\end{equation}
and find that this solves the VFE.
The focus is first on self-similar dilating solutions of VFE \[ \gamma(t,x) = \sqrt{t}G\Big(\frac{x}{\sqrt{t}}\Big). \] [21] showed that the family of self-similar solutions $ \{\gamma_{a}\}_{a\in\mathbf{R}^{+}} $ is characterized by explicit curvature and torsion $ c_{a} = \frac{a}{\sqrt{t}},\quad\tau_{a} = \frac{x}{2t} $. These self-similar solutions with a corner have been shown to be analogous with the 'delta-wing' vortex and serves as a good analogy of its behavior.
Using this we then look at perturbation solutions using Hasimoto's transform using the previous explicit torsion and curvature, giving the family of filament functions \begin{equation}
\psi_{a}(t,x) = \frac{a}{\sqrt{t}}e^{ix^{2}/4t}
\end{equation}
which solve the NLCSE\[ i\psi_{t}+\psi_{xx}+\frac{1}{2}\Big(|\psi|^{2}-\frac{a^2}{t}\Big)\psi = 0. \] The corner of the soliton corresponds to the initial condition $ \psi_{a}(0,x) = a\delta_{x = 0} $, where $ \delta_{x = 0} $ is the delta distribution centered at $ x =0 $.
Here we shall present verbatim the theorems Vega and Banica derived and afterwards we present their discussion on the results ([16]), but first we must give some definitions to provide sufficient background.
\begin{definition}[Filament Function]
We shall define as a filament function the function $ \psi $ that is obtained when carrying a solution of VFE through a transform, such as the Hasimoto transform, to create an equation that solves the NLCSE, i.e if it has curvature $ c $ and torsion $ \tau $ then the corresponding vortex filament function is \[ \psi = c(x,t)e^{-\int_{0}^{x}\tau ds} \]
\end{definition}
\begin{definition}[Weighted Space]
We define the weighted space $ X^{\alpha}, $\[ X^{\alpha}:= \{f\in L^{2}|\zeta^{\alpha}\hat{f}(\zeta)\in L^{\infty}(|\zeta|\leq 1) \}\] \end{definition}
\begin{theorem}[Continuation of Vortex after singularity $ \text{[16]} $]
Let $ \gamma(1) $ be a perturbation of a self-similar solution $ \gamma_{a} $ at time $ t = 1 $ in the sense that the filament function of $ \gamma(1) $ is $ (a+u(1,x))e^{ix^{2}/4} $, with $ \partial^{k}_{x}u(1) $ small in $ X^{\alpha} $ with respect to $ a $ for all $ 0 \leq k\leq4 $, for some $ \alpha<\frac{1}{2} $.
One can construct a solution $ \gamma\in C([-1,1], Lip)\cap C([-1,1]\backslash \{0\},C^{4}) $ for VFE on $ t\in[-1,1]\backslash\{0\} $ with a weak solution on the entire interval [-1,1]. The solution $ \gamma $ is unique on the subset of $ C([-1,1], Lip)\cap C([-1,1]\backslash \{0\},C^{4}) $ such that the associated filament functions at times $ \pm 1 $ can be written as $ (a+u(\pm1,x))e^{ix^{2}/4} $ with $ \partial^{k}_{x}u(\pm1) $ small in $ X^{\alpha} $ with respect to a for all $ 0\leq k\leq4 $.
This solution enjoys the following properties:
\begin{itemize}
\item there exists a limit of $ \gamma(t,x) $ and of its tangent vector $ T(x,t) $ at time zero, and \begin{equation}
\sup_{s}|\gamma(t,x)-\gamma(0,x)|\leq C\sqrt{|t|},\quad \sup_{|x|\geq\epsilon>0}|T(x,t)-T(0,x)|\leq C_{\epsilon}|t|^{\frac{1}{6}^{-}}
\end{equation}
\item $ \forall t_{1},t_{2}\in[-1,1]\backslash \{0\} $ the following asymptotic properties hold\begin{equation}
\gamma(t_{1},x)-\gamma(t_{2},x) =O(\frac{1}{x}),\quad T(t_{1},x)-T(t_{2},x) = O(\frac{1}{x})
\end{equation}
\item $ \exists T^{\infty}\in \mathbb{S}^{2}, N^{\infty}\in \mathbb{C}^{3} $ such that uniformly in $ -1\leq t\leq1 $\begin{equation}
T(t,x)-T^{\infty} = O\Big(\frac{1}{\sqrt{x}}\Big),\quad (N+iB)(t,x)-N^{\infty}e^{ia^{2}\log \frac{\sqrt{t}}{x}-ix^{2}/4t}=O(\frac{1}{\sqrt{x}}),
\end{equation}
\item modulo rotation and a translation, we recover at the singularity point (0,0) the same structure as for $ \gamma_{a} $:\begin{equation}
\lim_{x\rightarrow 0^{\pm}}T(0,x) = A^{\pm}_{a},\quad \lim_{x\rightarrow 0^{\pm}}\lim_{t\rightarrow 0}(N+iB)(t,x)e^{-ia^{2}log \frac{\sqrt{t}}{x}-ix^{2}/4t}=B^{\pm}_{a}.
\end{equation}
\end{itemize}
Where $ u $ in the previous proof is the solution to the following PDE \[ iu_{t}+u_{xx}+\frac{1}{2t}(|u+a|^{2}-a^{2})(u+a) = 0 \] which arises by taking perturbations of the filament function $ \psi_{\alpha} $ and proceeding to analyze the long term behavior of $ u $.
\end{theorem}
\begin{theorem}[VFE with values with a corner $ \text{[16]} $]
Let $ \gamma_{0} $ be a smooth $ C^{4} $ curve, except at $ \gamma_{0}(0) = 0 $ where a corner is located, i.e. that there exist $ A^{+} $ and $ A^{-} $ two distinct non-colinear unitary vectors in $ \mathbf{R}^{3} $ such that\begin{equation}
\gamma_{0}'(0^{+}) =A^{+},\quad\gamma_{0}'(0^{-}) = A^{-}.
\end{equation}
We set a to be the real number given by the unique self-similar solution of VFE with the same corner as $ \gamma_{0} $ at time $ t=0 $. We suppose that the curvature of $ \gamma_{0}(x) $ (for $ x \neq0 $) satisfies $ (1+|x^{4}|) c(x)\in L^{2}$ and $ |x|^{\zeta}c(x)\in L^{\infty}_{(|x|\leq1)} $, small with respect to a.
Then there exists $ \gamma(t,x)\in C([-1,1], Lip)\cap C([-1,1]\backslash\{0\},C^{4})
,$ regular solution of the VFE for $ t\in[-1,1]\backslash\{0\}, $ having $ \gamma_{0} $ as value at time $ t=0 $, and enjoying all properties from the previous theorem.
\end{theorem}
\begin{remark}
The first theorem constructs a solution for VFE that although presenting a discontinuity at time $ t=0 $ and show some convergence in large values in space and for small values in time, and shows that this discontinuity is in fact a corner singularity at the point $ (0,0) $. The second theorem applies these results to the evolution of curves with this corner singularity, and the proof of this theorem occurs naturally in the proof of 3.3.4.
Since we do not present explicit proofs of these results in these sections we direct the reader to Vega and Banica's papers [17], [18], [19], [20], and [21] which provide a rich exposition of the underlying structure of the equation as well as the spaces in which solutions to VFE exist.
\end{remark}
\subsection{Self-Similar Rotating Solutions of Vortex Filament Equation}
In this section we derive expressions of rotating self-similar solutions of VFE in 3 dimensional space as well as those that rotate around a singular plane. We begin first by deriving those rotating in 3-space, and those in a singular plane follow as a corollary.
\begin{theorem}
A derivation for the ODEs governing the motion of self-similar rotating solutions of the Vortex Filament Equation. We find, without loss of generality, that considering a self-similar curve in $ \mathbf{R}^3 $ of the form $ \Gamma(x) = (x, y(x), z(x)) $ that is rotating in space. Let $ \Gamma^{x}, \Gamma^{y}, \Gamma^{z} $ denote the components in the x, y, and z axes so as to not confuse them with the notation for partial derivatives. We obtain that rotating self-similar solutions around a single axis can be split into the following 2 cases:
(1) Rotation around the z or y axis: This is the case for rotation around the y-axis, for rotation around the z-axis simply switch the values for $ \Gamma^z $ and $ \Gamma^y $. In this case we find the following system of equations describing the vector $ \Gamma $:
\begin{align}
&\Gamma^{x} = x\\
&\Gamma^{y} = C_{1}x\\
&\Gamma^{z} = z(x)
\end{align}
where $ z(x) $ must be a solution of the nonlinear ODE\begin{equation}
-\frac{z'(x)}{\sqrt{1+C_{1}^{2}+z'(x)^{2}}}=\frac{1}{2}(1+C_{1}^{2})(x^{2}+2C_{2}).
\end{equation}
This gives a two parameter family of solutions for rotations of this type.
(2) Rotation around the x-axis: For this case the components of the curve in this situation are as follows\begin{align}
&\Gamma^{x} = x\\
&\Gamma^{y} = \lambda z(x)\\
&\Gamma^{z} = z(x)
\end{align}
where $ \lambda \in \mathbf{R}$ and $ z(x) $ solves the following ODE \begin{equation}
z'(x) = \pm\sqrt{ \frac{4-(1+\lambda^2)^2(z^2 + 2C_1)^2}{(1+\lambda^2)^3(z^2 + 2C_1)^2}}
\end{equation}
and with this equation we have a 3 parameter family of solutions for rotation around the x-axis.
\end{theorem}
\begin{proof}
Consider a solution of VFE of the form $\Gamma (x,t)=e^{tM}\Gamma_{0}(x)$, where $\Gamma_{0}$ is a curve in $\mathbb{R}^{3}$ and $M$ is the following skew-symmetric matrix:
$M=
\begin{Bmatrix}
0 &-\omega_{y} &\omega_{z}\\
\omega_{y} &0 &-\omega_{x}\\
-\omega_{z} &\omega_{x} &0
\end{Bmatrix}$.
Here the entries are scalars representing the angular velocities of each axis' rotation.
Applying the equations of VFE:
\begin{align}
\Gamma_{t} &= \Gamma_{s}\times\Gamma_{ss}\\
Me^{tM}\Gamma_{0} &= e^{tM}(\Gamma_{0}'\times\Gamma_{0}'')\\
M\Gamma_{0} &= \Gamma_{0}'\times\Gamma_{0}''
\end{align}
Suppose $\Gamma_{0} = \langle x, y(x), z(x)\rangle$. Taking the derivate with respect to arc-length of this curve gives $\frac{d \Gamma_{0}}{ds} = \frac{1}{\sqrt{1+y_{x}^{2}+z_{x}^{2}}}\frac{d\Gamma_{0}}{dx}$.
So then we give expressions for the first and second order derivatives with respect to arclength:
\begin{align}
\frac{d\Gamma_{0}}{ds} &= [1+y_{x}^{2}+z_{x}^{2}]^{-1/2}\langle 1,y'(x),z'(x)\rangle\\
\frac{d^{2}\Gamma_{0}}{ds^{2}} &= [1+y_{x}^{2}+z_{x}^{2}]^{-1/2}(\frac{d}{dx})(\frac{d\Gamma_{0}}{ds})\\
\frac{d^{2}\Gamma_{0}}{ds^{2}} &= [1+y_{x}^{2}+z_{x}^{2}]^{-2}\{-(y''y'+z''z')\langle 1, y',z'\rangle+[1+y_{x}^{2}+z_{x}^{2}]\langle 0,y'',z''\rangle\}
\end{align}
Plugging our results into (161) and using equations (162)-(164), as well as the definition of $M$ we have the following system of ODE's describing the behavior of solutions:
\begin{align}
\omega_{z}z-\omega_{y}y &=(y'(x)z''(x)-z'(x)y''(x))[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\\
\omega_{y}x-\omega_{x}z &= -z''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\\
\omega_{x}y-\omega_{z}x &= y''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}
\end{align}
\textbf{Rotation around the y-axis}:
In this situation the constants $\omega_{x} =\omega_{z} = 0$ and $\omega_{y} =1 $, where the latter is taken for simplicity. Then the equations (insert numbers here) become:
\begin{align}
-y &=(y'(x)z''(x)-z'(x)y''(x))[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\\
x &=-z''(x)[[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\\
0 &= y''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}
\end{align}
so that by the last equation we have that $ y(x) = C_{1}x $. This then transforms (165) and (166) into the second order nonlinear differential equation:\begin{align}
x = -z_{xx}[1+C_{1}^{2}+z_{x}^{2}]^{-3/2}.
\end{align}
This equation can be explicitly solved and reduced to a first order differential equation by making the substitutions $ a^{2} = 1+C_{1}^{2} $ and $ z'(x) = a\tan(\theta) $, we then proceed as follows:\begin{align}
\frac{1}{2}x^{2}+C_{2} &= -\int \frac{a\sec^{2}(\theta)}{a^{3}(1+\tan^{2}(\theta))^{3/2}}d\theta\\
&= -\int \frac{1}{a^{2}}\cos(\theta)d\theta\\
&= -\frac{1}{a^{2}}\sin(\theta)\\
\frac{a^{2}}{2}(x^{2}+2C_{2})&= -\frac{z'(x)}{\sqrt{a^{2}+z'(x)^{2}}}\\
\frac{1}{2}(1+C_{1}^{2})(x^{2}+2C_{2})&= -\frac{z'(x)}{\sqrt{1+C_{1}^{2}+z'(x)^{2}}}
\end{align}
and this is the ODE whose 2 parameter families of solutions provide rotating self-similar solutions around either the z or y-axis.
\textbf{Rotation around the x-axis:} For rotation around the x-axis we obtain the following system of equations:
\begin{align}
0 &=(y'(x)z''(x)-z'(x)y''(x))[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\\
z&=z''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}\\
y &= y''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}.
\end{align}
The first equation it gives us that $ y'(x) = \lambda z'(x) $. This transforms the other equations into the second order non-linear differential equation: \[ z = z''[1+(\lambda^{2}+1)z_{x}^{2}]^{-3/2}. \]
We now solve this equation and turn it into a first order nonlinear equation:\begin{align}
z'z &= \frac{z'z''}{(1+(1+\lambda^{2})(z')^{2})^{3/2}}\\
\frac{1}{2}z^{2}+C_{1} &= \int \frac{z'z''}{(1+(1+\lambda^{2})(z')^{2})^{3/2}}dx.\\ .
\end{align}
As in the previous problem we substitute $ u = (1+\lambda^{2})(z')^{2}(x) $, $ (1+\lambda^{2})^{-1}du = 2z'z'' dx$, continuing with the integration:\begin{align}
(1+\lambda^{2})(z^{2}+2C_{1}) &= \int {(1+u)^{-3/2}}du.\\
(1+\lambda^{2})(z^{2}+2C_{1}) &= -2(1+(1+\lambda^{2})(z')^{2})^{-1/2}\\
(z')^{2}&=\frac{4}{(1+\lambda^{2})^{3}(z^{2}+2C_{1})^{2}}-\frac{1}{(1+\lambda^{2})}\\
z'(x) &= \pm\sqrt{ \frac{4-(1+\lambda^2)^2(z^2 + 2C_1)^2}{(1+\lambda^2)^3(z^2 + 2C_1)^2}}
\end{align}
Finally, using this we can derive the expression for $ y'(x) $ using that $ y'(x) = \lambda z'(x) $, we have that $ y(x) = \lambda z(x)+C_{2} $. Plugging this into our previous expression gives:\begin{equation}
y'(x) = \lambda \sqrt{ \frac{4-(1+\lambda^2)^2((\frac{y}{\lambda}+C_{2})^2 + 2C_1)^2}{(1+\lambda^2)^3((\frac{y}{\lambda}+C_{2})^2 + 2C_1)^2}}
\end{equation}
Putting all results together gives a final expression for how the curves will behave if we assume the vortex filament ot be rotating around a particular axis.
\end{proof}
\begin{corollary}
We find that if a vortex filament rotates inside of a singular plane it must obey the following second order non-linear differential equation \begin{align}
f(x) +\frac{f''(x)}{(1+f'(x))^{3/2}} = 0
\end{align}
\end{corollary}
\begin{proof}
The general form for a curve rotating in a plane is of the form \begin{equation}
\Gamma(x,t) = (x, \cos(t)f(x),-\sin(t)f(x))
\end{equation}
We now take the appropriate time and space derivatives \begin{align}
\partial_{t}\Gamma &= (0, -\sin(t)f(x),-\cos(t)f(x))\\
\partial_{x}\Gamma &= (1, \cos(t)f'(x),-\sin(t)f'(x))\\
\partial_{xx}\Gamma&=(0, \cos(t)f''(x),-\sin(t)f''(x))
\end{align}
We can now plug these equations into our previous system of ODEs (insert equation numbers here) for rotating solutions in 3 dimensions by setting the left hand side as the components of $ \partial_{t}\Gamma(x,t) $ as well as taking $ y(x,t) = \cos(t)f(x) $ and $ z(x,t) = -\sin(t)f(x) $ and changing the right hand derivatives to partial derivatives with respect to x gives us:\begin{align}
0 &= 0\\
\partial_{t}y(x,t) &= \sin(t)f''(x)[1+f'(x)^{2}]^{-3/2}\\
\partial_{t}z(x,t) &= \cos(t)f''(x)[1+f'(x)^{2}]^{-3/2}
\end{align}
Substituting the values for $ y_{t}(x,t) $ and $ z_{t}(x,t) $ gives the nonlinear differential equation: \begin{equation}
f(x) +\frac{f''(x)}{(1+f'(x))^{3/2}} = 0
\end{equation}
We can further simplify this equation to turn it into a first degree ODE:\begin{align}
ff' &= \frac{-f'f''}{(1+f'(x)^{2})^{3/2}}\\
f(x)^{2}+2C_{1} &= -2(1+f'(x)^{2})^{-1/2}\\
f'(x) &= \sqrt{\frac{4}{(f(x)^{2}+2C_{1})^{2}}-1}
\end{align}
This then gives the 2 parameter family of solutions of rotating solutions in the plane.
\end{proof}
\begin{remark}
It is of interest to note that the previous result could have been achieved from theorem 3.4.1 by saying that it was a solution rotating around the x-axis with $ \lambda = 0 $. This then gives the same ODE governing the family of solutions for planar rotating vortex filaments. We also provide the following argument that demonstrates the solvability of equations (186), (187) and (199):
\end{remark}
Using the differentiability of $ z(x) $ we can create an inverse function such that now $ x $ is a function of z. Then from basic analysis we would have that the derivative with respect to z of the inverse function would then be \begin{equation}
\frac{dx}{dz} = \sqrt{\frac{(1+\lambda^{2})^{3}(z^{2}+2C_{1})^{2}}{4-(1+\lambda^{2})^{2}(z^{2}+2C_{1})^{2}}}
\end{equation}
so that then x exists as a smooth increasing function on the range \[-\frac{2}{(1+\lambda^{2})}-2C_{1}<z^{2}<\frac{2}{(1+\lambda^{2})}-2C_{1} \] so then we have $ z(x) $ to be a smooth, increasing and bounded function of x.
\section*{Acknowledgments}
I would like to thank my graduate student advisor Jiewon Park for the direction and support in the development of this paper, as well as directing the goal of this paper during its development. I would also like to extend my gratitude towards Professor Tobias Colding for providing the initial papers that led to the development of this paper. I would finally extend my gratitude towards the MIT mathematics department and UROP+ coordinators, as well as the Undergraduate Research Opportunity office for providing funding over the summer towards the creation of this paper.
\end{document} | arXiv |
Annie Raoult
Annie Raoult (born 14 December 1951)[1] is a French applied mathematician specializing in the mathematical modeling of cell membranes, graphene sheets, and other thin nanostructures. She is vice president of the Centre International de Mathématiques Pures et Appliquées and professor emerita at Paris Descartes University, where she directed the laboratory for applied mathematics.[2][3]
Education and career
Raoult was a student from 1971 to 1974 at the École normale supérieure de Fontenay-aux-Roses, and a maître de conférences at Pierre and Marie Curie University from 1975 to 1992,[2] earning a third-cycle doctorate there in 1980[4] and a state doctorate in 1988.[1] She became a professor at Joseph Fourier University in Grenoble in 1992, and remained there until 2005, also serving as deputy director of the computer science and applied mathematics laboratories UFR and UJF from 1996 to 2001. In 2005 she came to Paris Descartes University as a professor, and from 2009 to 2014 she directed the laboratory for applied mathematics (MAP5) at Paris Descartes University. In 2017 she became vice president of the Centre International de Mathématiques Pures et Appliquées (CIMPA), and later the same year retired from Paris Descartes University to become a professor emerita.[2]
Recognition
In 2000, Raoult won the Prix Paul Doistau–Émile Blutet of the French Academy of Sciences in the area of mechanical and computational sciences.[5]
References
1. Raoult, Annie (1951-....), French National Library, retrieved 2020-12-26
2. Curriculum vitae, retrieved 2020-12-26
3. "Annie Roualt", Governing board, CIMPA, retrieved 2020-12-26
4. "Raoult, Annie", ISNI, retrieved 2020-12-26
5. Prix Paul Doistau–Émile Blutet: Sciences Mécaniques et Informatiques (PDF), French Academy of Sciences, 2014, retrieved 2020-12-25
External links
• Home page
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
Academics
• Mathematics Genealogy Project
Other
• IdRef
| Wikipedia |
\begin{document}
\title{Intersection Graphs of Non-Crossing Paths\thanks{A preliminary version of this article appeared at WG 2019~\cite{wg2019}, and a preprint is available at~\href {http://arxiv.org/abs/1907.00272}{arxiv.org/abs/1907.00272}}}
\author{Steven Chaplick\orcidID{0000-0003-3501-4608} \thanks{Part of this research was conducted while the author was employed at Lehrstuhl f\"ur Informatik~I, Universit\"at W\"urzburg, and partially supported by DFG grant WO$\,$758/11-1.} }
\institute{Department of Data Science and Knowledge Engineering, Maastricht University, The Netherlands\\
\email{[email protected]}}
\maketitle
\begin{abstract} We study graph classes modeled by families of non-crossing (NC) connected sets. Two classic graph classes in this context are disk graphs and proper interval graphs. We focus on the cases when the sets are paths and the host is a tree (generalizing proper interval graphs). Forbidden induced subgraph characterizations and linear time certifying recognition algorithms are given for intersection graphs of NC paths of a tree (and related subclasses). A direct consequence of our certifying algorithms is a linear time algorithm certifying the presence/absence of an induced claw $(K_{1,3})$ in a chordal graph.
For the intersection graphs of NC paths of a tree, we characterize the minimum connected dominating sets (leading to a linear time algorithm to compute one). We further observe that there is always an independent dominating set which is a minimum dominating set, leading to the dominating set problem being solvable in linear time. Finally, each such graph $G$ is shown to have a Hamiltonian cycle if and only if it is 2-connected, and when $G$ is not 2-connected, a minimum-leaf spanning tree of $G$ has $\ell$ leaves if and only if $G$'s block-cutpoint tree has exactly $\ell$ leaves (e.g., implying that the block-cutpoint tree is a path if and only if the graph has a Hamiltonian path). \end{abstract}
\keywords{Clique Trees \and Non-crossing Models \and Dominating Sets \and Hamiltonian Cycles \and Minimum-Leaf Spanning Trees.}
\section{Introduction}
Intersection models of graphs are ubiquitous in graph theory and covered in many graph theory textbooks, see, e.g., \cite{Golumbic2004,MckeeM1999}. Generally, for a given graph $G$ with vertex set $V(G)$ and edge set $E(G)$, a collection $\mathcal{S}$ of sets, $\{S_v\}_{v \in V(G)}$, is an \emph{intersection model} of $G$ when $S_u \cap S_v \neq \emptyset$ if and only if $uv \in E(G)$. Similarly, we say that $G$ is the \emph{intersection graph} of~$\mathcal{S}$. One quickly sees that all graphs have intersection models (e.g., by choosing, for every $v \in V(G)$, $S_v$ to be the edges incident to $v$). Thus, one often considers restrictions either on the \emph{host} set (i.e., the domain from which the elements of the $S_v$'s can be chosen), collection $\mathcal{S}$, and/or on the individual sets $S_v$.
In this paper we consider classes of intersection graphs where the sets are taken from a topological space, are \emph{(path) connected}, and are pairwise \emph{non-crossing}. A set $S$ is \emph{(path) connected} when any two of its points can be connected by a \emph{curve} within the set (note: a \emph{curve} is a homeomorphic image of a closed interval). Notice that, when the topological space is a graph, connectedness is precisely the usual connectedness of a graph and curves are precisely paths. Two connected sets $S_1,S_2$ are called \emph{non-crossing} when both $S_1 \setminus S_2$ and $S_2 \setminus S_1$ are connected. Our focus will be on intersection graphs of non-crossing paths.
The most general case of intersection graphs of non-crossing sets which has been studied is the class of intersection graphs of non-crossing connected (NC-C) sets in the plane~\cite{Kratochvil1997}. These were considered together with another non-crossing class, the intersection graphs of disks in the plane or simply \emph{disk} graphs. The recognition of both NC-C graphs and disk graphs is \hbox{\textsf{NP}}-hard~\cite{Kratochvil1997}. More recently~\cite{Kang2012}, disk graph recognition was shown to complete for the \emph{existential theory of the reals} ($\exists\mathbb{R}$); note that all $\exists\mathbb{R}$-hard problems are \hbox{\textsf{NP}}-hard, see~\cite{Matousek2014} for an introduction to $\exists\mathbb{R}$.
One of the simplest cases of connected sets one can consider are those which reside in $\mathbb{R}$, i.e., the intervals of $\mathbb{R}$. The corresponding intersection graphs are precisely the well studied \emph{interval graphs}. Moreover, imposing the non-crossing property on these intervals leads to the \emph{proper} interval graphs -- which are usually (and equivalently) defined by restricting the guests intervals so that no interval strictly contain any other. It has often been considered how to generalize proper interval graphs to more complicated hosts, but
simple attempts to do so involving the property that the sets are \emph{proper} are often uninteresting. For example, the intersection graphs of proper paths in trees or proper subtrees of a tree are easily seen as the same as their non-proper versions. We will see that the non-crossing property leads to natural new classes which generalize proper interval graphs.
We formalize the setting as follows. For graph classes $\mathcal{S}$ and $\mathcal{H}$, a graph $G$ is an \emph{$\mathcal{S}$-$\mathcal{H}$} graph when each $v \in V(G)$ has an $S_v \in \mathcal{S}$ such that: \begin{itemize}[noitemsep,topsep=0.5\baselineskip] \item the graph $H = \bigcup_{v \in V(G)} S_v$ is in $\mathcal{H}$, and
\item $uv$ is an edge of $G$ if and only if $S_u \cap S_v \neq \emptyset$.
\end{itemize} Additionally, we say that $(\{S_v\}_{v \in V(G)},H)$ is an \emph{$\mathcal{S}$-$\mathcal{H}$ model} of $G$ where $H$ is the \emph{host} and each $S_v$ is a \emph{guest}, we will also refer to $S_v$ as the \emph{model of $v$}.
We further state that $G$ is a \emph{non-crossing}-$\mathcal{S}$-$\mathcal{H}$ (NC-$\mathcal{S}$-$\mathcal{H}$) graph when the sets $S_v$ are pairwise non-crossing. In this context the proper interval graphs are the NC-path-path graphs.
Many classes of $\mathcal{S}$-$\mathcal{H}$ graphs have been studied in the literature; see, e.g.,~\cite{MckeeM1999}. Some of these are described in the table below together with the complexity of their recognition problems and whether a \emph{forbidden induced subgraph characterization (FISC)} is known. The table utilizes the following terminology. A \emph{directed tree (d.tree)} is a tree in which every edge $uv$ has been assigned one direction.
A \emph{rooted tree (r.tree)} is a directed tree where there is exactly one source node. A survey of path-tree graph classes is given in \cite{MonmaW86}.
Two further key graph classes here are the chordal graphs and the split graphs, defined as follows. A graph is \emph{chordal} when it has no induced cycles of length four or more. A graph is a \emph{split} graph when its vertices can be partitioned into a clique and an independent set. The split graphs are easily seen as a subset of the chordal graphs.
\begin{center} \begin{tabular}{@{~~~}l@{~~~}l@{~~~}l@{~~~}l@{~~~}l@{~~~}l} \hline & Graph Class & Guest & Host~~~~ & Recognition & FISC? \\ \hline 1 & interval & path & path & $O(n+m)$ \cite{CorneilOS2009} & yes \cite{LB1962} \\ 2 & rooted path tree (RPT) & path & r.tree & $O(n+m)$ \cite{Dietz1984} & open \\ 3 & directed path tree (DPT) & path & d.tree & $O(nm)$ \cite{ChaplickGLT2010} & yes \cite{Panda1999} \\ 4 & path tree (PT) & path & tree & $O(nm)$ \cite{Schaffer1993} & yes \cite{LevequeMP2009} \\ 5 & chordal & tree & tree & $O(n+m)$ \cite{RoseTL1976} & by definition \\%\footnotemark \\
\hline \end{tabular} \end{center}
\paragraph{Results and outline.}
We study the non-crossing graph classes corresponding to graph classes 1--4 given in the table. Section~\ref{sec:prelim} contains background, terminology, and notation concerning intersection models.
In Section~\ref{sec:nc-path}, we provide forbidden induced subgraph characterizations for the non-crossing classes corresponding to 1--4 and certifying linear time recognition algorithms for them. Interestingly, this implies that one can test whether a chordal graph contains a claw in linear time.
(In contrast, for general graphs, the best deterministic claw-detection algorithms run in time $O(\min\{n^{3.252},m^{(\omega+1)/2}\})$~\cite{EisenbrandG04}, and $O(m^{\frac{2\omega}{\omega+1}})$~\cite{KloksKM00}, whereas the best randomized algorithm (succeeding with high probability) runs in time $O(n^\omega)$~\cite{WilliamsWWY15}; here $\omega$ is the exponent from square matrix multiplication and the $3.252$ is based on the time to compute the product of an $n \times n^2$ matrix and an $n^2 \times n$ matrix~\cite{GallU18}.)
The next two sections concern algorithmic results on domination and Hamiltonicity problems on NC-path-tree graphs. To obtain these results, we use the special structure of NC-path-tree models established in Section~\ref{sec:nc-path-tree-structure}. Note that the problems mentioned below are formalized in the corresponding sections.
In Section~\ref{sec:mds}, our main result is a characterization of the minimum connected dominating sets in NC-path-tree graphs. This leads to a linear time algorithm to solve the minimum connected dominating set problem, which also implies a linear time algorithm for the cardinality Steiner tree (ST) problem. In contrast, the minimum connected dominating set problem is known to be \hbox{\textsf{NP}}-hard on split graphs~\cite{WhiteFP85}, and, as such, on chordal graphs as well. We further discuss the relationship between the (standard) minimum dominating set problem and the minimum independent dominating set problem on these graphs, observing that both can be solved in linear time on NC-path-tree graphs.
Notably, the minimum dominating set problem is \hbox{\textsf{NP}}-hard on PT graphs~\cite{BoothJ1982}, and split graphs~\cite{CorneilP1984},
but it is polynomial time solvable on RPT graphs~\cite{BoothJ1982}.
Further references and background on domination problems are given in Section~\ref{sec:mds}.
In Section~\ref{sec:ham}, we again consider NC-path-tree graphs but now study the Hamiltonian cycle (HC) problem and minimum-leaf spanning tree problem (which generalizes the Hamiltonian path (HP) problem), on them. We show that 2-connectedness implies that each plane drawing of an NC-path-tree model leads to a distinct HC, and from each such plane drawing, an HC can be found in linear time. When such a graph is not 2-connected (i.e., when it has a cut-vertex) it cannot have an HC, but we can similarly observe a nice spanning substructure. Namely, we show that for any NC-path-tree graph $G$ containing a cut-vertex, its block-cutpoint tree having $\ell$ leaves characterizes the presence of an $\ell$-leaf spanning tree in $G$. For example, as a special case, we obtain that $G$'s block-cutpoint tree is a path (i.e., has at most two leaves) if and only if $G$ has an HP. Our characterization also leads to a linear time algorithm for the minimum-leaf spanning tree problem. Note that, the HC and HP problems are NP-complete on \emph{strongly chordal} split graphs~\cite{Muller96a}, and DPT graphs~\cite{Narasimhan89}, but easily solved (and similarly characterized) on proper interval graphs~\cite{Bertossi83}.
We conclude with avenues for further research.
\section{Preliminaries} \label{sec:prelim}
\paragraph{Notation.}
Unless explicitly stated otherwise, all the graphs we discuss in this work are connected, undirected, simple, and loopless. For a graph $G$ with a vertex $v$, we use $N_G(v)$ to denote the \emph{neighborhood} of $v$, and $N_G[v]$ to denote the \emph{closed neighborhood} of $v$, i.e., $N_G[v] = N_G(v) \cup \{v\}$. The subscript $G$ will be omitted when it is clear. For a subset $S$ of $V(G)$, we use $G[S]$ to denote the subgraph of $G$ induced by $S$. For a set of graphs $\mathcal{F}$, we say that a graph $G$ is $\mathcal{F}$-free when $G$ does not contain any $F \in \mathcal{F}$ as an induced subgraph.
For graph classes $\mathcal{S}$ and $\mathcal{H}$, and an $\mathcal{S}$-$\mathcal{H}$ model $(\{S_v\}_{v \in V(G)}, H)$ of a graph $G$, we use the following notation. We refer to elements of $V(G)$ as \emph{vertices} and use symbols $u$ and $v$ to refer them whereas we call elements of $V(H)$ \emph{nodes} and use $x$, $y$, and $z$ to refer to them.
For a node $x$ of $H$ we use $G_x$ to denote the set of vertices $v$ in $G$ where $S_v$ contains $x$. Observe that every set $G_x$ induces a clique in $G$. Note that Section~\ref{sec:nc-path-tree-structure} defines the terms \emph{terminal}, \emph{junction}, and \emph{mixed} that are also used in later sections of the paper.
Several special graphs are named and depicted in Fig.~\ref{fig:small-graphs} along with models of them. We will refer to these throughout this paper. Of particular note is the middle graph, $K_{1,3}$ aka the claw, where we will refer to its degree~3 vertex as its \emph{central vertex}.
\begin{figure}
\caption{Some small graphs and tree-tree models of them. In the models the nodes of the host graph are given as darkly shaded circles and its edges are lightly shaded corridors connecting them. Each subset $S_v$ is depicted by a tree (or single point) overlaid on the drawing of the host graph. }
\label{fig:small-graphs}
\end{figure}
\paragraph*{Twin-free Graphs.}
For a graph $G$, two vertices $x$ and $y$ are called \emph{twins} when they have the same closed neighborhood, i.e., $N[x] = N[y]$. Note that, for the MDS problem, it is an easy exercise to show that it suffices to consider twin-free graphs. Also, as the vertex set of a graph can be easily partitioned into its equivalence classes of twins in linear time, one can distill the relevant twin-free induced subgraph of $G$ in linear time.
\paragraph{Chordality and Clique Trees.}\label{sec:chordal}
This area is deeply studied and while there are many interesting results related to our work, we only pick out a few concepts and results which are useful in this paper.
The starting point is that the chordal graphs are well-known to be the tree-tree graphs~\cite{Buneman74,Gavril1974,Walter1978}.
For a chordal graph $G$, a \emph{clique tree} $T$ of $G$ has the maximal cliques of $G$ as its vertices, and for every vertex $v$ of $G$, the set $K_v$ of maximal cliques containing $v$ induces a subtree of $T$. In other words, a clique tree of $G$ is a tree-tree model of $G$ whose nodes are in bijection with the maximal cliques of $G$. Clique trees are very useful when discussing models where the host graph is a tree.
When a graph has a tree-tree~\cite{Buneman74,Gavril1974,Walter1978}, path-tree~\cite{Gavril1978}, path-d.tree~\cite{MonmaW86}, path-r.tree~\cite{Gavril1975}, or path-path~\cite{FG1965} model, then it also has one that is a clique tree. Such results are also summarized in~\cite{MckeeM1999}.
We establish similar clique tree results for the corresponding NC graphs when the guests are paths. However, we remark that when the guests are trees, we cannot rely on clique trees. For example, the claw ($K_{1,3}$) is an NC-tree-tree graph, but it does not have an NC-tree-tree model that is a clique tree; in particular, in the center of Fig.~\ref{fig:small-graphs}, we depict two tree-tree models of the claw: one is the only clique tree (which is readily seen to fail the NC condition due to the point in the ``middle'' node) and the other is an NC-tree-tree model.
Essential to the linear running time of our algorithms is the following property of maximal cliques of chordal graphs, and ultimately clique trees.
For a chordal graph $G$, $\sum_{v\in V(G)} |K_v| \in O(n+m)$~\cite{Golumbic2004}. This implies that the total size of a clique tree $T$ is $O(n+m)$. So, any algorithm that is linear in the size of $T$ is also linear in the size of $G$. Additionally, one can produce a clique tree of a chordal graph in linear time~\cite{BlairP1993,GalinierHP1995}
One final aspect of clique trees which is relevant for us is the study of chordal graphs with unique clique trees~\cite{Kumar2002}. One observation that they made is that claw-free chordal graphs have unique clique trees. This is relevant for us because, as we will see in Section~\ref{sec:nc-path}, the claw-free chordal graphs are precisely the NC-path-tree graphs. The uniqueness of the clique trees of proper interval graphs (a subclass of claw-free chordal graphs) was also observed (later) in~\cite{Ibarra2009}. In fact, a very recent paper~\cite{Grussien2019} specifically studies the claw-free chordal graphs, providing a logarithmic-space isomorphism test while also re-proving that claw-free chordal graphs have unique clique trees.
\section{Non-crossing Paths in Trees: Structure and Recognition} \label{sec:nc-path}
In this section we characterize and recognize classes of intersection graphs of non-crossing paths in trees; namely, the classes of the following types: NC-path-tree, NC-path-d.tree, NC-path-r.tree, and NC-path-path.
We first note that the \emph{claw} ($K_{1,3}$) is not an NC-path graph regardless of the host.
\begin{observation}\label{obs:nc-path=>claw-free} If $G$ is an NC-path graph, then $G$ is claw-free. \end{observation} \begin{proof} Suppose $G$ contains a claw with central vertex $u$ and pendant vertices $a$, $b$, $c$. Let $\mathcal{P}$ be a path-$\mathcal{H}$ model of $G$ where $\mathcal{P} = \{P_v\}_{v \in V(G)}$. Clearly, $P_a \cap P_u$, $P_b \cap P_u$ and $P_c \cap P_u$ are disjoint. As such, at most two of them include an endpoint of $P_u$. Thus, for some $d\in \{a,b,c\}$, $P_u \setminus P_d$ is disconnected. \end{proof}
This section proceeds as follows.
The NC-path-tree graphs are shown to be the claw-free chordal graphs and the structure of NC-path-tree models is described. From this structure, we then show that NC-path-d.tree $=$ NC-path-r.tree $=$ (claw,3-sun)-free chordal. This provides, as a nearly direct consequence, the classic result that proper interval graphs are precisely the (claw, 3-sun, net)-free chordal graphs~\cite{Roberts1970,Wegner1967}. We conclude with linear time certifying recognition algorithms for NC-path-tree and NC-path-r.tree graphs.
\subsection{The Structure of NC-path-tree Models} \label{sec:nc-path-tree-structure}
In this subsection we explore the structure of NC-path-tree models and prove our FISCs along the way. We first take a slight detour to claw-free chordal graphs and prove the FISC of NC-path-tree graphs. In doing so we obtain the first insight into NC-path-tree models. Namely, that it suffices to consider clique trees and that the clique trees of these graphs are unique (see Theorem~\ref{thm:nc-path-tree-fisc}). We then take a closer examination of these clique NC-path-tree models and carefully describe the nodes they contain -- the results of this examination will be used repeatedly in the rest of the paper.
\begin{theorem} \label{thm:nc-path-tree-fisc} A graph $G$ is claw-free chordal if and only if it is an NC-path-tree graph. Moreover, $G$ has a unique clique tree and this clique tree is an NC-path-tree model. \end{theorem} \begin{proof} \noindent $\Leftarrow$ Observation~\ref{obs:nc-path=>claw-free} and chordal graphs being tree-tree graphs imply this.
\noindent $\Rightarrow$ Let $T$ be a clique tree of a claw-free chordal graph $G$. In the two claims below, we first show that every subtree $T_v$ must be a path, and then we show that these paths are non-crossing. These two claims prove the characterization. The uniqueness of the clique tree of every claw-free chordal graph has been shown previously~\cite{Kumar2002}.
\claim{1: For every $v \in V(G)$, $T_v$ is a path.} \begin{claimproof} Suppose $T_v$ is not a path. Then $T_v$ contains some claw $x_0,x_1,x_2,x_3$ with central node $x_0$. However, since $G_{x_j}$ is a maximal clique (for each $j \in \{0,1,2,3\}$), for each $i \in \{1,2,3\}$, there is $v_i \in G_{x_i} \setminus G_{x_0}$. Thus $v, v_1, v_2, v_3$ induces a claw in $G$. \end{claimproof}
\claim{2: The set $\{T_v : v \in V(G)\}$ is non-crossing.} \begin{claimproof} Suppose that $T_u$ intersects $T_v$ but does not include either end of $T_v$. Let $x_1$ and $x_2$ be the endpoints of $T_v$. Now there must be $v_1 \in G_{x_1} \setminus N_G(u)$ and $v_2 \in G_{x_2} \setminus N_G(u)$. That is, $v, u, v_1, v_2$ induces a claw in $G$. \end{claimproof}
\end{proof}
We now study the structure of the clique NC-path-tree model $(\{P_v\}_{v\in V(G)},T)$ of a graph $G$. We introduce some terminology. A node $x$ of $T$ is called a \emph{terminal} when it is a leaf of every path which contains it, i.e., $x$ is not an internal node of any $P_v$. For example, the leaves of $T$ are terminals. Similarly, a node $x$ of $T$ is a \emph{junction} when it is an internal node of every path which contains it, i.e., $x$ is not a leaf of any $P_v$. A node of $T$ that is neither a terminal nor a junction is called \emph{mixed}. The remainder of this section consists~of \begin{itemize}
\item the main lemma describing $T$ in these terms (Lemma~\ref{lem:nc-path-tree.nodes}),
\item an observation connecting these terms with certain induced subgraphs of $G$ (Observation~\ref{obs:path-tree.nets.suns}), and
\item a corollary regarding how the terminals can be used to partition $T$ into ``simple'' subtrees (Corollary~\ref{cor:nc-path-tree.structure}). \end{itemize}
\begin{lemma}\label{lem:nc-path-tree.nodes} For an NC-path-tree graph $G$, let $(\{P_v\}_{v\in V(G)},T)$ be its clique NC-path-tree model. A node $x$ of $T$ must satisfy the following properties: \begin{enumerate}[noitemsep,topsep=0pt] \item If $x$ is mixed, then $x$ has degree two. \item If $x$ is a junction, then (i) $x$ has degree 3 and (ii) $x$'s neighbors are terminals. \item If $x$ has degree four or more, then $x$ is a terminal. \end{enumerate} \end{lemma} \begin{proof} We establish the claimed properties in order as follows.
\begin{claimproof} \textbf{1.:} Suppose that $x$ has degree at least 3, is a leaf of $P_v$, and is an internal node of $P_u$. Further, let $y$ be the unique neighbor of $x$ in $P_v$. We see that $P_u$ includes $y$ (otherwise, $P_v$ and $P_u$ cross). Let $y'$ be the neighbor of $x$ in $P_u \setminus P_v$ and let $y''$ be a neighbor of $x$ which is not in $P_u$. Since $G$ is connected, there exists $u' \in G_x \cap G_{y''}$. Furthermore, $x$ is not a leaf of $P_{u'}$ (otherwise, $P_u$ crosses $P_{u'}$). Thus, similarly to $P_u$, $y$ belongs to $P_{u'}$. Now, since $G_x$ and $G_{y}$ are maximal cliques, there is $u'' \in G_x \setminus G_{y}$. Thus for $P_{u''}$ to neither cross $P_{u}$ nor $P_{u'}$ it must include both $y'$ and $y''$. However, this means $P_{u''}$ and $P_v$ cross. \end{claimproof}
\begin{claimproof} \textbf{2.:} Suppose that $x$ is a junction and let $y_1, \ldots, y_k$ be the neighbors of $x$. Since $x$ is a junction, for every $v \in G_x$, $P_v$ contains exactly two $y_i$'s. Thus, if $k=2$, then $G_x \subseteq G_{y_1}$ -- contradicting $T$ being a clique tree. Now suppose $k \geq 3$ and consider $v \in G_x$ where (w.l.o.g.) $P_{v}$ contains $y_1$ and $y_2$. Since $G$ is connected, there must be $v' \in G_x \cap G_{y_3}$. Furthermore, (w.l.o.g.) $P_{v'}$ contains $y_1$ (otherwise, $P_v$ and $P_{v'}$ cross). Now, since $G_x$ and $G_{y_1}$ are maximal cliques, there is $v'' \in G_x \setminus G_{y_1}$. Notice that $P_{v''}$ must contain $y_2$ and $y_3$ in order for $P_{v''}$ to cross neither $P_{v}$ nor $P_{v'}$. Finally consider any $u \in G_x \setminus \{v,v',v''\}$. Notice that, in order for $P_u$ to not cross any of $P_{v}$, $P_{v'}$, or $P_{v''}$, it must contain at least two of $y_1,y_2,y_3$. In particular, if $k \geq 4$, then $G_x \cap G_{y_4} = \emptyset$ -- contradicting $G$ being connected. Thus, $k = 3$ (establishing (i)).
Now, suppose that $y_1$ is not a terminal. By 1. and 2.(i), $y_1$ is either a junction with degree 3 or mixed with degree 2.
\textbf{Case 1:} \textit{$y_1$ is a junction with neighbors $x$, $z_1$, $z_2$.} Notice that each of $P_{v}$ and $P_{v'}$ must contain exactly one of $z_1$ or $z_2$. Moreover, w.l.o.g. they both must contain $z_1$ otherwise they will cross. However, since $y_1$ is a junction, we have vertices $w,w',w''$ such that $P_w \supseteq \{x,y_1,z_1\}$, $P_{w'} \supseteq \{x,y_1,z_2\}$ and $P_{w''} \supseteq \{z_1,y_1,z_2\}$. Moreover, both $P_w$ and $P_{w'}$ must contain either $y_2$ or $y_3$. Regardless of this choice, we end up with a crossing between either $P_{w'}$ and $P_v$ or $P_{w'}$ and $P_{v'}$. Thus, junctions cannot be neighbors.
\textbf{Case 2:} \textit{$y_1$ has degree 2 and is mixed.} Let $z$ be the neighbor of $y_1$ other than $x$ and let $w$ be a vertex of $G$ where $y_1$ is not a leaf of $P_w$, i.e., w.l.o.g. $P_w \supseteq \{z,y_1,x, y_2\}$. Notice that, $P_{v'}$ must also contain $z$ otherwise $P_{v'}$ and $P_w$ would cross. Similarly, since $P_{v'}$ now contains $z$, $P_v$ must also contain $z$ otherwise $P_v$ and $P_{v'}$ would cross. However, now a vertex $u \in G_{y_1} \setminus G_{z}$ must have $P_u = \{y_1\}$ but then $P_u$ crosses $P_w$. Thus, no neighbor of a junction is mixed. \end{claimproof}
\begin{claimproof} \textbf{3.:} This follows immediately from 1. and 2.(i). \end{claimproof} \end{proof}
\begin{observation}\label{obs:path-tree.nets.suns} For an NC-path-tree graph $G$, let $(\{P_v : v\in V(G)\},T)$ be its clique NC-path-tree model.
Let $x$ be a node of $T$ of degree at least three. \begin{enumerate}[noitemsep,topsep=0pt]
\item\label{prop:twin-free-junction} If $x$ is a junction, then $G$ contains a 3-sun. Also, if $G$ is twin-free, $|G_x|=3$. \item If $x$ is a terminal, then $G$ contains a net. \end{enumerate} \end{observation} \begin{proof} We establish the claimed properties in order as follows.
\begin{claimproof} \textbf{1.:} As in the proof of Lemma~\ref{lem:nc-path-tree.nodes}.2.(i) a junction $x$ in $T$ has three neighbors $y_1,y_2,y_3$ and vertices $v,v',v'' \in G_x$ such that $P_v \supseteq \{y_1,x,y_2\}$, $P_{v'} \supseteq \{y_1,x,y_3\}$ and $P_{v''} \supseteq \{y_2,x,y_3\}$. Additionally, since $x,y_1,y_2,y_3$ are maximal cliques, there are vertices $u_1,u_2,u_3 \in V(G)$ such that $u_i \in G_x \setminus G_{y_i}$ for each $i \in \{1,2,3\}$. Moreover, all of these vertices are distinct due to their paths being incomparable. Thus, by considering the 3-sun and its clique tree model given in Fig.~\ref{fig:small-graphs}, it is now easy to see that $G[v,v',v'',u_1,u_2,u_3]$ is a 3-sun. Furthermore, since $y_1,y_2,y_3$ are terminals, the paths $P_v,P_{v'},P_{v''}$ are the only distinct paths which are possible for vertices in $G_x$. In other words, every vertex in $G_x \setminus \{v,v',v''\}$ is a twin of one of $v$, $v'$, or $v''$. \end{claimproof}
\begin{claimproof} \textbf{2.:} Let $y_1,y_2,y_3$ be distinct neighbors of $x$. Since $G$ is connected and $x,y_1,y_2,y_3$ are maximal cliques, we have $v_i \in G_x \cap G_{y_i}$ and $u_i \in G_{y_i} \setminus G_x$ for each $i \in \{1,2,3\}$. The $v_i$'s are distinct since $x$ is a terminal, and the $u_i$'s are distinct since their paths are disjoint. Thus, by considering the net and its clique tree model given in Fig.~\ref{fig:small-graphs}, it is easy to see that $G[v_1,v_2,v_3,u_1,u_2,u_3]$ is a net. \end{claimproof} \end{proof}
\begin{corollary} \label{cor:nc-path-tree.structure} For an NC-path-tree graph $G$ and its clique NC-path-tree model $T$, the edges of $T$ uniquely partition into connected subtrees so that each subtree $T'$ has one of the following two types. \begin{enumerate} \item\label{prop:structure.junction} $T'$ consists of the three edges incident to a junction $x$, i.e., $T'$ is the $K_{1,3}$ formed by $x$ together with its three neighbors $y_1, y_2, y_3$ (all of which are terminals). \item\label{prop:structure.mixed-path} $T'$ is a path where the two end nodes are terminals and each inner node (if there are any) has degree 2 and is mixed. \end{enumerate} \end{corollary} \begin{proof} This follows from Lemma~\ref{lem:nc-path-tree.nodes} and by simply partitioning the edges of $T$ into maximal connected sets delimited by the terminals of $T$. \end{proof}
\subsection{Restricted Host Trees}
Here we relate and characterize the classes of NC-path-d.tree, NC-path-r.tree, and NC-path-path graphs as stated in the next two theorems. While Theorem~\ref{thm:nc-path-path.fisc} (concerning NC-path-path graphs) is well known~\cite{Roberts1970,Wegner1967}, it also follows directly from our study of NC-path-tree graphs.
\begin{theorem}\label{thm:nc-path-r.tree-fisc} A graph $G$ is (claw,3-sun)-free chordal if and only if it is NC-path-r.tree. Moreover, a graph has an NC-path-\textbf{d}.tree model if and only if it has a clique NC-path-\textbf{r}.tree. \end{theorem} \begin{proof} \
$\Leftarrow$ It is known and easy to see that the 3-sun is not a path-d.tree graph \cite{ChaplickGLT2010}. Thus, NC-path-d.tree is a subclass of (3-sun)-free NC-path-tree = (claw,3-sun)-free chordal by Theorem~\ref{thm:nc-path-tree-fisc}.
$\Rightarrow$ By Theorem~\ref{thm:nc-path-tree-fisc} and Observation~\ref{obs:path-tree.nets.suns}, for every (3-sun,claw)-free chordal graph $G$, there are no junctions in the clique NC-path-tree model $(\{P_v\}_{v \in V(G)},T)$ of $G$. Thus, since every node of $T$ with degree at least three is a terminal, rooting $T$ at any terminal results in an NC-path-r.tree model. \end{proof}
\begin{theorem}\label{thm:nc-path-path.fisc} A graph $G$ is (claw,3-sun,net)-free chordal if and only if it is NC-path-path, i.e., proper interval. \end{theorem} \begin{proof} \
$\Leftarrow$ It is known and easy to see that the net is not a path-path (interval) graph~\cite{LB1962}. Thus, NC-path-path is a subclass of (net)-free NC-path-r.tree = (claw,3-sun,net)-free chordal by Theorem~\ref{thm:nc-path-r.tree-fisc}.
$\Rightarrow$ As in the proof of Theorem~\ref{thm:nc-path-r.tree-fisc}, we note that since $G$ is a (net,3-sun)-free NC-path-tree graph, by Observation~\ref{obs:path-tree.nets.suns}, its unique clique NC-path-tree model has maximum degree two. Thus, the host is a path. \end{proof}
\subsection{Recognition Algorithms} \label{sec:nc-path.recog}
From our characterizations, there are straightforward polynomial-time certifying algorithms for the classes of NC-path-tree and NC-path-r.tree graphs. Specifically, since these classes are characterized as chordal graphs with an additional finite set of forbidden induced subgraphs, we can apply a linear time certifying algorithm for chordal graphs~\cite{RoseTL1976}, and then apply brute-force search for our additional forbidden induced subgraphs. If no forbidden induced subgraph is found, we can simply construct the unique clique tree of the given graph (e.g., using~\cite{GalinierHP1995}) and it will be an NC-path-tree (or NC-path-r.tree) model as needed to positively certify membership in our classes. However, we can do this more carefully and obtain linear time certifying algorithms as in the next theorem. A direct consequence of our certifying algorithm is that one can determine whether a chordal graph contains an induced claw in linear time. As we mentioned before, this stands in contrast to the case of general graphs where the best deterministic algorithms run in time $O(\min\{n^{3.252},m^{(\omega+1)/2}\})$~\cite{EisenbrandG04}, and $O(m^{\frac{2\omega}{\omega+1}})$~\cite{KloksKM00}.
\begin{theorem}\label{thm:nc-path-tree-recog} The classes NC-path-tree and NC-path-r.tree (= NC-path-d.tree) have linear-time certifying algorithms. In particular, one can certify the presence/absence of an induced claw in a chordal graph in linear time. \end{theorem} \begin{proof}
Recall that the size $\sum_{v \in V(G)} |K_v|$ of a clique tree is $O(n+m)$ (we use this implicitly throughout the following). First, we run a linear-time certifying algorithm for chordal graphs, e.g.,~\cite{RoseTL1976}. Then, we construct a clique tree $T$ in linear-time~\cite{GalinierHP1995}. We then annotate the clique tree to mark, for each vertex, for each maximal clique $K$ in $K_v$, if $K$ is a leaf or an internal node of the model of $v$. This annotation can be carried out in linear time because the total size of the clique tree is $O(n+m)$. If some vertex $v$ uses $\geq 3$ cliques as leaves, we produce a claw as in Claim~1 of the proof of Theorem~\ref{thm:nc-path-tree-fisc}. If there is a mixed node $x$ of degree $\geq 3$, then we proceed as in the proof of Lemma~\ref{lem:nc-path-tree.nodes}.1. This provides us with a pair of paths that cross in linear time. Then, proceeding as in Claim~2 in the proof of Theorem~\ref{thm:nc-path-tree-fisc}, we identify a claw. Now all of the nodes of degree $\geq 3$ are either terminals or junctions, and we mark them as such. So, if there is a junction $x$ with degree $\geq 4$, we proceed as in Lemma~\ref{lem:nc-path-tree.nodes}.2.(i) to identify a pair of paths that cross and, as before, report a corresponding claw. Furthermore, if a junction $x$ neighbors a non-terminal $y$, we proceed as in Lemma~\ref{lem:nc-path-tree.nodes}.2.(i) to identify a pair of paths that cross and (again) a corresponding claw.
Now, no crossing between two paths can involve a node of degree $\geq 3$. So, it remains just to ensure no crossings occur on a path between such nodes. In particular, since the neighbors of all junctions are terminals, such a crossing must occur on a path connecting two terminals (where all of the inner nodes are mixed, and, by Lemma~\ref{lem:nc-path-tree.nodes}, have degree two in $T$). Let $x_1, \ldots, x_k$ be such a path. Clearly, this path of cliques represents an interval graph. Moreover, we will find a pair of crossing paths on it precisely when this interval graph is not a proper interval graph. Conveniently, this problem is known to be solvable in linear time~\cite{DengHH1996}. However, to obtain linear time in total (when processing all such paths) we need to be a bit careful. Namely, rather than simply checking whether each $G[\bigcup_{i=1}^k G_{x_i}]$ is a proper interval graph, for each such path we create the following auxiliary graph $G'$.
\paragraph{\textbf{The graph} $\mathbf{G'}$ built from a path $x_1, \ldots, x_k$ in $T$ where $x_1$ and $x_k$ are terminals and each $x_i$ ($i \in \{2, \ldots, k-1\}$) is mixed.} The vertex set of $G'$ is $\{u_1,u_k\} \cup \bigcup_{i=2}^{k-1} G_{x_i}$. In $G'$, for each $i \in \{2, \ldots, k-1\}$, we make $G_{x_i}$ a clique. Also, we make $u_1$ adjacent to $G_{x_1} \cap G_{x_2}$ and $u_k$ is adjacent to $G_{x_{k-1}} \cap G_{x_k}$. In this way, the size of $G'$ can easily be seen as linear in the size of $G[\bigcup_{i=2}^{k-1} G_{x_i}]$. Moreover, since we only consider paths connecting terminals, each vertex and edge of $G$ is contained in at most one $G'$. Finally, observe that $G'$ is interval and is a proper interval graph if and only if $G[\bigcup_{i=2}^{k-1} G_{x_i}]$ is as well.
Thus, running the certifying recognition algorithm for proper interval graphs on $G'$ will provide a claw when $G'$ is not a proper interval graph, and such a claw is easily mapped back to a claw in $G$.
This completes the case of NC-path-tree graphs. For NC-path-r.tree graphs, we additionally check if $T$ contains junctions and proceed as in Observation~\ref{obs:path-tree.nets.suns}.1. In particular, if there are no junctions, we have an NC-path-r.tree model, and if a junction is present, we easily report a 3-sun to certify that the graph is not an NC-path-r.tree graph as described in Observation~\ref{obs:path-tree.nets.suns}.1. \end{proof}
\section{Domination Problems} \label{sec:mds}
A \emph{dominating set} in a graph $G$ is a subset $D$ of $V(G)$ such that every vertex is either in $D$ or adjacent to a vertex in $D$. In the \emph{minimum dominating set (MDS)} problem a graph $G$ is given and the goal is to determine a dominating set in $G$ with the fewest vertices. The MDS problem is \hbox{\textsf{NP}}-complete on PT graphs~\cite{BoothJ1982}, and split graphs~\cite{CorneilP1984}, and line graphs of planar graphs~\cite{YannakakisG1980} (which are of course claw-free).
A dominating set in a graph is \emph{independent} when the subgraph it induces is edgeless. Interestingly, the \emph{minimum independent dominating set (MIDS)} problem (defined analgously to the MDS problem) can be solved on chordal graphs in linear time~\cite{Farber1982}. For NC-path-tree graphs, the size of an MIDS is the same as the size of an MDS as shown in the conference version of this paper~\cite{wg2019}. However, this is also true for claw-free graphs~\cite{AllanL78,FaudreeFR97} (which, due to our characterization, trivially form a superclass of the NC-path-tree graphs). This implies the following theorem.
\begin{theorem}\label{thm:mds->mids} For any NC-path-tree graph $G$, there is an independent dominating set that is also a minimum dominating set. Moreover, such an independent dominating set can be found in linear time. \end{theorem}
So, we turn to another natural domination problem on NC-path-tree graphs.
A dominating set in a graph is \emph{connected} when the subgraph it induces is connected. In the \emph{minimum connected dominating set (MCDS)} problem, the input is a graph $G$, and the goal is to find a connected dominating set with the fewest vertices.
The MCDS problem is \hbox{\textsf{NP}}-hard even on \emph{line graphs} of planar graphs of maximum degree four~\cite[Lemma~46]{Munaro17}\cite[Theorem 10.5]{HermelinMLW19} (a quite restricted subclass of claw-free graphs) but fixed-parameter tractable on claw-free graphs~\cite{HermelinMLW19}. (In fact, under the \emph{Exponential Time Hypothesis}~\cite{ImpagliazzoPZ01}, there is no constant $c$ such that there is a $2^{o(k)}n^c$ time algorithm to decide whether a line graph has a connected dominating set of size $k$~\cite[Corollary 10.9]{HermelinMLW19}.) This problem is also \hbox{\textsf{NP}}-hard on split graphs but can be solved in polynomial time on strongly chordal graphs~\cite{WhiteFP85}. Note that the strongly chordal graphs include the NC-d.path-tree graphs, but do not include the NC-path-tree graphs (since, e.g., the 3-sun is an NC-path-tree graph but it is not strongly chordal~\cite{Farber1983}). Interestingly, it has also been shown~\cite[Corollary 4.3, Theorem 4.4]{WhiteFP85} that, for chordal graphs, the MCDS problem and the (cardinality) Steiner tree problem, defined next, are equivalent under linear time reductions.
For a graph $G$ and subset $X$ of $V(G)$, a \emph{Steiner tree (ST) of $X$} is a subtree of $G$ that includes $X$. In the ST problem, the input is a graph $G$ and a subset $X$ of $V(G)$, and the aim to find a ST of $X$ with the fewest vertices.
We will now establish the following theorem and corollary regarding the MCDS and ST problems on NC-path-tree graphs (note that the corollary follows simply from the theorem and~\cite[Theorem 4.4]{WhiteFP85}).
\begin{theorem} \label{thm:mcds} For any connected NC-path-tree graph $G$, a minimum connected dominating set of $G$ can be produced in linear time. \end{theorem}
\begin{corollary} \label{cor:st} For any NC-path-tree graph $G$ and subset $X$ of the vertices of a connected component of $G$, a minimum cardinality Steiner tree can be produced in linear time. \end{corollary}
To establish Theorem~\ref{thm:mcds} (and Corollary~\ref{cor:st}), we design an algorithm based on the following lemma. This lemma characterizes every MCDS in an NC-path-tree graph via the terminals and junctions of its clique NC-path-tree model. Recall that, as formalized in Corollary~\ref{cor:nc-path-tree.structure}, by thinking of the terminals in the clique NC-path-tree model $T$ of any NC-path-tree graph $G$ as delimiters, the edges of $T$ partition into the following two special types of subtrees. \begin{itemize} \item A junction $x$ together with its three neighbors $y_1, y_2, y_3$ (all of which are terminals). \item A path $P$ where the two end nodes are terminals and each inner node (if there are any) has degree 2 and is mixed. (In the lemma below, we will also differentiate the case when the $P$ is a single edge). \end{itemize}
\begin{lemma} \label{lem:mcds-structure} Let $(\{P_v\}_{v \in V(G)},T)$ be a clique NC-path-tree model a graph $G$ where $G$ is not a clique. A subset $D$ of $V(G)$ is an MCDS of $G$ if and only if properties 1--3 below are satisfied. \begin{enumerate}
\item\label{prop:mcds-junction} For each junction $x$ in $T$, $D$ contains exactly two vertices $u$ and $v$ from $G_x$ so that $P_u \cup P_v$ includes the three (terminal) neighbors of $x$ (i.e., $u$ and $v$ are not twins).
\item\label{prop:mcds:terminal-termainal} For each edge $xy$ in $T$ where both $x$ and $y$ are terminals, $D$ contains exactly one vertex of $G_x \cap G_y$.
\item\label{prop:mcds-mixed-path} For each path $(z_1, \ldots, z_k)$ in $T$ where $k \geq 3$, both $z_1$ and $z_k$ are terminals, and each $z_i$ ($i \in \{2, \ldots, k-1\}$) has degree 2, we have that the subgraph of $G$ induced by $D \cap \bigcup_{i=2}^{k-1} G_{z_i})$ is a shortest path connecting each $u \in G_{z_1}\setminus G_{z_2}$ to each $v \in G_{z_k} \setminus G_{z_{k-1}}$. \end{enumerate} \end{lemma} \begin{proof} The key to this proof is the next simple claim regarding connected dominating sets.
\claim{$\star$: A subset $S$ of $G$'s vertices is a connected dominating set if and only if for every edge $xy$ of $T$, $S$ contains at least one vertex of $G_x \cap G_y$. } \begin{claimproof} \noindent\textit{Proof of Claim~$\star$.}
$\Leftarrow$ First, since $G$ is not a clique, $T$ contains at least one edge. Therefore, since every node $x$ of $T$ is incident to some edge, $S$ contains a vertex of $G_x$ for every node $x$ of $T$, i.e., $S$ dominates $G$. Second, we have that $\bigcup_{v \in S} P_v$ is connected (since it is equal to $T$). In particular, $G[S]$ is connected.
$\Rightarrow$ Suppose that there is an edge $xy$ of $T$ where for every $v \in S$, $xy$ does not belong to~$P_v$. Let $T'$ and $T''$ be the two subtrees of $T$ obtained by deleting the edge $x$ from~$T$. Now, since $S$ is dominating, it contains a vertex $u$ whose model (path) in $T$ is contained in~$T'$ and a vertex $v$ whose model (path) in $T$ is contained in~$T''$. However, every $(u,v)$-path in $G$ must contain a vertex of $G_x \cap G_y$: This contradicts $G[S]$ being connected. \end{claimproof}
\noindent The key consequence of Claim~$\star$ is that an MCDS is, equivalently, a smallest set $S$ of vertices where every edge of $T$ is included in the path of some vertex in $S$. In particular, to characterize the MCDSs, it suffices to independently consider each subtree of $T$ as in the edge-partition with respect to terminals stated in Corollary~\ref{cor:nc-path-tree.structure} (and, also as slightly more finely enumerated in the statement of this lemma). With this in mind, we proceed with the proof for each item of the enumeration separately.
\begin{claimproof} \textbf{1.:} Here we have to cover the three edges incident to a junction $x$. Clearly, doing so requires at least two paths arising from non-twin vertices. Moreover, the only vertices whose paths contain edges incident to $x$, are those in $G_x$. Thus, we must pick two non-twin vertices of $G_x$, and, since $x$ is a junction, by doing so we indeed obtain two paths that cover all three edges. \end{claimproof}
\begin{claimproof} \textbf{2.:} Here, we just need to ensure the edge $xy$ is covered. Since, $G$ is connected, there must be at least one vertex whose path includes this edge. In particular, $G_x \cap G_y \neq \emptyset$ and it suffices to just take any such vertex. \end{claimproof}
\begin{claimproof} \textbf{3.:} Finally, we arrive at the somewhat non-trivial case concerning a path $P = (z_1, \ldots, z_k)$ in $T$ where $z_1$ and $z_k$ are terminals, and each $z_i$ ($i \in \{2, \ldots, k-1\}$) is mixed (and as such has degree 2).
Let $u$ be a vertex in $G_{z_1} \setminus G_{z_2}$, and let $v$ be a vertex in $G_{z_k} \setminus G_{z_{k-1}}$. In claims~(a) and~(b) below, we establish that (a) for any induced $(u,v)$-path in $G$, the models (paths) of the inner vertices cover the edges of $P$ (in $T$); and, (b) that in any MCDS, the vertices whose models (paths) include edges of $P$ constitute the inner vertices of an induced $(u,v)$-path. Together, these claims indeed imply Property~\ref{prop:structure.mixed-path} of this lemma since shortest paths are the smallest induced paths.
\claim{(a) For any induced $(u,v)$-path $Q = (u, w_1, \ldots, w_\ell, v)$ in $G$, the models (paths) of the inner vertices $(w_i, i \in \{1, \ldots, k\})$ cover precisely the edges of $P$ (in $T$).}
First, observe that $u$ and $v$ are not adjacent, i.e., $\ell \geq 1$ and $Q$ contains inner vertices. Second, observe that, for any inner vertex $w_i$ ($i \in \{1, \ldots, \ell\}$ of $Q$, $P_{w_i}$ is a subpath of~$P$ since $Q$ is an induced path and $z_1$ and $z_k$ are terminals. Finally, similarly to the proof of~$\Rightarrow$ for Claim~$\star$, since $Q$ is connected and includes $u$ and $v$ where $P_u$ and $P_v$ are separated by the path $P$ in $T$, we have that $\bigcup_{i=1}^{k} P_{w_i} \supseteq P$; thus, $\bigcup_{i=1}^{k} P_{w_i} = P$, completing the proof of this claim.
\claim{(b) In any MCDS $D$ of $G$, the vertices whose models (paths) include edges of $P$ constitute the inner vertices of an induced $(u,v)$-path.}
Let $D_P$ be the subset of $D$ where $w \in D_P$ if and only if $P_w$ contains an edge of $P$. By Claim~$\star$, $D_P \neq \emptyset$ and $P \supseteq \bigcup_{w \in D_P} P_w$. Moreover, since $z_1$ and $z_k$ are terminals, the union $\bigcup_{w \in D_P} P_w$ is contained in $P$, i.e., $P = \bigcup_{w \in D_P} P_w$. In particular, $D_P$ induces a proper interval subgraph of $G$.
We now show that $D_P$ induces a path $w_1, \ldots, w_\ell$ in $G$ such that $w_1$ is adjacent to $v$ and $w_\ell$ is adjacent to $u$.
We first establish the adjacency to $u$ and $v$. By Claim~$\star$, $D_P \cap G_{z_1} \cap G_{z_2} \neq \emptyset$, and we pick $w_1$ as any vertex in $D_P \cap G_{z_1} \cap G_{z_2}$. Since $w_1 \in G_{z_1}$, we indeed have that $uw_1$ is an edge. Notice that, if there is a vertex $w \in D_P \cap G_{z_1} \cap G_{z_2}$ such that $w \neq w_1$, then either $P_w \supseteq P_{w_1}$ or $P_{w_1} \supseteq P_w$, i.e., this would contradict the fact that $D$ is an MCDS. Thus, $w_1$ is the only vertex of $D$ in $G_{z_1} \cap G_{z_2}$. (Symmetrically, we have a vertex $w_\ell$ adjacent to $v$ such that $\{w_\ell\} = D_P \cap G_{z_{k-1}} \cap G_{z_k}$.)
To establish that $D_P$ really induces a path, we remark that it suffices to show that $G[D_P]$ is triangle-free. In particular, it is known~\cite{Eckhoff93} that a triangle-free interval graph is a caterpillar. Thus, since $G[D_P]$ is a proper interval graph (and as such claw-free---recall Theorem~\ref{thm:nc-path-path.fisc}), if $G[D_P]$ is triangle-free, then it is indeed a path. Moreover, the $(w_1,w_\ell)$ subpath of $G[D_P]$ constitutes a set of inner vertices of an induced $(u,v)$-path in $G$, and thus, by Claim~(a) and the minimality of $D$, $G[D_P]$ is precisely this subpath.
We now establish that $G[D_P]$ is triangle-free to show that it is indeed a path, completing the proof of Claim~(b). Suppose (for a contradiction) that $G[D_P]$ contains a triangle $w,w',w''$. By the Helly property of subtrees of a tree, we have that there is $z_j$ such that $z_j \in P_w \cap P_{w'} \cap P_{w''}$. However, since each of $P_w$, $P_{w'}$, and $P_{w''}$ is a subpath of $P$, without loss of generality, we have that $P_w \subseteq P_{w'} \cup P_{w''}$. This contradicts the minimality of $D$, and establishes that $G[D_P]$ is indeed triangle-free. Therefore, $G[D_P]$ is indeed a path and we have established Claim~(b).
\noindent Finally, as remarked above, combining Claims (a) and (b) establishes Property~\ref{prop:mcds-mixed-path}. \end{claimproof}
\end{proof}
Based on the above lemma we will now prove Theorem~\ref{thm:mcds}, establishing our linear time algorithm for the MCDS problem.
\begin{proof}[Proof of Theorem~\ref{thm:mcds}] In essence, this is just describing how to efficiently determine the vertices of an MCDS as described by the properties established in Lemma~\ref{lem:mcds-structure}. First, as in our certifying recognition algorithm (in the proof of Theorem~\ref{thm:nc-path-tree-recog}), we construct the clique NC-path-tree model $(\{P_v\}_{v \in V(G)}, T)$ of $G$ and mark each node as mixed, terminal, or junction. This allows us to partition $T$ according to its terminals as in Corollary~\ref{cor:nc-path-tree.structure}.
Now, as justified by Property~\ref{prop:mcds-junction}, for each junction, we simply pick any two non-twin vertices. Let $D_1$ be this set of vertices.
Similarly, as justified by Property~\ref{prop:mcds:terminal-termainal}, for each edge connecting two terminals, we simply pick any vertex whose path contains this edge (actually, the path of any such vertex will be precisely this edge). Let $D_2$ be this set of vertices.
As justified by Property~\ref{prop:mcds-mixed-path}, for each path of mixed nodes connecting two terminals in $T$, we will compute an appropriate shortest path in $G$. Of course, here, to obtain a linear running time, we have to be a bit careful. Let $(z_1, \ldots, z_k)$ be a path in $T$ where $z_1$ and $z_k$ are terminals and for each $i \in \{2, \ldots, k-1\}$, $z_i$ is mixed (and, as such, has degree 2). Here, we again construct the auxiliary graph $G'$ (as described in the proof of Theorem~\ref{thm:nc-path-tree-recog}) for this path $(z_1, \ldots, z_k)$, and compute a shortest path between the special vertices $u_1$ and $u_k$. Note that, since $G'$ is an interval graph, such a shortest path can be computed in linear time~\cite{AtallahCL95}. After doing so, we simply keep the inner vertices of such a path for our MCDS. Moreover, as remarked before, the total size of all of these $G'$ graphs is linear in the size of $G$, thus we can compute a shortest path for each such $G'$ graph in linear time in total. This gives us the set $D_3$ consisting of the inner vertices from this collection of paths.
Finally, we output the set $D_1 \cup D_2 \cup D_3$ as our MCDS. \end{proof}
\section{Hamiltonian Cycles and Minimum Leaf Spanning Trees} \label{sec:ham}
As mentioned earlier, the HC and HP problems are \hbox{\textsf{NP}}-complete on DPT graphs and split graphs. They are also \hbox{\textsf{NP}}-complete on line graphs of biparite graphs, i.e., (claw, diamond, odd-hole)-free graphs~\cite{LaiW93}, where the \emph{diamond} is the graph obtained by removing one edge from $K_4$. In contrast, we show that, like proper interval graphs~\cite{Bertossi83}, 2-connectivity suffices for Hamiltonicity in NC-path-tree graphs, but additionally, every \emph{tracing} of a clique NC-path-tree model provides a distinct HC of its graph. We similarly characterize the presence of an~HP via an obvious necessary condition in Theorem~\ref{thm:hamilton-path} below. This characterization of HPs directly allows us to characterize the number of leaves in a minimum-leaf spanning tree, see Corollary~\ref{cor:min-leaf-spanning-tree}, and ultimately provide a linear time algorithm for the minimum-leaf spanning tree problem on NC-path-tree graphs.
\begin{theorem}\label{thm:hamiltonicity} An NC-path-tree graph $G$ has a Hamiltonian cycle if and only if it is 2-connected and has at least three vertices. Also, for each plane layout of $G$'s clique NC-path-tree model $T$, a distinct a Hamiltonian cycle of $G$ can be constructed in linear time. \end{theorem} \begin{proof} We build on the fact that 2-connected proper interval graphs are not only Hamiltonian but have an HC with quite special structure, established in~\cite{Bertossi83}, and described as follows. Consider a proper interval graph $G$. Let $x_1, \ldots, x_k$ be the maximal cliques $G$ ordered according to the clique NC-path-path model of $G$. Further, let $u_1$ be a vertex of $G_{x_1} \setminus G_{x_2}$ and let $u_k$ be a vertex of $G_{x_k} \setminus G_{x_{k-1}}$. When $G$ is 2-connected there are internally disjoint $(u_1,u_k)$-paths $Q_1$ and $Q_2$ such that every vertex of $G$ belongs to either $Q_1$ or $Q_2$. Importantly for our claimed time bound is that these two paths can be found in linear time~\cite{Bertossi83}. In essence, we will see (through an auxiliary multigraph $X$ constructed below) that such paths also occur in 2-connected NC-path-tree graphs by considering the proper interval graphs occurring between terminals.
Now consider a 2-connected NC-path-tree graph $G$ and its clique NC-path-tree model $T$. Recall that, as we noted when designing our certifying algorithm for NC-path-tree graphs, for a path $x_1, \ldots, x_k$ in $T$ where $x_1$ and $x_k$ are terminals and each inner node is mixed (and consequently of degree 2), the graph $G[\bigcup_{i=1}^k G_{x_i}]$ is a proper interval graph. Moreover, since $G$ is 2-connected, each such subgraph is also 2-connected. Additionally, the graph $G'$ created from $G[\bigcup_{i=1}^k G_{x_i}]$ as before is also 2-connected. However, there is one special case where we use a slightly different auxiliary graph (otherwise we simply use the $G'$ defined before). When $k=2$, the graph $G'$ is the clique $G_{x_1} \cap G_{x_k}$ together with new vertices $u_1$ and $u_k$ where $N(u_1) = N(u_k) = G_{x_1} \cap G_{x_k}$. Now, it is easy to see that each such graph $G'$ is 2-connected and proper interval, and since $u_1$ and $u_k$ are not adjacent, we have two non-empty disjoint paths that both start with a vertex of $G_{x_1} \cap G_{x_2}$, and end with a vertex of $G_{x_{k-1}} \cap G_{x_k}$. Moreover, as remarked above these two paths can be built in linear time.
We now consider the case when a neighbor $y$ of $x$ is a junction before completing our construction of the HC. Let the other two neighbors of the junction $y$ be $x'$ and $x''$. Due to the fact that $x,x',x''$ are all terminals, the vertices of $G_y$ form three equivalence classes $A,A',A''$ of twins, where: \begin{itemize} \item each vertex in $A$ is represented by the path $x,y,x'$, \item each vertex in $A'$ is represented by the path $x',y,x''$, and \item each vertex in $A''$ is represented by the path $x'',y,x$. \end{itemize} Namely, using $A, A', A''$ we can ``traverse'' $T$ from $x$ to $x'$, from $x'$ to $x''$, and from $x''$ back to $x$. Due to the simple structure here, and the fact that partitioning into equivalence classes of twins is a linear time task, it is easy to construct the three steps of such a traversal ``around'' all junctions in linear time in total.
Based on the above observations, we can now build our HCs. The intuition here is to consider the tree $T$ to be drawn crossing-free in the plane, and trace the outline of $T$ terminal-to-terminal by using the paths guaranteed by the above arguments. We will encode the family of all such traces by a multigraph $M$ formed on the terminals of $T$ where each Eulerian tour of $X$ will correspond to a distinct HC of $G$. Namely, for each terminal $x$, and each neighbor $y$ of $x$ in $T$: \begin{itemize}[noitemsep,topsep=0pt] \item if $y$ is a terminal, then in $M$, $x$ and $y$ are connected by two edges (representing the two paths present in the corresponding $G'$). \item if $y$ is a mixed node and $z$ is the terminal so that $y$ occurs on the $(x,z)$-path in $T$, then, in $M$, $x$ and $z$ are connected by two edges (representing the two paths present in the corresponding $G'$). \item if $y$ is a junction and $x'$ and $x''$ are its two other neighbors, then in $M$, we have the edges $xx'$ and $xx''$. \item finally, if $G_x$ contains vertices that do not belong to any other $G_{x'}$ (e.g., when $x$ is a leaf of $T$), we also add a self-loop on $x$ in $M$ and map to this self-loop the vertices of $G_x \setminus (\bigcup_{x' \in N(x)}G_{x'})$. \end{itemize} We note the following properties of $M$ to complete the proof. The edges of $M$ partition the vertices of $G$ and each edge $xy$ corresponds to a path in $G$ where one end vertex belongs $G_x$ and the other end vertex belongs to $G_y$. Furthermore, $X$ is Eulerian, each Eulerian cycle $C$ provides an HC, and $C$ describes a plane layout of $T$, i.e., a cyclic order of the edges around each node of $T$ so that $C$ traces the outline of this plane layout of $T$. Note that, each such plane layout will often arise from multiple Eulerian cycles in $M$, but no two distinct layouts arise from the same cycle. \end{proof}
We now turn to HPs, and ultimately to minimum-leaf spanning trees. Note that, an \emph{$\ell$-leaf spanning tree} of a graph $G$ is simply a spanning tree of $G$ with exactly $\ell$ leaves, and a \emph{minimum-leaf spanning tree} is a spanning tree having the fewest leaves. Clearly, checking for an HP is a special case of finding a minimum-leaf spanning tree. A natural lower bound on the number of leaves in a minimum-leaf spanning tree comes from looking at the block-cutpoint tree (defined next).
The \emph{block-cutpoint tree} $BC(G)$ of a graph $G$ contains a node for each cut-vertex of $G$, a node for each maximal 2-connected subgraph (\emph{block}) of $G$, and its edge set is $\{ cB$ : $c$ is a cut-vertex, and $B$ is a block of $G$ containing $c$\}. It is well-known that $BC(G)$ can be computed in linear time~\cite{Hopcroft:1973}, and is indeed a tree. Clearly, if $G$ has an HP, $BC(G)$ is a path. In the next theorem, we show the converse is also true in NC-path-tree graphs, and further below we generalize this to $\ell$-leaf spanning trees. The main idea is to observe where the cut-vertices occur in the model and then reuse our Eulerian structure $M$ from the previous proof. More generally, if $G$ has a spanning tree with at most $\ell$ leaves, then $BC(G)$ also can have at most $\ell$ leaves. Here, we observe that once we have the characterization for the presence of an HP, the converse of this easily follows, see Lemma~\ref{lem:hp-to-min-leaf}. In particular, it holds for NC-path-tree graphs, see Corollary~\ref{cor:min-leaf-spanning-tree}.
\begin{theorem}\label{thm:hamilton-path} An NC-path-tree graph $G$ contains a Hamiltonian path if and only if its block-cutpoint tree is a path. Moreover, when the block-cutpoint tree of $G$ is a path, a Hamiltonian path can be produced in linear time. \end{theorem} \begin{proof} As noted above, it suffices to prove the $\Leftarrow$ direction. Let $G$ be an NC-path-tree graph and let $(\{P_v\}_{v \in V(G)},T)$ be its clique NC-path-tree model. Recall that, by Theorem~\ref{thm:hamiltonicity}, if $G$ has no cut-vertices, it has an HC, and thus also an HP. Therefore, we suppose $G$ contains a cut-vertex $v$. Note that $P_v$ must contain an edge $xy$ such that, for every vertex $u$ distinct from $v$, $xy$ is not an edge of $P_u$ (otherwise, $v$ would not be a cut-vertex). The next claim is the key to the proof. \claim{: $P_v$ is precisely the edge $xy$ and both $x$ and $y$ are terminals.} \begin{claimproof} Note that $P_v$ cannot contain any junctions since the vertices whose paths use junctions cannot be cut-vertices (recall that, by Observation~\ref{obs:path-tree.nets.suns}, if $P_v$ contains a junction, $v$ is a central vertex of a 3-sun such that the two other central vertices are adjacent and dominate $N(v)$). Suppose that $y$ is not an end-node of $P_v$, and let $z$ be the neighbor of $y$ distinct from $x$ (note: $y$ is mixed and as such has degree 2 by Lemma~\ref{lem:nc-path-tree.nodes}.1). Now, since $G_z$ and $G_y$ are maximal cliques, we have a vertex $u \in G_y \setminus G_z$, but $u$ cannot belong to $G_x$ since $u \neq v$ and $P_v$ is the only path containing $xy$. Thus, $P_u = y$ and $P_u$ and $P_v$ cross. Furthermore, since $P_v$ is the only path that uses $xy$, both $x$ and $y$ must be terminals. \end{claimproof}
Note that, since $BC(G)$ is a path (and $G$ is not 2-connected), it consists of the two \emph{end} blocks (containing a single cut-vertex each) and (possibly) some \emph{inner} blocks containing exactly two cut-vertices each. Clearly, an HP must consist of one path in each block where, in the two end blocks, the cut-vertex is an end vertex of the path, and, in each inner block, the two cut-vertices are the two end vertices of the path.
Let $B$ be an end block of $G$, i.e., a leaf of the block-cutpoint tree $BC(G)$ which contains one cut-vertex $v$. Since $B$ is a 2-connected induced subgraph of $G$, $B$ has a Hamiltonian cycle $C_B$ by Theorem~\ref{thm:hamiltonicity}. So, to obtain a Hamiltonian path that ends at $v$, we just delete one edge incident to $v$ from $C_B$.
To complete the proof, we will now argue that each inner block $B$ of $G$ containing two cut-vertices $v$ and $v'$ has a Hamiltonian path that connects $v$ to $v'$ within $B$.
By the claim above, in the clique NC-path-tree model $(\{P^B_v\}_{v \in V(B)}, T^B)$ of $B$, each of $P^B_{v}$ and $P^B_{v'}$ is a single terminal node. Let these nodes be $x$ and $x'$ (respectively) in $T^B$. Consider the path $x = x_1, x_2, \ldots, x_k = x'$ in $T^B$. Now, consider the Eulerian multigraph $M$ as in the proof of Theorem~\ref{thm:hamiltonicity}. Note that, since $x_1$ is a terminal, we can use $M$ to construct a path $Q_1$ that starts with $v$ and ends with a vertex of $G_{x_1} \setminus G_{x_2}$ and visits precisely the vertices in the connected component of $B \setminus (G_{x_1} \cap G_{x_2}$ that contains $v$. The path $Q_k$ is defined analogously. Similarly, for each terminal $x_i$ $(i \in \{2, \ldots, k-1\})$, we can use $M$ to craft a path $Q_i$ that visits precisely the vertices whose paths occur strictly within the subtree of $T^B \setminus \{x_{i-1}, x_{i+1}\}$ that contains $x_i$. Moreover, this path will start and end with vertices whose paths contain $x_i$. When $x_i$ is a junction, let $x'_i$ be the terminal that neighbors $x_i$ and is distinct from $x_{i-1}$ and $x_{i+1}$. Similarly to the case of $x_1$, we note that there is a path $Q'_i$ that visits all the vertices $U_i$ ``hanging below'' $x'_i$ and starts and ends with a vertex of $x'_i$. Additionally, due to the three equivalence classes of twins whose paths contain the junction $x_i$, we can extend this path $Q'_i$ to a path $Q_{i-1,i+1}$ that starts in a vertex of $G_{x_{i-1}} \cap G_{x_i}$, ends in a vertex of $G_{x_i} \cap G_{x_{i+1}}$, and visits every vertex of $U_i \cup G_{x_i}$.
Finally, consider two terminals $x_i$ and $x_j$ ($i < j$) where, for each $l \in \{i+1,\ldots, j-1\}$, $x_l$ is mixed. As in the proof of Theorem~\ref{thm:hamiltonicity}, we again consider the auxiliary graph $G'$ corresponding to this path. Here, we instead need a Hamiltonian path in $G'$ that starts and ends in our special vertices $u_1$ and $u_k$. Fortunately, it is known~\cite{Bertossi83}, that such a path does exist and actually only requires that $G'$ is connected. Namely, we have the path $Q_{i,j}$ which starts in a vertex of $G_{x_i} \cap G_{x_{i+1}}$, ends in a vertex of $G_{x_{k-1}} \cap G_{x_k}$, and visits every vertex of $\bigcup_{l=i+1}^{k-1}G_{x_l}$.
Thus, to form a desired Hamiltonian path $Q_B$ of $B$ that starts with $v$ and ends with $v'$, we simply concatenate the paths $Q_1, Q_{1,i_1}, Q_{i_1}, Q_{i_1,i_2}, Q_{i_2}, \ldots, Q_{i_t}, Q_{i_t,k}, Q_k$ where $x_{i_1}, \ldots, x_{i_t}$ are the terminals that occur between $x_1$ and $x_k$. In particular, by forming such a Hamiltonian path $Q_B$ for each inner block $B$ of $G$, we are done.
We conclude by remarking that the construction of an HP here can be completed in linear time. In particular, it suffices to describe how to obtain linear time on each block separately. For each end block $B$, we simply invoke the HC algorithm (leading to linear time in the the size of $B$). For each inner block, the paths $Q_1, Q_{i_1}, \ldots, Q_{i_t}$, and $Q_k$ can similarly be constructed by invoking the HC algorithm, leading to linear time in the size of $B$ in total. Each of the other paths $Q_{1, i_1}, Q_{i_1, i_2}, \ldots, Q_{i+{t-1}, i_t},$ and $Q_{i_t, k}$ can also be constructed in linear time by a simple greedy algorithm~\cite{Bertossi83}. Thus since these paths, which we concatenate in order to make $Q_B$, are constructed from edge-disjoint induced subgraphs of $B$, the total time to construct $Q_B$ is also linear in the size of $B$. \end{proof}
We now show how Theorem~\ref{thm:hamilton-path} can be generalized to minimum leaf spanning trees. While we expect that the following straightforward lemma has been observed before, we could not find an explicit proof of it, and so we include it here.
\begin{lemma} \label{lem:hp-to-min-leaf} For any graph class $\mathcal{G}$ closed under taking induced subgraphs, if every graph $G\in \mathcal{G}$ whose block-cutpoint tree is a path has a Hamiltonian path, then every graph $G \in \mathcal{G}$ whose block-cutpoint tree has $\ell$ leaves ($\ell \geq 2$) has a spanning tree with exactly $\ell$ leaves. \end{lemma} \begin{proof} We proceed by induction on the number of leaves in $BC(G)$. In the base case $BC(G)$ has two leaves, and the result follows trivially. So, suppose $BC(G)$ has $\ell \geq 3$ leaves. Let $Q$ be a path in $BC(G)$ that starts and ends in distinct leaf-blocks $B_1$ and $B_2$ where $B_1$ and $B_2$ share a cut vertex $v$, i.e., $Q$ is the path $(B_1,v,B_2)$. Further, let $G_Q$ be the subgraph of $G$ induced by the vertices occurring in the blocks on this path, and let $G'$ be the graph obtained by deleting every vertex of $B_1$ except $v$ from $G$. Now, by induction $G_Q$ has a Hamiltonian path $P$ and $G'$ has a spanning tree $T'$ with $\ell-1$ leaves. Moreover, since $v$ is a cut-vertex of $G_Q$, the path $P$ contains a subpath $P_1$ whose vertices are precisely the vertices of $B_1$ so that the vertex $v$ is an end vertex of $P_1$. Therefore, by gluing the path $P_1$ to $T'$ by identifying the occurrence of $v$ in both, we obtain a spanning tree $T$ of $G$ with precisely $\ell$ leaves. \end{proof}
\begin{corollary} \label{cor:min-leaf-spanning-tree} For any NC-path-tree graph $G$ that is not 2-connected (i.e., containing at least one cut-vertex), the number of leaves in a minimum-leaf spanning tree of $G$ is $\ell$ if and only if its block-cutpoint tree has exactly $\ell$ leaves. \end{corollary} \begin{proof} $\Rightarrow$ Clearly, when the block-cutpoint tree has more than $\ell$ leaves, $G$ cannot have an $\ell$-leaf spanning tree.
$\Leftarrow$ This follows directly from Theorem~\ref{thm:hamilton-path} and Lemma~\ref{lem:hp-to-min-leaf}. \end{proof}
\section{Concluding Remarks}
In this paper we have studied intersection graph classes of non-crossing paths in trees. We have provided forbidden induced subgraph characterizations and recognition algorithms for the natural classes of such graphs. We have further studied and provided efficient algorithms for variations of domination and Hamiltonicity problems on intersection graphs of non-crossing paths in a tree.
It might be interesting to investigate further algorithmic questions on this class that similarly have efficient algorithms on proper interval graphs, but are \hbox{\textsf{NP}}-hard on chordal graphs. A few problems in this context include: role assignment (aka locally surjective homomorphism) testing~\cite{Heggernes2012}, the simple max-cut problem~\cite{BodlaenderJ00} (here, the problem is still open even for proper interval graphs, see~\cite{BoyaciES2020-abs-2006-03856}), and the minimum outer-connected dominating set problem~\cite{KeilP13}.
Regarding further NC classes of graphs, a natural next step would be to study the NC-tree-tree graphs. But, as we mentioned before, it is not safe to simply work with clique trees in this case as the claw requires the use of a non-clique tree model.
We conjecture that the NC-tree-tree graphs can be characterized as chordal graphs avoiding a finite set of forbidden induced subgraphs.
It would also be interesting to see if similar algorithmic results on domination and Hamiltonicity problems can be obtained on this class.
Other host domains have been considered in the literature. Notice that, similar to proper interval graphs being NC-path-path graphs, the proper circular arc graphs are precisely the NC-path-cycle graphs. A simple host graph class that generalizes both trees and cycles is that of \emph{cacti}. A \emph{cactus} is a connected graph in which every 2-connected component is a single vertex, a single edge, or a chordless cycle. The intersection graphs of
subtrees of a cactus were studied by Gavril~\cite{Gavril1996}. So, one might consider the NC-path/tree/cactus-cactus graphs.
Finally, an alternative view of host domains has been considered quite recently through the notion of \emph{$H$-graphs}~\cite{ChaplickFGKZ19,Chaplick0VZ17,ChaplickZ17,FominGR20}, i.e., for a fixed graph $H$, a graph $G$ is an $H$-graph when it is an intersection graph of connected subgraphs of a subdivision of $H$. Here, interval graphs are the $K_2$-graphs and circular-arc graphs are the $K_3$-graphs. While there is a natural notion of proper $H$-graphs~\cite{ChaplickFGKZ19} (which indeed restrict $H$-graphs for every $H$), the more restrictive non-crossing $H$-graphs might have a nicer structure and lead to easier (and faster) algorithms.
{}
\appendix
\end{document}
\section*{Proofs omitted from the main text.}
\end{document} | arXiv |
Sargan–Hansen test
The Sargan–Hansen test or Sargan's $J$ test is a statistical test used for testing over-identifying restrictions in a statistical model. It was proposed by John Denis Sargan in 1958,[1] and several variants were derived by him in 1975.[2] Lars Peter Hansen re-worked through the derivations and showed that it can be extended to general non-linear GMM in a time series context.[3]
The Sargan test is based on the assumption that model parameters are identified via a priori restrictions on the coefficients, and tests the validity of over-identifying restrictions. The test statistic can be computed from residuals from instrumental variables regression by constructing a quadratic form based on the cross-product of the residuals and exogenous variables.[4]: 132–33 Under the null hypothesis that the over-identifying restrictions are valid, the statistic is asymptotically distributed as a chi-square variable with $(m-k)$ degrees of freedom (where $m$ is the number of instruments and $k$ is the number of endogenous variables).
See also
• Durbin–Wu–Hausman test
References
1. Sargan, J. D. (1958). "The Estimation of Economic Relationships Using Instrumental Variables". Econometrica. 26 (3): 393–415. doi:10.2307/1907619. JSTOR 1907619.
2. Sargan, J. D. (1988) [1975]. "Testing for misspecification after estimating using instrumental variables". Contributions to Econometrics. New York: Cambridge University Press. ISBN 0-521-32570-6.
3. Hansen, Lars Peter (1982). "Large Sample Properties of Generalized Method of Moments Estimators". Econometrica. 50 (4): 1029–1054. doi:10.2307/1912775. JSTOR 1912775.
4. Sargan, J. D. (1988). Lectures on Advanced Econometric Theory. Oxford: Basil Blackwell. ISBN 0-631-14956-2.
Further reading
• Davidson, Russell; McKinnon, James G. (1993). Estimation and Inference in Econometrics. New York: Oxford University Press. pp. 616–620. ISBN 0-19-506011-3.
• Verbeek, Marno (2004). A Guide to Modern Econometrics (2nd ed.). New York: John Wiley & Sons. pp. 142–158. ISBN 0-470-85773-0.
• Kitamura, Yuichi (2006). "Specification Tests with Instrumental Variable and Rank Deficiency". In Corbae, Dean; et al. (eds.). Econometric Theory and Practice: Frontiers of Analysis and Applied Research. New York: Cambridge University Press. pp. 59–124. ISBN 0-521-80723-9.
| Wikipedia |
Nonlinear XUV signal generation probed by transient grating spectroscopy with attosecond pulses
Ashley P. Fidler ORCID: orcid.org/0000-0001-6539-29041,2,
Seth J. Camp3,
Erika R. Warrick1,2,
Etienne Bloch1,
Hugo J. B. Marroux1,2,
Daniel M. Neumark ORCID: orcid.org/0000-0002-3762-94731,2,
Kenneth J. Schafer3,
Mette B. Gaarde3 &
Stephen R. Leone1,2,4
Nature Communications volume 10, Article number: 1384 (2019) Cite this article
Atomic and molecular interactions with photons
Attosecond science
Nonlinear optics
Nonlinear spectroscopies are utilized extensively for selective measurements of chemical dynamics in the optical, infrared, and radio-frequency regimes. The development of these techniques for extreme ultraviolet (XUV) light sources facilitates measurements of electronic dynamics on attosecond timescales. Here, we elucidate the temporal dynamics of nonlinear signal generation by utilizing a transient grating scheme with a subfemtosecond XUV pulse train and two few-cycle near-infrared pulses in atomic helium. Simultaneous detection of multiple diffraction orders reveals delays of ≥1.5 fs in higher-order XUV signal generation, which are reproduced theoretically by solving the coupled Maxwell–Schrödinger equations and with a phase grating model. The delays result in measurable order-dependent differences in the energies of transient light induced states. As nonlinear methods are extended into the attosecond regime, the observed higher-order signal generation delays will significantly impact and aid temporal and spectral measurements of dynamic processes.
The development of wave-mixing techniques in the extreme ultraviolet (XUV) and X-ray regimes represents the next frontier of nonlinear spectroscopy1. In optical2, infrared3, and radio-frequency4 nonlinear spectroscopies, highly selective multiphoton interactions are routinely employed to probe the structure and evolution of complex chemical systems dominated by rotational and vibrational dynamics5. Using femtosecond pulses, time-domain nonlinear techniques effectively reveal sub-picosecond transition states in chemical reactions6, mechanisms of energy transfer within photosynthetic complexes7, and the timescales of relaxation dynamics in semiconductors8. The extension of similar techniques to XUV/X-ray wavelengths and subfemtosecond pulse durations will provide critical insights into the fundamental dynamics associated with valence and core-level electronic transitions9,10.
High harmonic generation (HHG) and free electron laser (FEL)-based sources promise to exploit nonlinear processes to access ultrafast dynamics in the XUV. Transient grating schemes with intense 80 fs XUV pulses at FELs have successfully generated wave-mixing signals11,12,13,14,15. Similar schemes with XUV light produced by HHG paired with visible or near-infrared (NIR) pulses have probed acoustic and optical processes on timescales of tens to hundreds of femtoseconds16,17. Although table-top HHG can produce XUV pulses as short as 43 as18, low conversion efficiencies of 10−6 to 10−8 result in pulse energies insufficient to support nonlinear processes independently19,20,21,22,23. Nonetheless, pioneering experiments with subfemtosecond HHG XUV pulse trains and moderately intense femtosecond NIR pulses have isolated long-lived XUV four-wave-mixing signals through spectral filtering and noncollinear beam geometries. These methods have achieved background-free measurements of wavepacket dynamics in Rydberg and valence excited states of atomic24,25 and diatomic systems26,27. In conjunction with recent progress in subfemtosecond pulse generation at FELs28,29, these advances provide the foundation necessary for selective measurements on electronic timescales as well as for the direct probing of electronic correlations and strong-field effects in excited states. However, as these initial experiments focused primarily on the coupling of long-lived states, the impact of the nonlinear signal generation process itself has yet to be explored.
Here, we elucidate the temporal dynamics of nonlinear signal generation with broadband, ultrashort pulses. A beam geometry in which a NIR transient grating probes an XUV-induced coherence30 facilitates the simultaneous detection of up to five orders of resonance-enhanced nonlinear signals from 1snp Rydberg states in atomic helium between 20 and 24.6 eV. The time-domain characteristics of these signals reveal few-femtosecond time delays between the formation of distinct grating orders emitting at the energies of both the 1snp resonances and dressed states that appear in a strong infrared (IR) field, called light-induced states (LISs)31,32,33,34,35. Furthermore, we demonstrate that these delays lead to a less pronounced AC Stark shift of higher-order LIS features relative to lower-order features. Both the delays in emission and LIS energy shifts are directly reproduced by numerically solving the coupled Maxwell–Schrödinger equations for helium gas. We attribute the order-dependent delays to the accumulation of the AC Stark phase over the NIR pulse duration, increasing the modulation of a phase grating and thereby enhancing the efficiency of higher grating order formation with time. These results demonstrate a fundamental accumulation time inherent in the generation of nonlinear signals from electronically excited states, which represents an opportunity for increased selectivity and control in few-femtosecond and attosecond spectroscopy. Experimental observation of the fastest processes in nature, including electron correlation36 and transfer37,38,39, directly in the time domain is ultimately limited by the duration of the pulses utilized in these spectroscopic techniques. In the transient grating results described here, different transient grating orders emerge at different times within the duration of a few-cycle pulse and thus report on different time periods during ultrafast dynamics, providing a route towards measuring the timescales of dynamic processes occurring within the pulse duration. Furthermore, because of their distinct and predictable spectral characteristics, analysis of multiple grating orders may allow for the identification and isolation of desirable short-lived signals obfuscated in congested spectra. Utilization of this technique will therefore facilitate the observation and measurement of dynamics on electronic timescales in both the temporal and spectral domains.
Transient XUV holography in atomic helium
To generate wave-mixing emission, we utilize a transient grating geometry with a subfemtosecond XUV pulse train and two 6 fs NIR pulses centered at 800 nm (Fig. 1a, see Methods and Supplementary Note 1 for more details). The XUV pulse train initiates a coherent superposition of the ground state and the lowest-lying excited state manifold in gas-phase helium atoms at a density of ~ 3 × 1017 cm−3 in a 3 mm long cell. The time-coincident NIR pulses intersect the XUV at angles of approximately +1° and −0.75° to capitalize on the phase-matching conditions inherent in wave-mixing processes, producing spatially distinct higher-order grating signals. Multiple orders of nonlinear signals are imaged simultaneously as a function of XUV-NIR delay by a flat-field imaging spectrometer. Positive delays indicate that the XUV precedes the combined peak intensity of the NIR pulses. A charge-coupled device (CCD) camera image taken 3.5 fs after time overlap exhibits up to five distinct orders of resonance-enhanced nonlinear signals (Fig. 1b). In the zeroth order at a divergence angle of 0 mrad, depletion features provide clear evidence of XUV absorption by helium 1snp resonances. These long-lived states possess large absorption cross-sections40, facilitating the generation of multiple orders of nonlinear signals at the photon energies of the one-photon dipole-allowed (bright) np states. The slight distortion of the 2p state's absorption and emission profiles is due to resonant pulse propagation effects41,42.
Experimental characterization and demonstration of extreme ultraviolet (XUV) transient grating spectroscopy. a Simplified experimental set-up depicting the formation of a transient grating in the helium sample (green) and the spectrometer employed to image the diffracted signals as a function of the delay between an subfemtosecond XUV pulse train and two few-cycle, time-coincident near-infrared (NIR) pulses. b A charge-coupled device (CCD) camera image taken 3.5 fs after pulse overlap. Positive delays indicate that the XUV pulse precedes the NIR grating. A number indicating the diffraction order, m, is provided to the right of the figure. Resonances and light-induced states (LISs) are assigned at the top of the image. c Energy level schematic depicting the three-photon pathway that emits at the energy of the LIS in the first grating order. Solid black lines correspond to np bright states, dashed lines correspond to ns or nd dark states, and solid red lines correspond to LISs
Notably, we also detect broad higher-order emission features at energies (e.g., ~21.8 eV) distinct from any np state. These features correspond to LISs, which, in a simple picture, can be described as intermediate dressed states in Raman-like two-photon transitions to one-photon dipole-forbidden (dark) states31,32,33,34,35. Enabled by strong coupling between a bright and dark state, LISs typically exist only when the XUV and NIR pulses overlap in time and space, and appear at energies approximately one NIR photon (~1.5 eV) above or below the associated dark state (Supplementary Note 2). The dominant LIS feature here originates from the 3d dark state located 1.5 eV above it in energy, and is referred to as the 3d− LIS (Fig. 1c, identification procedure in Supplementary Note 3)33,34,35. In a moderately intense NIR field, the 3d dark state and its LIS adiabatically blueshift in energy with the pulse envelope due to the AC Stark effect, as illustrated by the shift between the predicted and observed LIS energetic position in Fig. 1b. Importantly, LIS features are less blueshifted by ~ 0.1 eV with increasing order, indicating that the AC Stark effect is less pronounced for higher-order signals.
A time-domain holographic picture can be readily employed to conceptualize the observed spatial dependence of the nonlinear signals30. The intersection of the two noncollinear NIR beams causes spatial modulations in the electric field experienced by the helium gas, leading to periodically varying frequency-dependent changes in both the real and imaginary components of its refractive index. The spatial periodicity in the refractive index forms a grating that diffracts the XUV-induced free induction decay, the temporal characteristics of which map to the coherence time of the excited state43. According to the Bragg diffraction equation, the fringe spacing, aNIR, of this transient grating is given by \(\lambda _{{\mathrm{NIR}}}{\mathrm{/}}\left( {{\mathrm{2}}\;{\mathrm{sin}}\left( {\theta {\mathrm{/2}}} \right)} \right)\), where λNIR is the NIR wavelength and θ is the crossing angle of the two NIR pulses. From this fringe spacing, it follows that each diffraction order, m, will appear at an angle defined by \(m \cdot \lambda\)NIR/aNIR. Alternatively, the diffracted signals can be described using the wavevector phase-matching requirements intrinsic to the perturbative interaction of one XUV photon and an even number of noncollinear NIR photons (additional details in Supplementary Note 4)44. However, as demonstrated in Supplementary Fig. 9, the NIR intensity dependence of the nonlinear signals described here is indicative of a nonperturbative regime in which distinct grating orders are composed of multiple orders of wave-mixing signal emitting at the same spatial location.
Time- and energy-domain evolution of distinct grating orders
Figure 2 examines emission signals in different transient grating orders as a function of XUV-NIR delay, providing compelling insights into the origin of the order-dependent LIS energy shifts and, more generally, the temporal dynamics of nonlinear signal generation. In Fig. 2b–d, false color plots produced by integrating vertically over 0.7 mrad (10 pixels) for each of the features specified in Fig. 2a illustrate the evolution of LIS signals in both energy and time delay. The delay dependence of the features after pulse overlap (>8 fs) can be attributed to pulse reshaping in the dense He gas45, coupling to longer-lived states, and population accumulation in the dark state. These secondary considerations are detailed in Supplementary Note 5. In the three grating orders considered here, the LIS features initially appear centered around 21.66 ± 0.05 eV and subsequently broaden and blueshift in energy with increasing delay. A similar LIS shift has been documented extensively in collinear transient absorption geometries and is attributed to the AC Stark effect34,35. Interestingly, lower grating orders exhibit a greater relative energy shift than higher orders. The maximum of the first-order feature shifts more than 0.2 eV higher in energy (Fig. 2b), while the second- and third-order features shift only just over 0.1 eV (Fig. 2c) and 0.05 eV (Fig. 2d) respectively. Thus, LISs in higher grating orders experience a reduced AC Stark shift relative to those in lower orders. Another striking difference between grating orders is their delay dependence, with lower-order features emerging at earlier delays than higher-order features.
Order-dependent AC Stark shift of the 3d− light-induced state. a The experimental charge-coupled device (CCD) camera image taken 3.5 fs after overlap and cropped to emphasize the redshift of the light-induced state (LIS) spectra with grating order. The energy and delay dependence of the LIS emission features associated with the b m = −1, c m = −2, and d m = −3 grating orders are obtained by integrating vertically (0.7 mrad) over the three highest energy black boxes in a. The selected regions were chosen to avoid contamination from the distorted 2p state and allow for the full breadth of the energy shift
Calculation of order-specific spectral and temporal profiles
To better understand the experimental observations, we calculated the observable spatio-spectral profile resulting from the transient grating interaction using the procedure detailed in ref. 46 (see Methods for more details). Using the single active electron (SAE) approximation, the coupled time-dependent Schrödinger equation (TDSE) and Maxwell's equations were solved numerically using experimentally derived parameters. These calculations utilize a pseudopotential that accurately reproduces the energy levels for a singly excited He atom. The XUV pulse is assumed to be a single 330 as pulse centered at 23 eV, allowing for the simultaneous excitation of the entire manifold of 1snp states. Two identical 6 fs NIR pulses, each with a central frequency of 800 nm and an intensity of 2 × 1012 W cm−2, are crossed at angles of +1° and −0.75° with respect to the XUV beam to replicate the noncollinear experimental geometry. To accommodate the resulting non-cylindrical symmetry, the response is calculated in one transverse direction only47. These calculations yield the space- and time-dependent electric field at the end of a thin helium gas jet, ignoring the resonant pulse propagation effects observed experimentally for the 2p state. Figure 3a shows the spatio-spectral intensity profile after transforming to the far field for an XUV-NIR delay of 4 fs. The profile is plotted on a log scale to highlight weak higher-order features. The delay and energy dependence of the LIS features designated in the higher energy windows of Fig. 3a is shown in Fig. 3b–d. The calculation successfully reproduces the multiple diffraction orders visible at the 2p and LIS energies as well as the order-dependent variations in energy and delay for the LIS features observed experimentally.
Far and near-field results of the time-dependent Schrödinger and Maxwell's equations. a Calculated log spectral intensity for the 1s2p resonance and 3d− light-induced state (LIS) of helium gas propagated to the far field. The selected regions indicate distinct orders of either 2p or 3d− features. The delay dependence of the LIS emission features associated with the b m = −1, c m = −2, and d m = −3 grating orders are obtained by integrating vertically over the three highest energy black boxes in a in the near field. The data in b–d have been smoothed to reflect the lack of carrier envelope phase control in the experiment
To further quantify the delay dependence of different grating orders and characterize the dynamics of nonlinear signal generation, the experimental LIS features indicated in Fig. 2b–d are integrated over an energy bandwidth of 0.5 eV (21.6–22.1 eV). Integration areas were chosen to minimize spurious effects from nearby states and to accommodate the energy shift of the features. The resulting delay dependence demonstrates that while the temporal profiles associated with the LISs in each grating order are similar, higher orders indeed emerge at later XUV-NIR delays (Fig. 4a). The second and third diffraction order LIS features peak at 1.5 ± 0.6 fs and 2.4 ± 0.6 fs, respectively, after the first order. To verify that these delays are not unique to the transient LISs, the same analysis was performed for the long-lived 2p state (Fig. 4b). Integrating over 0.3 eV (21.1–21.4 eV) results in measured delays of 1.8 ± 0.8 fs between first and second orders and 3.6 ± 0.8 fs between the first and third orders. The local maximum at −10 fs may be due to propagation effects. The delays observed for distinct grating orders of long-lived np states are consistent with those for the transient 3d− LIS. Additional examples are provided in Supplementary Note 6.
Experimental and calculated delay in the emergence of higher-order signals. a Experimental extreme ultraviolet and near-infrared pulse delay dependence of the lowest three nonlinear grating orders of the 3d− light-induced state (LIS) and of b the 2p state obtained by integrating over the higher (21.6–22.1 eV; 0.7 mrad) and lower energy windows (21.0–21.4 eV; 0.7 mrad), respectively, in Fig. 2a. c Real-time dependence calculated for distinct orders of 3d− LIS features obtained by integrating over the higher energy windows in Fig. 3a and of d 2p features obtained by integrating over the lower energy windows after transforming the results into the nearfield. The decay of the 2p signals is due to a decoherence lifetime of 20 optical cycles added to the time-dependent Schrödinger equation calculation to simulate the experimentally observed coherence time
In the calculations, we access the evolution of distinct diffraction orders as a function of real time, rather than XUV-NIR delay, by transforming the windows shown in Fig. 3a into the time domain and the near field. Time zero corresponds to the peak of the NIR pulses, which arrive 4 fs after the initial XUV excitation. For the 3d− LIS, calculated delays of 1.8 fs between first and second orders and 3.0 fs between the first and third orders compare well to the experimental results (Fig. 4c). Similar delays of 2.1 fs and 3.3 fs are observed for the 2p state (Fig. 4d). The comparison between delay and real time is particularly appropriate for the earliest delays when the time-integrated emission signals are not yet dominated by cumulative effects. As shown in Supplementary Note 7, the calculated delay dependence agrees with Fig. 4. The agreement between the delay-dependent experimental and time-dependent theoretical results suggests that the experimentally observed delays between diffraction orders originate in real-time differences in the temporal dynamics of higher-order signal generation: different diffraction orders are formed at different times during the nonlinear interaction.
AC Stark phase grating model
To better understand the origin of the time delay between the emergence of different grating orders in the Rydberg states and LISs, we compare the results with a simple model based on the formation of a transient phase grating due to the crossed NIR pulses. In a noncollinear geometry, the crossing of the NIR beams results in an interference pattern in the NIR intensity (Fig. 5a) that generates both a spatial grating in the AC Stark phase and an amplitude grating in the XUV-excited population. The time- and space-dependent dipole, d(x,t), of an excited atom oscillating at XUV frequency, ω, that interacts with perturbing NIR pulses can be expressed as:
$$d\left( {x,t} \right)\,{\mathrm{ = }}\,f\left( {x,t} \right){\mathrm{cos}}^2(k_{{\mathrm{NIR}}}x)e^{{i}\phi (x,t)}e^{ - i\omega t}{\mathrm{ + }}\,{\mathrm{c}}{\mathrm{.c}}{\mathrm{.,}}\;{\mathrm{for}}\;t \ge t_0.$$
Here, f(x,t) describes the envelopes of the XUV field in space and the NIR field in time, and \({\mathrm{cos}}^2\left( {k_{{\mathrm{NIR}}}x} \right)\) represents an amplitude grating where \(2k_{{\mathrm{NIR}}} = 2\pi /a_{{\mathrm{NIR}}}\) is the wavevector associated with the NIR intensity grating. The NIR-induced phase shift, \(\phi (x,t)\), can be expressed as:
$$\phi {(x,t) = }{\int _{t_0}^{t}} \frac{{\Delta {E}\left( {x,t \prime } \right)}}{\hbar }{{\mathrm{d}}t}\prime,$$
where \(\Delta E\left( {x,t\prime } \right)\) is the AC Stark shift of the excited state energy and t0 is the arrival time of the XUV pulse. This phase shift accumulates over the course of the NIR pulse (Fig. 5b) and can therefore be approximated by the relationship:
$$\phi \left( {x,t} \right) \approx \Delta \left( {x,t} \right){\mathrm{cos}}^2\left( {k_{{\mathrm{NIR}}}x} \right),$$
where \(\Delta (x,t)\) is the phase shift due to the NIR envelope and is modulated by the sinusoidal grating pattern.
Model of AC Stark phase grating accumulation and nonlinear signal generation. a Spatially modulated intensity profile generated by two crossed near-infrared (NIR) pulses. b The calculated NIR-induced AC stark shift plotted as a function of real time, where time zero corresponds to the time at which the extreme ultraviolet (XUV) pulse interacts with the system and the peak of the NIR pulse. c A false color plot shows the amplitude of the dipole moment modulated by both the AC Stark phase grating and an amplitude grating in the nearfield as a function of time. Later times exhibit an increased modulation depth. d In the far field, different grating orders are plotted as a function of time during the NIR pulse
This combined phase and amplitude grating leads to diffraction of the near-field XUV electric field into multiple orders in the far field. The diffracted XUV field is proportional to the dipole moment. Figure 5c demonstrates that the dipole becomes more strongly modulated with time due to the accumulation of the AC Stark phase, meaning that the efficiency with which higher-order diffraction signals are generated will increase with time. This evolution can be quantified by sampling the near-field dipole moment at specific points in real time and transforming it into the far field, thereby providing a view into grating order formation (Fig. 5d). At early times, only the zeroth and first orders are generated due to the population grating. However, at later times, as higher-order modulations appear in the dipole moment because of the accumulating phase grating, higher diffraction order peaks arise in the far field. The excellent agreement of this simple model's time dependence with both the delay-dependent experiment and the full calculations suggests that the observed delays in higher-order signal generation arise due to the time associated with the build-up of the AC Stark phase over the NIR pulse and validates the use of a phase grating model as a physical picture for nonlinear signal generation in this experimental configuration. Contrary to a perturbative picture in which the highest order signals peak at pulse overlap48, the highest order signals here are delayed relative to the peak of the NIR pulse (0 fs). The grating orders' intensity dependence also must be described as nonperturbative and can be explained via the phase grating model. Supplementary Note 8 provides additional details about both the phase grating model and its nonperturbative characteristics.
Finally, the phase grating model also provides insight into the attenuation of the LISs' AC Stark shift with increasing grating order. As shown in Fig. 5d, higher-order signals arise only after accumulation of the AC Stark phase. Thus, features in higher diffraction orders emerge later in the NIR pulse and subsequently have less time to accumulate AC Stark phase after generation. Since the AC Stark phase originates from the energy shift of the state (Eq. 2), these higher-order features will therefore exhibit a less pronounced energy shift relative to lower-order features that interact with the NIR pulse over a longer timeframe. This effect is only evident for the transient LIS features because their emission exists only when the NIR pulse is present, whereas the dipole moment from the comparatively long-lived np states persists well after the NIR-induced Stark shift has passed.
In summary, the few-femtosecond temporal dynamics of nonlinear signal generation is investigated in gas-phase helium using a transient grating geometry between two noncollinear few-cycle NIR pulses and a subfemtosecond XUV pulse train. Simultaneous measurements of multiple diffraction orders of nonlinear signals reveal significant delays in the emergence of higher-order signals associated with both Rydberg states and LISs. Calculations using the coupled Maxwell–Schrödinger equations in the SAE approximation reproduce these delays and demonstrate that they originate in real-time differences in signal generation between distinct grating orders. Furthermore, because higher grating orders arise later within the NIR pulse as a consequence of this delay, higher-order LIS features exhibit a less prominent AC Stark shift than those in lower orders. Finally, we introduce a conceptual model to relate the delay times to the accumulation of an AC Stark phase grating over the course of the NIR pulse, explicitly defining the observed delays as a fundamental property of nonlinear signal generation. Given the intense interest in developing nonlinear spectroscopies in the HHG and FEL communities, these delays will be important to the design and interpretation of nonlinear experiments that probe subfemtosecond dynamics. Experimentally obtainable pulse durations can obfuscate the temporal signatures of short-lived processes, limiting the dynamics that can be studied in the time domain. This issue is amplified in nonlinear spectroscopic techniques requiring noncollinear beam geometries that degrade the time resolution dictated by the pulse duration. These transient grating results provide a compelling alternative. The knowledge that different grating orders emerge at different times in the NIR pulse can be utilized systematically to unravel dynamics occurring within the NIR pulse duration. In the spectral domain, the grating order-dependent energy shift of short-lived features suggests a mechanism by which features associated with ultrafast processes can be discriminated from spectra with overlapping and complex features. This work represents one of the potentially many experiments in which these intrinsic delays will impact the behavior of nonlinear signals.
Experimental scheme
The experimental apparatus employed for these measurements shown as Supplementary Fig. 1 has been described previously27. Briefly, 22 fs NIR pulses produced by a 1 kHz, 2 mJ commercial laser system (Femtopower HE, Femtolasers) are spectrally broadened in a neon-filled hollow core fiber with an inner diameter of 400 µm and subsequently compressed by 6 pairs of broadband chirped mirrors (Ultrafast Innovations) to produce 6 fs pulses with a spectral bandwidth extending between 550 and 950 nm. The pulses transmitted through a 50:50 beamsplitter are focused by 50 cm focal length mirror into the vacuum chamber (10−6 Torr) containing a sample cell with xenon gas to produce a pulse train of 2–3 subfemtosecond pulses in the XUV via high harmonic generation. A 0.2 µm Sn filter (Lebow, 17–24 eV transmission) spectrally filters the XUV and co-propagating NIR light to include only the 13th and15th harmonics. A gold-plated toroidal mirror focuses the XUV through an annular mirror into a second 3 mm long cell containing the helium gas target at densities of approximately 3 × 1017 cm−3. The XUV intensity is estimated to be between 108 and 1010 W cm−2.
The NIR pulses reflected from the 50:50 beamsplitter before high harmonic generation are delayed relative to the XUV by a piezoelectric stage and then divided into two arms by a second 50:50 beamsplitter. The reflected and transmitted beams are displaced above and below the hole of the annular mirror in the XUV beam path such that they are recombined with the XUV in the target cell at angles of approximately 1° and 0.75° respectively and focused to roughly 100 µm FWHM (full width at half maximum) spots with an intensity of 2 × 1012 W cm−2. The delay between these two NIR arms is controlled by a second stage positioned in the transmitted beam path. For these experiments, the position of the second stage is set to ensure the NIR arms are time-coincident. Temporal and spatial overlap of the pulses is determined by the appearance of fringes on a CCD camera positioned at the focus due to the interference between the NIR beam used for HHG and the noncollinear probe beams.
After propagating through 3 mm of approximately 3 × 1017 cm−3 of helium gas in the target cell, a 0.2 µm Al filter (Lebow, 20–76 eV transmission) removes any residual NIR light. The transmitted XUV spectrum is dispersed by a flat-field grating (01–0464, Hitachi) and recorded by an x-ray CCD camera (Pixis XO 400B, Princeton Instruments). A step size of 300 as was chosen for the delay between the XUV and noncollinear NIR pulses in order to resolve time dynamics on the few-femtosecond timescale. At each delay, 1500 laser pulses were accumulated three times to obtain an appropriate signal-to-noise ratio. The delay at which the transient 2s+ zeroth grating order feature at 22.1 eV obtains its peak absorbance is assigned a value of 0 fs (Supplementary Fig. 4). This feature was chosen to avoid contamination from 1s2p pulse propagation effects. The energy axis of the CCD camera was calibrated daily using atomic transition line data available from the National Institute of Standards and Technology (NIST). Camera image data, false color plots, and lineouts of higher-order signals are presented in terms of raw counts (flux) on the CCD camera.
Calculated spatio-spectral profile and time dependence
The coupled TDSE and the Maxwell wave equation are solved numerically in the SAE approximation to generate spatio-spectral profiles in the far field. First, an initial frequency-domain electric field is defined at the beginning of a helium jet by specifying each input field as a focused Gaussian beam. The time-domain transform of this field provides a two-color source term for TDSE, which results in a space- and time-dependent dipole moment that can be transformed back into the frequency domain. The calculations proceed by space-marching the frequency-domain driving and the generated fields in the propagation (z) direction. At each point in z, we transform the total electric field to the time domain and calculate the space- and time-dependent polarization field (proportional to the time-dependent dipole moment) by solving the TDSE at a number of different points across the (transverse, x) laser profile. The polarization field is then transformed to the frequency domain and used as the source term in propagating to the next z-plane. At the end of the medium we transform the resulting space- and frequency-dependent electric field to the far field. The two NIR beams both have Gaussian transverse profiles with waists of 64 µm, and the XUV beam waist is 22 µm. Although a train of two to three attosecond pulses was employed experimentally, the use of single XUV pulse in the calculations is justified given the relative timescales of the pulse train and the NIR pulses. The density of the He gas is 1019 cm−3, and we work in the thin-medium limit and propagate through only 0.01 mm of gas, corresponding to only space-marching one step in the forward direction. This means that we incorporate all aspects of the self-consistent interactions that lead to absorption of the light propagating along the axis, and the emission of the light propagating (diffracting) in off-axis directions, but we ignore effects such as resonant pulse propagation that occur in longer/denser media. In this limit, our medium is equivalent to a 1 mm long gas with a density of 1017 cm−3, almost an order of magnitude lower than the estimated experimental gas density. A lower gas density was chosen for the calculations to demonstrate that the delays observed between nonlinear grating orders in the experimental data do not originate from resonant propagation effects. An increase in gas density is not expected to significantly modify the conclusions drawn here.
The data generated during and analyzed during the current study are available from the corresponding author on reasonable request.
Mukamel, S., Healion, D., Zhang, Y. & Biggs, J. D. Multidimensional attosecond resonant X-ray spectroscopy of molecules: lessons from the optical regime. Annu. Rev. Phys. Chem. 64, 101–127 (2013).
ADS CAS Article Google Scholar
Mukamel, S. Femtosecond optical spectroscopy: a direct look at elementary chemical events. Annu. Rev. Phys. Chem. 41, 647–681 (1990).
ADS MathSciNet CAS Article Google Scholar
Hamm, P. & Zanni, M. T. Concepts and Methods of 2D Infrared Spectroscopy (Cambridge University Press, Cambridge, 2011).
Sakakibara, D. et al. Protein structure determination in living cells by in-cell NMR spectroscopy. Nature 458, 102–105 (2009).
Dantus, M. Coherent nonlinear spectroscopy: from femtosecond dynamics to control. Annu. Rev. Phys. Chem. 52, 639–679 (2001).
Zewail, A. H. Femtochemistry: recent progress in studies of dynamics and control of reactions and their transition states. J. Phys. Chem. 100, 12701–12724 (1996).
Engel, G. S. et al. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 446, 782–786 (2007).
Lange, C. et al. Ultrafast nonlinear optical response of photoexcited Ge/SiGe quantum wells: evidence for a femtosecond transient population inversion. Phys. Rev. B Condens. Matter Mater. Phys. 79, 201306 (2009).
Krausz, F. & Ivanov, M. Attosecond physics. Rev. Mod. Phys. 81, 163–234 (2009).
Beck, A. R., Neumark, D. M. & Leone, S. R. Probing ultrafast dynamics with attosecond transient absorption. Chem. Phys. Lett. 624, 119–130 (2015).
Bencivenga, F. et al. Four-wave mixing experiments with extreme ultraviolet transient gratings. Nature 520, 205–208 (2015).
Bencivenga, F. et al. Four-wave-mixing experiments with seeded free electron lasers. Faraday Discuss. 194, 283–303 (2016).
Bencivenga, F. et al. Nanoscale dynamics by short-wavelength four wave mixing experiments. New J. Phys. 15, 123023 (2013).
Foglia, L. et al. First evidence of purely extreme-ultraviolet four-wave mixing. Phys. Rev. Lett. 120, 263901 (2018).
Glover, T. E. et al. X-ray and optical wave mixing. Nature 488, 603–608 (2012).
Tobey, R. I. et al. Transient grating measurement of surface acoustic waves in thin metal films with extreme ultraviolet radiation. Appl. Phys. Lett. 89, 091108 (2006).
Sistrunk, E. et al. Broadband extreme ultraviolet probing of transient gratings in vanadium dioxide. Opt. Express 23, 4340–4347 (2015).
Gaumnitz, T. et al. Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver. Opt. Express 25, 27506 (2017).
Ding, T. et al. Time-resolved four-wave-mixing spectroscopy for inner-valence transitions. Opt. Lett. 41, 709 (2016).
Chang, Z., Corkum, P. B. & Leone, S. R. Attosecond optics and technology: progress to date and future prospects [Invited]. J. Opt. Soc. Am. B 33, 1081 (2016).
Corkum, P. B. & Krausz, F. Attosecond science. Nat. Phys. 3, 381–387 (2007).
Chang, Z. Fundamentals of Attosecond Optics (CRC Press, Boca Raton, 2011).
Pfeifer, T., Spielmann, C. & Gerber, G. Femtosecond x-ray science. Rep. Prog. Phys. 69, 443–505 (2006).
Cao, W., Warrick, E. R., Fidler, A., Leone, S. R. & Neumark, D. M. Near-resonant four-wave mixing of attosecond extreme-ultraviolet pulses with near-infrared pulses in neon: detection of electronic coherences. Phys. Rev. A 94, 021802 (2016).
Cao, W., Warrick, E. R., Fidler, A., Neumark, D. M. & Leone, S. R. Noncollinear wave mixing of attosecond XUV and few-cycle optical laser pulses in gas-phase atoms: toward multidimensional spectroscopy involving XUV excitations. Phys. Rev. A At. Mol. Opt. Phys. 94, 053846 (2016).
Cao, W., Warrick, E. R., Fidler, A., Leone, S. R. & Neumark, D. M. Excited-state vibronic wave-packet dynamics in H2 probed by XUV transient four-wave mixing. Phys. Rev. A 97, 023401 (2018).
Warrick, E. R. et al. Multiple pulse coherent dynamics and wave packet control of the N 2 a" 1 Σ g+dark state by attosecond four wave mixing. Faraday Discuss. 212, 157–174 (2018).
Huang, S. et al. Generating single-spike hard X-ray pulses with nonlinear bunch compression in free-electron lasers. Phys. Rev. Lett. 119, 154801 (2017).
Marinelli, A. et al. Experimental demonstration of a single-spike hard-X-ray free-electron laser starting from noise. Appl. Phys. Lett. 111, 151101 (2017).
Brown, E. J., Zhang, Q. & Dantus, M. Femtosecond transient-grating techniques: Population and coherence dynamics involving ground and excited states. J. Chem. Phys. 110, 5772–5788 (1999).
Chen, S. et al. Light-induced states in attosecond transient absorption spectra of laser-dressed helium. Phys. Rev. A At. Mol. Opt. Phys. 86, 063408 (2012).
Bell, M. J., Beck, A. R., Mashiko, H., Neumark, D. M. & Leone, S. R. Intensity dependence of light-induced states in transient absorption of laser-dressed helium measured with isolated attosecond pulses. J. Mod. Opt. 60, 1506–1516 (2013).
Reduzzi, M. et al. Polarization control of absorption of virtual dressed states in helium. Phys. Rev. A At. Mol. Opt. Phys. 92, 033498 (2015).
Wu, M., Chen, S., Camp, S., Schafer, K. J. & Gaarde, M. B. Theory of strong-field attosecond transient absorption. J. Phys. B At. Mol. Opt. Phys. 49, 062003 (2016).
Chini, M. et al. Sub-cycle oscillations in virtual states brought to light. Sci. Rep. 3, 1105 (2013).
Foumouo, E., Antoine, P., Bachau, H. & Piraux, B. Attosecond timescale analysis of the dynamics of two-photon double ionization of helium. New J. Phys. 10, 025017 (2008).
Wurth, W. & Menzel, D. Ultrafast electron dynamics at surfaces probed by resonant Auger spectroscopy. Chem. Phys. 251, 141–149 (2000).
Föhlisch, A. et al. Direct observation of electron dynamics in the attosecond domain. Nature 436, 373–376 (2005).
Brühwiler, P. A., Karis, O. & Mårtensson, N. Charge-transfer dynamics studied using resonant core spectroscopies. Rev. Mod. Phys. 74, 703–740 (2002).
Chan, W. F., Cooper, G. & Brion, C. E. Absolute optical oscillator strengths of the electronic excitaiton of atoms at high resolution: experiemental methods and measurements for helium. Phys. Rev. A At. Mol. Opt. Phys. 44, 186–204 (1991).
Pfeiffer, A. N. et al. Alternating absorption features during attosecond-pulse propagation in a laser-controlled gaseous medium. Phys. Rev. A At. Mol. Opt. Phys. 88, 051402(R) (2013).
Liao, C. T., Sandhu, A., Camp, S., Schafer, K. J. & Gaarde, M. B. Beyond the single-atom response in absorption line shapes: probing a dense, laser-dressed helium gas with attosecond pulse trains. Phys. Rev. Lett. 114, 143002 (2015).
Bengtsson, S. et al. Space–time control of free induction decay in the extreme ultraviolet. Nat. Photonics 11, 252–258 (2017).
Mukamel, S. Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, Oxford, 1995).
Strasser, D. et al. Coherent interaction of femtosecond extreme-uv light with He atoms. Phys. Rev. A At. Mol. Opt. Phys. 73, 021805(R) (2006).
Gaarde, M. B., Buth, C., Tate, J. L. & Schafer, K. J. Transient absorption and reshaping of ultrafast XUV light by laser-dressed helium. Phys. Rev. A At. Mol. Opt. Phys. 83, 013419 (2011).
Farrell, J. P. et al. Strongly dispersive transient Bragg grating for high harmonics. Opt. Lett. 35, 2028–2030 (2010).
Herrmann, J. et al. Multiphoton transitions for robust delay-zero calibration in attosecond transient absorption. Springer Proc. Phys. 162, 83–86 (2015).
This work was supported by the Director, Office of Science, Office of Basic Energy Sciences through the Atomic, Molecular, and Optical Sciences Program of the Division of Chemical Sciences, Geosciences, and Biosciences of the US Department of Energy at LBNL under contract no. DE-AC02-05CH11231 and at LSU under contract no. DE-SC0010431. A.P.F. acknowledges funding from the National Science Foundation Graduate Research Fellowship Program. Portions of this research were conducted with high-performance computing resources provided by Louisiana State University (LSU HPC) and the Louisiana Optical Network Initiative (LONI).
Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA
Ashley P. Fidler, Erika R. Warrick, Etienne Bloch, Hugo J. B. Marroux, Daniel M. Neumark & Stephen R. Leone
Department of Chemistry, University of California, Berkeley, Berkeley, CA, 94720, USA
Ashley P. Fidler, Erika R. Warrick, Hugo J. B. Marroux, Daniel M. Neumark & Stephen R. Leone
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, 70803, USA
Seth J. Camp, Kenneth J. Schafer & Mette B. Gaarde
Department of Physics, University of California, Berkeley, Berkeley, CA, 94720, USA
Stephen R. Leone
Ashley P. Fidler
Seth J. Camp
Erika R. Warrick
Etienne Bloch
Hugo J. B. Marroux
Daniel M. Neumark
Kenneth J. Schafer
Mette B. Gaarde
A.P.F., E.R.W., and E.B. designed and built the experimental apparatus. A.P.F., E.R.W., E.B., and H.J.B.M. conducted the experiments and examined the collected data. S.R.L. and D.M.N. supervised the experiments and provided guidance as necessary. S.J.C., M.B.G., and K.J.S. performed the calculations and developed the model to aid in the interpretation of experimental results. A.P.F. wrote the manuscript. M.B.G. wrote major components of Supplementary Notes 2 and 7. All authors contributed to the discussion of results and commented on the manuscript.
Correspondence to Stephen R. Leone.
Journal peer review information: Nature Communications thanks Filippo Bencivenga and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Fidler, A.P., Camp, S.J., Warrick, E.R. et al. Nonlinear XUV signal generation probed by transient grating spectroscopy with attosecond pulses. Nat Commun 10, 1384 (2019). https://doi.org/10.1038/s41467-019-09317-4
Received: 13 January 2019
Non-linear self-driven spectral tuning of Extreme Ultraviolet Femtosecond Pulses in monoatomic materials
Carino Ferrante
Emiliano Principi
Tullio Scopigno
Light: Science & Applications (2021)
Extreme-ultraviolet spectral compression by four-wave mixing
L. Drescher
O. Kornilov
B. Schütte
Nature Photonics (2021) | CommonCrawl |
A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm | springerprofessional.de Skip to main content
PDF-Version jetzt herunterladen
vorheriger Artikel Automated Reasoning with Power Maps
nächster Artikel Formalizing the Cox–Ross–Rubinstein Pricing of ...
PatentFit aktivieren
Weitere Artikel dieser Ausgabe durch Wischen aufrufen
Tipp schließen
17.06.2019 | Ausgabe 4/2020 Open Access
A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm
Journal of Automated Reasoning > Ausgabe 4/2020
Jose Divasón, Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada
» Zur Zusammenfassung PDF-Version jetzt herunterladen
This research was supported by the Austrian Science Fund (FWF) Project Y757. Jose Divasón is partially funded by the Spanish Projects MTM2014-54151-P and MTM2017-88804-P. Sebastiaan is now working at University of Twente, The Netherlands, and supported by the NWO VICI 639.023.710 Mercedes project. Akihisa is now working at National Institute of Informatics, Japan, and supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST. The authors are listed in alphabetical order regardless of individual contributions or seniority.
Modern algorithms to factor univariate integer polynomials—following Berlekamp and Zassenhaus—first preprocesses the input polynomial to extract the content and detect duplicate factors. Afterwards, the main task is to factor primitive square-free integer polynomials, first over prime fields \(\mathrm {GF}(p)\), then over quotient rings \({\mathbb {Z}}/{p^k}{\mathbb {Z}}\), and finally over integers \({\mathbb {Z}}\) [ 5 , 8 ]. Algorithm 1 illustrates the basic structure of such a method for factoring polynomials. 1
In earlier work on algebraic numbers [ 31 ] we implemented Algorithm 1 in Isabelle/HOL [ 29 ]. There, however, the algorithm was not formally proven correct and thus followed by certification, i.e., a validity check on the result factorization. Moreover, there was no guarantee on the irreducibility of resulting factors. During our formalization we indeed found an error in the implementation of Line 7 of this earlier work. Since in several experiments with algebraic numbers this error was not exposed, this clearly shows the advantage of verification over certification.
In this work we fully formalize the correctness of our implementation. It delivers a factorization into the content and a list of irreducible factors.
(Factorization of Univariate Integer Polynomials)
Here, means that \(f = c \cdot f_1^{m_1+1} \cdot \ldots \cdot f_n^{m_n+1}\), c is a constant, each \(f_i\) is square-free, and \(f_i\) and \(f_j\) are coprime whenever \(i \ne j\).
To obtain Theorem 1 we perform the following tasks.
In Sect. 3 we introduce three Isabelle/HOL definitions of \({\mathbb {Z}}/{m}{\mathbb {Z}} \) and \(\mathrm {GF}(p)\). We first define a type to represent these domains, which allows us to reuse many algorithms for rings and fields from the Isabelle distribution and the AFP (Archive of Formal Proofs). At some points in our development, however, the type-based setting becomes too restrictive. Hence we also introduce the second integer representation, which explicitly applies the remainder operation modulo m. For efficient implementation we also introduce the third representation, which allows us to employ machine integers [ 24 ] for reasonably small m. Between the representations we transform statements using transfer [ 15 ] and local type definitions [ 21 ].
The first part of the algorithm is square-free factorization over integer polynomials. In Sect. 4 we adapt Yun's square-free factorization algorithm [ 32 , 35 ] from \({\mathbb {Q}}\) to \({\mathbb {Z}}\).
The prime p in step 5 must be chosen so that \(f_i\) remains square-free in \(\mathrm {GF}(p)\). Therefore, in Sect. 5 we prove the crucial property that such a prime always exists.
In Sect. 6, we formalize Berlekamp's algorithm, which factors polynomials over prime fields, using the type-based representation. Since Isabelle's code generation does not work for the type-based representation of prime fields, we follow the steps presented in Sect. 3 to define a record-based implementation of Berlekamp's algorithm and prove its soundness.
In Sect. 7 we formalize Mignotte's factor bound and Graeffe's transformation used in step 7, where we need to find bounds on the coefficients and degrees of the factors of a polynomial. During this formalization task we detected a bug in our previous oracle implementation, which computed improper bounds on the degrees of factors.
In Sect. 8 we formalize Hensel's algorithm, lifting a factorization modulo p into a factorization modulo \(p^k\). The basic operation there is lifting from \(p^i\) to \(p^{i+1}\), which we formalize in the type-based setting. Unfortunately, iteratively applying this basic operation to lift p to \(p^k\) cannot be done in the type-based setting. Therefore, we remodel the Hensel lifting using the integer representation. We moreover formalize the quadratic Hensel lifting and consider several approaches to efficiently lift p to \(p^k\).
Details on step 9 are provided in Sect. 9 where we closely follow the brute-force algorithm as it is described by Knuth [ 18 , p. 452]. Here, we use the same representation of polynomials over \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) as for the Hensel lifting.
In Sect. 10 we illustrate how to assemble all the previous results in order to obtain the verified algorithm. This includes some optimizations for improving the runtime of the algorithm, such as the use of reciprocal polynomials and Karatsuba's multiplication algorithm.
Finally, we compare the efficiency of our factorization algorithm with the one in Mathematica 11.2 [ 34 ] in Sect. 11 and give a summary in Sect. 12.
Since the soundness of each sub-algorithm has been formalized separately, our formalization is easily reusable for other related verification tasks. For instance, the polynomial-time factorization algorithm of Lenstra et al. [ 23 ] has been verified [ 11 ], and that formalization could directly use the results about steps 4–8 of Algorithm 1 from this paper without requiring any adaptations.
Our formalization is available in the AFP. The following website links theorems in this paper to the Isabelle sources. Moreover, it provides details on the experiments.
https://doi.org/10.5281/zenodo.2525350
The formalization as described in this paper corresponds to the AFP 2019 version which compiles with the Isabelle 2019 release.
1.1 Related Work
To our knowledge, the current work provides the first formalization of a modern factorization algorithm based on Berlekamp's algorithm. Indeed, it is reported that there is no formalization of an efficient factorization algorithm over \(\mathrm {GF}(p)\) available in Coq [ 4 , Sect. 6, note 3 on formalization].
Kobayashi et al. [ 19 ] provide an Isabelle formalization of Hensel's lemma. They define the valuations of polynomials via Cauchy sequences, and use this setup to prove the lemma. Consequently, their result requires a 'valuation ring' as a precondition in their formalization. While this extra precondition is theoretically met in our setting, we did not attempt to reuse their results, because the type of polynomials in their formalization (from HOL-Algebra) differs from the polynomials in our development (from HOL/Library). Instead, we formalize a direct proof for Hensel's lemma. The two formalizations are incomparable: On the one hand, Kobayashi et al. did not restrict to integer polynomials as we do. On the other hand, we additionally formalize the quadratic Hensel lifting [ 36 ], extend the lifting from binary to n-ary factorizations, and prove a uniqueness result, which is required for proving Theorem 1. A Coq formalization of Hensel's lemma is also available. It is used for certifying integral roots and 'hardest-to-round computation' [ 26 ].
If one is interested in certifying a factorization, rather than in a certified algorithm that performs it, it suffices to test that all the found factors are irreducible. Kirkels [ 17 ] formalized a sufficient criterion for this test in Coq: when a polynomial is irreducible modulo some prime, it is also irreducible in \({\mathbb {Z}}\). These formalizations are in Coq, and we did not attempt to reuse them, in particular since there are infinitely many irreducible polynomials which are reducible modulo every prime.
This work is a revised and extended version of our previous conference paper [ 10 ]. The formalization has been improved by adding over 7000 lines of new material, which are detailed through different sections of this paper. This new material has been developed to improve the performance of the verified factorization algorithm and includes among others:
Integration of unsigned-32/64-bit integer implementation, cf. Sect. 3.
Formalization of distinct-degree factorization and integration of it as an optional preprocessing step for Berlekamp's factorization, cf. Sect. 6.3.
Integration of Graeffe's transformation for tighter factor bounds, cf. Sect. 7.
Formalization of a fast logarithm algorithm, required for Graeffe's transformation, cf. Sect. 7.
Formalization of balanced multifactor Hensel lifting based on factor trees, cf. Sect. 8.
Formalization of Karatsuba's polynomial multiplication algorithm, cf. Sect. 10.
Improvements on the GCD algorithm for integer polynomials, cf. Sect. 10.
Integration of reciprocal polynomial before factoring, cf. Sect. 10.
Overall, the runtime of our verified factorization algorithm has improved significantly. The new implementation is more than 4.5 times faster than the previous version [ 10 ] when factoring 400 random polynomials, and the new version is only 2.5 times slower than Mathematica's factorization algorithm.
2 Preliminaries
Our formalization is based on Isabelle/HOL. We state theorems, as well as certain definitions, following Isabelle's syntax. For instance, is the ring homomorphism from integers to type \(\alpha \), which is of class . Isabelle's type classes are similar to Haskell; a type class is defined by a collection of operators (over a single type variable \(\alpha \)) and premises over them. The type class is provided by the HOL library, representing the algebraic structure of ring with a multiplicative unit. We also often use the extension of the above function to polynomials, denoted by . Isabelle's keywords are written in . Other symbols are either clear from their notation, or defined on their appearance. We only assume the HOL axioms and local type definitions, and ensure that Isabelle can build our theories. Consequently, a sceptical reader that trusts the soundness of Isabelle/HOL only needs to validate the definitions, as the proofs are checked by Isabelle.
We also expect basic familiarity with algebra, and use some of its standard notions without further explanation. The notion of polynomial in this paper always means univariate polynomial. Concerning notation, we write for the leading coefficient of a polynomial f and \(\mathsf {res}_{}(f,g)\) for the resultant of f and another polynomial g.
The derivative of a polynomial \(f = \sum _{i=0}^n a_i x^i\) is \(f' = \sum _{i=1}^n i a_i x^{i-1}\). A factorization of a polynomial f is a decomposition into irreducible factors \(f_1,\ldots ,f_n\) such that \(f = f_1 \cdot \ldots \cdot f_n\). The irreducibility of a ring element x is defined via divisibility (denoted by the binary relation following Isabelle):
We also define the degree-based irreducibility of a polynomial f as
Note that ( 1) and ( 2) are not equivalent on integer polynomials; e.g., a factorization of \(f = 10x^2-10\) in terms of ( 1) will be \(f = 2 \cdot 5 \cdot (x-1) \cdot (x+1)\), where the prime factorization of the content, i.e., the GCD of the coefficients, has to be performed. In contrast, ( 2) does not demand a prime factorization, and a factorization may be \(f = (10x-10)\cdot (x+1)\). Note that definitions ( 1) and ( 2) are equivalent on primitive polynomials, i.e., polynomials whose contents are 1, and in particular for field polynomials.
In a similar way to irreducibility w.r.t. ( 2), we also define that a polynomial f is square-free if there does not exist a polynomial g of non-zero degree such that \(g^2\) divides f. In particular, the integer polynomial \(2^2 x\) is square-free. A polynomial f is separable if f and its derivative \(f'\) are coprime. Every separable polynomial is square-free, and in fields of characteristic zero, also the converse direction holds.
3 Formalizing Prime Fields
Our development requires several algorithms that work in the quotient ring \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) and the prime field \(\mathrm {GF}(p)\). Hence, we will need a formalization of these fundamental structures.
We will illustrate and motivate different representations of these structures with the help of a heuristic to ensure that two integer polynomials f and g are coprime [ 18 , p. 453ff]: If f and g are already coprime in \(\mathrm {GF}(p)[x]\) then f and g are coprime over the integers, too. In particular if f and its derivative \(f'\) are coprime in \(\mathrm {GF}(p)[x]\), i.e., f is separable modulo p, then f is separable and square-free over the integers. Hence, one can test whether f is separable modulo p for a few primes p, as a quick sufficient criterion to ensure square-freeness.
The informal proof of the heuristic is quite simple and we will discuss its formal proof in separate sections.
If f is separable modulo p, then f is square-free modulo p (Sect. 3.1).
If f is square-free modulo p then f is square-free in \({\mathbb {Z}}[x]\), provided that and p are coprime (Sect. 3.2).
Testing separability (i.e., coprimality) modulo p is implemented via the Euclidean algorithm in the ring \(\mathrm {GF}(p)[x]\) (Sect. 3.3).
Moreover, we will describe the connection of the separate steps, which is nontrivial since these steps use different representations (Sect. 3.4).
3.1 Type-Based Representation
The type system of Isabelle/HOL allows concise theorem statements and good support for proof automation [ 21 ]. In our example, we formalize the first part of the proof of the heuristic conveniently in a type-based setting for arbitrary fields, which are represented by a type variable \(\tau \) with sort constraint . All the required notions like separability, coprimality, derivatives and square-freeness are implicitly parametrized by the type.
In order to apply Lemma 1 to a finite field \(\mathrm {GF}(p)\) we need a type that represents \(\mathrm {GF}(p)\). To this end, we first define a type to represent \({\mathbb {Z}}/{p}{\mathbb {Z}} \) for an arbitrary \(p>0\), which forms the prime field \(\mathrm {GF}(p)\) when p is a prime. Afterwards we can instantiate the lemma, as well as polymorphic functions that are available for , e.g., the Gauss–Jordan elimination, GCD computation for polynomials, etc.
Since Isabelle does not support dependent types, we cannot directly use the term variable p in a type definition. To overcome the problem, we reuse the idea of the vector representation in HOL analysis [ 13 ]: types can encode natural numbers. We encode p as , i.e., the cardinality of the universe of a (finite) type represented by a type variable \(\alpha \). The keyword introduces a new type whose elements are isomorphic to a given set, along with the corresponding bijections.
Given a finite type \(\alpha \) with p elements, is a type with elements 0, ..., \(p - 1\). With the help of the lifting and transfer package, we naturally define arithmetic in based on integer arithmetic modulo ; for instance, multiplication is defined as follows:
Here the keyword applies the bijections from our type definition via such that is defined on through a definition on the type of the elements of the set used in the , namely natural numbers. It is straightforward to show that forms a commutative ring:
Note that does not assume the existence of the multiplicative unit 1. If , then is not an instance of the type class , for which \(0 \ne 1\) is required. Hence we introduce the following type class:
and derive the following instantiation: 2
Now we enforce the modulus to be a prime number, using the same technique as above, namely introducing a corresponding type class.
The key to being a field is the existence of the multiplicative inverse \(x^{-1}\). This follows from Fermat's little theorem: for any nonzero integer x and prime p,
$$\begin{aligned} x \cdot x^{p-2} \equiv x^{p-1} \equiv 1 \quad (mod p) \end{aligned}$$
that is, if is a prime. The theorem is already available in the Isabelle distribution for the integers, and we just apply the transfer tactic [ 15 ] to lift the result to .
In the rest of the paper, we write instead of . 3
3.2 Integer Representation
The type-based representation becomes inexpressive when, for instance, formalizing a function which searches for a prime modulus p such that a given integer polynomial f is separable modulo p and hence square-free modulo p. Isabelle does not allow us to state this in the type-based representation: there is no existential quantifier on types, so in particular the expression
is not permitted.
Hence we introduce the second representation. This representation simply uses integers (type ) for elements in \({\mathbb {Z}}/{m}{\mathbb {Z}} \) or \(\mathrm {GF}(p)\), and uses for polynomials over them. To conveniently develop formalization we utilize Isabelle's locale mechanism [ 3 ], which allows us to locally declare variables and put assumptions on them in a hierarchical manner. We start with the following locale that fixes the modulus:
For prime fields we additionally assume the modulus to be a prime.
Degrees, divisibility and square-freeness for polynomials modulo m are defined by 4
The integer representations have an advantage that they are more expressive than the typed-based ones. For instance, the soundness statement of the aforementioned function can be stated like " ". Another advantage of the integer representation is that one can easily state theorems which interpret polynomials in different domains like \({\mathbb {Z}}[x]\) and \(\mathrm {GF}(p)[x]\). For instance, the second part of the soundness proof of the heuristic is stated as follows:
Note that there is no type conversion like needed.
A drawback of this integer representation is that many interesting results for rings or fields are only available in the Isabelle library and AFP in type-based forms. To overcome the problem, we establish a connection between the type-based representation and the locale . This is achieved by first introducing the intermediate locale
for \({\mathbb {Z}}/{m}{\mathbb {Z}} \) and its sublocale for prime fields:
Second, we import type-based statements into these intermediate locales by means of transfer [ 15 ]. The mechanism allows us to translate facts proved in one representation into facts in another representation. To apply this machinery we first define the representation relation describing when an integer polynomial represents a polynomial of type . Then we prove a collection of transfer rules, stating the correspondences between basic notions in one representation and those in the other representation. For instance,
relates multiplication of polynomials of type with multiplication of polynomials of type . Concretely, it states that, if polynomials f and g of type are related to polynomials \({{\overline{f}}}\) and \({{\overline{g}}}\) of type respectively (via ), then \(f \cdot g\) is related to \({{\overline{f}}} \cdot {{\overline{g}}}\), again, via . Note that the same syntax is used to represent the polynomial multiplication operation in both worlds ( and ). The symbol represents the relator for function spaces. That is, related functions map related inputs to related outputs. Then facts about rings and fields are available via transfer; e.g., from
of standard library, we obtain
Finally, we migrate Lemma 5 from locale to . It is impossible to declare the former as a sublocale of the latter, since the locale assumption can be satisfied only for certain \(\alpha \). Instead, we see Lemma 5 from the global scope; then the statement is prefixed with assumption . In order to discharge this assumption we use the local type definition mechanism [ 21 ], an extension of HOL that allows us to define types within proofs.
3.3 Record-Based Implementation
The integer representation from the preceding section does not speak about how to implement modular arithmetic. For instance, although Lemma 3 can be interpreted as that one can implement multiplication of polynomials in \({\mathbb {Z}}/{m}{\mathbb {Z}} [x]\) by that over \({\mathbb {Z}}[x]\), there are cleverer implementations that occasionally take remainder modulo m to keep numbers small.
Hence, we introduce another representation.
3.3.1 Abstraction Layer
This third representation introduces an abstraction layer for the implementation of the basic arithmetic in \({\mathbb {Z}}/{m}{\mathbb {Z}} \) and \(\mathrm {GF}(p)\), and builds upon it various algorithms over (polynomials over) \({\mathbb {Z}}/{m}{\mathbb {Z}} \) and \(\mathrm {GF}(p)\). Such algorithms include the computation of GCDs, which is used for the heuristic when checking, for various primes p, whether the polynomial f is separable modulo p, i.e., the GCD of f and \(f'\) in \(\mathrm {GF}(p)[x]\) is 1 or not.
The following datatype, which we call dictionary, encapsulates basic arithmetic operations. Here the type variable \(\rho \) represents Isabelle/HOL's types for executable integers: , , and . 5
Given a dictionary ops, we build more complicated algorithms. For instance, following is the Euclidean algorithm for GCD computation, which is adjusted from the type-based version from the standard library.
Here and often we use [ 20 ], since and others terminate only if ops contains a correct implementation of the basic arithmetic functions. Obviously, these algorithms are sound only if ops is correct. Correct means that the functions zero, plus etc. implement the ring operations and indeed form a euclidean semiring, a ring, or a field, depending on the algorithm in which the operations are used.
So we now consider proving the correctness of derived algorithms, assuming the correctness of ops in form of locales. The following locale assumes that ops is a correct implementation of a commutative ring \(\tau \) using a representation type \(\rho \), where correctness assumptions are formulated in the style of transfer rules, and locale parameter R is the representation relation.
The second assumption just states that the output of the addition operation of the ops record ( ops) is related to the output of the addition operation \((+)\) of elements of type \(\tau \) via R, provided that the input arguments are also related via R.
We need more locales for classes other than . For instance, for the Isabelle/HOL class , which admits the Euclidean algorithm, we need some more operations to be correctly implemented.
In this locale we prove the soundness of , again in form of a transfer rule. The proof is simple since the definition of is a direct translation of the definition of .
For class moreover the inverse operation has to be implemented. Since in our application p is usually small, we compute \(x^{-1}\) as \(x^{p-2}\), using the binary exponentiation algorithm.
3.3.2 Defining Implementations
Here we present three record-based implementations of \(\mathrm {GF}(p)\) using integers, 32-bit integers, and 64-bit integers. This means to instantiate \(\tau \) by , and the representation type \(\rho \) by , , and .
We first define the operations using , which is essentially a direct translation of the definitions in Sect. 3.1. For example, \(x \cdot y\) is implemented as as in , and the inverse of x is computed via \(x^{p-2}\). The soundness of the implementation, stated as follows, is easily proven using the already established soundness proofs for the type-based version.
Hereafter, denotes the dictionary of basic arithmetic operations for \(\mathrm {GF}(p)\) (where the representation type \(\rho \) should be clear), and denotes the representation relation.
The implementations using and have the advantage that generated code will be more efficient as they can be mapped to machine integers [ 24 ]. It should be taken into account that they work only for sufficiently small primes, so that no overflows occur in multiplications: e.g., \(65535 \cdot 65535 < 2^{32}\). The corresponding soundness statements look as follows, and are proven in a straightforward manner using the native words library [ 24 ].
Lemma 10
To obtain an implementation of GCD for polynomials over \(\mathrm {GF}(p)\), we need further work: instantiating \(\tau \) by . So we define a dictionary implementing polynomial arithmetic. Here polynomials are represented by their coefficient lists: the representation relation between and is defined pointwise as follows.
We define by directly translating the implementations of polynomial arithmetic from the standard library; it is thus straightforward to prove the following correctness statement.
Finally we can instantiate Lemma 7 for polynomials as follows.
3.4 Combination of Results
Let us shortly recall what we have achieved at this point. We formalized Lemma 1 in a type-based setting, and the type variable \(\tau \) can be instantiated by the type , where the cardinality of \(\alpha \) encodes the prime p. Moreover, we have a connection between square-freeness in \(\mathrm {GF}(p)[x]\) and \({\mathbb {Z}}[x]\), all represented via integer polynomials in Lemma 2. Finally, we rewrote the type-based GCD-algorithm into a record-based implementation, and we provide three different records that implement basic arithmetic operations in \(\mathrm {GF}(p)\) and \(\mathrm {GF}(p)[x]\).
Let us now assemble all of the results. In the implementation layer we just define a test on separability of f using the existing functions like from the implementation layers. In the following definition, corresponds to the implementation of the one polynomial based on the element provided by the arithmetic operations record.
Since requires as input the polynomial in the internal representation type \(\rho \), we write a wrapper which converts an input integer polynomial into the internal type. Here, heavily relies upon the function from the arithmetic operations record.
The soundness of this function as a criterion for square-freeness modulo p is proven in a locale which combines the locale — ops is a sound implementation of —with the requirement that locale parameter p is equal to the cardinality of \(\alpha \).
The proof goes as follows: Consider the polynomial . The soundness of states that and are related by . In combination with the soundness of (via ) we know that the GCD of g and \(g'\) is 1, i.e., . Then Lemma 1 concludes . Using the premise , we further prove , thus concluding .
Since we are still in a locale that assumes arithmetic operations, we next define a function of type which is outside any locale. It dynamically chooses an implementation of \(\mathrm {GF}(p)\) depending on the size of p.
Although the soundness statement in Lemma 14 is quite similar to the one of Lemma 13, there is a major obstacle in formally proving it in Isabelle/HOL: Lemma 13 was proven in a locale which fixes a type \(\alpha \) such that . In order to discharge this condition we have to prove that such a type \(\alpha \) exists for every . This claim is only provable using the extension of Isabelle that admits local type definitions [ 21 ].
Having proven Lemma 14, which solely speaks about integer polynomials, we can now combine it with Lemma 2 to have a sufficient criterion for integer polynomials to be square free.
The dynamic selection of the implementation of \(\mathrm {GF}(p)\) in —32-bit or 64-bit or arbitrary precision integers—is also integrated in several other algorithms that are presented in this paper. This improves the performance in comparison to a static implementation which always uses arbitrary precision integers, as it was done in our previous version [ 10 ], cf. Sect. 11.
4 Square-Free Factorization of Integer Polynomials
Algorithm 1 takes an arbitrary univariate integer polynomial f as input. As the very first preprocessing step, we extract the content—a trivial task. We then detect and eliminate multiple factors using a square-free factorization algorithm, which is described in this section. As a consequence, the later steps of Algorithm 1 can assume that \(f_i\) is primitive and square-free.
Consider the input polynomial \(48 + 1128 x + 6579 x^2 - 1116 x^3 - 6042 x^4 + 5592 x^5 + 4191 x^6 - 2604 x^7 - 408 x^8 + 1080 x^9 + 300 x^{10}\). In step 4 of Algorithm 1 this polynomial will be decomposed into
$$\begin{aligned} 3 \cdot (\underbrace{4 + 47 x - 2 x^2 - 23 x^3 + 18 x^4 + 10 x^5}_f)^2. \end{aligned}$$
The square-free primitive polynomial f will be further processed by the remaining steps of Algorithm 1 and serves as a running example throughout this paper.
We base our verified square-free factorization algorithm on the formalization [ 32 , Sect. 8] of Yun's algorithm [ 35 ]. Although Yun's algorithm works only for polynomials over fields of characteristic 0, it can be used to assemble a square-free factorization algorithm for integer polynomials with a bit of post-processing and the help of Gauss' Lemma as follows: Interpret the integer polynomial f as a rational one, and invoke Yun's algorithm. This will produce the square-free factorization \(f = \ell \cdot f_{1,{\mathbb {Q}}}^1 \cdot \ldots \cdot f_{n,{\mathbb {Q}}}^n\) over \({\mathbb {Q}}\). Here, \(\ell \) is the leading coefficient of f, and all \(f_{i,{\mathbb {Q}}}\) are monic and square-free. Afterwards eliminate all fractions of each \(f_{i,{\mathbb {Q}}}\) via a multiplication with a suitable constant \(c_i\), i.e., define \(f_{i,{\mathbb {Z}}} := c_i \cdot f_{i,{\mathbb {Q}}}\), such that \(f_{i,{\mathbb {Z}}}\) is primitive. Define \(c:= \ell \div (c_1^1 \cdot \ldots \cdot c_n^n)\). Then \(f = c \cdot f_{1,{\mathbb {Z}}}^1 \ldots \cdot f_{n,{\mathbb {Z}}}^n\) is a square-free factorization of f over the integers, where c is precisely the content of f because of Gauss' Lemma, i.e., in particular \(c \in {\mathbb {Z}}\).
The disadvantage of the above approach to perform square-free factorization over the integers is that Yun's algorithm over \({\mathbb {Q}}\) requires rational arithmetic, where after every arithmetic operation a GCD is computed to reduce fractions. We therefore implement a more efficient version of Yun's algorithm that directly operates on integer polynomials. To be more precise, we adapt certain normalization operations of Yun's algorithm from field polynomials to integer polynomials, and leave the remaining algorithm as it is. For instance, instead of dividing the input field polynomial by its leading coefficient to obtain a monic field polynomial, we now divide the input integer polynomial by its content to obtain a primitive integer polynomial. Similarly, instead of using the GCD for field polynomials, we use the GCD for integer polynomials, etc.
To obtain the soundness of the integer algorithm, we show that all polynomials \(f_{\mathbb {Z}}\) and \(f_{\mathbb {Q}}\) that are constructed during the execution of the two versions of Yun's algorithm on the same input are related by a constant factor. In particular \(f_{i,{\mathbb {Z}}} = c_i \cdot f_{i,{\mathbb {Q}}}\) is satisfied for the final results \(f_{i,{\mathbb {Z}}}\) and \(f_{i,{\mathbb {Q}}}\) of the two algorithms for suitable \(c_i \in {\mathbb {Q}}\). In this way, we show that the outcome of the integer variant of Yun's algorithm directly produces the square-free factorization \(f = c \cdot f_{1,{\mathbb {Z}}}^1 \ldots \cdot f_{n,{\mathbb {Z}}}^n\) from above, so there even is no demand to post-process the result. The combination of the integer version of Yun's algorithm together with the heuristic of Sect. 3 is then used to assemble the function .
(Yun Factorization and Square-Free Heuristic)
5 Square-Free Polynomials in \(\mathrm {GF}(p)\)
Step 5 in Algorithm 1 mentions the selection of a suitable prime p, where two conditions have to be satisfied: First, p must be coprime to the leading coefficient of the input polynomial f. Second, f must be square-free in \(\mathrm {GF}(p)\), required for Berlekamp's algorithm to work. Here, for the second condition we use separability as sufficient criterion to ensure square-freeness.
Continuing Example 1, we need to process the polynomial
$$\begin{aligned} f = 4 + 47x - 2x^2 - 23x^3 + 18x^4 + 10x^5. \end{aligned}$$
Selecting \(p = 2\) or \(p = 5\) is not admissible since these numbers are not coprime to 10, the leading coefficient of f. Also \(p = 3\) is not admissible since the GCD of f and \(f'\) is \(2 + x\) in \(\mathrm {GF}(3)\). Finally, \(p = 7\) is a valid choice since the GCD of f and \(f'\) is 1 in \(\mathrm {GF}(7)\), and 7 and 10 are coprime.
In the formalization we must prove that a suitable prime always exists and provide an algorithm which returns such a prime. Whereas selecting a prime that satisfies the first condition is in principle easy—any prime larger than the leading coefficient will do—it is actually not so easy to formally prove that the second condition is satisfiable. We split the problem of computing a suitable prime into the following steps.
Prove that if f is square-free over the integers, then f is separable (and therefore square-free) modulo p for every sufficiently large prime p.
Develop a prime number generator which returns the first prime such that f is separable modulo p.
The prime number generator lazily generates all primes and aborts as soon as the first suitable prime is detected. This is easy to model in Isabelle by defining the generator ( ) via .
Our formalized proof of the existence of a suitable prime proceeds along the following line. Let f be square-free over \({\mathbb {Z}}\). Then f is also square-free over \({\mathbb {Q}}\) using Gauss' Lemma. For fields of characteristic 0, f is square-free if and only if f is separable. Separability of f, i.e., coprimality of f and \(f'\) is the same as demanding that the resultant is non-zero, so we get \(\mathsf {res}_{}(f,f') \ne 0\). The advantage of using resultants is that they admit the following property: if p is larger than \(\mathsf {res}_{}(f,f')\) and the leading coefficients of f and \(f'\), then \(\mathsf {res}_{p}(f,f')\ne 0\), where \(\mathsf {res}_{p}(f,g)\) denotes the resultant of f and g computed in \(\mathrm {GF}(p)\). Now we go back from resultants to coprimality, and obtain that f and \(f'\) are coprime in \(\mathrm {GF}(p)\), i.e., f is separable modulo p.
Whereas the reasoning above shows that any prime larger than \(\mathsf {res}_{}(f,f')\), and is admitted, we still prefer to search for a small prime p since Berlekamp's algorithm has a worst case lower bound of operations. The formal statement follows:
(Suitable prime)
6 Berlekamp's Algorithm
In this section we will describe step 6 of Algorithm 1, i.e., our verified implementation of Berlekamp's Algorithm to factor square-free polynomials in \(\mathrm {GF}(p)\).
6.1 Informal Description
Algorithm 2 briefly describes Berlekamp's algorithm [ 5 ]. It focuses on the core computations that have to be performed. For a discussion on why these steps are performed we refer to Knuth [ 18 , Sect. 4.6.2].
We illustrate the algorithm by continuing Example 2.
In Algorithm 1, step 6, we have to factor f in \(\mathrm {GF}(7)[x]\). To this end, we first simplify f by
$$\begin{aligned} f \equiv 4 + 5x + 5x^2 + 5x^3 + 4x^4 + 3x^5 \quad (mod 7) \end{aligned}$$
before passing it to Berlekamp's algorithm.
Step 1 now divides this polynomial by its leading coefficient \(c = 3\) in \(\mathrm {GF}(7)\) and obtains the new \(f:= 6 + 4x + 4x^2 + 4x^3 + 6x^4 + x^5\).
Step 2 computes the Berlekamp matrix as
$$\begin{aligned} B_f = \left( \begin{array}{ccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 4 &{} \quad 6 &{}\quad 2 &{}\quad 4 &{} \quad 3 \\ 2 &{}\quad 3 &{}\quad 6 &{}\quad 1 &{}\quad 4 \\ 6 &{}\quad 3 &{}\quad 5 &{}\quad 3 &{}\quad 1 \\ 1 &{}\quad 5 &{}\quad 5 &{}\quad 6 &{}\quad 6 \end{array}\right) \end{aligned}$$
$$\begin{aligned} \begin{array}{rcll} x^0 \mathbin {mod }f &{} \equiv &{} 1 &{}(mod 7) \\ x^7 \mathbin {mod }f &{} \equiv &{} 4 + 6x + 2x^2 + 4x^3 + 3x^4 &{}(mod 7) \\ x^{14} \mathbin {mod }f &{} \equiv &{} 2 + 3x + 6x^2 + x^3 + 4x^4 &{}(mod 7) \\ x^{21} \mathbin {mod }f &{} \equiv &{} 6 + 3x + 5x^2 + 3x^3 + x^4 &{}(mod 7) \\ x^{28} \mathbin {mod }f &{} \equiv &{} 1 + 5x + 5x^2 + 6x^3 + 6x^4 &{}(mod 7). \end{array} \end{aligned}$$
Step 3 computes a basis of the left null space of \(B_f - I\), e.g., by applying the Gauss–Jordan elimination to its transpose \((B_f -I)^\mathrm {T}\):
$$\begin{aligned} \left( \begin{matrix} 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 4&{}\quad 5&{}\quad 2&{}\quad 4&{}\quad 3\\ 2&{}\quad 3&{}\quad 5&{}\quad 1&{}\quad 4\\ 6&{}\quad 3&{}\quad 5&{}\quad 2&{}\quad 1\\ 1&{}\quad 5&{}\quad 5&{}\quad 6&{}\quad 5 \end{matrix}\right) ^{\!\!\mathrm {T}} = \left( \begin{array}{ccccc} 0 &{}\quad 4 &{}\quad 2 &{}\quad 6 &{}\quad 1 \\ 0 &{}\quad 5 &{}\quad 3 &{}\quad 3 &{}\quad 5 \\ 0 &{}\quad 2 &{}\quad 5 &{}\quad 5 &{}\quad 5 \\ 0 &{}\quad 4 &{}\quad 1 &{}\quad 2 &{}\quad 6 \\ 0 &{}\quad 3 &{}\quad 4 &{}\quad 1 &{}\quad 5 \end{array}\right) \hookrightarrow \left( \begin{array}{ccccc} 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 2 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 2 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{array}\right) \end{aligned}$$
We determine \(r = 2\), and extract the basis vectors \(b_1 = (1\ 0\ 0\ 0\ 0)\) and \(b_2 = (0\ 5\ 6\ 5\ 1)\). Step 4 converts them into the polynomials \(h_1 = 1\) and \(h_2 = 5x + 6x^2 + 5x^3 + x^4\), and step 5 initializes \(H = \{h_2\}\), \(F = \{f\}\), and \(F_I = \emptyset \).
The termination condition in step 6 does not hold. So in step 7 we pick \(h = h_2\) and compute the required GCD s.
$$\begin{aligned} \gcd (f, h_2 - 1)&= 6 + 5x + 6x^2 + 5x^3 + x^4 =: f_1\\ \gcd (f, h_2 - 4)&= 1 + x =: f_2\\ \gcd (f, h_2 - i)&= 1 \qquad \qquad \text {for all }i \in \{0,2,3,5,6\} \end{aligned}$$
Afterwards, we update \(F := \{f_1,f_2\}\) and \(H := \emptyset \).
Step 8 is just an optimization. For instance, in our implementation we move all linear polynomials from F into \(F_I\), so that in consecutive iterations they do not have to be tested for further splitting in step 7. Hence, step 8 updates \(F_I := \{f_2\}\), \(F := \{f_1\}\), and \(r := 1\).
Now we go back to step 6, where both termination criteria fire at the same time ( \(|F| = 1 = r \wedge H = \emptyset \)). We return \(c \cdot f_1 \cdot f_2\) as final factorization, i.e.,
$$\begin{aligned} f \equiv 3 \cdot (1 + x) \cdot (6 + 5x + 6x^2 + 5x^3 + x^4) \quad (mod 7) \end{aligned}$$
All of the arithmetic operations in Algorithm 2 have to be performed in the prime field \(\mathrm {GF}(p)\). Hence, in order to implement Berlekamp's algorithm, we basically need the following operations: arithmetic in \(\mathrm {GF}(p)\), polynomials over \(\mathrm {GF}(p)\), the Gauss–Jordan elimination over \(\mathrm {GF}(p)\), and GCD-computation for polynomials over \(\mathrm {GF}(p)\).
6.2 Soundness of Berlekamp's Algorithm
Our soundness proof for Berlekamp's algorithm is mostly based on the description in Knuth's book.
We first formalize the equations (7, 8, 9, 10, 13, 14) in the textbook [ 18 , p. 440 and 441]. To this end, we also adapt existing proofs from the Isabelle distribution and the AFP; for instance, to derive (7) in the textbook, we adapted a formalization of the Chinese remainder theorem, which we could find only for integers and naturals, to be applicable to polynomials over fields. For another example, (13) uses the equality \((f + g)^p = f^p + g^p\) where f and g are polynomials over \(\mathrm {GF}(p)\), which we prove using some properties about binomial coefficients that were missing in the library. Having proved these equations, we eventually show that after step 3 of Algorithm 2, we have a basis \(b_1,\dots ,b_r\) of the left null space of \(B_f - I\).
Now, step 4 transforms the basis into polynomials. We define an isomorphism between the left null space of \(B_f - I\) and the Berlekamp subspace
so that the isomorphism transforms the basis \(b_1,\dots ,b_r\) into a Berlekamp basis \(H_b := \{h_1,\dots ,h_r\}\), a basis of \(W_f\). Then we prove that every factorization of f has at most r factors.
In this proof we do not follow Knuth's arguments, but formalize our own version of the proof to reuse some results which we have already proved in the development. Our proof is based on another isomorphism between the vector spaces \(W_f\) and \(\mathrm {GF}(p)^r\) as well as the use of the Chinese remainder theorem over polynomials and the uniqueness of the solution.
Every factorization of a square-free monic polynomial \(f \in \mathrm {GF}(p)[x]\) has at most \(\dim W_f\) factors.
Let \(f \equiv f_1 \cdot \ldots \cdot f_r \quad (mod p)\) be a monic irreducible factorization in \(\mathrm {GF}(p)[x]\), which exists and is unique up to permutation since \(\mathrm {GF}(p)[x]\) is a unique factorization domain. We show that there exists an isomorphism between the vector spaces \(W_f\) and \(\mathrm {GF}(p)^r\). Then they have the same dimension and thus every factorization of f has at most \(\dim W_f = \dim \mathrm {GF}(p)^r = r\) factors, which is the desired result.
First, the following equation holds for any polynomial \(g \in W_f\). It corresponds to equation (10) in the textbook [ 18 , p. 440].
$$\begin{aligned} g^p - g\ = \prod _{a \in \mathrm {GF}(p)} (g - a). \end{aligned}$$
From this we infer that each \(f_i\) divides \(\prod _{a \in \mathrm {GF}(p)}(g - a)\). Since \(f_i\) is irreducible, \(f_i\) divides \(g-a\) for some \(a \in \mathrm {GF}(p)\) and thus, \((g \mathbin {mod }f_i) = -a\) is a constant.
Now we define the desired isomorphism \(\phi \) between \(W_f\) and \(\mathrm {GF}(p)^r\) as follows:
$$\begin{aligned} \phi :&W_f \rightarrow \mathrm {GF}(p)^r\\&g \mapsto (g \mathbin {mod }f_1, \dots , g \mathbin {mod }f_r) \end{aligned}$$
To show that \(\phi \) is an isomorphism, we start with proving that \(\phi \) is injective. Let us assume that \(\phi \, g = 0\) for some \(g \in W_f\). It is easy to show and \(\forall i<r.\ g \equiv \phi \, g \quad (mod f_i)\). Since \(v = 0 \in W_f\) satisfies these properties, the uniqueness result of the Chinese remainder theorem guarantees that \(g = 0\). This implies the injectivity of \(\phi \), since any linear map is injective if and only if its kernel is {0} [ 2 , Proposition 3.2].
To show that \(\phi \) is surjective, consider an arbitrary \(x = (x_1, \dots , x_r) \in \mathrm {GF}(p)^r\). We show that there exists a polynomial \(g \in W_f\) such that \(\phi \, g = x\). The Chinese remainder theorem guarantees that there exists a polynomial g such that:
$$\begin{aligned}&\forall i<r.\ g \equiv x_i \quad (mod f_i) \end{aligned}$$
Then, for each \(i < r\) we have , and so \(g^{p} \equiv g \quad (mod f_i)\). Since each \(f_i\) is irreducible and f is square-free, we have \(g^p \equiv g \quad (mod \prod f_i)\). As \(\prod f_i = f\), we conclude \(g \in W_f\). Finally, \(\phi \, g = x\) follows from ( 4) and the fact that \(g \mathbin {mod }f_i\) is a constant. \(\square \)
As expected, the proof in Isabelle requires more details and it takes us about 300 lines (excluding any previous necessary result and the proof of the Chinese remainder theorem). We define a function for indexing the factors, we prove that both \(W_f\) and \(\mathrm {GF}(p)^r\) are finite-dimensional vector spaces and also that \(\phi \) is a linear map. Since each equation of the proof involves polynomials over \(\mathrm {GF}(p)\) (so everything is modulo p), we also proved facts like and so on. In addition, we also extend an existing AFP entry [ 22 ] about vector spaces for some necessary results about linear maps, isomorphisms between vector spaces, dimensions, and bases.
Once having proved that \(H_b\) is a Berlekamp basis for f and that the number of irreducible factors is \(|H_b|\), we prove (14); for every divisor \(f_i\) of f and every \(h \in H_b\), we have
$$\begin{aligned} f_i = \prod _{0 \le j < p} \gcd (f_i, h - j). \end{aligned}$$
Finally, it follows that every non-constant reducible divisor \(f_i\) of f can be properly factored by \(\gcd (f_i, h - j)\) for suitable \(h \in H_b\) and \(0 \le j < p\).
In order to prove the soundness of steps 5–9 in Algorithm 2, we use the following invariants—these are not stated by Knuth as equations. Here, \(H_{old }\) represents the set of already processed polynomials of \(H_b\).
\(f = \prod (F \cup F_I)\).
All \(f_i \in F \cup F_I\) are monic and non-constant.
All \(f_i \in F_I\) are irreducible.
\(H_b = H \cup H_{old }\).
\(\gcd (f_i,h - j) \in \{1,f_i\}\) for all \(h \in H_{old }\), \(0 \le j < p\) and \(f_i \in F \cup F_I\).
\(|F_I| + r = |H_b|\).
It is easy to see that all invariants are initially established in step 5 by picking \(H_{old } = \{1\} \cap H_b\). In particular, invariant 5 is satisfied since the GCD of the monic polynomial f and a constant polynomial c is either 1 (if \(c\ne 0\)) or f (if \(c=0\)).
It is also not hard to see that step 7 preserves the invariants. In particular, invariant 5 is satisfied for elements in \(F_I\) since these are irreducible. Invariant 1 follows from ( 14).
The irreducibility of the final factors that are returned in step 6 can be argued as follows. If \(|F| = r\), then by invariant 6 we know that \(|H_b| = |F \cup F_I|\), i.e., \(F \cup F_I\) is a factorization of f with the maximum number of factors, and thus every factor is irreducible. In the other case, \(H = \emptyset \) and hence \(H_{old } = H_b\) by invariant 4. Combining this with invariant 5 shows that every element \(f_i\) in \(F \cup F_I\) cannot be factored by \(\gcd (f_i,h - j)\) for any \(h \in H_b\) and \(0 \le j < p\). Since \(H_b\) is a Berlekamp basis, this means that \(f_i\) must be irreducible.
Putting everything together we arrive at the formalized main soundness statement of Berlekamp's algorithm. As in Sect. 6.3 we will integrate the distinct-degree factorization [ 18 , p. 447 and 448], the algorithm takes, besides the monic polynomial f to be factored, an extra argument \(d \in {\mathbb {N}}\) such that any degree- d factor of f is known to be irreducible. Fixing \(d=1\) yields the usual Berlekamp's algorithm. The final statement looks as follows.
(Berlekamp's Algorithm for monic polynomials)
In order to prove the validity of the output factorization, we basically use the invariants mentioned before. However, it still requires some tedious reasoning.
6.3 Formalizing the Distinct-Degree Factorization Algorithm
The distinct-degree factorization (cf. [ 18 , p. 447 and 448]) is an algorithm that splits a square-free polynomial into (possibly reducible) factors, where irreducible factors of each factor have the same degree. It is commonly used before applying randomized algorithms to factor polynomials, and can also be used as a preprocessing step before Berlekamp's algorithm. Algorithm 3 briefly describes how it works.
We implement the algorithm in Isabelle/HOL as . Termination follows from the fact that difference between d and the degree of v decreases in every iteration. The key to the soundness of the algorithm is the fact that any irreducible polynomial f of degree d divides the polynomial \(x^{p^d} - x\) and does not divide \(x^{p^c}-x\) for \(1 \le c < d\). The corresponding Isabelle's statement looks as follows where the polynomial x is encoded as , i.e., \(1 \cdot x^1\).
Knuth presents such a property as a consequence of an exercise in his book, whose proof is sketched in prose in just 5 lines [ 18 , Exercise 4.6.2.16]. In comparison, our Isabelle proof required more effort: it took us about 730 lines, above all because we proved several facts and subproblems: 6
Given a degree- n irreducible polynomial \(f \in \mathrm {GF}(p)[x]\), the \(p^n\) polynomials of degree less than n form a field under arithmetic modulo f and p.
Any field with \(p^n\) elements has a generator element \(\xi \) such that the elements of the field are \(\{0,1,\xi , \xi ^2, \dots , \xi ^{p^n-2}\}\). We do not follow Knuth's short argument in this step, but we reuse some theorems of the Isabelle library to provide a proof based on the existence of an element in the multiplicative group of the finite field with the adequate order.
Given a degree- n irreducible polynomial \(f \in \mathrm {GF}(p)[x]\), \(x^{p^m} - x\) is divisible by f if and only if m is a multiple of n. Essentially, we are proving that \(\mathrm {GF}(p^n)\) is a subfield of \(\mathrm {GF}(p^m)\) if and only if n divides m.
The difference between the sizes of Knuth's and our proofs is also due to some properties which Knuth leaves as exercises. For instance, we show that \(a^{p^n} = a\) for any element \(a\in \mathrm {GF}(p)\), also that \((f+g)^{p^n} = f^{p^n} + g^{p^n}\) in the ring \(\mathrm {GF}(p)[x]\), for natural numbers \(x>1\), \(a>0\) and \(b>0\) we demonstrate and some other properties like these ones which cause the increase in the number of employed lines. The whole formalization of these facts, the termination-proof of the algorithm and its soundness can be seen in the file Distinct_Degree_Factorization.thy of our development.
Once we have the distinct-degree factorization formalized, it remains to find a way to split each factor that we have found into the desired irreducible factors, but this can just be done by means of the Berlekamp's algorithm. This way, we have two ways of factoring polynomials in \(\mathrm {GF}(p)[x]\):
Using Berlekamp's algorithm directly.
Preprocessing the polynomial using the distinct-degree factorization and then apply Berlekamp's algorithm to the factors.
We verified both variants as a single function where a Boolean constant is used to enable or disable the preprocessing via distinct-degree factorization. Our experiments revealed that currently the preprocessing slows down the factorization algorithm, so the value of the Boolean constant is set to disable the preprocessing. However, since distinct degree factorization heavily depends on polynomial multiplication, the preprocessing might pay off, once more efficient polynomial multiplication algorithms become available in Isabelle.
Independent of the value of the Boolean constant, the final type-based statement for the soundness of is as follows.
(Finite Field Factorization)
Here, converts a list into a multiset, and demands that the given factorization is the unique factorization of f, i.e., c is the leading coefficient of f and \( fs \) a list of irreducible and monic factors such that \(f = c \cdot {\prod \, fs }\). Uniqueness follows from the general theorem that the polynomials over fields form a unique factorization domain.
6.4 Implementing Finite Field Factorization
The soundness of Theorem 4 is formulated in a type-based setting. In particular, the function has type
In our use case, recall that Algorithm 1 first computes a prime number p, and then invokes a factorization algorithm (such as Berlekamp's algorithm) on \(\mathrm {GF}(p)\). This requires Algorithm 1 to construct a new type \(\tau \) with depending on the value of p, and then invoke for type .
Unfortunately, this is not possible in Isabelle/HOL. Hence, Algorithm 1 requires a finite field factorization algorithm to have a type like
where the first argument is the dynamically chosen prime p.
The final goal is to prove Theorem 4 but just involving integers, integer polynomials and integer lists, and then avoiding statements and definitions that require anything of type (or in general, anything involving the type ).
The solution is to follow the steps already detailed in Sect. 3. We briefly recall the main steps here:
We implement a record-based copy of all necessary algorithms like Gauss–Jordan elimination, and where the type-based arithmetic operations are replaced by operations in the record.
In a locale that assumes a sound implementation of the record-based arithmetic and that fixes p such that , we develop transfer rules to relate the new implementation of all subalgorithms that are invoked with the corresponding type-based algorithms.
Out of the locale, we define a function which dynamically selects an efficient implementation of \(\mathrm {GF}(p)\) depending on p, by means of . This function has the desired type. Its soundness statement can be proven by means of the transfer rules, but the resulting theorem still requires that .
Thanks to local type definitions, such a premise is replaced by .
As the approach is the same as the presented in Sect. 3, we omit here the details. We simply remark that the diagnostic commands and were helpful to see why certain transfer rules could initially not be proved automatically; these commands nicely pointed to missing transfer rules.
Most of the transfer rules for non-recursive algorithms were proved mainly by unfolding the definitions and finishing the proof by . For recursive algorithms, we often perform induction via the algorithm. To handle an inductive case, we locally declare transfer rules (obtained from the induction hypothesis), unfold one function application iteration, and then finish the proof by .
Still, problems arose in case of underspecification. For instance it is impossible to prove an unconditional transfer rule for the function that returns the head of a list using the standard relator for lists, ; when the lists of type and are empty, we have to relate with . To circumvent this problem, we had to reprove invariants that is invoked only on non-empty lists.
Similar problems arose when using matrix indices where transfer rules between matrix entries \(A_{ij}\) and \(B_{ij}\) are available only if i and j are within the matrix dimensions. So, again we had to reprove the invariants on valid indices—just unfolding the definition and invoking was not sufficient.
Although there is some overhead in this approach—namely by copying the type-based algorithms into record-based ones, and by proving the transfer rules for each of the algorithms—it still simplifies the overall development: once this setup has been established, we can easily transfer statements about properties of the algorithms, without having to copy or adjust their proofs.
This way, we obtain a formalized and executable factorization algorithm for polynomials in finite fields where the prime number p can be determined at runtime, and where the arithmetic in \(\mathrm {GF}(p)\) is selected dynamically without the risk of integer overflow. The final theorem follows, which is the integer-based version of Theorem 4.
(Finite Field Factorization on Integers)
In summary, the development of the separate implementation is some annoying overhead, but still a workable solution. In numbers: Theorem 4 requires around 4300 lines of difficult proofs whereas Theorem 5 demands around 600 lines of easy proofs.
7 Mignotte's Factor Bound
Reconstructing the polynomials proceeds by obtaining factors modulo \(p^k\). The value of k should be large enough, so that any coefficient of any factor of the original integer polynomial can be determined from the corresponding coefficients in \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \). We can find such k by finding a bound on the coefficients of the factors of f, i.e., a function such that the following statement holds:
(Factor Bound)
Clearly, if b is a bound on the absolute value of the coefficients, and \(p^k > 2\cdot b\) then we can encode all required coefficients: In \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) we can represent the numbers \(\{-\lfloor \frac{p^k-1}{2}\rfloor ,\dots ,{\lceil }{\frac{p^k-1}{2}}{\rceil }\} \supseteq \{-b,\dots ,b\}\).
The Mignotte bound [ 27 ] provides a bound on the absolute values of the coefficients. The Mignotte bound is obtained by relating the Mahler measure of a polynomial to its coefficients. The Mahler measure is defined as follows:
where and \(r_1, \ldots , r_n\) are the complex roots of f taking multiplicity into account. For nonzero f, is a nonzero integer. It follows that . The equality easily follows by the definition of the Mahler measure. We conclude that if g is a factor of f.
The Mahler measure is bounded by the coefficients from above through Landau's inequality:
Mignotte showed that the coefficients also bound the measure from below: whenever . Putting this together we get:
Consequently, we could define as follows:
Such a definition of was the one used in our previous work [ 10 ]. However, we have introduced an important improvement at this point to get tighter factor bounds by means of integrating Graeffe transformations.
Given a complex polynomial \(f = c \prod _i (x-r_i)\), we can define its m-th Graeffe transformation as the polynomial \(f_m = c^{2^m} \prod _{i} (f-r_i^{2^m})\).
These polynomials are easy to compute, since
$$\begin{aligned} f_m={\left\{ \begin{array}{ll} f, &{} \text {if }m=0.\\ c \cdot (g^2-xh^2), &{} \text {otherwise}\\ \end{array}\right. } \end{aligned}$$
where g and h are the polynomials that separates \(f_{m-1}\) into its even and odd parts such that \(f_{m-1}(x) = g(x^2)+xh(x^2)\). For instance, if \(f_{m-1} = ax^4+bx^3+cx^2+dx+e\) then \(g=ax^2 + cx + e\) and \(h = bx+d\).
We implement both the definition of Graeffe transformation and ( 5) and then we show they are equivalent. The former one makes proofs easier, whereas the latter one is devoted for computational purposes and thus used during code generation. At this point we introduce functions involving lists, e.g. (to obtain the odd and even parts of a polynomial) and (to split a list into another two ones in which elements are alternated). For a polynomial f of degree n, we then prove three important facts:
The first one follows from the definition of Mahler measure and Graeffe transformation, the second one follows from the first property and the Landau's inequality and the third one is obtained from the definition of Mahler measure and the Mignotte's inequality.
The implementation of an approximation for the Mahler measure based on Graeffe transformations requires the computation of n-th roots, which already can be done thanks to previous work based on the Babylonian method [ 30 ]. That work implements functions to decide whether \(\root n \of {a} \in {\mathbb {Q}}\) and compute the ceiling and floor of \(\root n \of {a}\). The computation of the n-th root of a number is based on a variant of Newton iteration, but involving integer divisions instead of floating point or rational divisions, i.e., each occurrence of in the algorithm has been substituted by . We must also choose a starting value in the iteration, which must be larger than the n-th root. This property is essential, since the algorithm will abort as soon as we fall below the n-th root. Thus, the starting value is defined as \(2 ^ {\lceil \lceil \log _2 {a}\rceil / n\rceil }\).
This of course requires a function to approximate logarithms. At first, the development [ 30 ] implemented this approximation in a naive way, i.e., multiplying the base until we exceed the argument, which causes an impact on the efficiency and avoid an improvement on the performance if Graeffe transformations are integrated.
To tackle this, we implement the discrete logarithm function in a manner similar to a repeated squaring exponentiation algorithm. This way, we get a fast logarithm algorithm, as required for Graeffe transformations. This algorithm allows us to derive the floor- and ceiling-logarithm functions. We also connect them to the function working on real numbers.
Once we have a fast logarithm algorithm implemented, we can now define a function which returns an upper bound for the Mahler measure, based on the Graeffe transformations. We refer to the sources and [ 9 ] for the details of the implementation. The function receives three parameters: the number m of Graeffe transformations which are performed, a scalar c and the polynomial f. Using the previous properties, we can now prove the following important fact:
Putting all together, for a polynomial g of we have:
Consequently, we can define based on , but firstly it remains to decide the number of iterations (the value of m), in a balance between the precision of the bound and the computational time needed to get it. First we tried too high numbers which gave good results for small polynomials but have been too expensive to compute for larger polynomials, i.e., where the factor-bound computation resulted in a timeout. After some experiments we finally selected a value of \(m = 2\) and defined in Isabelle as follows, which is a function that satisfies the statement presented at the beginning of this section:
For \(m = 2\) we get quite some decrease in the estimation of the Mahler measure. Let us show two examples of it. Consider the polynomials \(f = x^8+ 8x^7+ 47x^6+ 136x^5+ 285x^4+ 171x^3-20x^2-21x+ 2\) and \(g=2x^8-16x^7+ 26x^6-10x^5-41x^4+ 89x^3-87x^2+ 52x-10\) that appear in [ 1 , Sects. 3.6.1 and 3.6.2].
The paper estimates a Mahler measure of 197 for f and 33.4 for g, Our results are presented in Table 1. They clearly illustrate an improved precision when applying Graeffe's transformation a few times.
Interestingly, even with the slightly worse estimation of 200 for f when \(m=2\), we result in better factor bounds: they report 1181 and 200 for the largest coefficient for a factor of degree 4 of f and g, respectively, whereas our results in 604 and .
Approximating the Mahler measure of the polynomials f and g
So in both cases, the Mahler measure estimations are close to the ones in [ 1 ] (with \(m = 2\)), but we manage to get much smaller coefficient bounds via the Mignotte bound (roughly a factor of 2).
In order to compute a factor bound via Theorem 18 it remains to choose a bound d on the degrees of factors of f that we require for reconstruction. A simple choice is , but we can do slightly better. After having computed the Berlekamp factorization, we know the degrees of the factors of f in \(\mathrm {GF}(p)\). Since the degrees will not be changed by the Hensel lifting, we also know the degrees of the polynomials \(h_i\) in step 8 of Algorithm 1.
Since in step 9 of Algorithm 1 we will combine at most half of the factors, it suffices to take , where we assume that the sequence \(h_1,\ldots ,h_m\) is sorted by degree, starting with the smallest. In the formalization this gives rise to the following definition:
Note also that in the reconstruction step we actually compute factors of . Thus, we have to multiply the factor bound for f by .
At the end of Example 3 we have the factorization \(f = 4 + 47x - 2x^2 - 23x^3 + 18x^4 + 10x^5 \equiv 3 \cdot (1 + x) \cdot (6 + 5x + 6x^2 + 5x^3 + x^4) \quad (mod 7)\).
We compute . With the bound used in our previous work [ 10 ], we have to be able to represent coefficients of at most \(10 \cdot \lfloor \sqrt{{\left( {\begin{array}{c}4\\ 2\end{array}}\right) }^2 \cdot (4^2 + 47^2 + 2^2 + 23^2 + 18^2 + 10^2)}\rfloor = 3380\), i.e., the numbers \(\{-3380, \ldots , 3380\}\). In contrast, using the new estimations we can reduce the bound, and compute that it suffices to represent coefficients of at most 1730. Thus the modulus has to be larger than \(2 \cdot 1730 = 3460\). Hence, in step 7 of Algorithm 1 we choose \(k = 5\), since this is the least number k such that \(p^k = 7^k > 3460\).
Finally, we report that our previous oracle implementation [ 31 , Sect. 4] had a flaw in the computation of a suitable degree bound d, since it just defined d to be the half of the degree of f. This choice might be insufficient: 7 Consider the list of degree of the \(h_i\) to be [1, 1, 1, 1, 1, 5]. Then the product \(h_1 \cdot h_6\) of degree 6 might be a factor of f, but the degree bound in the old implementation was computed as \(\frac{1+1+1+1+1+5}{2} = 5\), excluding this product. This wrong choice of d was detected only after starting to formalize the required degree bound.
8 Hensel Lifting
Given a factorization in \(\mathrm {GF}(p)[x]\):
which Berlekamp's algorithm provides, the task of the Hensel lifting is to compute a factorization in \({\mathbb {Z}}/{p^k}{\mathbb {Z}} [x]\)
Hensel's lemma, following Miola and Yun [ 28 ], is stated as follows.
(Hensel) Consider polynomials f over \({\mathbb {Z}}\), \(g_1\) and \(h_1\) over \(\mathrm {GF}(p)\) for a prime p, such that \(g_1\) is monic and \(f \equiv g_1\cdot h_1 \quad (mod p)\). For any \(k \ge 1\), there exist polynomials \(g_k\) and \(h_k\) over \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) such that \(g_k\) is monic, \(f \equiv g_k \cdot h_k\, (mod p^k)\), \(g_k \equiv g_1 \quad (mod p)\), \(h_k \equiv h_1 \quad (mod p)\). Moreover, if f is monic, then \(g_k\) and \(h_k\) are unique (mod \({p^k}\)).
The lemma is proved inductively on k where there is a one step lifting from \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) to \({\mathbb {Z}}/{p^{k+1}}{\mathbb {Z}} \). To be more precise, the one step lifting assumes polynomials \(g_k\) and \(h_k\) over \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) satisfying the conditions, and computes the desired \(g_{k+1}\) and \(h_{k+1}\) over \({\mathbb {Z}}/{p^{k+1}}{\mathbb {Z}} \).
As explained in Sect. 3, it is preferable to carry on the proof in the type-based setting whenever possible, and indeed we proved the one-step lifting in this way.
(Hensel lifting–one step)
Here, represents \(p^{k+1}\), represents p, and represents \(p^k\). The prefix " \(\#\)" denotes the function that converts polynomials over integer modulo m into those over integer modulo n, where the type inference determines n.
Unfortunately, we could not see how to use Lemma 21 in the inductive proof of Lemma 20 in a type-based setting. A type-based statement of Lemma 20 would have an assumption like . Then the induction hypothesis would look like
and the goal statement would be . There is no hope to be able to apply the induction hypothesis ( 6) for this goal, since the assumptions are clearly incompatible. A solution to this problem seems to require extending the induction scheme to admit changing the type variables, and produce an induction hypothesis like where \(?\alpha \) can be instantiated. Unfortunately this is not possible in Isabelle/ HOL. A rule that seems useful for this problem is the cross-type induction schema [ 6 ], which is a general-purpose axiom for cross-type well-founded induction and recursion. However, it is not admissible in current HOL.
We therefore formalized most of the reasoning for Hensel's lemma on integer polynomials in the integer-based setting (cf. Sect. 3.2), so that the modulus (the k in the \(p^k\)) can be easily altered within algorithms and inductive proofs. 8 The binary version of Hensel's lemma is formalized as follows, and internally one step of the Hensel lifting is applied over and over again, i.e., the exponents are p, \(p^2\), \(p^3\), \(p^4\), ... [ 28 , Sect. 2.2]. In the statement, Isabelle's syntax \(\exists !\) represents the unique existential quantification.
(Hensel lifting–multiple steps, binary)
It is also possible to lift in one step from \(p^k\) to \(p^{2k}\), which is called the quadratic Hensel lifting, cf. [ 28 , Sect. 2.3]. In this paper we consider several combinations of one-step and quadratic Hensel lifting.
In the following we use the symbols \(\rightarrow \), \(\Rightarrow \), and \(\searrow \) to indicate a one-step Hensel lifting step, a quadratic Hensel lifting step, and the operation which decreases the modulus from \(p^{i+j}\) to \(p^i\), respectively. For each alternative, we immediately illustrate the sequence of operations that are performed to produce a factorization modulo \(p^{51}\).
Repeated one-step lifting:
$$\begin{aligned} p^1 \rightarrow p^2 \rightarrow p^3 \rightarrow \dots \rightarrow p^{51} \end{aligned}$$
Repeated quadratic lifting [ 28 , Sect. 2.3], which applies the quadratic Hensel lifting until \(p^{2^\ell } \ge k\) and then finally take remainder operation modulo \(p^k\) in order to convert the \({\mathbb {Z}}/{p^{2^\ell }}{\mathbb {Z}} \) factorization into a \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) factorization. Hence, the operations for \(k = 51\) are:
$$\begin{aligned} p^1 \Rightarrow p^2 \Rightarrow p^4 \Rightarrow p^8 \Rightarrow p^{16} \Rightarrow p^{32} \Rightarrow p^{64} \searrow p^{51} \end{aligned}$$
Combination of one-step and quadratic liftings. Lifting to \(p^k\) proceeds by recursively computing the lifting to \(p^{\lfloor \frac{k}{2} \rfloor }\), then perform a quadratic Hensel lifting to \(p^{2 \cdot \lfloor \frac{k}{2} \rfloor }\), and if k is odd, do a final linear Hensel lifting to \(p^k\). Hence, the operations are:
$$\begin{aligned} p^1 \Rightarrow p^2 \rightarrow p^3 \Rightarrow p^6 \Rightarrow p^{12} \Rightarrow p^{24} \rightarrow p^{25} \Rightarrow p^{50} \rightarrow p^{51} \end{aligned}$$
Although the numbers stay smaller than in the second approach, this approach has the disadvantage that the total number of Hensel liftings is larger.
Combination of quadratic lifting and modulus decrease. To obtain a lifting for \(p^k\), we recursively compute the lifting to \(p^{\lceil \frac{k}{2} \rceil }\), then do a quadratic Hensel lifting to \(p^{2 \cdot \lceil \frac{k}{2} \rceil }\), and if k is odd, do a final decrease operation to \(p^k\).
$$\begin{aligned} p^1 \Rightarrow p^2 \Rightarrow p^4 \Rightarrow p^8 \searrow p^7 \Rightarrow p^{14} \searrow p^{13} \Rightarrow p^{26} \Rightarrow p^{52} \searrow p^{51} \end{aligned}$$
In comparison to the third approach, we have slightly larger numbers, but we can replace (expensive) one-step Hensel liftings by the cheap modulus decrease.
In our experiments, it turned out that alternative 4 is faster than 2, which in turn is faster than 3. Alternative 2 is faster than 1 in contrast to the result of Miola and Yun [ 28 , Sect. 1]. 9 Therefore, the current formalization adopts alternative 4, whereas our previous version [ 10 ] implemented alternative 2.
We further extend the binary (quadratic) lifting algorithm to an n-ary lifting algorithm. It inputs a list \( fs \) of factors modulo p of a square-free polynomial f, splits it into two groups \( fs _1\) and \( fs _2\), then applies the binary Hensel lifting to \(\left( \prod fs _1\right) \cdot \left( \prod fs _2\right) \equiv f \quad (mod p)\) obtaining \(g_1 \cdot g_2 \equiv f \quad (mod p^k)\), and finally calls the algorithm recursively to both \(\prod fs _1 \equiv g_1\) and \(\prod fs _2 \equiv g_2 \quad (mod p)\).
Since the runtime of the binary Hensel lifting is nonlinear to the degree, the lists \( fs _1\) and \( fs _2\) should better be balanced so that their products have similar degrees. To this end, we define the following instead of lists:
We implement operations involving this datatype, such as obtaining the multiset of factors of a factor tree, subtrees and product of factor trees modulo p. This change from lists to trees allows us to implement the multifactor Hensel lifting [ 33 , Chapter 15.5] as well as easily balance the involved trees with respect to the degree, that is, we construct the tree so that the sum of the degrees of the factors of f modulo p which are stored in the left-branch is similar to the sum of the degrees of the factors stored in the right-branch of the tree. This way, we avoid expensive computations of Hensel lifting steps involving high-degree polynomials. We refer to the 1st edition of the textbook [ 33 ] for further details on factor trees and to the Isabelle sources for our implementation.
The final lemma that states the soundness of the Hensel lifting.
(Hensel Lifting–general case)
Note that uniqueness follows from the fact that the preconditions already imply that f is uniquely factored in \({\mathbb {Z}}/{p}{\mathbb {Z}} \)—just apply Theorem 5.
We do not go into details of the proofs, but briefly mention that also here local type definitions have been essential. The reason is that the computation relies upon the extended Euclidean algorithm applied on polynomials over \(\mathrm {GF}(p)\). Since the soundness theorem of this algorithm is available only in a type-based version in the Isabelle distribution, we first convert it to the integer representation of \(\mathrm {GF}(p)\) and a record-based implementation as in Sect. 3.
We end this section by proceeding with the running example, without providing details of the computation.
Applying the Hensel lifting on the factorization of Example 3 with \(k = 5\) from Example 4 yields
$$\begin{aligned} f \equiv&\ 3 \cdot (2885 + x) \cdot (14\,027 + 7999x + 13\,691x^2 + 7201x^3 + x^4) \quad (mod p^k) \end{aligned}$$
9 Reconstructing True Factors
For formalizing step 9 of Algorithm 1, we basically follow Knuth, who describes the reconstruction algorithm briefly and presents the soundness proof in prose [ 18 , steps F2 and F3, p. 451 and 452]. At this point of the formalization the De Bruijn factor is quite large, i.e., the formalization is by far more detailed than the intuitive description given by Knuth.
The following definition presents (a simplified version of) the main worklist algorithm, which is formalized in Isabelle/HOL via the command. 10
Here, \( rf \) is supposed to be the number of remaining factors, i.e., the length of ; denotes the list of length- d sublists of \( hs \); and is the inverse modulo function, which converts a polynomial with coefficients in \(\{0,\ldots ,m\}\) into a polynomial with coefficients in \(\{-\lfloor \frac{m-1}{2}\rfloor ,\ldots , {\lceil }{\frac{m-1}{2}}{\rceil }\}\), where the latter set is a superset of the range of coefficients of any potential factor of , cf. Sect. 7.
Basically, for every sublist \( gs \) of \( hs \) we try to divide by the reconstructed potential factor g. If this is possible then we store \(f_i\), the primitive part of g, in the list \( res \) of resulting integer polynomial factors and update the polynomial f and its factorization \( hs \) in \({\mathbb {Z}}/{p^k}{\mathbb {Z}} \) accordingly. When the worklist becomes empty or a factor is found, we update the number \( rf \) of remaining factors \( hs \) and the length d of the sublists we are interested in. Finally, when we have tested enough sublists ( \( rf < 2d\)) we finish.
For efficiency, the actual formalization employs three improvements over the simplified version presented here.
Values which are not frequently changed are passed as additional arguments. For instance is provided via an additional argument and not recomputed in every invocation of .
For the divisibility test we first test whether the constant term of the candidate factor g divides that of . In our experiments, in over 99% of the cases this simple integer divisibility test can prove that g is not a factor of . This test is in particular efficient, since the constant term of g is just the product of the constant terms of the polynomials in gs, so that one can execute the test without computing g itself.
The enumeration of sublists is made parametric, and we developed an efficient generator of sublists which reuses results from previous iterations. Moreover, the sublist generator also shares computations to calculate the constant term of g.
Continuing Example 5, we have only two factors, so it suffices to consider \(d = 1\). We obtain the singleton sublists \([g_1] = [2885 + x]\) and \([g_2] = [14 027 + 7999x + 13 691x^2 + 7201x^3 + x^4]\). The constant term of is the inverse modulo of \((10 \cdot 2885) \mathbin {mod }p^k\), i.e., \(-4764\), and similarly, for \(g_2\) we obtain 5814. Since neither of them divides 40, the constant term of , the algorithm returns [ f], i.e., f is irreducible.
The formalized soundness proof of is much more involved than the paper proof; it is proved inductively with several invariants, for instance
correct input:
corner cases: \(2d \le rf \), \( todo \ne [\,] \longrightarrow d < rf \), \(d = 0 \longrightarrow todo = [\,]\)
irreducible result:
properties of prime: ,
factorization mod \(p^k\):
normalized input:
factorization over integers: the polynomial \(f \cdot \prod \! res \) stays constant throughout the algorithm
all factors of with degree at most have coefficients in the range \(\{-\lfloor \frac{p^k-1}{2}\rfloor ,\dots ,{\lceil }{\frac{p^k-1}{2}}{\rceil }\}\)
all non-empty sublists \( gs \) of \( hs \) of length at most d which are not present in \( todo \) have already been tested, i.e., these \( gs \) do not give rise to a factor of f
The hardest parts in the proofs were to ensure the validity of all invariants after a factor g has been detected—since then nearly all parameters are changed—and to ensure that the final polynomial f is irreducible when the algorithm terminates.
In total, we achieve the following soundness result, which already integrates many of the results from the previous sections. Here, is a simple composition of the finite field factorization algorithm (that is, the function which internally uses the Berlekamp factorization) and the Hensel lifting, and invokes with the right set of starting parameters.
(Zassenhaus Reconstruction of Factors)
The worst-case runtime of this factor-reconstruction algorithm is known to be exponential. We also have a polynomial-time version based on the lattice reduction algorithm [ 7 , 11 ], but this contribution goes beyond the scope of this paper.
10 Assembled Factorization Algorithm
At this point, it is straightforward to combine the algorithms presented in Sects. 5 to 9 to get a factorization algorithm for square-free polynomials.
Here, just computes an exponent k such that \(p^k > bnd \).
It satisfies the following soundness theorem.
(Berlekamp–Zassenhaus Algorithm)
Putting this together with the square-free factorizaton algorithm presented in Sect. 4, we now assemble a factorization algorithm for integer polynomials
and prove its soundness:
(Factorization of Integer Polynomials)
So, we get a factorization algorithm that works for any integer polynomial. But we can do it even better: Performance improves if we include reciprocal polynomials when , since then the values of and are swapped, and thus the value of \( bnd \) in the definition of decreases.
The reciprocal polynomial of polynomial \(f = \sum _{i=0}^n a_ix^i\) is \(\sum _{i=0}^n a_{n-i}x^i\), and is defined in Isabelle as . Reciprocal polynomials satisfy some important properties that we have proved in Isabelle, among others:
Using these properties and some others already present in the library, we prove that it is possible to factor a polynomial by factoring its reciprocal and then taking reciprocal of its irreducible factors. To avoid unnecessary computations, we define a function of type to do this step for a polynomial which does not have zero as constant part and then assemble everything in a function of the same type to get a full factorization of any integer polynomial as follows. It satisfies the soundness Theorem 1 from the introduction.
By using Gauss' lemma we also assembled a factorization algorithm for rational polynomials which just converts the input polynomial into an integer polynomial by a scalar multiplication and then invokes . The algorithm has exactly the same soundness statement as Theorem 1 except that the type changes from integer polynomials to rational polynomials.
Finally, it is worth noting that several of the presented algorithms require polynomial multiplications. However, there is no fast polynomial multiplication algorithm implemented in Isabelle. Indeed, just the naive one is present in the standard library, which is \({{{\mathcal {O}}}}(n^2)\). Thus, we decided to formalize Karatsuba's multiplication algorithm, which is an algorithm of complexity \(\mathcal{O}(n^{\log _2 3})\), to improve the performance of our verified version of the Berlekamp–Zassenhaus algorithm. Karatsuba's algorithm performs multiplication operation by replacing some multiplications with subtraction and addition operations, which are less costly [ 16 ]. We provide a verified implementation for type-based polynomials, e.g., integer polynomials, but we also implement a record-based one for polynomials over \(\mathrm {GF}(p)\), cf. Sect. 3. The type-based formalization is valid for arbitrary polynomials over a commutative ring, so we fully replace Isabelle's polynomial multiplication algorithm by it.
We also tune the GCD algorithm for integer polynomials, so that it first tests whether f and g are coprime modulo a few primes. If so, we are immediately done, otherwise the GCD of the polynomials is computed. Our experiments shows that this preprocessing is faster than a direct computation of the GCD. Since this heuristic involves a few small primes, all operations in the heuristic are carried out using 64-bit integers.
11 Experimental Evaluation
We evaluate the performance of our algorithm in comparison to a modern factorization algorithm—here we choose the factorization algorithm of Mathematica 11.2 [ 34 ]. To evaluate the runtime of our algorithm, we use Isabelle's code generation mechanism [ 12 ] to extract Haskell code for . The code generator is designed for partial correctness, i.e., if an execution of the generated code terminates, then the answer will be correct, but termination itself is not guaranteed. Another restriction is that we rely upon soundness of Haskell's arithmetic operations on integers, since we map Isabelle's integer types ( , , and ) to Haskell's integer types ( Data.Word.Word32, Data.Word.Word64, and Integer). The resulting code was compiled with GHC version 8.2.1 using the O2 switch to turn on most optimizations. All experiments have been conducted under macOS Mojave 10.14.1 on an 8-core Intel Xeon W running at 3.2 GHz.
Figure 1 shows the runtimes of our implementation compared to that of Mathematica on a logarithmic scale. We also include a comparison between the version presented in our previous work [ 10 ] and the new one which includes the optimizations explained through this paper. The runtimes are given in seconds (including the 0.5 s startup time of Mathematica), and the horizontal axis shows the number of coefficients of the polynomial. The test suite consists of 400 polynomials with degrees between 100 and 499 and coefficients are chosen at random between \(-100\) and 100.
Runtimes compared with Mathematica and the version with no improvements
Impact of individual optimizations
Total runtime (%)
New without GCD heuristic
\(+\) 1.2
New without reciprocal polynomials
New without dynamic selection of \(\mathrm {GF}(p)\) implementation
\(+\) 15.5
New without balanced multifactor Hensel lifting
New without Karatsuba's multiplication algorithm
As these polynomials have been randomly generated, they are typically irreducible. In this case using a fast external factorization algorithm as a preprocessing step will not improve the performance, as then the preprocessing does not modify the polynomial. We conjecture that the situation could be alleviated by further incorporating an efficient irreducibility test.
Besides making a global comparison between the old and the new algorithm, we also evaluate several different optimizations separately. The results are presented in Table 2, where a row "new without opt" indicates a configuration, where only optimization opt has been disabled in the new implementation. The time is given relative to the implementation "new" which includes all optimizations and requires around 14 min to factor all 400 example polynomials. The table does not list all optimizations of this paper, since some of them could not easily be disabled in the generated code. In particular, all configurations use the same variant of the binary Hensel lifting algorithm, which considerably differs from the binary Hensel lifting of the old implementation. The results show, that in particular the dynamic selection of the \(\mathrm {GF}(p)\) implementation, the balancing of multifactor Hensel lifting, and the improved polynomial multiplication algorithm are significant improvements.
Profiling revealed that for the 400 random example polynomials, most of the time is spent in the Berlekamp factorization, i.e., in step 6 of Algorithm 1, or more precisely in Step 3 of Algorithm 2, the computation of the basis via Gauss–Jordan elimination. Interestingly, the exponential reconstruction algorithm in step 9 does not have any significance on these random polynomials, cf. Table 3.
Nevertheless we remark that this situation can dramatically change on non-random polynomials, e.g., on polynomials constructed via algebraic numbers. For instance when computing the minimal integer polynomial that has \(\sum _{i=1}^6 \root 3 \of {i}\) as root, 87.3% of the overall time is spent in the reconstruction algorithm; and for \(\sum _{i=1}^7 \root 3 \of {i}\) we had to abort the computation within the reconstruction phase. Note that even Mathematica does not finish the computation of the latter minimal polynomial within a day. As a possible optimization, the exponential reconstruction phase can be replaced by van Hoeij's fast reconstruction algorithm based on lattice-reduction [ 14 ], which is implemented in Maple 2017.3 [ 25 ]. Although Maple is only 20 % faster than Mathematica when factoring the 400 random polynomials, it can compute the minimal polynomial within a second, in contrast to the timeout of Mathematica. However, a soundness proof of van Hoeij's algorithm is much more involved.
Profiling results
Amount of total runtime (%)
Berlekamp factorization
Hensel lifting
Square-free factorization
Find suitable prime
Determine factor bound
Remaining parts
We formalized the Berlekamp–Zassenhaus algorithm for factoring univariate integer polynomials. To this end we switched between different representations of finite fields and quotient rings with the help of locales, the transfer package and local type definitions. The generated code can factor large polynomials within seconds. The whole formalization consists of 21320 lines of Isabelle and took about 17 person months of Isabelle experts. As far as we know, this is the first formalization of an efficient polynomial factorization algorithm in a theorem prover.
Most of the improvements mentioned as potential future work in our previous conference paper [ 10 ] have now been formalized and are integrated in the development, but there still remain some possibilities to extend the current formalization for optimizing the factorization algorithm even further. For instance, one can consider using the Cantor–Zassenhaus algorithm [ 8 ] for finite-field factorization, although its formalization would be more intricate (indeed, it is a probabilistic algorithm).
Open access funding provided by Austrian Science Fund (FWF). We thank Florian Haftmann for integrating our changes in the polynomial library into the Isabelle distribution; we thank Manuel Eberl for discussions on factorial rings in Isabelle; and we thank the anonymous reviewers for their helpful remarks, which led us to include many of their suggestions into the current article.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Our algorithm starts with step 4, so that section numbers and step-numbers coincide.
A formalization of the ring \({\mathbb {Z}}/{p}{\mathbb {Z}} \) is already present in as a locale . In principle we could reuse results from the library by proving a connection between the locale and our class; however, as the resulting proofs became slightly longer than direct proofs, we did not use this library.
We would like to have introduced this abbreviation also in Isabelle. However, we are not aware of how to do this, since the keyword does not allow specifying type constraints such as .
In this paper we use the conventional notations \(f \equiv g \quad (mod m)\) and \(f \mathbin {mod }m\). In the formalization these notions are defined as and respectively.
The preliminary version [ 10 ] of this paper does not require such an abstraction layer since there we always implement \(\mathrm {GF}(p)\) via integers.
Knuth gives a brief outline of a proof, but he also classifies the exercise as a problem of moderate complexity that may involve more than two hours' work to solve it on paper.
Indeed, one can reduce the degree bound to half of the degree of f if one uses a slightly more complex reconstruction algorithm which sometimes considers the complement of the selected factors. We did not investigate the trade-off between the two alternatives.
One might transfer the type-based Lemma 21 to integer polynomials, in order to use it within the inductive proof of Lemma 22. However, the current proof of Lemma 22 does not rely upon Lemma 21.
Perhaps our quadratic version of Hensel lifting is faster than the iterated one-step version since we did not integrate (and prove) optimizations (iii) and (iv) of Miola and Yun [ 28 , Sect. 2.4].
Although does not support pattern matching, we prefer to use pattern matching in the presentation.
Premium-Abo der Gesellschaft für Informatik
Sie erhalten uneingeschränkten Vollzugriff auf alle acht Fachgebiete von Springer Professional und damit auf über 45.000 Fachbücher und ca. 300 Fachzeitschriften.
Zurück zum Zitat Abbott, J.: Bounds on factors in \(Z[x]\). J. Symb. Comput. 50, 532–563 (2013) MathSciNetCrossRef Abbott, J.: Bounds on factors in \(Z[x]\). J. Symb. Comput. 50, 532–563 (2013) MathSciNetCrossRef
Zurück zum Zitat Axler, S.J.: Linear Algebra Done Right. Undergraduate Texts in Mathematics. Springer, Berlin (1997) CrossRef Axler, S.J.: Linear Algebra Done Right. Undergraduate Texts in Mathematics. Springer, Berlin (1997) CrossRef
Zurück zum Zitat Ballarin, C.: Locales: a module system for mathematical theories. J. Autom. Reason. 52(2), 123–153 (2014) MathSciNetCrossRef Ballarin, C.: Locales: a module system for mathematical theories. J. Autom. Reason. 52(2), 123–153 (2014) MathSciNetCrossRef
Zurück zum Zitat Barthe, G., Grégoire, B., Heraud, S., Olmedo, F., Béguelin, S.Z.: Verified indifferentiable hashing into elliptic curves. In: Degano, P., Guttman, J.D. (eds.) Principles of Security and Trust. POST 2012, Volume 7215 of LNCS, pp. 209–228. Springer, Berlin (2012) Barthe, G., Grégoire, B., Heraud, S., Olmedo, F., Béguelin, S.Z.: Verified indifferentiable hashing into elliptic curves. In: Degano, P., Guttman, J.D. (eds.) Principles of Security and Trust. POST 2012, Volume 7215 of LNCS, pp. 209–228. Springer, Berlin (2012)
Zurück zum Zitat Berlekamp, E.R.: Factoring polynomials over finite fields. Bell Syst. Tech. J. 46(8), 1853–1859 (1967) MathSciNetCrossRef Berlekamp, E.R.: Factoring polynomials over finite fields. Bell Syst. Tech. J. 46(8), 1853–1859 (1967) MathSciNetCrossRef
Zurück zum Zitat Blanchette, J.C., Meier, F., Popescu, A., Traytel, D.: Foundational nonuniform (co)datatypes for higher-order logic. In: ACM/IEEE Symposium on Logic in Computer Science, LICS 32, pp. 1–12. IEEE Computer Society (2017). Cross-type induction is explained in Appendix D of the extended report version at http://matryoshka.gforge.inria.fr/pubs/nonuniform_report.pdf Blanchette, J.C., Meier, F., Popescu, A., Traytel, D.: Foundational nonuniform (co)datatypes for higher-order logic. In: ACM/IEEE Symposium on Logic in Computer Science, LICS 32, pp. 1–12. IEEE Computer Society (2017). Cross-type induction is explained in Appendix D of the extended report version at http://matryoshka.gforge.inria.fr/pubs/nonuniform_report.pdf
Zurück zum Zitat Bottesch, R., Haslbeck, M.W., Thiemann, R.: A verified efficient implementation of the LLL basis reduction algorithm. In: Barthe, G., Sutcliffe, G., Veanes, M. (eds.) Logic for Programming, Artificial Intelligence and Reasoning. LPAR 22, Volume 57 of EPiC Series in Computing, pp. 164–180. EasyChair (2018) Bottesch, R., Haslbeck, M.W., Thiemann, R.: A verified efficient implementation of the LLL basis reduction algorithm. In: Barthe, G., Sutcliffe, G., Veanes, M. (eds.) Logic for Programming, Artificial Intelligence and Reasoning. LPAR 22, Volume 57 of EPiC Series in Computing, pp. 164–180. EasyChair (2018)
Zurück zum Zitat Cantor, D.G., Zassenhaus, H.: A new algorithm for factoring polynomials over finite fields. Math. Comput. 36(154), 587–592 (1981) MathSciNetCrossRef Cantor, D.G., Zassenhaus, H.: A new algorithm for factoring polynomials over finite fields. Math. Comput. 36(154), 587–592 (1981) MathSciNetCrossRef
Zurück zum Zitat Cerlienco, L., Mignotte, M., Piras, F.: Computing the measure of a polynomial. J. Symb. Comput. 4(1), 21–33 (1987) MathSciNetCrossRef Cerlienco, L., Mignotte, M., Piras, F.: Computing the measure of a polynomial. J. Symb. Comput. 4(1), 21–33 (1987) MathSciNetCrossRef
Zurück zum Zitat Divasón, J., Joosten, S.J.C., Thiemann, R., Yamada, A.: A formalization of the Berlekamp–Zassenhaus factorization algorithm. In: Bertot, Y., Vafeiadis, V. (eds.) Certified Programs and Proofs. CPP 2017, pp. 17–29. ACM (2017) Divasón, J., Joosten, S.J.C., Thiemann, R., Yamada, A.: A formalization of the Berlekamp–Zassenhaus factorization algorithm. In: Bertot, Y., Vafeiadis, V. (eds.) Certified Programs and Proofs. CPP 2017, pp. 17–29. ACM (2017)
Zurück zum Zitat Divasón, J., Joosten, S.J.C., Thiemann, R., Yamada, A.: A formalization of the LLL basis reduction algorithm. In: Avigad, J., Mahboubi, A. (eds.) Interactive Theorem Proving. ITP 2018, Volume 10895 of LNCS, pp. 160–177. Springer, Berlin (2018) Divasón, J., Joosten, S.J.C., Thiemann, R., Yamada, A.: A formalization of the LLL basis reduction algorithm. In: Avigad, J., Mahboubi, A. (eds.) Interactive Theorem Proving. ITP 2018, Volume 10895 of LNCS, pp. 160–177. Springer, Berlin (2018)
Zurück zum Zitat Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) Functional and Logic Programming. FLOPS 2010, Volume 6009 of LNCS, pp. 103–117. Springer, Berlin (2010) Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) Functional and Logic Programming. FLOPS 2010, Volume 6009 of LNCS, pp. 103–117. Springer, Berlin (2010)
Zurück zum Zitat Harrison, J.: The HOL light theory of Euclidean space. J. Autom. Reason. 50(2), 173–190 (2013) MathSciNetCrossRef Harrison, J.: The HOL light theory of Euclidean space. J. Autom. Reason. 50(2), 173–190 (2013) MathSciNetCrossRef
Zurück zum Zitat van Hoeij, M.: Factoring polynomials and the knapsack problem. J. Number Theory 95(2), 167–189 (2002) MathSciNetCrossRef van Hoeij, M.: Factoring polynomials and the knapsack problem. J. Number Theory 95(2), 167–189 (2002) MathSciNetCrossRef
Zurück zum Zitat Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Certified Programs and Proofs. CPP 2013, Volume 8307 of LNCS, pp. 131–146. Springer, Berlin (2013) Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Certified Programs and Proofs. CPP 2013, Volume 8307 of LNCS, pp. 131–146. Springer, Berlin (2013)
Zurück zum Zitat Karatsuba, A., Ofman, Y.: Multiplication of multidigit numbers on automata. Sov. Phys. Dokl. 7(7), 595–596 (1963) Karatsuba, A., Ofman, Y.: Multiplication of multidigit numbers on automata. Sov. Phys. Dokl. 7(7), 595–596 (1963)
Zurück zum Zitat Kirkels, B.: Irreducibility certificates for polynomials with integer coefficients. Master's thesis, Radboud Universiteit Nijmegen (2004) Kirkels, B.: Irreducibility certificates for polynomials with integer coefficients. Master's thesis, Radboud Universiteit Nijmegen (2004)
Zurück zum Zitat Knuth, D.E.: The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edn. Addison-Wesley, Reading (1998) MATH Knuth, D.E.: The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edn. Addison-Wesley, Reading (1998) MATH
Zurück zum Zitat Kobayashi, H., Suzuki, H., Ono, Y.: Formalization of Hensel's lemma. In: Hurd, J., Smith, E., Darbari, A. (eds.) Theorem Proving in Higher Order Logics: Emerging Trends Proceedings, pp. 114–118. Oxford University Computing Laboratory (2005) Kobayashi, H., Suzuki, H., Ono, Y.: Formalization of Hensel's lemma. In: Hurd, J., Smith, E., Darbari, A. (eds.) Theorem Proving in Higher Order Logics: Emerging Trends Proceedings, pp. 114–118. Oxford University Computing Laboratory (2005)
Zurück zum Zitat Krauss, A.: Recursive definitions of monadic functions. In: Bove, A., Komendantskaya, E., Niqui, M. (eds.) Partiality and Recursion in Interactive Theorem Provers. PAR 2010, Volume 43 of EPTCS, pp. 1–13 (2010) Krauss, A.: Recursive definitions of monadic functions. In: Bove, A., Komendantskaya, E., Niqui, M. (eds.) Partiality and Recursion in Interactive Theorem Provers. PAR 2010, Volume 43 of EPTCS, pp. 1–13 (2010)
Zurück zum Zitat Kunčar, O., Popescu, A.: From types to sets by local type definition in higher-order logic. J. Autom. Reason. 62(2), 237–260 (2019) MathSciNetCrossRef Kunčar, O., Popescu, A.: From types to sets by local type definition in higher-order logic. J. Autom. Reason. 62(2), 237–260 (2019) MathSciNetCrossRef
Zurück zum Zitat Lee, H.: Vector spaces. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/VectorSpace.html (2014) Lee, H.: Vector spaces. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/VectorSpace.html (2014)
Zurück zum Zitat Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982) MathSciNetCrossRef Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982) MathSciNetCrossRef
Zurück zum Zitat Lochbihler, A.: Fast machine words in Isabelle/HOL. In: Avigad, J., Mahboubi, A. (eds.) Interactive Theorem Proving. ITP 2018, Volume 10895 of LNCS, pp. 388–410. Springer, Berlin (2018) Lochbihler, A.: Fast machine words in Isabelle/HOL. In: Avigad, J., Mahboubi, A. (eds.) Interactive Theorem Proving. ITP 2018, Volume 10895 of LNCS, pp. 388–410. Springer, Berlin (2018)
Zurück zum Zitat Maple 2017.3. Maplesoft, a division of Waterloo Maple Inc. Waterloo (2017) Maple 2017.3. Maplesoft, a division of Waterloo Maple Inc. Waterloo (2017)
Zurück zum Zitat Martin-Dorel, É., Hanrot, G., Mayero, M., Théry, L.: Formally verified certificate checkers for hardest-to-round computation. J. Autom. Reason. 54(1), 1–29 (2015) MathSciNetCrossRef Martin-Dorel, É., Hanrot, G., Mayero, M., Théry, L.: Formally verified certificate checkers for hardest-to-round computation. J. Autom. Reason. 54(1), 1–29 (2015) MathSciNetCrossRef
Zurück zum Zitat Mignotte, M.: An inequality about factors of polynomials. Math. Comput. 28(128), 1153–1157 (1974) MathSciNetCrossRef Mignotte, M.: An inequality about factors of polynomials. Math. Comput. 28(128), 1153–1157 (1974) MathSciNetCrossRef
Zurück zum Zitat Miola, A., Yun, D.Y.: Computational aspects of Hensel-type univariate polynomial greatest common divisor algorithms. ACM SIGSAM Bull. 8(3), 46–54 (1974) CrossRef Miola, A., Yun, D.Y.: Computational aspects of Hensel-type univariate polynomial greatest common divisor algorithms. ACM SIGSAM Bull. 8(3), 46–54 (1974) CrossRef
Zurück zum Zitat Nipkow, T., Paulson, L., Wenzel, M.: Isabelle/HOL—A Proof Assistant for Higher-Order Logic, Volume 2283 of LNCS. Springer, Berlin (2002) MATH Nipkow, T., Paulson, L., Wenzel, M.: Isabelle/HOL—A Proof Assistant for Higher-Order Logic, Volume 2283 of LNCS. Springer, Berlin (2002) MATH
Zurück zum Zitat Thiemann, R.: Computing n-th roots using the Babylonian method. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/Sqrt_Babylonian.html (2013) Thiemann, R.: Computing n-th roots using the Babylonian method. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/Sqrt_Babylonian.html (2013)
Zurück zum Zitat Thiemann, R., Yamada, A.: Algebraic numbers in Isabelle/HOL. In: Blanchette, J., Merz, S. (eds.) Interactive Theorem Proving. ITP 2016, Volume 9807 of LNCS, pp. 391–408. Springer, Berlin (2016) Thiemann, R., Yamada, A.: Algebraic numbers in Isabelle/HOL. In: Blanchette, J., Merz, S. (eds.) Interactive Theorem Proving. ITP 2016, Volume 9807 of LNCS, pp. 391–408. Springer, Berlin (2016)
Zurück zum Zitat Thiemann, R., Yamada, A.: Formalizing Jordan normal forms in Isabelle/HOL. In: Avigad, J., Chlipala, A. (eds.) Certified Programs and Proofs. CPP 2016, pp. 88–99. ACM (2016) Thiemann, R., Yamada, A.: Formalizing Jordan normal forms in Isabelle/HOL. In: Avigad, J., Chlipala, A. (eds.) Certified Programs and Proofs. CPP 2016, pp. 88–99. ACM (2016)
Zurück zum Zitat von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 3rd edn. Cambridge University Press, Cambridge (2013) CrossRef von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 3rd edn. Cambridge University Press, Cambridge (2013) CrossRef
Zurück zum Zitat Mathematica Version 11.2. Wolfram Research, Inc. Champaign (2017) Mathematica Version 11.2. Wolfram Research, Inc. Champaign (2017)
Zurück zum Zitat Yun, D.Y.: On square-free decomposition algorithms. In: Symbolic and Algebraic Computation. SYMSAC 1976, pp. 26–35. ACM (1976) Yun, D.Y.: On square-free decomposition algorithms. In: Symbolic and Algebraic Computation. SYMSAC 1976, pp. 26–35. ACM (1976)
Zurück zum Zitat Zassenhaus, H.: On Hensel factorization, I. J. Number Theory 1(3), 291–311 (1969) MathSciNetCrossRef Zassenhaus, H.: On Hensel factorization, I. J. Number Theory 1(3), 291–311 (1969) MathSciNetCrossRef
Jose Divasón
Sebastiaan J. C. Joosten
René Thiemann
Akihisa Yamada
https://doi.org/10.1007/s10817-019-09526-y
Journal of Automated Reasoning
OriginalPaper
Formalizing the Cox–Ross–Rubinstein Pricing of European Derivatives in Isabelle/HOL
An Assertional Proof of Red–Black Trees Using Dafny
Automated Reasoning with Power Maps
A Formalized General Theory of Syntax with Bindings: Extended Version
Homogeneous Length Functions on Groups: Intertwined Computer and Human Proofs
ec4u, Neuer Inhalt/© ITandMEDIA | CommonCrawl |
Temperature rise after a doubling of CO2
Frequency distribution of equilibrium climate sensitivity, based on simulations of doubling CO
2.[1] Each model simulation has a different guess at processes that scientists don't understand sufficiently well. Few of the simulations result in less than 2 °C of warming or significantly more than 4 °C.[1] However the positive skew, which is also found in other studies,[2] suggests that if carbon dioxide concentrations double, the probability of very large increases in temperature is greater than the probability of very small increases.[1]
Climate sensitivity is the globally averaged temperature change in response to changes in radiative forcing, which can occur, for instance, due to increased levels of carbon dioxide (CO
2).[3] Although the term climate sensitivity is usually used in the context of radiative forcing by CO2, it is thought of as a general property of the climate system: the change in surface air temperature following a unit change in radiative forcing, and the climate sensitivity parameter[note 1] is therefore expressed in units of °C/(W/m2). The measure is approximately independent of the nature of the forcing (e.g. from greenhouse gases or solar variation).[4] When climate sensitivity is expressed for a doubling of CO2, its units are degrees Celsius (°C).
In the context of global warming, different measures of climate sensitivity are used. The equilibrium climate sensitivity (ECS) is the temperature increase that would result from sustained doubling of the concentration of carbon dioxide in Earth's atmosphere, after the Earth's energy budget and the climate system reach radiative equilibrium.[5] The transient climate response (TCR) is the amount of temperature increase that might occur at the time when CO2 doubles, having increased gradually by 1% each year. The earth system sensitivity (ESS) includes the effects of very-long-term Earth system feedback loops, such as changes in ice sheets or changes in the distribution of vegetative cover.[6]
Climate sensitivity is typically estimated in three ways; by using observations taken during the industrial age, by using temperature and other data from the Earth's past and by modelling the climate system in computers.[6] For coupled atmosphere-ocean global climate models the climate sensitivity is an emergent property; rather than being a model parameter it is a result of a combination of model physics and parameters. By contrast, simpler energy-balance models may have climate sensitivity as an explicit parameter.
1 Different forms of climate sensitivity
1.1 Equilibrium climate sensitivity
1.1.1 Effective climate sensitivity
1.2 Transient climate response
1.3 Earth system sensitivity
2 Radiative forcing
3 Sensitivity to nature of the forcing
4 State dependence
5 Estimating climate sensitivity
5.1 Historical estimates
5.1.1 Intergovernmental Panel on Climate Change
5.2 Using industrial-age data
5.2.1 Other strategies
5.3 Using data from Earth's past
5.4 Using climate models
5.4.1 Constrained models
6 Socio-economic implications
Different forms of climate sensitivity[edit]
Schematic of how different measures of climate sensitivity relate to one another
A component of climate sensitivity is directly due to radiative forcing, for instance by CO
2, and a further contribution arises from climate feedback, both positive and negative.[7] Without feedbacks the radiative forcing of approximately 3.7 W/m2, due to doubling CO
2 from the pre-industrial 280 ppm, would eventually result in roughly 1 °C global warming. This is easy to calculate[note 2][8] and undisputed.[9] The uncertainty is due entirely to feedbacks in the system: the water vapor feedback, the ice-albedo feedback, the cloud feedback, and the lapse rate feedback.[9] Due to climate inertia, the climate sensitivity depends upon the timescale. The transient response is defined by scientists as the temperature response over human time scales of around 70 years, the equilibrium climate sensitivity over centuries, and finally the Earth system sensitivity after multiple millennia.[10]
Equilibrium climate sensitivity[edit]
The equilibrium climate sensitivity (ECS) refers to the equilibrium change in global mean near-surface air temperature that would result from a sustained doubling of the atmospheric equivalent CO
2 concentration (ΔT2×). A comprehensive model estimate of equilibrium sensitivity requires a very long model integration; fully equilibrating ocean temperatures requires the integration of thousands of model years, although it is possible to produce an estimate more quickly using the method of Gregory et al. (2004).[11] As estimated by the IPCC Fifth Assessment Report (AR5), "there is high confidence that ECS is extremely unlikely less than 1°C and medium confidence that the ECS is likely between 1.5°C and 4.5°C and very unlikely greater than 6°C".[12]
Effective climate sensitivity[edit]
The effective climate sensitivity is an estimate of equilibrium climate sensitivity using data from a climate system, either in a model or real-world observations, that is not yet in equilibrium.[13] Estimation is done by using the assumption that the net effect of feedbacks as measured after a period of warming remains constant afterwards.[14] This is not necessarily true, as feedbacks can change with time, or with the particular starting state or forcing history of the climate system.[15][13]
Transient climate response[edit]
The transient climate response (TCR) is defined as the average temperature response over a twenty-year period centered at CO
2 doubling in a transient simulation with CO
2 increasing at 1% per year (compounded), i.e., 60 to 80 years following initiation of the increase in CO
2.[16] The transient response is lower than the equilibrium sensitivity because the deep ocean, which takes many centuries to reach a new steady state after a perturbation, continues to serve as a sink for heat from the upper ocean.[17] The IPCC literature assessment estimates that TCR likely lies between 1 °C and 2.5 °C.[18] A related concept is the transient climate response to cumulative carbon emissions, which is the globally averaged surface temperature change per unit of CO
2 emitted.[19]
Earth system sensitivity[edit]
The Earth system sensitivity (ESS) includes the effects of slower feedback loops, such as the change in Earth's albedo from the melting of large ice sheets that covered much of the northern hemisphere during the last glacial maximum. These extra feedback loops make the ESS larger than the ECS – possibly twice as large. Data from Earth's history is used to estimate ESS, but climatic conditions were quite different which makes it difficult to infer information for future ESS.[20] ESS includes the entire system except the carbon cycle.[21] Changes in albedo as a result of vegetation changes are included.[22]
Radiative forcing[edit]
Main article: Radiative forcing
Radiative forcing is the imbalance between incoming and outgoing radiation at the top of the atmosphere, resulting from a change in atmospheric composition or other changes in radiation budget, prior to long-term changes in global temperature due to the forcing.[23] A number of inputs can give rise to radiative forcing: the extra downwelling radiation due to the greenhouse effect, solar radiation variability due to orbital changes, changes in solar irradiance, direct aerosol effects (for example changes in albedo due to cloud cover), indirect aerosol effects, and changes in land use.[24]
Radiative forcing by greenhouse gases is well understood but, as of 2013[update], large uncertainties remain for aerosols.[25] In time-dependent estimates of climate sensitivity, the concept of the effective radiative forcing, which includes rapid adjustments in the stratosphere and the troposphere to the instantaneous radiative forcing, is usually used.[26]
Sensitivity to nature of the forcing[edit]
Radiative forcing from sources other than CO
2 can cause a higher or lower surface warming than a similar radiative forcing due to CO
2; the amount of feedback varies, mainly because these forcings are not uniformly distributed over the globe. Forcings that initially warm the northern hemisphere, land, or polar regions more strongly; are systematically more effective at changing temperatures than an equivalent amount of CO2 whose forcing is more uniformly distributed over the globe. Several studies indicate that aerosols are more effective than CO
2 at changing global temperatures while volcanic forcing is less effective.[27] Ignoring these factors causes lower estimates of climate sensitivity when using radiative forcing and temperature records from the historical period.[28]
State dependence[edit]
While climate sensitivity is defined as the sensitivity to any doubling of CO
2, there is evidence that the sensitivity of the climate system is not always constant. Until the world's ice has melted, for instance, a positive ice-albedo feedback loop makes the system more sensitive overall.[29] Thus the climate system may warm by a different amount after a second doubling of CO
2 than after the first doubling. The effect of this is small or negligible in the first century after CO
2 is released into the atmosphere.[29] Furthermore, the climate may become more sensitive if tipping points are crossed. It is unlikely that climate sensitivity increases instantly; rather, it changes at the time scale of the subsystem that is undergoing the tipping point.[30] The more sensitive a climate system is to increased greenhouse gases, the more likely it is to have decades when temperatures are much higher or much lower than the longer-term average.[31][32]
Estimating climate sensitivity[edit]
Climate sensitivity is often evaluated in terms of the change in equilibrium temperature due to radiative forcing caused by the greenhouse effect. The radiative forcing, and hence the change in temperature, is proportional to the logarithm of the concentration of infrared-absorbing ("greenhouse") gases in the atmosphere, as quantified by Svante Arrhenius in the 19th century.[33] The sensitivity of temperature to atmospheric gasses, most notably CO
2, is often expressed in terms of the change in temperature per doubling of the concentration of the gas.
Historical estimates[edit]
Arrhenius was the first person to quantify global warming as a consequence of a doubling of CO
2. In his first paper on the matter, he estimated that global temperature would rise by around 5 to 6 °C (9.0 to 10.8 °F) if the quantity of CO
2 was doubled. In later work he revised this estimate to 4 °C (7.2 °F).[34] Arrhenius used the observations of radiation emitted by the full moon made by the astronomer Samuel Pierpont Langley to estimate the amount of radiation that was absorbed by water vapour and CO
2. To account for water vapour feedback he assumed relative humidity would stay the same under global warming.[35][36]
The first calculation of climate sensitivity using detailed measurements of absorption spectra, and the first to use a computer to numerically integrate the radiative transfer through the atmosphere, was by Manabe and Wetherald in 1967.[37] For constant humidity they computed a climate sensitivity of 2.3 °C per doubling of CO2 (which they rounded to 2, the value most often quoted from their work, in the abstract of the paper). This work has been called "arguably the greatest climate-science paper of all time"[38] and "the most influential study of climate of all time."[39]
A committee on anthropogenic global warming, convened in 1979 by the United States National Academy of Sciences and chaired by Jule Charney,[40] estimated climate sensitivity to be 3 °C (5.4 °F), give or take 1.5 °C (2.7 °F). Apart from the Manabe and Wetherald model, with a climate sensitivity of 2 °C (3.6 °F), the only other available was from James E. Hansen, with 4 °C (7.2 °F). According to Manabe, "Charney chose 0.5 °C (0.90 °F) as a reasonable margin of error, subtracted it from Manabe's number, and added it to Hansen's, giving rise to the 1.5 to 4.5 °C (2.7 to 8.1 °F) range of likely climate sensitivity that has appeared in every greenhouse assessment since ..."[41]
In 2008 climatologist Stefan Rahmstorf wrote, regarding the Charney report's original range of uncertainty; "At that time, this range was on very shaky ground. Since then, many vastly improved models have been developed by a number of climate research centers around the world. Current state-of-the-art climate models span a range of 2.6 to 4.1 °C (4.7 to 7.4 °F), most clustering around 3 °C (5.4 °F)."[9]
Intergovernmental Panel on Climate Change[edit]
Historical estimates of climate sensitivity from the IPCC assessments. The first three reports gave a qualitative likely range, while the fourth and fifth assessment report formally quantified the uncertainty. The dark blue range is judged as being more than 66% likely.[42][43]
After the publication of the Charney report, despite considerable progress in the understanding of the climate system, further assessments reported a similar range in climate sensitivity.[44] The 1990 IPCC First Assessment Report estimated that equilibrium climate sensitivity to a doubling of CO
2 lay between 1.5 and 4.5 °C (2.7 and 8.1 °F), with a "best guess in the light of current knowledge" of 2.5 °C (4.5 °F).[45] This report used models that had simplified representations of ocean dynamics. The IPCC supplementary report, 1992, which used full-ocean circulation models, saw "no compelling reason to warrant changing" from this estimate;[46] and the IPCC Second Assessment Report said, "No strong reasons have emerged to change" these estimates.[47] In these reports, much of the uncertainty was attributed to cloud processes. The 2001 IPCC TAR also retained this likely range.[48]
Authors of the IPCC Fourth Assessment Report[42] stated that confidence in estimates of equilibrium climate sensitivity had increased substantially since the Third Annual Report.[49] IPCC authors concluded ECS is very likely to be greater than 1.5 °C (2.7 °F) and likely to lie in the range 2 to 4.5 °C (4 to 8.1 °F), with a most likely value of about 3 °C (5 °F). For fundamental physical reasons and data limitations, the IPCC stated a climate sensitivity higher than 4.5 °C (8.1 °F) could not be ruled out, but that agreement for these values with observations and "proxy" climate data is generally worse compared with values within the likely range.[49]
The IPCC Fifth Assessment Report reverted to the earlier range of 1.5 to 4.5 °C (2.7 to 8.1 °F) (high confidence) because some estimates using industrial-age data came out low.[6] They also stated that ECS is extremely unlikely to be less than 1 °C (1.8 °F) (high confidence), and is very unlikely to be greater than 6 °C (11 °F) (medium confidence). These values are estimated by combining the available data with expert judgement.[43]
Using industrial-age data[edit]
Climate sensitivity can be estimated using observed temperature rise, observed ocean heat uptake, and modeled or observed radiative forcing. These data are linked though a simple energy-balance model to calculate climate sensitivity.[50] Radiative forcing is often modeled, because Earth observation satellites that measure it have not existed for the entire period. Estimates of climate sensitivity calculated from these global energy constraints have consistently been lower than those calculated using other methods;[51] estimates calculated using this method have been around 2 °C (3.6 °F) or lower (e.g.[50][52][53][54]).
Estimates of transient climate response (TCR) calculated from models and observational data can be reconciled if it is taken into account that fewer temperature measurements are taken in the polar regions, which warm more quickly than average. If only regions for which measurements are available are used in evaluating the model, differences in TCR estimates almost disappear.[6][55]
Rahmstorf (2008)[9] provides an informal example of the estimation of climate sensitivity using observations made since the pre-industrial era, from which the following is modified. Denote the sensitivity, i.e. the equilibrium increase in global mean temperature including the effects of feedbacks due to a sustained forcing by doubled CO
2 (F2 × {\displaystyle \times } CO2; taken as 3.7 W/m2), as S (°C). If Earth was to experience an equilibrium temperature change of ΔT (°C) due to a sustained forcing of ΔF (W/m2), then:
S = Δ T × F 2 × C O 2 / Δ F {\displaystyle S=\Delta T\times F_{2\times CO_{2}}/\Delta F} .
The global temperature increase since the beginning of the industrial period (taken as 1750) is about 0.8 °C (1.4 °F), and the radiative forcing due to CO
2 and other long-lived greenhouse gases – mainly methane, nitrous oxide, and chlorofluorocarbons – emitted since that time is about 2.6 W/m2. Neglecting other forcings and considering the temperature increase to be an equilibrium increase would lead to a sensitivity of about 1.1 °C (2.0 °F). However, ΔF also contains contributions from solar activity (+0.3 W/m2), aerosols (−1.0 W/m2), ozone (0.3 W/m2), and other smaller influences, bringing the total forcing over the industrial period to 1.6 W/m2 according to the best estimate of the IPCC AR4, with substantial uncertainty. The absence of equilibrium of the climate system must be accounted for by subtracting the planetary heat uptake rate H from the forcing; i.e.,
S = Δ T × F 2 × C O 2 / ( Δ F − H ) . {\displaystyle S=\Delta T\times F_{2\times CO_{2}}/(\Delta F-H).}
Taking planetary heat uptake rate as the rate of ocean heat uptake estimated by the IPCC AR4 as 0.2 W/m2, yields a value for S of 2.1 °C (3.8 °F).
Other strategies[edit]
In theory, industrial-age temperatures could also be used to determine a timescale of the climate system, and thus climate sensitivity.[56] If the effective heat capacity of the climate system is known and the timescale estimated by using the autocorrelation of the measured temperature, an estimate of climate sensitivity can be derived. However, in practice determination of both the timescale and heat capacity is difficult.[57][58][59]
Attempts to use the 11-year solar cycle to constrain the transient climate response have been made.[60] Solar irradiance is about 0.9 W/m2 brighter during solar maximum than during solar minimum, which correlated in measured average global temperature over the period 1959-2004.[61] The solar minima in this period coincided with volcanic eruptions, which have a cooling effect on the global temperature. Because this causes a larger radiative forcing than the solar variations, it is questionable whether much information can be derived from the temperature variations.[62]
Volcanic eruptions have also been used to try to estimate climate sensitivity. But as the aerosols from a single volcanic eruption only last a couple of years in the atmosphere and the climate system's response to radiative forcing has inertia, only a lower bound on the transient climate sensitivity can be found.[63]
Using data from Earth's past[edit]
Climate sensitivity can be estimated by using reconstructions of Earth's past temperatures and CO
2 levels. Different geological periods, for instance the warm Pliocene and the colder Pleistocene, are studied.[64] Scientist seek periods that are in some sense analogous or informative to current climate change. As more information about them becomes available; recent periods, such as the Mid-holocene that occurred about 6,000 years ago, and the Last Glacial Maximum (LGM) that took place about 21,000 years ago, are often chosen.[65]
As the name suggests, the LGM was a lot colder than today; also scientists have a good idea of the CO
2 concentration and radiative forcing during that period.[66] While orbital forcing was different from the present, this had little effect on mean annual temperatures.[67] Different approaches to the task of estimating climate sensitivity from the LGM are taken.[66] One way is to use estimates of global radiative forcing and temperature directly. The set of feedbacks active during the LGM, however, may be different than the feedbacks due to doubling CO
2, introducing additional uncertainty.[67][68] In a different approach, a single model of intermediate complexity is run using a set of parameters so that each version has a different ECS. Those model versions that can best simulate the cooling during the LGM are thought to have the best ECS values.[69] Other researchers use an ensemble of different models.[70][66]
Over the last 800,000 years, climate sensitivity has been found to be greater in cold periods than in warm periods.[71] Although climates further back in Earth's history are also used; an additional difficulty is that CO
2 concentrations cannot be readily obtained from ice cores so they must be estimated less directly. An estimate of sensitivity made using data from a major part of the Phanerozoic is consistent with sensitivities of current climate models and with other determinations.[72] The Paleocene–Eocene Thermal Maximum provides a good opportunity to study the climate system when it is in a warm state.[73]
Using climate models[edit]
Climate models of earth, for example the Coupled model intercomparison project (CMIP), are used to simulate the quantity of warming that will occur with rising CO
2 concentrations. The models are based on physical laws and represent the biosphere. Because of limited computer power, the physical laws have to be approximated, which leads to a wide range of estimates of climate sensitivity. Climate sensitivity is an emergent property of these models.[6]
In preparation for the 2021 6th IPCC report, a new generation of climate models are being developed:[74][75] some show climate sensitivity around 5 °C (9.0 °F), meaning temperature can rise by 6.5 - 7 degree by 2100 in the worst socio-economic scenario ("SSP5 8.5 – rapid economic growth driven by fossil fuels without mitigation"). However the CMIP6 models have yet to be thoroughly independently analysed and researchers do not yet fully understand why some show this higher sensitivity.[76][77][78]
Constrained models[edit]
Bottom-up modelling of the climate system can lead to a wide range of outcomes. Models are often run using different plausible parameters in their approximation of physical laws and the behaviour of the biosphere; a so-called perturbed physics ensemble. Alternatively, structurally different models developed at different institutions are put together, creating an ensemble. By selecting only those simulations that can simulate some part of the historical climate well, a constrained estimate of climate sensitivity can be made. One strategy is the placing of more trust in climate models that perform well in general.[79]
Alternatively, specific metrics that are directly and physically linked to climate sensitivity are sought; examples of this are the global patterns of warming,[80] the ability of the models to reproduce observed relative humidity in the tropics and sub-tropics,[81] patterns of radiation,[82] and the variability of temperature about long term historical warming.[83][84][85] When using ensemble climate models developed in different institutions, many of these constrained estimates of ECS are slightly higher than 3 °C (5.4 °F); as the models with ECS slightly above 3 °C (5.4 °F) perform better in these metrics than models with a low climate sensitivity.[86]
Socio-economic implications[edit]
Because the economics of climate change mitigation depend a lot on how quickly carbon neutrality needs to be achieved, climate sensitivity is very important economically: one study suggests that halving the uncertainty of the transient climate response could save trillions of dollars.[87] It has been argued that uncertainty in the value of climate sensitivity implies that it becomes more prudent to take climate action as there will be higher tail risks.[88]
^ Here the IPCC definition is used. In some other sources, the climate sensitivity parameter is simply called the climate sensitivity. The inverse of this parameter, is called the climate feedback parameter and is expressed in (W/m2)/°C.
^ This calculation goes as follows. In equilibrium, the energy of incoming and outgoing radiation have to balance. The outgoing radiation F {\displaystyle F} is given by the Stefan-Boltzmann law: F = − σ T 4 {\displaystyle F=-\sigma T^{4}} . When incoming radiation increases, the outgoing radiation, and therefore temperature has to increase as well. The temperature rise Δ T 2 × C O 2 {\displaystyle \Delta T_{2\times CO_{2}}} for the additional radiative forcing Δ F 2 × C O 2 {\displaystyle \Delta F_{2\times CO_{2}}} due to doubling of CO2 is then given by
Δ F 2 × C O 2 = d F d T Δ T 2 × C O 2 = 4 σ T 3 Δ T 2 × C O 2 {\displaystyle \Delta F_{2\times CO_{2}}={\frac {dF}{dT}}\Delta T_{2\times CO_{2}}=4\sigma T^{3}\Delta T_{2\times CO_{2}}} .
Given an effective temperature of 255 K, a constant lapse rate, the value of the Stefan-Boltzmann constant σ {\displaystyle \sigma } of 5.67 × 10 − 8 {\displaystyle \times 10^{-8}} W/m2 K-4 and F 2 × C O 2 {\displaystyle F_{2\times CO_{2}}} around 4 W/m2, this gives a climate sensitivity of a world without feedbacks of approximately 1 K.
^ a b c Edited quote from public-domain source: Lindsey, Rebecca (3 August 2010), What if global warming isn't as severe as predicted? : Climate Q&A : Blogs, NASA Earth Observatory, part of the EOS Project Science Office, located at NASA Goddard Space Flight Center
^ Roe, Gerald H.; Baker, Marcia B. (2007). "Why Is Climate Sensitivity So Unpredictable?". Science. 318 (5850): 629–632. Bibcode:2007Sci...318..629R. doi:10.1126/science.1144735. ISSN 0036-8075. PMID 17962560.
^ PALAEOSENS Project Members (2012). "Making sense of palaeoclimate sensitivity" (PDF). Nature. 491 (7426): 683–91. Bibcode:2012Natur.491..683P. doi:10.1038/nature11574. hdl:2078.1/118863. PMID 23192145.
^ Modak, Angshuman; Bala, Govindasamy; Cao, Long; Caldeira, Ken (2016). "Why must a solar forcing be larger than a CO2forcing to cause the same global mean surface temperature change?". Environmental Research Letters. 11 (4): 044013. Bibcode:2016ERL....11d4013M. doi:10.1088/1748-9326/11/4/044013. ISSN 1748-9326.
^ Held, Isaac and Winton, Mike. Transient and Equilibrium Climate Sensitivity, Geophysical Fluid Dynamics Laboratory, U.S. National Oceanic & Atmospheric Administration. Retrieved 2019-06-19.
^ a b c d e "Explainer: How scientists estimate climate sensitivity". Carbon Brief. 2018-06-19. Retrieved 2019-03-14.
^ Lenton, Timothy M.; Rockström, Johan; Gaffney, Owen; Rahmstorf, Stefan; Richardson, Katherine; Steffen, Will; Schellnhuber, Hans Joachim (2019-11-27). "Climate tipping points — too risky to bet against". Nature. 575 (7784): 592–595. Bibcode:2019Natur.575..592L. doi:10.1038/d41586-019-03595-0. PMID 31776487.
^ Roe, Gerard (2009). "Feedbacks, Timescales, and Seeing Red". Annual Review of Earth and Planetary Sciences. 37 (1): 93–115. Bibcode:2009AREPS..37...93R. doi:10.1146/annurev.earth.061008.134734.
^ a b c d Rahmstorf, Stefan (2008). "Anthropogenic Climate Change: Revisiting the Facts". In Zedillo, Ernesto (ed.). Global Warming: Looking Beyond Kyoto (PDF). Brookings Institution Press. pp. 34–53.
^ Knutti, Reto; Rugenstein, Maria A. A.; Knutti, Reto (2017). "Beyond equilibrium climate sensitivity". Nature Geoscience. 10 (10): 727–736. Bibcode:2017NatGe..10..727K. doi:10.1038/ngeo3017. hdl:20.500.11850/197761. ISSN 1752-0908.
^ Gregory, J. M.; et al. (2004). "A new method for diagnosing radiative forcing and climate sensitivity". Geophysical Research Letters. 31 (3): L03205. Bibcode:2004GeoRL..31.3205G. doi:10.1029/2003GL018747.
^ Bindoff, Nathaniel L.; Stott, Peter A. (2013). "10.8.2 Constraints on Long-Term Climate Change and the Equilibrium Climate Sensitivity" (PDF). Climate Change 2013: The Physical Science Basis - IPCC Working Group I Contribution to AR5. Geneva, Switzerland: Intergovernmental Panel on Climate Change.
^ a b Solomon, S. D.; et al., eds. (2007). "Glossary A-D, Climate sensitivity" (PDF). Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, 2007. Cambridge University Press.
^ Bitz, C. M.; Shell, K. M.; Gent, P. R.; Bailey, D. A.; Danabasoglu, G.; Armour, K. C.; Holland, M. M.; Kiehl, J. T. (2011). "Climate Sensitivity of the Community Climate System Model, Version 4". Journal of Climate. 25 (9): 3053–3070. doi:10.1175/JCLI-D-11-00290.1. ISSN 0894-8755.
^ Prentice, I. C.; et al. (2001). "9.2.1 Climate Forcing and Climate Response, in chapter 9. Projections of Future Climate Change" (PDF). In Houghton, J. T.; et al. (eds.). Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press. ISBN 9780521807678.
^ Randall, D. A.; et al. (2007). "8.6.2 Interpreting the Range of Climate Sensitivity Estimates Among General Circulation Models, In: Climate Models and Their Evaluation.". In Solomon, S. D.; et al. (eds.). Climate Change 2007: Synthesis Report. Contribution of Working Groups I, II and III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press.
^ Hansen, James; Sato, Makiko; Kharecha, Pushker; von Schuckmann, Karina (2011). "Earth's energy imbalance and implications". Atmospheric Chemistry and Physics. 11 (24): 13, 421–49. arXiv:1105.1140. Bibcode:2011ACP....1113421H. doi:10.5194/acp-11-13421-2011.
^ Collins et al. 2013, Executive Summary; p.1033
^ Matthews, H. D.; Gillett, N. P.; Stott, P. A.; Zickfeld, K. (2009). "The proportionality of global warming to cumulative carbon emissions". Nature. 459 (7248): 829–832. Bibcode:2009Natur.459..829M. doi:10.1038/nature08047. PMID 19516338.
^ "Target CO
2". RealClimate. 7 April 2008. Archived from the original on 24 August 2017.
^ "On sensitivity: Part I". RealClimate.org.
^ Previdi, M.; et al. (2013). "Climate sensitivity in the Anthropocene". Quarterly Journal of the Royal Meteorological Society. 139 (674): 1121–31. Bibcode:2013QJRMS.139.1121P. CiteSeerX 10.1.1.434.854. doi:10.1002/qj.2165.
^ "Explained: Radiative forcing". MIT News. Retrieved 2019-03-30.
^ Climate Change: The IPCC Scientific Assessment (1990), Report prepared for Intergovernmental Panel on Climate Change by Working Group I, J. T. Houghton, G. J. Jenkins and J. J. Ephraums (eds.), chapter 2, Radiative Forcing of Climate Archived 2018-08-08 at the Wayback Machine, pp. 41–68
^ Myhre et al. 2013
^ Gregory, J. M.; Andrews, T. (2016). "Variation in climate sensitivity and feedback parameters during the historical period" (PDF). Geophysical Research Letters. 43 (8): 3911–3920. Bibcode:2016GeoRL..43.3911G. doi:10.1002/2016GL068406.
^ Marvel, Kate; Schmidt, Gavin A.; Miller, Ron L.; Nazarenko, Larissa S. (2016). "Implications for climate sensitivity from the response to individual forcings". Nature Climate Change. 6 (4): 386–389. Bibcode:2016NatCC...6..386M. doi:10.1038/nclimate2888. hdl:2060/20160012693. ISSN 1758-6798.
^ Robert Pincus; Mauritsen, Thorsten (2017). "Committed warming inferred from observations". Nature Climate Change. 7 (9): 652–655. Bibcode:2017NatCC...7..652M. doi:10.1038/nclimate3357. ISSN 1758-6798.
^ a b Pfister, Patrik L.; Stocker, Thomas F. (2017). "State-Dependence of the Climate Sensitivity in Earth System Models of Intermediate Complexity". Geophysical Research Letters. 44 (20): 10, 643–10, 653. Bibcode:2017GeoRL..4410643P. doi:10.1002/2017GL075457. ISSN 1944-8007.
^ Lontzek, Thomas S.; Lenton, Timothy M.; Cai, Yongyang (2016). "Risk of multiple interacting tipping points should encourage rapid CO2 emission reduction". Nature Climate Change. 6 (5): 520–525. Bibcode:2016NatCC...6..520C. doi:10.1038/nclimate2964. hdl:10871/20598. ISSN 1758-6798.
^ "Opinion: Europe is burning just as scientists offer a chilling truth about climate change". The Independent. 2019-07-24. Retrieved 2019-07-26.
^ Nijsse, Femke J. M. M.; Cox, Peter M.; Huntingford, Chris; Williamson, Mark S. (2019). "Decadal global temperature variability increases strongly with climate sensitivity". Nature Climate Change. 9 (8): 598–601. Bibcode:2019NatCC...9..598N. doi:10.1038/s41558-019-0527-4. ISSN 1758-6798.
^ Walter, Martin E. (2010). "Earthquakes and Weatherquakes: Mathematics and Climate Change" (PDF). Notices of the American Mathematical Society. 57 (10): 1278–84.
^ Lapenis, Andrei G. (1998). "Arrhenius and the Intergovernmental Panel on Climate Change". Eos, Transactions American Geophysical Union. 79 (23): 271. Bibcode:1998EOSTr..79..271L. doi:10.1029/98EO00206. ISSN 2324-9250.
^ Sample, Ian (2005-06-30). "The father of climate change". The Guardian. ISSN 0261-3077. Retrieved 2019-03-18.
^ Anderson, Thomas R.; Hawkins, Ed; Jones, Philip D. (2016). "CO
2, the greenhouse effect and global warming: from the pioneering work of Arrhenius and Callendar to today's Earth System Models". Endeavour. 40 (3): 178–187. doi:10.1016/j.endeavour.2016.07.002. PMID 27469427.
^ Manabe S.; Wetherald R. T. (May 1967). "Thermal Equilibrium of the Atmosphere with a Given Distribution of Relative Humidity". Journal of the Atmospheric Sciences. 24 (3): 241–259. Bibcode:1967JAtS...24..241M. doi:10.1175/1520-0469(1967)024<0241:teotaw>2.0.co;2.
^ Forster, Piers (18 May 2017). "In retrospect: Half a century of robust climate models" (PDF). Nature. 545 (7654): 296–297. Bibcode:2017Natur.545..296F. doi:10.1038/545296a. PMID 28516918. Retrieved 2019-10-19.
^ Pidcock, Roz (July 6, 2015) "The most influential climate change papers of all time", CarbonBrief. Retrieved 2019-10-19.
^ Ad Hoc Study Group on Carbon Dioxide and Climate (1979). Carbon Dioxide and Climate: A Scientific Assessment (PDF). National Academy of Sciences. doi:10.17226/12181. ISBN 978-0-309-11910-8. Archived from the original (PDF) on 13 August 2011.
^ Kerr, Richard A. (2004). "Three Degrees of Consensus". Science. 305 (5686): 932–4. doi:10.1126/science.305.5686.932. PMID 15310873.
^ a b Meehl, G. A.; et al., "Ch. 10: Global Climate Projections; Box 10.2: Equilibrium Climate Sensitivity", IPCC Fourth Assessment Report WG1 2007
^ a b Solomon, S.; et al., "Technical summary", Climate Change 2007: Working Group I: The Physical Science Basis, Box TS.1: Treatment of Uncertanties in the Working Group I Assessment , in IPCC AR4 WG1 2007
^ Forster, Piers M. (2016). "Inference of Climate Sensitivity from Analysis of Earth's Energy Budget". Annual Review of Earth and Planetary Sciences. 44 (1): 85–106. Bibcode:2016AREPS..44...85F. doi:10.1146/annurev-earth-060614-105156.
^ Climate Change: The IPCC Scientific Assessment (1990), Report prepared for Intergovernmental Panel on Climate Change by Working Group I, J. T. Houghton, G. J. Jenkins and J. J. Ephraums (eds.), chapter 5, Equilibrium Climate Change — and its Implications for the Future Archived 2018-04-13 at the Wayback Machine, pp. 138–9
^ IPCC '92 p118 section B3.5
^ IPCC SAR p 34, technical summary section D.2
^ Albritton, D. L.; et al. (2001). "Technical Summary: F.3 Projections of Future Changes in Temperature". In Houghton J. T.; et al. (eds.). Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press. Archived from the original on 12 January 2012.
^ a b This article incorporates public domain material from the US Environmental Protection Agency (US EPA) document: US EPA (7 December 2009), "Ch. 6: Projected Future Greenhouse Gas Concentrations and Climate Change: Box 6.3: Climate sensitivity" (PDF), Technical Support Document for Endangerment and Cause or Contribute Findings for Greenhouse Gases under Section 202(a) of the Clean Air Act, Washington, DC, USA: Climate Change Division, Office of Atmospheric Programs, US EPA , p.66 (78 of PDF file)
^ a b Skeie, R. B.; Bernsten, T.; Aldrin, M.; Holden, M.; Myhre, G. (2014). "A lower and more constrained estimate of climate sensitivity using updated observations and detailed radiative forcing time series". Earth System Dynamics. 5 (1): 139–75. Bibcode:2014ESD.....5..139S. doi:10.5194/esd-5-139-2014.
^ Armour, Kyle C. (2017). "Energy budget constraints on climate sensitivity in light of inconstant climate feedbacks". Nature Climate Change. 7 (5): 331–335. Bibcode:2017NatCC...7..331A. doi:10.1038/nclimate3278. ISSN 1758-6798.
^ Forster, Piers M. de F.; Gregory, Jonathan M. (2006). "The Climate Sensitivity and Its Components Diagnosed from Earth Radiation Budget Data". Journal of Climate. 19 (1): 39–52. Bibcode:2006JCli...19...39F. doi:10.1175/JCLI3611.1.
^ Lewis, Nicholas; Curry, Judith A. (2014). "The implications for climate sensitivity of AR5 forcing and heat uptake estimates". Climate Dynamics. 45 (3–4): 1009–23. Bibcode:2015ClDy...45.1009L. doi:10.1007/s00382-014-2342-y.
^ Otto, Alexander; Otto, Frederieke, E. L.; Boucher, Olivier; Church, John; Hegerl, Gabi; Forster, Piers M.; Gillet, Nathan P.; Gregory, Jonathan; Johnson, Gregory C.; Knutti, Reto; Lewis, Nicholas; Lohmann, Ulrike; Marotzke, Jochem; Myhre, Gunnar; Shindell, Drew; Stevens, Bjorn; Allen, Myles R. (2013). "Energy budget constraints on climate response" (PDF). Nature Geoscience. 6 (6): 415–416. Bibcode:2013NatGe...6..415O. doi:10.1038/ngeo1836. ISSN 1752-0908.
^ Stolpe, Martin B.; Ed Hawkins; Cowtan, Kevin; Richardson, Mark (2016). "Reconciled climate response estimates from climate models and the energy budget of Earth" (PDF). Nature Climate Change. 6 (10): 931–935. Bibcode:2016NatCC...6..931R. doi:10.1038/nclimate3066. ISSN 1758-6798.
^ Schwartz, Stephen E. (2007). "Heat capacity, time constant, and sensitivity of Earth's climate system". Journal of Geophysical Research: Atmospheres. 112 (D24): D24S05. Bibcode:2007JGRD..11224S05S. CiteSeerX 10.1.1.482.4066. doi:10.1029/2007JD008746.
^ Knutti, R.; Kraehenmann, S.; Frame, D. J.; Allen, M. R. (2008). "Comment on "Heat capacity, time constant, and sensitivity of Earth's climate system" by SE Schwartz". Journal of Geophysical Research: Atmospheres. 113 (D15): D15103. Bibcode:2008JGRD..11315103K. doi:10.1029/2007JD009473.
^ Foster, G.; Annan, J. D.; Schmidt, G. A.; Mann, M. E. (2008). "Comment on "Heat capacity, time constant, and sensitivity of Earth's climate system" by SE Schwartz". Journal of Geophysical Research: Atmospheres. 113 (D15): D15102. Bibcode:2008JGRD..11315102F. doi:10.1029/2007JD009373.
^ Scafetta, N. (2008). "Comment on "Heat capacity, time constant, and sensitivity of Earth's climate system" by SE Schwartz". Journal of Geophysical Research: Atmospheres. 113 (D15): D15104. Bibcode:2008JGRD..11315104S. doi:10.1029/2007JD009586.
^ Tung, Ka Kit; Zhou, Jiansong; Camp, Charles D. (2008). "Constraining model transient climate response using independent observations of solar-cycle forcing and response" (PDF). Geophysical Research Letters. 35 (17): L17707. Bibcode:2008GeoRL..3517707T. doi:10.1029/2008GL034240.
^ Camp, Charles D.; Tung, Ka Kit (2007). "Surface warming by the solar cycle as revealed by the composite mean difference projection" (PDF). Geophysical Research Letters. 34 (14): L14703. Bibcode:2007GeoRL..3414703C. doi:10.1029/2007GL030207. Archived from the original (PDF) on 13 January 2012. Retrieved 20 January 2012.
^ Rypdal, K. (2012). "Global temperature response to radiative forcing: Solar cycle versus volcanic eruptions". Journal of Geophysical Research: Atmospheres. 117 (D6): n/a. Bibcode:2012JGRD..117.6115R. doi:10.1029/2011JD017283. ISSN 2156-2202.
^ Merlis, Timothy M.; Held, Isaac M.; Stenchikov, Georgiy L.; Zeng, Fanrong; Horowitz, Larry W. (2014). "Constraining Transient Climate Sensitivity Using Coupled Climate Model Simulations of Volcanic Eruptions". Journal of Climate. 27 (20): 7781–7795. Bibcode:2014JCli...27.7781M. doi:10.1175/JCLI-D-14-00214.1. hdl:10754/347010. ISSN 0894-8755.
^ McSweeney, Robert (2015-02-04). "What a three-million year fossil record tells us about climate sensitivity". Carbon Brief. Retrieved 2019-03-20.
^ Hargreaves, Julia C.; Annan, James D. (2009). "On the importance of paleoclimate modelling for improving predictions of future climate change" (PDF). Climate of the Past. 5.
^ a b c Masson-Delmotte et al. 2013
^ a b Hopcroft, Peter O.; Valdes, Paul J. (2015). "How well do simulated last glacial maximum tropical temperatures constrain equilibrium climate sensitivity?: CMIP5 LGM TROPICS AND CLIMATE SENSITIVITY". Geophysical Research Letters. 42 (13): 5533–5539. doi:10.1002/2015GL064903.
^ Ganopolski, Andrey; von Deimling, Thomas Schneider (2008). "Comment on "Aerosol radiative forcing and climate sensitivity deduced from the Last Glacial Maximum to Holocene transition" by Petr Chylek and Ulrike Lohmann". Geophysical Research Letters. 35 (23): L23703. Bibcode:2008GeoRL..3523703G. doi:10.1029/2008GL033888.
^ Schmittner, Andreas; Urban, Nathan M.; Shakun, Jeremy D.; Mahowald, Natalie M.; Clark, Peter U.; Bartlein, Patrick J.; Mix, Alan C.; Rosell-Melé, Antoni (2011). "Climate Sensitivity Estimated from Temperature Reconstructions of the Last Glacial Maximum". Science. 334 (6061): 1385–8. Bibcode:2011Sci...334.1385S. CiteSeerX 10.1.1.419.8341. doi:10.1126/science.1203513. PMID 22116027.
^ Hargreaves, J. C.; Annan, J. D.; Yoshimori, M.; Abe‐Ouchi, A. (2012). "Can the Last Glacial Maximum constrain climate sensitivity?". Geophysical Research Letters. 39 (24): L24702. Bibcode:2012GeoRL..3924702H. doi:10.1029/2012GL053872. ISSN 1944-8007.
^ von der Heydt, A. S.; Köhler, P.; van de Wal, R. S. W.; Dijkstra, H. A. (2014). "On the state dependency of fast feedback processes in (paleo) climate sensitivity". Geophysical Research Letters. 41 (18): 6484–6492. arXiv:1403.5391. doi:10.1002/2014GL061121. ISSN 1944-8007.
^ Royer, Dana L.; Berner, Robert A.; Park, Jeffrey (2007). "Climate sensitivity constrained by CO
2 concentrations over the past 420 million years". Nature. 446 (7135): 530–2. Bibcode:2007Natur.446..530R. doi:10.1038/nature05699. PMID 17392784.
^ Kiehl, Jeffrey T.; Shields, Christine A. (2013). "Sensitivity of the Palaeocene–Eocene Thermal Maximum climate to cloud properties". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 371 (2001): 20130093. Bibcode:2013RSPTA.37130093K. doi:10.1098/rsta.2013.0093. PMID 24043867.
^ "The CMIP6 landscape". Nature Climate Change. 9 (10): 727. 2019-09-25. Bibcode:2019NatCC...9..727.. doi:10.1038/s41558-019-0599-1. ISSN 1758-6798.
^ "New climate models suggest Paris goals may be out of reach". France 24. 2020-01-14. Retrieved 2020-01-18.
^ "Two French climate models consistently predict a pronounced global warming". CNRS. Retrieved 19 September 2019.
^ "New climate models predict a warming surge". American Association for the Advancement of Science. 16 April 2019.
^ Hood, Marlowe (2019-09-17). "Earth to warm more quickly, new climate models show". phys.org. Retrieved 2019-09-17.
^ Sanderson, Benjamin M.; Knutti, Reto; Caldwell, Peter (2015). "Addressing Interdependency in a Multimodel Ensemble by Interpolation of Model Properties". Journal of Climate. 28 (13): 5150–5170. Bibcode:2015JCli...28.5150S. doi:10.1175/JCLI-D-14-00361.1. ISSN 0894-8755.
^ Forest, Chris E.; Stone, Peter H.; Sokolov, Andrei P.; Allen, Myles R.; Webster, Mort D. (2002). "Quantifying uncertainties in climate system properties with the use of recent observations" (PDF). Science. 295 (5552): 113–7. Bibcode:2002Sci...295..113F. CiteSeerX 10.1.1.297.1145. doi:10.1126/science.1064419. PMID 11778044.
^ Fasullo, John T.; Trenberth, Kevin E. (2012), "A Less Cloudy Future: The Role of Subtropical Subsidence in Climate Sensitivity", Science, 338 (6108): 792–94, Bibcode:2012Sci...338..792F, doi:10.1126/science.1227465, PMID 23139331 . Referred to by: ScienceDaily (8 November 2012), Future warming likely to be on high side of climate projections, analysis finds, ScienceDaily
^ Caldeira, Ken; Brown, Patrick T. (2017). "Greater future global warming inferred from Earth's recent energy budget". Nature. 552 (7683): 45–50. Bibcode:2017Natur.552...45B. doi:10.1038/nature24672. ISSN 1476-4687. PMID 29219964.
^ Cox, Peter M.; Huntingford, Chris; Williamson, Mark S. (2018). "Emergent constraint on equilibrium climate sensitivity from global temperature variability" (PDF). Nature. 553 (7688): 319–322. Bibcode:2018Natur.553..319C. doi:10.1038/nature25450. ISSN 0028-0836. PMID 29345639.
^ Brown, Patrick T.; Stolpe, Martin B.; Caldeira, Ken (2018). "Assumptions for emergent constraints". Nature. 563 (7729): E1–E3. Bibcode:2018Natur.563E...1B. doi:10.1038/s41586-018-0638-5. ISSN 0028-0836. PMID 30382203.
^ Cox, Peter M.; Williamson, Mark S.; Nijsse, Femke J. M. M.; Huntingford, Chris; et al. (2018). "Cox et al. reply". Nature. 563 (7729): E10–E15. Bibcode:2018Natur.563E..10C. doi:10.1038/s41586-018-0641-x. ISSN 0028-0836. PMID 30382204.
^ Caldwell, Peter M.; Zelinka, Mark D.; Klein, Stephen A. (2018). "Evaluating Emergent Constraints on Equilibrium Climate Sensitivity". Journal of Climate. 31 (10): 3921–3942. Bibcode:2018JCli...31.3921C. doi:10.1175/JCLI-D-17-0631.1. ISSN 0894-8755.
^ Hope, Chris (2015). "The $10 trillion value of better information about the transient climate response". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 373 (2054): 20140429. Bibcode:2015RSPTA.37340429H. doi:10.1098/rsta.2014.0429. ISSN 1364-503X. PMID 26438286.
^ Freeman, Mark C.; Wagner, Gernot; Zeckhauser, Richard J. (2015-11-28). "Climate sensitivity uncertainty: when is good news bad?". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 373 (2055): 20150092. Bibcode:2015RSPTA.37350092F. doi:10.1098/rsta.2015.0092. PMID 26460117.
IPCC AR4 WG1 (2007), Solomon, S.; Qin, D.; Manning, M.; Chen, Z.; Marquis, M.; Averyt, K. B.; Tignor, M.; Miller, H. L. (eds.), Climate Change 2007: The Physical Science Basis, Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, ISBN 978-0-521-88009-1 (pb: 978-0-521-70596-7).
IPCC (2013). Stocker, T. F.; Qin, D.; Plattner, G.-K.; Tignor, M.; et al. (eds.). Climate Change 2013: The Physical Science Basis (PDF). Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press. ISBN 978-1-107-05799-9. (pb: 978-1-107-66182-0).
Masson-Delmotte, V.; Schulz, M.; Abe-Ouchi, A.; Beer, J.; et al. (2013). "Chapter 5: Information from Paleoclimate Archives" (PDF). IPCC AR5 WG1 2013. pp. 383–464.
Myhre, G.; Shindell, D.; Bréon, F.-M.; Collins, W.; et al. (2013). "Chapter 8: Anthropogenic and Natural Radiative Forcing" (PDF). IPCC AR5 WG1 2013. pp. 659–740.
Collins, M.; Knutti, R.; Arblaster, J. M.; Dufresne, J.-L.; et al. (2013). "Chapter 12: Long-term Climate Change: Projections, Commitments and Irreversibility" (PDF). IPCC AR5 WG1 2013. pp. 1029–1136.
What is 'climate sensitivity'? Met Office
How scientists estimate 'climate sensitivity' Carbon Brief
Attribution of recent climate change
Effects of global warming
By country and region
Scientific consensus on climate change
Deforestation and climate change
History of climate change science
Svante Arrhenius
Charles David Keeling
Effects and issues
Abrupt climate change
Anoxic event
Arctic methane emissions
Physical impacts
Retreat of glaciers since 1850
Runaway climate change
Season creep
Shutdown of thermohaline circulation
Climate change and ecosystems
Effects on plant biodiversity
Effects on marine mammals
Effects on terrestrial animals
Extinction risk from global warming
Forest dieback
Climate change and gender
Climate change and poverty
Economics of global warming
Effects on health
Effects on humans
Environmental migrant
Fisheries and climate change
By country & region
Society and climate change
Politics of global warming
Public opinion on global warming
Climate change opinion by country
Soft climate change denial
Climate movement
Media coverage of global warming
Global warming in popular culture
Carbon credit
Low-carbon energy
Individual action on climate change
Carbon neutrality
Carbon dioxide removal
Climate change mitigation scenarios
Climate engineering
Reducing emissions from deforestation and forest degradation
Background and theory
Satellite measurements
Carbon sink
Climate change (general concept)
Cloud forcing
Effective temperature
Earth's energy budget
Global warming potential
Radiative forcing
Paleotempestology
Retrieved from "https://en.wikipedia.org/w/index.php?title=Climate_sensitivity&oldid=936453372" | CommonCrawl |
\begin{document}
\renewcommand{\arabic{section}}{\arabic{section}}
\newcommand{\partial}{\partial} \newcommand{\vskip 1pc}{\vskip 1pc} \newcommand{\mbox{co}}{\mbox{co}}
\newcommand{\calT} {\mathcal T}
\title{Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations}
\author{Kyungkeun Kang and Jae-Myoung Kim} \date{} \maketitle
\begin{abstract} We present some new regularity criteria for suitable weak solutions of magnetohydrodynamic equations near boundary in dimension three. We prove that suitable weak solutions are H\"older continuous near boundary provided that either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\le 2$, $2<q<\infty$, or the scaled $L^{p,q}_{x,t}$-norm of the vorticity with $3/p+2/q\le 3$, $2<q<\infty$ are sufficiently small near the boundary.
\end{abstract}
\section{Introduction}
We study the regularity problem for {\it suitable }weak solutions $(u, b, \pi): Q_T\rightarrow { \mathbb{R} }^3\times{ \mathbb{R} }^3\times{ \mathbb{R} }$ of the three-dimensional incompressible magnetohydrodynamic (MHD) equations \begin{equation}\label{MHD}
\left\{ \begin{array}{ll} \displaystyle u_t- \triangle u +(u \cdot \nabla) u - (b\cdot \nabla) b +\nabla \pi= 0\\
\\ \displaystyle b_t - \triangle b +(u \cdot \nabla) b - (b\cdot \nabla) u = 0\\
\\ \displaystyle \text{div} \ u =0 \quad \text{and}\quad \text{div} \ b=0 ,\\
\\ \displaystyle u(x,0)=u_0(x), \quad b(x,0)=b_0(x) \end{array}\right. \,\,\, \mbox{ in } \,\,Q_T:={ \mathbb{R} }^3_{+}\times [0,\, T). \end{equation} Here $u$ is the flow velocity vector, $b$ is the magnetic vector and $\displaystyle\pi=p+ \frac{\abs{b}^2}{2}$ is the magnetic pressure. The boundary conditions of $u$ and $b$ are given as no-slip and slip conditions, respectively, namely \begin{equation}\label{noslip-slip} u=0 \quad \text{and}\quad b\cdot \nu=0,\, \ (\nabla \times b)\times \nu=0,\qquad \mbox{ on }\,\,\partial{ \mathbb{R} }^3_+, \end{equation} where $\nu=(0,0,-1)$ is the outward unit normal vector along boundary $\partial{ \mathbb{R} }^3_+$. By suitable weak solutions we mean solutions that solve MHD equations in the sense of distribution and satisfy the local energy inequality (see Definition \ref{sws-3dnse} in section 2 for details).
The MHD equations describe the dynamics of the interaction of moving conducting fluids with electro-magnetic fields which are frequently observed in nature and industry, e.g., plasma liquid metals, gases, two-phase mixtures (see e.g. \cite{D} and \cite{DL}).
Let $x=(x_1,x_2,0)\in\partial{ \mathbb{R} }^3_+$. For a point $z=(x,t)\in \partial{ \mathbb{R} }^3_+\times (0,T)$, we denote \[ B_{x,r}=:\{y\in{ \mathbb{R} }^3: \abs{y-x}<r\},\quad B^{+}_{x,r}:=\{y \in B_{x,r}: y_3>0 \}, \] \[ Q_{z,r}=:B_{x,r}\times (t-r^2,t),\quad Q^{+}_{z,r}:=\{(y,t) \in Q_{z,r}: y_3>0 \},\quad r<\sqrt{t}. \] We say that solutions $u$ and $b$ are regular at $z\in \overline{{ \mathbb{R} }^3_+}\times (0,T)$ if $u$ and $b$ are H\"older continuous for some $Q^+_{z,r}$, $r>0$. Otherwise, it is said that $u$ and $b$ are singular at $z$.
\begin{comment}
It was shown in \cite{DL} that weak solutions for MHD exist globally
in time and in the two-dimensional case weak solutions become
regular. In the three-dimensional case, as shown in \cite{ST83}, if
a weak solution pair $(u,\, b)$ are additionally in $L^{\infty}(0,
\, T; H^1({\mathbb{R}}^3))$, they become regular. As in the
Navier-Stokes equations, regularity question, however, remains open
in dimension three. Although many significant contributions have
been made on the existence, uniqueness and regularity of weak
solutions to the MHD equations, we list only some results relevant
to our concern.
In case that $\Omega={ \mathbb{R} }^3$, He and Xin proved in \cite{He-Xin05-jde}
that a weak solution pair $(u, b)$ become regular in the presence of
a certain type of scaling invariant integral conditions for velocity
field, often referred as Serrin's condition. Recently, the authors
have obtained similar results in the case that $\Omega$ is a bounded
domain or half space (see \cite[Theorem 1]{KK12}). For a local
case, various types of $\epsilon$ regularity criteria for suitable
weak solutions have been also established in terms of scaled norms
(see e.g. \cite{He-Xin05-jfa}, \cite{KL09} and \cite{Vya08} for
interior case and \cite{Vya10} for boundary case). Here we emphasize
that for the global case, additional conditions are imposed on only
velocity field but not on the magnetic field,
however, require control of some scaled norms of magnetic fields as
well as those of the velocity field.
If there was no magnetic field $b$ in MHD equations, it seems to be
Navier-Stokes equations. Concerning this equations, we can see the
results (see e.g., \cite{L34}, \cite{P59},
\cite{JS},
\cite{L67}, \cite{FJR72}, \cite{VS76}, \cite{VS77}, \cite{VS80},
\cite{VS82}, \cite{CKN}, \cite{Sohr83}, \cite{S88}, \cite{T90},
\cite{Lin98}, \cite{LS99}, \cite{GAS02},\cite{ESS03}, \cite{K04}, \cite{VAS},
\cite{GKT06}, \cite{SSS06},
\cite{GKT07}, \cite{AS07}) for one's information
The motivation of our study is to establish new regularity criteria
for MHD depending only on velocity fields for local cases. To be
more precise, main objective of this paper is to present new
sufficient conditions, not relying on magnetic fields, for the
regularity of suitable weak solutions to the MHD near boundary as
well as in the interior.
While preparing this paper, the authors have become to know that,
very recently, Wang and Zhang showed that local interior regularity
can be ensured by the control of only scaled norm of velocity
fields. More precisely, interior regularity criteria shown in
\cite{WZ12} is the following:
\begin{equation}\label{wz-int-rc}
\limsup_{r\rightarrow 0}r^{-(\frac{3}{p}+\frac{2}{q}-1)}
\norm{\norm{u}_{L^p(B_{x,r})}}_{L^q(t-r^2,t)}<\epsilon,
\end{equation}
where $ 1\leq \frac{3}{p}+\frac{2}{q}\leq 2$ with $1\leq q\leq
\infty$. We have also proved independently the same result as in
\cite{WZ12} and since we think that our proof is a different version
to that in \cite{WZ12}, its details are given in Appendix. Our main
concern is, however, for local boundary case and we obtain similar
results near boundary as in the interior case. Let
$x_0\in\partial{ \mathbb{R} }^3_+$ be a boundary point, which lies on the
boundary of a half-space in $R^3$. We expect that our analysis would also hold
in a smooth boundary as in the case of flat boundary, but our study
is restricted, in this paper, to the flat boundary of a half space.
\end{comment}
We list some known results for MHD equations relevant to our concern, in particular regarding regularity conditions in terms of scaled invariant quantities.
It was shown in \cite{DL} that weak solutions for MHD equations exist globally in time and in the two-dimensional case weak solutions become regular (compare to \cite{L34} and \cite{H} for the NSE). In the three-dimensional case, as shown in \cite{ST83}, if a weak solution pair $(u,\, b)$ are additionally in $L^{\infty}(0,
\, T; H^1({\mathbb{R}}^3))$, $(u,\, b)$ become regular. Although many significant contributions have been made on the existence, uniqueness and regularity of weak solutions to the MHD equations, as in the NSE, regularity question, however, remains open in dimension three.
In case that $\Omega={ \mathbb{R} }^3$, it was proved in \cite{He-Xin05-jde} that a weak solution pair $(u, b)$ become regular if a certain type of scaling invariant integral conditions for velocity field, often referred as Serrin's condition, is additionally assumed (see e.g. \cite{P59}, \cite{JS62}, \cite{L67}, \cite{FJR72}, \cite{Sohr83} for the NSE). Recently, the authors have obtained similar results in the case that $\Omega$ is a bounded domain or half space (see \cite[Theorem 1]{KK12}) (refer to \cite{G86} for the NSE). The local interior case of Serrin's condition including limiting case $L^{3,\infty}_{x,t}$ was treated for MHD equations in \cite{MNS07} (compare to \cite{ESS03}, \cite{AS07} for the NSE).
For a local case, various types of $\epsilon-$regularity criteria for suitable weak solutions have been also established in terms of scaled norms. Among others, it was shown in \cite{Vya10} that suitable weak solutions become regular near a boundary point $z$ if the following conditions are satisfied: There exists $\epsilon>0$ such that \[
\limsup_{r\rightarrow 0}\frac{1}{r} \int_{Q^{+}_{z,r}}|\nabla b(y,s)|^{2}dyds<\infty,\qquad \limsup_{r\rightarrow 0}\frac{1}{r}
\int_{Q^{+}_{z,r}}|\nabla u(y,s)|^{2}dyds<\epsilon. \] Other types of conditions in terms of scaled invariant norms near boundary are also found in \cite{Vya11} (compare to \cite{VS82}, \cite{GAS}, \cite{K04}, \cite{VAS}, \cite{GKT06}, \cite{SSS06}, \cite{W10} for the NSE). We also refer to \cite{He-Xin05-jfa}, \cite{KL09} and \cite{Vya08} in the interior case for MHD equations (compare to \cite{VS76}, \cite{CKN}, \cite{S88}, \cite{T90}, \cite{Lin98}, \cite{LS99}, \cite{GKT07} for the NSE).
Here we emphasize that for the global case, i.e. $\Omega={ \mathbb{R} }^3$, additional conditions are imposed on only velocity field but not on the magnetic field. For local interior and boundary cases, however, known results require control of some scaled norms with scaled factors of magnetic fields as well as those of the velocity fields.
The motivation of our study is to establish new regularity criteria for MHD equations depending only on velocity fields for local cases. To be more precise, main objective of this paper is to present new sufficient conditions, not relying on magnetic fields, for the regularity of suitable weak solutions to the MHD equations near boundary as well as in the interior.
While preparing this paper, the authors have become to know that, very recently, Wang and Zhang showed that local interior regularity can be ensured by the control of only scaled norm of velocity fields. More precisely, interior regularity criteria shown in \cite{WZ12} is the following: \begin{equation}\label{wz-int-rc} \limsup_{r\rightarrow 0}r^{-(\frac{3}{p}+\frac{2}{q}-1)} \norm{\norm{u}_{L^p(B_{x,r})}}_{L^q(t-r^2,t)}<\epsilon, \end{equation} where $ 1\leq \frac{3}{p}+\frac{2}{q}\leq 2$ with $1\leq q\leq \infty$. We have also proved independently the same result as in \cite{WZ12} and since we think that our proof is a different version to that in \cite{WZ12}, its details are given in Appendix (see Theorem \ref{main-theorem-interior}). Our main concern is, however, to obtain new regularity conditions near boundary. Let $x_0\in\partial{ \mathbb{R} }^3_+$ be a boundary point in a half space. We expect that our analysis would also hold in a smooth boundary as in the case of flat boundary, but our study is restricted, in this paper, to the case of ${ \mathbb{R} }^3$, whose boundary is flat.
Now we are ready to state the first part of our main results.
\begin{theorem}\label{main-thm-boundary-v} Let $(u,b,\pi)$ be a suitable weak solution of the MHD equations \eqref{MHD} according to Definition \ref{sws-3dnse}. Suppose that for every pair $p,q$ satisfying $\frac{3}{p}+\frac{2}{q}\leq 2$, $\ 2< q \le\infty$ and $(p,q)\neq (\frac{3}{2}, \infty)$,
there exists $\epsilon>0$ depending only on $p,q$ such that for some point $z=(x,t)\in\partial { \mathbb{R} }_{+}^3\times(0,T)$ $u$ is locally in $L_{x,t}^{p,q}$ near $z$ and \begin{equation}\label{b-con2} \limsup_{r\rightarrow 0}r^{-(\frac{3}{p}+\frac{2}{q}-1)} \norm{\norm{u}_{L^p(B^{+}_{x,r})}}_{L^q(t-r^2,t)}<\epsilon. \end{equation} Then, $u$ and $b$ are regular at $z$. \end{theorem}
\begin{remark} The result in Theorem \ref{main-thm-boundary-v} is also valid in the interior. In fact, the range of $q$ in \eqref{b-con2} in the interior is wider than that of boundary case. To be more precise, the pair $(p,q)$ can be relaxed in the interior as follows: \[ \frac{3}{p}+\frac{2}{q}\leq 2,\qquad 1\le q \le\infty,\quad (p,q)\neq (\frac{3}{2}, \infty). \] As mentioned earlier, in \cite{WZ12} Wang and Zhang showed interior regularity criteria depending only on the control of velocity fields and we also obtain the same result independently. Since the method of proof is a bit different to that in \cite{WZ12}, we present its details in the Appendix for a variety of proof. \end{remark}
Next corollaries are direct consequences of Theorem \ref{main-thm-boundary-v}.
\begin{corollary}\label{Cor-KK1} Let $(u,b,\pi)$ be a suitable weak solution of the MHD equations \eqref{MHD} according to Definition \ref{sws-3dnse}. Suppose that for some point $z=(x,t)\in\partial { \mathbb{R} }_{+}^3\times(0,T)$ $u$ is locally in $L_{x,t}^{p,q}$ near $z$, where $\frac{3}{p}+\frac{2}{q}=1$ with $3<p<\infty$. Then, $u$ and $b$ are regular at $z$. \end{corollary}
It is straightforward to prove Corollary \ref{Cor-KK1} by H\"older's inequality, and thus we skip its details (compare to \cite{MNS07} for local interior case). Next corollary is due to Poincar\'e-Sobolev inequality and the details is again omitted.
\begin{corollary}\label{cor-boundary-v} The same statement of Theorem \ref{main-thm-boundary-v} remains true if the hypothesis including condition \eqref{b-con2} is replaced by the following:\,\, Suppose that for every pair $p,q$ satisfying $2\leq \frac{3}{p}+\frac{2}{q}\leq 3$, $2< q \le\infty$, and $(p,q)\neq (1, \infty)$, there exists $\epsilon>0$ depending only on $p,q$ such that for some point $z=(x,t)\in\partial { \mathbb{R} }_{+}^3\times(0,T)$ $u$ is locally in $L_{x,t}^{p,q}$ near $z$ and \begin{equation}\label{b-con44} \limsup_{r\rightarrow 0}r^{-(\frac{3}{p}+\frac{2}{q}-1)} \norm{\norm{\nabla u}_{L^p(B^{+}_{x,r})}}_{L^q(t-r^2,t)}<\epsilon. \end{equation} \end{corollary}
Considering scaling invariant quantities of vorticity, we can also establish other regularity criteria for vorticity near boundary.
\begin{theorem}\label{main-thm-boundary-w} Let $(u,b,\pi)$ be a suitable weak solution of the MHD equations \eqref{MHD} according to Definition \ref{sws-3dnse}. Suppose that for every pair $p,q$ satisfying $2\leq \frac{3}{p}+\frac{2}{q}\leq 3$, $2< q \le\infty$, and $(p,q)\neq (1, \infty)$, there exists $\epsilon>0$ depending only on $p,q$ such that for some point $z=(x,t)\in\partial { \mathbb{R} }_{+}^3\times(0,T)$ $\omega=\nabla\times u$ is locally in $L_{x,t}^{p,q}$ near $z$ and \begin{equation}\label{b-con3} \limsup_{r\rightarrow 0}r^{-(\frac{3}{\tilde{p}}+\frac{2}{q}-2)} \norm{\norm{\omega}_{L^{p}(B^{+}_{x,r})}}_{L^{q}(t-r^2,t)}<\epsilon. \end{equation} Then, $u$ and $b$ are regular at $z$. \end{theorem}
This paper is organized as follows. In Section 2 we introduce some scaling invariant functionals and the notion of suitable weak solutions. In Section 3 we present the proofs of Theorem \ref{main-thm-boundary-v} and Theorem \ref{main-thm-boundary-w}. In the Appendix, the interior case will be treated and give a detailed a proof with respect to $\epsilon$-regularity criteria for the modified suitable weak solution for MHD equations.
\section{Preliminaries}
In this section we introduce some scaling invariant functionals and suitable weak solutions, and recall
an estimation of the Stokes system.
We first start with some notations. Let $\Omega$ be an open domain in ${ \mathbb{R} }^3$
\ and $I$ be a finite time interval. For $1\le q\le \infty$, we denote the usual Sobolev spaces by $W^{k,q}(\Omega) = \{ u \in L^{q}( \Omega )\,:\, D^{ \alpha }u \in L^{q}( \Omega ), 0 \leq |
\alpha | \leq k \}$. As usual, $W^{k,q}_0(\Omega)$ is the completion of ${\mathcal C}^{\infty}_0(\Omega)$ in the $W^{k,q}(\Omega)$ norm. We also denote by $W^{-k,q'}(\Omega)$ the dual space of $W^{k,q}_0(\Omega)$, where $q$ and $q'$ are H\"older conjugates. We write the average of $f$ on $E$ as $\Xint\diagup_{E} f$, that is $\Xint\diagup_{E}f =\int_{E} f/\abs{E}$. For a function $f(x,t)$, we denote
$\|f\|_{L^{p,q}_{x,t}(\Omega\times I)}=\|f\|_{L^{q}_{t}(I; L^p_x(\Omega))}=\|\|f\|_{L^p_x(\Omega)}\|_{L^q_t(I)}$. For vector fields $u,v$ we write $(u_iv_j)_{i,j=1,2,3}$ as $u\otimes v$. We denote by $C=C(\alpha,\beta,...)$ a constant depending on the prescribed quantities $\alpha,\beta,...$, which may change from line to line.
In this paper, we consider the case that $\Omega={ \mathbb{R} }^3_+$, i.e. a half space in dimension three. For convenience, we denote the boundary of ${ \mathbb{R} }^3_+$ by $\Gamma={ \mathbb{R} }^3\cap\{x_3=0\}$. Next, we introduce scaling invariant quantities near boundary. Let $z=(x,t) \in \Gamma \times I$ and we set \[
A_{u}(r):=\sup_{t-r^{2}\leq s<t}\frac{1}{r}\int_{B^{+}_{x,r}}|u(y,s)|^{2}dy,\quad E_{u}(r):=\frac{1}{r} \int_{Q^{+}_{z,r}}|\nabla u(y,s)|^{2}dyds, \] \[
A_{b}(r):=\sup_{t-r^{2}\leq s<t}\frac{1}{r}\int_{B^{+}_{x,r}}|b(y,s)|^{2}dy,\quad E_{b}(r):=\frac{1}{r} \int_{Q^{+}_{z,r}}|\nabla b(y,s)|^{2}dyds, \] \[
M_{u}(r):=\frac{1}{r^2}\int_{Q^{+}_{z,r}}|u(y,s)|^{3}dyds,\quad M_{b}(r):=\frac{1}{r^2}\int_{Q^{+}_{z,r}}|b(y,s)|^{3}dyds,\quad \] \[
K_{b}(r):=\frac{1}{r^3} \int_{Q^{+}_{z,r}}|b(y,s)|^{2}dyds, \] \[ (u)_r(s):= \Xint\diagup_{B^{+}_{x,r}}u(\cdot,s) dy ,\quad (b)_r(s):= \Xint\diagup_{B^{+}_{x,r}}b(\cdot,s) dy, \quad (\pi)_{r}(s)=\Xint\diagup_{B^{+}_{x,r}}\pi(y,s)dy, \] \[ G_{u,p,q}(r):=r^{1-\frac{3}{p}-\frac{2}{q}}\norm{u(y,s)}_{L^{p,q}_{y,s}(Q^{+}_{z,r})}, \quad D_{u,\tilde{p},q}(r):=r^{2-\frac{3}{\tilde{p}}-\frac{2}{q}}\norm{\nabla u(y,s)}_{L^{\tilde{p},q}_{y,s}(Q^{+}_{z,r})}, \] \[ V_{u,\tilde{p},q}(r):=r^{2-\frac{3}{\tilde{p}}-\frac{2}{q}} \norm{\omega(y,s)}_{L^{\tilde{p},q}_{y,s}(Q^{+}_{z,r})},\qquad \omega=\nabla\times u, \] where $1 \leq p, q \leq \infty$, $3/\tilde{p}+2/q=3$ and $1/p=1/\tilde{p}-1/3$, \begin{equation*}
\tilde{Q}(r):=\frac{1}{r}\biggl(\int^{t}_{t-r^{2}}\Bigl(\int_{B^{+}_{x,r}}|\pi(y,s)-(\pi)_{r}(s)|^{\kappa^{*}}dy \Bigr)^{\frac{\lambda}{\kappa^{*}}}ds\biggr)^{\frac{1}{\lambda}},
\end{equation*} \begin{equation*}
Q(r):=\frac{1}{r}\biggl(\int^{t}_{t-r^{2}}\Bigl(\int_{B^{+}_{x,r}}|\pi(y,s)|^{\kappa^{*}}dy \Bigr)^{\frac{\lambda}{\kappa^{*}}}ds\biggr)^{\frac{1}{\lambda}},
\end{equation*} \begin{equation*} Q_{1}(r):=\frac{1}{r}\biggl(\int^{t}_{t-r^2}\Bigl(\int_{B^{+}_{x,r}}
|\nabla\pi(y,s)|^{\kappa}dy\Bigr)^{\frac{\lambda}{\kappa}}ds\biggr)^{\frac{1}{\lambda}}, \end{equation*} where $\kappa, \kappa^{*}$ and $\lambda$ are numbers satisfying \begin{equation}\label{pq} \frac{3}{\kappa}+\frac{2}{\lambda}=4,\quad\frac{1}{\kappa^{*}} =\frac{1}{\kappa}-\frac{1}{3},
\quad 1<\lambda<2. \end{equation}
\begin{comment} Note that \begin{equation}\label{pq1} \frac{1}{p}+\frac{1}{\kappa^{*}}=1, \quad \frac{1}{q}+\frac{1}{\lambda}=1, \quad \frac{3}{p}+\frac{2}{q}=2, \quad 2<q<\infty. \end{equation} \end{comment}
Next we recall suitable weak solutions for the MHD equations \eqref{MHD} in three dimensions.
\begin{definition}\label{sws-3dnse} Let $\Omega={ \mathbb{R} }^3_+$ and $Q_T={ \mathbb{R} }^3_+\times [0,T)$. A triple of $(u,b,\pi)$ is a suitable weak solution to \eqref{MHD} if the following conditions are satisfied: \begin{itemize} \item[(a)] The functions $u,b : Q_T\rightarrow \mathbb{R}^3$ and $\pi : Q_T \rightarrow \mathbb{R}$ satisfy \begin{equation*} u,b\in L^{\infty}\big(I;L^{2}(\Omega)\big)\cap L^{2}\big(I;W^{1,2}(\Omega)\big),\quad \pi\in L^{\lambda}\big(I;L^{\kappa^*}(\Omega)\big), \end{equation*} \begin{equation*} \nabla^{2}u,\nabla^{2}b\in L^{\lambda}\big(I;L^{k}(\Omega)\big),\quad \nabla \pi\in L^{\lambda}\big(I;L^{\kappa}(\Omega)\big), \end{equation*} where $\kappa,\kappa^*$ and $\lambda$ are numbers in \eqref{pq}.
\item[(b)] ($u,b,\pi$) solves the MHD equations in $Q_T$ in the sense of distributions and $u$ and $b$ satisfy the boundary conditions \eqref{noslip-slip} in the sense of traces.
\item[(c)] $u,b$ and $\pi$ satisfy the local energy inequality \begin{equation*} \int_{B^{+}_{x,r}}(\abs{u(x,t)}^2+\abs{b(x,t)}^2) \phi(x,t) dx \end{equation*} \begin{equation*} +2\int_{t_0}^t\int_{B^{+}_{x,r}} (\abs{\nabla u(x,t')}^{2}+\abs{\nabla b(x,t')}^{2}) \phi(x,t') dx dt' \end{equation*} \begin{equation*} \leq \int_{t_0}^t\int _{B^{+}_{x,r}}(\abs{u}^{2}+\abs{b}^{2})
(\partial_t\phi+\Delta \phi)dxdt'+ \int_{t_0}^t\int _{B^{+}_{x,r}}\bke{ |u|^2 +|b|^2+ 2\pi} u\cdot\nabla \phi dxdt' \end{equation*} \begin{equation}\label{local-energy}
-2\int_{t_0}^t\int _{B^{+}_{x,r}}(b \cdot u)(b \cdot\nabla\phi)dxdt'. \end{equation} for all $t\in I=(0,T)$ and for all nonnegative function $\phi \in C_0^{\infty}({ \mathbb{R} }^3\times R)$.
\end{itemize} \end{definition}
We consider the following Stokes system, which is the linearized Navier-Stokes equations: \begin{equation}\label{stokes-eqn} v_t-\Delta v+\nabla p=f, \quad {\rm{div}} \,v=0 \qquad \mbox{in }\,\,Q_T:=\Omega\times (0,T) \end{equation} with initial data $v(x,0)=v_0(x)$. As in \eqref{noslip-slip}, boundary condition of $v$ is assumed to be no-slip, namely $v(x,t)=0$ for $x\in\partial\Omega$. We recall maximal estimates of the Stokes system in terms of mixed norms (see e.g. \cite[Theorem 5.1]{G86}). \begin{lemma}\label{lem1} Let $1<l,m<\infty$. Suppose that $f\in L^{l,m}_{x,t}(Q_T)$ and $v_0\in D_l^{1-\frac{1}{m},m}$, where $D_l^{1-\frac{1}{m},m}$ is a Banach space with the following norm :
\begin{equation*} D_l^{1-\frac{1}{m},m}(\Omega):= \ \bket{w \in L_{\sigma}^{l}(\Omega) ; \norm{w}_{D_l^{1-\frac{1}{m},m}}=\norm{w}_{L^{l}}+\bke{\int_0^{\infty} \norm{t^{\frac{1}{m}} A_l e^{-t A_l}w}^m_{L^{l}}\frac{dt}{t} }^{\frac{1}{m}}<\infty }, \end{equation*} where $A_l$ is the Stokes operator(see \cite{GS91} for the details). If $(v,p)$ is the solution of the Stokes system \eqref{stokes-eqn} with no-slip boundary conditions, then the following estimate is satisfied: \[ \norm{v_t}_{L^{l,m}_{x,t}(Q_T)}+\norm{\nabla^2 v}_{L^{l,m}_{x,t}(Q_T)}+\norm{\nabla p}_{L^{l,m}_{x,t}(Q_T)} \] \begin{equation}\label{stokes-estimate} \leq C\norm{f}_{L^{l,m}_{x,t}(Q_T)}+\norm{v_0}_{D_l^{1-\frac{1}{m},m}(\Omega)}. \end{equation} \end{lemma}
\section{Boundary regularity}
In this section, we prove a local regularity criterion for MHD equations near the boundary and present the proofs of Theorem \ref{main-thm-boundary-v} and \ref{main-thm-boundary-w}.
For simplicity, we write $\Psi(r):=A_{u}(r)+A_{b}(r)+E_{u}(r)+E_{b}(r)$. Let $z=(x, t)\in\Gamma\times I$ and from now on, without loss of generality, we assume $x=0$ by translation. We first recall that the local energy estimate.
\begin{equation}\label{local-energy-estimate} \Psi(\frac{r}{2})\leq C\bigg(M^{\frac{2}{3}}_u(r)+K_b(r)+M_u(r)+ \frac{1}{r^2}\int_{Q^+_{z,r}}\abs{u}\abs{b}^2dz+\frac{1}{r^2}\int_{Q^+_{z,r}}\abs{u}\abs{\pi}dz\biggr). \end{equation}
Next we prove a local regularity condition near boundary for MHD equations (compare to \cite[Lemma 7]{GKT06} for the Navier-Stokes equations).
\begin{proposition}\label{ep-regularity} There exist $\epsilon^*>0$ and $r_0>0$ such that if $(u,b,\pi)$ is a suitable weak solution of MHD equations satisfying Definition \ref{sws-3dnse}, $z=(x,t) \in \Gamma \times I$, and \begin{equation} M_u(r)+M_b(r)+\tilde{Q}(r)<\epsilon^* \qquad \mbox{ for some }r\in (0,r_0), \end{equation} then $z$ is regular point. \end{proposition}
The proof of Proposition \ref{ep-regularity} is based on the following lemma, which shows a decay property of $(u,b,\pi)$ in a Lebesgue spaces. Although the method of proof is in principle the similar as in \cite[Lemma 7, Lemma 8]{GKT06}, we present its details for clarity (the proof will be given in Appendix).
\begin{lemma}\label{mhd-decay} Let $0<\theta<\frac{1}{2}$. There exist $\epsilon_1>0$ and $r_*$ depending on $\lambda$ and $\theta$ such that if $(u,b,\pi)$ is a suitable weak solution of the MHD equations satisfying Definition \ref{sws-3dnse}, $z=(x,t)\in\Gamma\times(0,T)$, and $M_{u}^{\frac{1}{3}}(r)+M_{b}^{\frac{1}{3}}(r)+\tilde{Q}(r)<\epsilon_1$ for some $r\in (0,r_*)$, then \[ M_{u}^{\frac{1}{3}}(\theta r)+M_{b}^{\frac{1}{3}}(\theta r)+\tilde{Q}(\theta r)< C\theta^{1+\alpha}\bke{M_{u}^{\frac{1}{3}}(r)+M_{b}^{\frac{1}{3}}(r) +\tilde{Q}(r)}, \] where $0<\alpha<1$ and $C>0$ are constants. \end{lemma}
Next lemma is estimates of the scaled integral of cubic term of $u$ and multiple of $u$ and square of $b$.
\begin{lemma}\label{estimate-ub} Let $z=(x, t)\in\Gamma\times I$. Suppose that $u\in L^{p,q}_{x,t}(Q^+_{z,r})$ with $3/p+2/q=2$, $3/2 \leq p \leq \infty$. Then for $0<r<\rho/4$, \begin{equation}\label{estimate-Mu} M_{u}(r)\leq CG_{u,p,q}(r)\Psi(r) \leq C\bigg(\frac{\rho}{r}\bigg)\Psi(\rho)G_{u,p,q}(r) , \end{equation} \begin{equation}\label{estimate-Mb} \frac{1}{r^2}\int_{Q^+_{z,r}}\abs{u}\abs{b}^2dz\leq CG_{u,p,q}(r)\Psi(r) \leq C\bigg(\frac{\rho}{r}\bigg)\Psi(\rho)G_{u,p,q}(r). \end{equation} \end{lemma} \begin{proof} It is sufficient to show estimate \eqref{estimate-Mb} because \eqref{estimate-Mu} can be proved in the same way as \eqref{estimate-Mb}. We note first that via H\"older's inequality \begin{equation}\label{proof-est-mb} \frac{1}{r^2}\int _{Q^+_{z,r}}\abs{u}\abs{b}^{2}dxds\leq
\frac{1}{r}\norm{u}_{L^{p,q}_{x,t}(Q^+_{z,r})}\frac{1}{r}\norm{b}^2_{L^{2p^{*},2q^{*}}_{x,t}(Q^+_{z,r})}, \end{equation} where $p^*$ and $q^*$ are H\"older conjugates of $p$ and $q$. For $\alpha:=(3-p^{*})/2p^{*}$ we see that \[ \norm{b}_{L^{2p^{*}}_x(B^+_{x,r})}\leq
\norm{b}^{\alpha}_{L^2_{x}(B^+_{x,r})}\norm{b-(b)_r}^{1-\alpha}_{L^6_{x}(B^+_{x,r})} +\norm{b}^{\alpha}_{L^2_{x}(B^+_{x,r})}\norm{(b)_r}^{1-\alpha}_{L^6_{x}(B^+_{x,r})} \] \[ \leq C\norm{b}^{\alpha}_{L^2_{x}(B^+_{x,r})}\norm{\nabla b}^{1-\alpha}_{L^2_{x}(B^+_{x,r})} +\norm{b}_{L^2_{x}(B^+_{x,r})}r^{-\frac{1}{2}+\frac{\alpha}{2}}, \] where we used Poincar\'{e} inequality. Taking $L^{2q^*}$ norm in temporal variable and using Young's inequality, \[ \norm{b}^2_{L^{2p^{*},2q^{*}}_{x,t}(Q^+_{z,r})} \leq C\norm{b}^{2}_{L^{2,\infty}_{x,t}(Q^+_{z,r})}+C\norm{\nabla b}^{2}_{L^{2,2}_{x,t}(Q^+_{z,r})}.
\] Recalling \eqref{proof-est-mb}, we can have \[ \frac{1}{r^2}\int _{Q^+_{z,r}}\abs{u}\abs{b}^{2}dxds\leq
CG_{u,p,q}(r)\Psi(r) \leq C(\frac{\rho}{r})\Psi(\rho)G_{u,p,q}(r). \] This completes the proof. \end{proof}
Next, we may continue with scaled norm of $L^{2,2}_{x,t}(Q^{+}_{z_0,r})$ estimate of $b$.
\begin{lemma}\label{lem3.2} Let $z=(x, t)\in\Gamma\times I$. Suppose that $u\in L^{p,q}_{x,t}(Q^{+}_{z,r})$ with $3/p+2/q=2$ and $3/2\leq p <3$. Then for $0<r<\rho/4$ \begin{equation}\label{boundary-b} K_{b}(r)\leq C\bigg(\frac{\rho}{r}\bigg)^3G_{u,p,q}^2(\rho)\Psi(\rho) +C\bigg(\frac{r}{\rho}\bigg)^2K_{b}(\rho). \end{equation} \end{lemma}
\begin{proof} For convenience, we write $x=(x_1, x_2, x_3)=(x',x_3)$ and by translation, we assume that without loss of generality, $z=(0,0)\in \Gamma\times I$. Let $\zeta(x,t)$ be a standard cut off function supported in $Q_{\rho}$ such that $\zeta(x,t)=1$ in $Q_{\rho/2}$. We set $g(x,t):= -\nabla\cdot([u\otimes b-b\otimes u]\zeta)$ in $Q^+_{z, \rho}$ and we then define $\tilde{g}(x,t)$, an extension of $g$ from $Q^+_{\rho}$ onto $Q_{\rho}$, in the following way: $\tilde{g}(x,t)=g(x,t)$ if $x_3\geq 0$. On the other hand, if $x_3< 0$, then \[ \tilde{g}_i(x',x_3,t)=g_i(x', -x_3, t),\qquad i=1,2 \] \[ \tilde{g}_3(x',x_3,t)=-g_3(x', -x_3, t). \] This can be done by extending tangential components of $u$ and $b$ as even functions and normal components of $u$ and $b$ as odd functions, respectively. We denote such extensions by $\tilde{u}$ and $\tilde{b}$ for simplicity. Here we also used the fact that $\zeta$ and $\nabla' \zeta$ are even and $\partial_{x_3}\zeta$ is odd with respect to $x_3-$variable, where $\nabla'=(\partial_{x_1}, \partial_{x_2})$.
Next, we define $\tilde{w}(x,t)$ for $(x,t)\in { \mathbb{R} }^3\times (-\infty,0)$ by \[ \tilde{w}(x,t)=\int_{-\infty}^{t}\int_{{\mathbb R}^3}
\frac{1}{(4\pi(t-s))^{\frac{3}{2}}}e^{-\frac{|x-y|^2}{4(t-s)}}\tilde{g}(y,s) dyds, \] namely, $\tilde{w}$ satisfies \[ \tilde{w}_t-\Delta \tilde{w}=\tilde{g} \qquad\mbox{in }\,\, { \mathbb{R} }^3\times (-\infty,0). \] Moreover, we can see that $\partial_{x_3}\tilde{w}_i=0$ for $i=1,2$ and $\tilde{w}_3=0$ on $\{x_3=0\}$. Let $h=b-\tilde{w}$ in $Q^+_{\rho}$. Then $h$ satisfies \[ h_t-\Delta h=0\qquad \mbox{in} \ Q^{+}_{\frac{\rho}{2}} \] and $\partial_{x_3}h_i=0$ for $i=1,2$ and $h_3=0$ on $\{x_3=0\}\cap Q_{\rho}$. Now we extend $h$ by the same manner as $g$, denoted by $\tilde{h}$, from $Q^+_{\rho/2}$ onto $Q_{\rho/2}$. We then see that \[ \tilde{h}_t-\Delta \tilde{h}=0 \qquad \mbox{in} \ Q_{\frac{\rho}{2}}. \] Via classical regularity theory, we have \begin{equation}\label{tilde-h}
\int_{Q_r}|\tilde{h}|^2 dz \leq C(\frac{r}{\rho})^5
\int_{Q_{\frac{\rho}{2}}}|\tilde{h}|^2 dz. \end{equation} On the other hand, due to Sobolev embedding, we have $ \norm{\tilde{w}}_{L^2(B_{\rho})}\leq
\norm{\tilde{w}}_{L^2({ \mathbb{R} }^3)}\leq C\|\nabla
\tilde{w}\|_{L^{\frac{6}{5}}({ \mathbb{R} }^3)}$ and we then take $L^2$ integration for the above in time interval $(-\rho^2, 0)$ such that we obtain
\[
\|\tilde{w}\|_{L^{2,2}_{x,t}(Q_{\rho})}\leq C\|\nabla
\tilde{w}\|_{L^{\frac{6}{5},2}_{x,t}({ \mathbb{R} }^3\times (-\rho^2,0))}\leq C\|\tilde{u}\tilde{b}\zeta\|_{L^{\frac{6}{5},2}_{x,t}({ \mathbb{R} }^3\times (-\rho^2,0))} \] \begin{equation}\label{tilde-w}
\leq C\|\tilde{u}\tilde{b}\|_{L^{\frac{6}{5},2}_{x,t}(Q_{\rho})}\leq C\|\tilde{u}\|_{L^{p,q}_{x,t}(Q_{\rho})}\|\tilde{b}\|_{L^{\alpha,\beta}_{x,t}(Q_{\rho})}, \end{equation} where $3/\alpha+2/\beta=3/2$ and $2\le \alpha < 6$, since $3/2\le p<3$.
Using the estimate \eqref{tilde-w} and Sobolev inequality, we have \[
\frac{1}{\rho^{3}}\|\tilde{w}\|^2_{L^{2,2}_{x,t}(Q_{\frac{\rho}{2}})}\leq C\frac{1}{\rho^2}\|\tilde{u}\|^2_{L^{p,q}_{x,t}(Q_{\rho})}\frac{1}{\rho}\|\tilde{b}\|^2_{L^{\alpha,\beta}_{x,t}(Q_{\rho})}
\leq C\frac{1}{\rho^2}\|u\|^2_{L^{p,q}_{x,t}(Q^+_{\rho})}
\frac{1}{\rho}\|b\|^2_{L^{\alpha,\beta}_{x,t}(Q^+_{\rho})} \] \begin{equation}\label{estimate-bar_w1} \leq
\frac{C}{\rho^2}\|u\|^2_{L^{p,q}_{x,t}(Q^+_{\rho})}(A_{b}(\rho)+E_{b}(\rho)) \leq CG^2_{u,p,q}(\rho)\Psi(\rho). \end{equation} Combining estimates \eqref{tilde-h} and \eqref{estimate-bar_w1}, we obtain \[ K_{b}(r)=\frac{1}{r^3}\norm{b}_{L^{2,2}_{x,t}(Q^+_r)}^2
\leq\frac{1}{r^3}\|\tilde{w}\|_{L^{2,2}_{x,t}(Q_r)}^2+\frac{1}{r^3}\|\tilde{h}\|_{L^{2,2}_{x,t}(Q_r)}^2 \] \[ \leq C(\frac{\rho}{r})^3G_{u,p,q}^2(\rho)\Psi(\rho)
+C(\frac{r}{\rho})^2\frac{1}{\rho^3}\|b\|_{L^{2,2}_{x,t}(Q^+_{\frac{\rho}{2}})}^2 \] \[ \leq C(\frac{\rho}{r})^3G_{u,p,q}^2(\rho)\Psi(\rho) +C(\frac{r}{\rho})^2K_{b}(\rho). \] This completes the proof. \end{proof}
In next lemma we show an estimate of the gradient of pressure (compare to \cite[Lemma 11]{GKT06}). \begin{lemma}\label{lem4.2} Let $z=(x, t)\in\Gamma\times I$. Then for $0<r<\rho/4$,
\[ Q_{1}(r)\leq C\bigg(\frac{\rho}{r}\biggr)\Big(A_u^{\frac{3-2\kappa}{2\kappa}}(\rho) E_u^{\frac{1}{\lambda}}(\rho) + A_b^{\frac{3-2\kappa}{2\kappa}}(\rho) E_b^{\frac{1}{\lambda}}(\rho)\Big) \] \begin{equation}\label{estimate-local-Q} +C\bigg(\frac{r}{\rho}\bigg)\Big(E_u^{\frac{1}{2}}(\rho)+Q_{1}(\rho)\Big), \end{equation} where $\kappa$ and $\lambda$ are numbers in \eqref{pq}. \end{lemma} \begin{proof} We assume, via translation, that $z=(x, t)=(0, 0)$. We choose a domain $\tilde{B}^{+}$ with a boundary such that $B^{+}_{\frac{\rho}{2}} \subset \tilde{B}^{+} \subset B^{+}_{\rho}$, and we denote $\tilde{Q}^{+}:=\tilde{B}^{+} \times (-\rho^2,0)$. Let $(v,\pi_1)$ be the unique solution of the following the Stokes system \[ v_t-\Delta v+\nabla \pi_1=-(u\cdot \nabla)u+(b\cdot \nabla)b, \quad \text{div}\, v=0 \ \ \text{in} \ \tilde{Q}^{+}, \] \[ (\pi_1)_{\tilde{B}^{+}}=\Xint\diagup_{\tilde{B}^{+}}\pi_1(y,t)dy=0, \quad t\in(-\rho^2,0), \] \[ v=0 \quad \partial\tilde{B}^{+} \times [-\rho^2,0], \quad v=0 \quad \tilde{B}^{+} \times \{t=-\rho^2\}. \] Using the Stokes estimate \eqref{stokes-estimate}, we have the following estimate \[
\frac{1}{\rho^2}\|v\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})}
+\frac{1}{\rho}\|\nabla v\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})}
+\|v_t\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})}+\|\nabla^{2}v\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})} \] \[
+\frac{1}{\rho}\|\pi_{1}\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})}
+\|\nabla \pi_{1}\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})} \] \[
\leq C\Big(\|(u\cdot\nabla)u\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})}+\|(b\cdot\nabla)b\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})} \Big) \] \[
\leq C\Big(\|(u\cdot\nabla)u\|_{L_{x,t}^{\kappa,\lambda}(Q_{\rho}^{+})}
+\|(b\cdot\nabla)b\|_{L_{x,t}^{\kappa,\lambda}(Q_{\rho}^{+})}\Big) \] \[ \leq C\big(\rho A_u^{\frac{3-2\kappa}{2\kappa}}(\rho) E_u^{\frac{1}{\lambda}}(\rho) +\rho A_b^{\frac{3-2\kappa}{2\kappa}}(\rho) E_b^{\frac{1}{\lambda}}(\rho)\big), \] where we used the following estimates in last inequality above: \[ \norm{(u\cdot \nabla)u}_{L^{\kappa,\lambda}_{x,t}(Q_{\rho}^{+})} +\norm{(b\cdot \nabla)b}_{L^{\kappa,\lambda}_{x,t}(Q_{\rho}^{+})} \] \[ \leq \norm{u}^{\frac{3-2\kappa}{\kappa}}_{L^{2,\infty}_{x,t}(Q_{\rho}^{+})}\norm{\nabla u}^{\frac{2}{\lambda}}_{L^{2,2}_{x,t}(Q_{\rho}^{+})} +\norm{b}^{\frac{3-2\kappa}{\kappa}}_{L^{2,\infty}_{x,t}(Q_{\rho}^{+})}\norm{\nabla b}^{\frac{2}{\lambda}}_{L^{2,2}_{x,t}(Q_{\rho}^{+})} \] \[ \leq C\rho A_u^{\frac{3-2\kappa}{\kappa}}(\rho)E^{\frac{2}{\lambda}}_u(\rho) +C\rho A_b^{\frac{3-2\kappa}{\kappa}}(\rho)E^{\frac{2}{\lambda}}_b(\rho). \]
Next, let $w=u-v$ and
$\pi_2=\pi-(\pi)_{B_{\frac{\rho}{2}}^{+}}-\pi_1$. Then
$(w,\pi_2)$ solves the following the boundary value problem: \[ w_t-\Delta w+\nabla \pi_2=0, \quad \text{div}\, w=0 \qquad\text{in} \ \tilde{Q}^{+}, \] \[ w=0 \quad \ \text{on}\ (\partial\tilde{B}^{+}\cap\{x_3=0\}) \times [-\rho^2,0]. \] Now we take $\kappa'$ such that $3/\kappa'+2/\lambda=2$. Then from the local estimate near the boundary for the Stokes systems (see \cite{GAS02}), we obtain \[
\|\nabla^{2}w\|_{L_{x,t}^{\kappa',\lambda}(Q^{+}_{\frac{\rho}{4}})}
+\|\nabla\pi_{2}\|_{L_{x,t}^{\kappa',\lambda}(Q^{+}_{\frac{\rho}{4}})}\\ \] \[
\leq\frac{C}{\rho^{2}}\bigg(\frac{1}{\rho^{2}}\|w\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\frac{\rho}{2}})}
+\frac{1}{\rho}\|\nabla w\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\frac{\rho}{2}})}
+\frac{1}{\rho}\|\pi_{2}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\frac{\rho}{2}})}\bigg) \]
\[\leq\frac{C}{\rho^{2}}\bigg(\frac{1}{\rho}\|\nabla u\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\frac{\rho}{2}})}
+\|\nabla\pi\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\frac{\rho}{2}})}
+\frac{1}{\rho}\|\nabla v\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\frac{\rho}{2}})}
+\frac{1}{\rho}\|\pi_{1}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\frac{\rho}{2}})}\bigg), \]
where Poincar\'e-Sobolev inequality is used. Since $\|\nabla u\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\rho})}\leq C\rho^{2}E_u^{\frac{1}{2}}(\rho)$, we have \[
\|\nabla\pi_{2}\|_{L_{x,t}^{\kappa',\lambda}(Q^{+}_{\frac{\rho}{4}})}\leq \frac{C}{\rho^2}\bke{\rho E_u^{\frac{1}{2}}(\rho)+\rho Q_{1}(\rho) +\rho A_u^{\frac{3-2\kappa}{2\kappa}}(\rho) E_u^{\frac{1}{\lambda}}(\rho) +\rho A_b^{\frac{3-2\kappa}{2\kappa}}(\rho) E_b^{\frac{1}{\lambda}}(\rho)} \] \[ =\frac{C}{\rho}\bigl(E_u^{\frac{1}{2}}(\rho)+ Q_{1}(\rho)+ A_u^{\frac{3-2\kappa}{2\kappa}}(\rho) E_u^{\frac{1}{\lambda}}(\rho) + A_b^{\frac{3-2\kappa}{2\kappa}}(\rho) E_b^{\frac{1}{\lambda}}(\rho)\bigr). \] Let $0\leq r\leq\rho/4$. Noting that
$\|\nabla\pi_{2}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{r})}\leq Cr^{2}\|\nabla\pi_{2}\|_{L_{x,t}^{\kappa',\lambda}(Q^{+}_{r})}$, we have \begin{equation*}
Q_{1}(r)=\frac{1}{r}\|\nabla\pi\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{r})}
\leq\frac{1}{r}\bigl(\|\nabla\pi_{1}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{r})}
+\|\nabla\pi_{2}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{r})}\bigr) \end{equation*} \begin{equation*}
\leq\frac{1}{r}\bigl(\|\nabla\pi_{1}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{\rho})}
+r^2\|\nabla\pi_{2}\|_{L_{x,t}^{\kappa',\lambda}(Q^{+}_{r})}\bigr) \end{equation*} \begin{equation*} \leq C(\frac{\rho}{r})\bigl( A_u^{\frac{3-2\kappa}{2\kappa}}(\rho)E^{\frac{1}{\lambda}}_u(\rho)+ A_b^{\frac{3-2\kappa}{2\kappa}}(\rho)E^{\frac{1}{\lambda}}_b(\rho)\bigr) \end{equation*} \begin{equation*} \quad +C(\frac{r}{\rho})\bigl(E_u^{\frac{1}{2}}(\rho)+ Q_{1}(\rho)+ A_u^{\frac{3-2\kappa}{2\kappa}}(\rho) E_u^{\frac{1}{\lambda}}(\rho) + A_b^{\frac{3-2\kappa}{2\kappa}}(\rho) E_b^{\frac{1}{\lambda}}(\rho)\bigr) \end{equation*} \begin{equation*} \leq C(\frac{\rho}{r})\Big(A_u^{\frac{3-2\kappa}{2\kappa}}(\rho) E_u^{\frac{1}{\lambda}}(\rho) + A_b^{\frac{3-2\kappa}{2\kappa}}(\rho) E_b^{\frac{1}{\lambda}}(\rho)\Big) +C(\frac{r}{\rho})\Big(E_u^{\frac{1}{2}}(\rho)+Q_{1}(\rho)\Big). \end{equation*} This completes the proof. \end{proof} We remark that, via Young's inequality, \eqref{estimate-local-Q} can be estimated as follows: \begin{equation}\label{est-local-Q1} Q_{1}(r)\leq C\bke{\bigg(\frac{\rho}{r}\bigg)+\bigg(\frac{r}{\rho}\bigg)}\Psi(\rho) +C\bke{\frac{r}{\rho}}\bke{Q_{1}(\rho)+1}. \end{equation}
Next lemma shows an estimate of a scaled norm of pressure. \begin{lemma}\label{lem4.3} Let $z=(x, t)\in\Gamma\times I$. Suppose that $\nabla \pi \in L^{\kappa,\lambda}_{x,t}(Q_{\rho})$ and $\pi \in L^{\kappa^*,\lambda}_{x,t}(Q_{\rho})$, where $3/\kappa+2/\lambda=4$, $1/\kappa^*=1/\kappa-1/3$ and $1 < \lambda < 2$. Then for $0<r<\rho/4$, \begin{equation}\label{boundary-pressure} Q(r)\leq C\bke{\frac{\rho}{r}}Q_1(\rho)+C\bke{\frac{r}{\rho}}^{\frac{3}{\kappa^*}-1}Q(\rho). \end{equation} \end{lemma} \begin{proof} Since $1<\lambda<2$, we note that $\frac{3}{2}<\kappa^*<3$. We first observe that due to H\"{o}lder inequality \[
\|(\pi)_{\rho }\|_{L_x^{\kappa^*}(B^+_{x,r})}\leq C(\frac{r}{\rho})^{\frac{3}{\kappa^*}}\|\pi\|_{L_x^{\kappa}(B^+_{x,\rho})}. \] Therefore, due to Poincar\'e-Sobolev inequality, we have \[ \norm{\pi}_{L^{\kappa^*,\lambda}_{x,t}(Q^+_{z,r})}\leq \norm{\pi-(\pi)_{\rho}}_{L^{\kappa^*,\lambda}_{x,t}(Q^+_{z,r})} +\norm{(\pi)_{\rho}}_{L^{\kappa^*,\lambda}_{x,t}(Q^+_{z,r})} \] \[
\leq C\|\nabla \pi\|_{L^{\kappa,\lambda}_{x,t}(Q^+_{z,r})} +C(\frac{r}{\rho})^{\frac{3}{\kappa^*}}
\|\pi\|_{L^{\kappa^*,\lambda}_{x,t}(Q^+_{z,\rho})}. \] Dividing both sides by $r$, we have \[
\frac{1}{r}\|\pi\|_{L^{\kappa^*,\lambda}_{x,t}(Q^+_{z,r})}\leq C(\frac{\rho}{r})\frac{1}{\rho}\|\nabla
\pi\|_{L^{\kappa,\lambda}_{x,t}(Q^+_{z,\rho})} +C(\frac{r}{\rho})^{\frac{3}{\kappa^*}-1}\frac{1}{\rho}
\|\pi\|_{L^{\kappa^*,\lambda}_{x,t}(Q^+_{z,\rho})}. \] This completes the proof. \end{proof}
We are ready to present the proof of Theorem \ref{main-thm-boundary-v}. \begin{pfthm1} We note first that via H\"older's inequality, it suffices to show the case that $3/p+2/q=2$, $2<q<\infty$. Recalling Lemma \ref{lem4.2} and Lemma \ref{lem4.3}, we have \[ \frac{1}{r^2}\int_{Q^+_{z,r}}\abs{u}\abs{\pi} dz \leq \frac{1}{r}\norm{u}_{L^{p,q}_{x,t}(Q^+_{z,r})}\frac{1}{r}\norm{\pi}_{L^{\kappa^{*},\lambda}_{x,t}(Q^+_{z,r})} \] \[ \leq C\epsilon\Bigg( (\frac{\rho}{r})Q_1(\frac{\rho}{2}) +(\frac{r}{\rho})^{\frac{3}{\kappa^{*}}-1}Q(\frac{\rho}{2})\Biggr) \] \begin{equation}\label{estimate-u-b} \leq C\epsilon \bke{(\frac{\rho}{r})^2+1}\Psi(\rho) +C\epsilon \bke{Q_{1}(\rho)+1} +C\epsilon(\frac{r}{\rho})^{\frac{3}{\kappa^{*}}-1}Q(\rho), \end{equation} where \eqref{est-local-Q1} is also used. With aid of Lemma \ref{estimate-ub}, Lemma \ref{estimate-Mu}, \eqref{est-local-Q1} and \eqref{estimate-u-b}, we have \begin{equation*} \Psi(\frac{r}{2}) \leq C\epsilon^{\frac{2}{3}}(\frac{\rho}{r})^{\frac{2}{3}}\Psi^{\frac{2}{3}}(\rho) +C\epsilon(\frac{\rho}{r})\Psi(\rho) +C\epsilon^2(\frac{\rho}{r})^3\Psi(\rho)+C(\frac{r}{\rho})^2K_{b}(\rho) \end{equation*} \begin{equation*} +C\epsilon \big((\frac{\rho}{r})^2+1)\big)\Psi(\rho) +C\epsilon (Q_{1}(\rho)+1) +C\epsilon(\frac{r}{\rho})^{\frac{3}{\kappa^{*}}-1}Q(\rho) \end{equation*} \begin{equation*} \leq C\bkt{\epsilon^2(\frac{\rho}{r})^3+\epsilon(\frac{\rho}{r})^2+(\epsilon^{\frac{1}{2}}+\epsilon) (\frac{\rho}{r})+\epsilon+(\frac{r}{\rho})^2}\Psi(\rho) \end{equation*} \begin{equation}\label{estimate-psi} +C\epsilon Q_{1}(\rho)+C\epsilon +C\epsilon(\frac{r}{\rho})^{\frac{3}{\kappa^{*}}-1}Q(\rho), \end{equation} where we used the Young's inequality and $K_b(\rho)\leq \Psi(\rho)$. Let $\epsilon_1$ and $\epsilon_2$ be small positive numbers, which will be specified later.
Adding $\epsilon_1 Q_1(\frac{r}{2})$ and $\epsilon_2 Q(\frac{r}{2})$ to both sides in \eqref{estimate-psi}, and using \eqref{est-local-Q1} and Lemma \ref{lem4.3}, we obtain \begin{equation*} \Psi(\frac{r}{2})+\epsilon_1Q_1(\frac{r}{2})+\epsilon_2Q(\frac{r}{2}) \end{equation*} \begin{equation*} \leq C\bkt{\epsilon^2\bke{\frac{\rho}{r}}^3+\epsilon(\frac{\rho}{r})^2+(\epsilon^{\frac{1}{2}}+\epsilon) (\frac{\rho}{r})+\epsilon+(\frac{r}{\rho})^2}\Psi(\rho) \end{equation*} \begin{equation*} +C\epsilon Q_{1}(\rho)+C\epsilon +C\epsilon(\frac{r}{\rho})^{\frac{3}{\kappa^{*}}-1}Q(\rho) +C\epsilon_1\bkt{(\frac{\rho}{r})+(\frac{r}{\rho})}\Psi(\rho), \end{equation*} \begin{equation*} +C\epsilon_1(\frac{r}{\rho})(Q_{1}(\rho)+1) +C\epsilon_2(\frac{\rho}{r})Q_1(\rho)+C\epsilon_2(\frac{r}{\rho} )^{\frac{3}{\kappa^{*}}-1}Q(\rho) \end{equation*} \begin{equation*} \leq C\bkt{\epsilon^2(\frac{\rho}{r})^3+\epsilon(\frac{\rho}{r})^2+(\epsilon^{\frac{1}{2}}+\epsilon+\epsilon_1) (\frac{\rho}{r})+\epsilon+(\frac{r}{\rho})^2+\epsilon_1(\frac{r}{\rho})}\Psi(\rho) \end{equation*} \begin{equation*} +C\bkt{\epsilon+\epsilon_1(\frac{r}{\rho})+\epsilon_2(\frac{\rho}{r})} Q_{1}(\rho)+C(\epsilon+\epsilon_2)\frac{r}{\rho}^{\frac{3}{\kappa^{*}}-1}Q(\rho) +C\bkt{\epsilon+\epsilon_1(\frac{r}{\rho})}. \end{equation*} We fix $\theta \in (0,\frac{1}{4})$ with $C(\theta+\theta^{\frac{3}{\kappa^{*}}-1})<\frac{1}{4}$ and then choose $\epsilon_1, \epsilon_2$ and $\epsilon$ satisfying \[ 0<\epsilon_1<\frac{\theta}{16C},\qquad 0<\epsilon_2<\frac{\epsilon_1\theta}{8C},\qquad 0<\epsilon<\min\bket{\frac{\epsilon^*}{16C}, \,\,\epsilon_2,\,\, \frac{\theta^6}{16C^2}}. \] Therefore, we have \begin{equation}\label{K1K} \Psi(\theta r)+\epsilon_1 Q_1(\theta r)+\epsilon_2 Q(\theta r) \leq \frac{\epsilon^*}{8}+\frac{1}{2}\Big(\Psi(r)+\epsilon_1 Q_1(r)+\epsilon_2 Q( r) \Big). \end{equation} Iterating \eqref{K1K}, we can see that there exists a sufficiently small $r_0>0$ such that for all $r<r_0$ \[ \Psi(r)+\epsilon_1 Q_1(r)+\epsilon_2 Q(r) \leq \frac{\epsilon^*}{4}. \] Therefore, we conclude that $\Psi(r)\leq \epsilon^*/8$. Next, we use the estimates \eqref{est-local-Q1} and \eqref{boundary-pressure} to obtain that there is $r_1>0$ such that $Q(r)\leq \epsilon^*/4$ for all $r<r_1$. This can be shown by the method of iterations as in \eqref{K1K}. Summing up, we obtain $\Psi(r)+Q(r)\leq \epsilon^*/2$ for all $r<r_1$, which implies the regularity condition in Proposition \ref{ep-regularity}. This completes the proof.
\end{pfthm1}
The proof of Theorem \ref{main-thm-boundary-w} is given below. \begin{pfthm2} As mentioned earlier, it suffices to show the case that $3/p+2/q=3$, $2<q<\infty$. We first show that the gradient of velocity is controlled by vorticity. To be more precise, we prove the following estimate (compare to \cite[Lemma 3.6]{GKT07}): \begin{equation}\label{boundary-vorticity-estimate1} D_{u,\tilde{p},q}(r)\leq C(\frac{\rho}{r})V_{u,\tilde{p},q}(\rho)+C(\frac{r}{\rho} )^{\frac{3}{\tilde{p}}-1}D_{u,\tilde{p},q}(\rho). \end{equation} Indeed, let $\xi$ be a cut off supported in $Q_{\rho}$ and $\xi=1$ in $Q_{\frac{\rho}{2}}$. We consider \begin{equation*} \quad \left\{ \begin{array}{ll} \displaystyle -\Delta v =\nabla \times ( \omega \xi) \quad \text{in}\quad \mathbb{R}^3_{+}\\
\\ \displaystyle v =0 \quad \text{on}\quad \{x_3=0\} \end{array}\right. \end{equation*} and set $h=u-v$. So $h$ is harmonic function in $B^{+}_{\frac{r}{2}}$ with $ h=0$ on $\partial B^{+}_{\frac{r}{2}}\cap\{x_3=0\}.$ By the mean value theorem of harmonic functions and $L^p$ estimates of elliptic equations for each fixed time $t$ \[
\|\nabla h\|_{L_x^{\tilde{p}}(B^{+}_r)}\leq C(\frac{r}{\rho}
)^{\frac{3}{\tilde{p}}}\|\nabla h\|_{L_x^{\tilde{p}}(B^{+}_{\frac{\rho}{2}})} \leq C(\frac{r}{\rho})^{\frac{3}{\tilde{p}}}\bke{\|\nabla u\|_{L_x^{\tilde{p}}(B^{+}_{\rho})}+\|\nabla v\|_{L_x^{\tilde{p}}(B^{+}_{\rho})}} \] \[
\leq C(\frac{r}{\rho} )^{\frac{3}{\tilde{p}}}\bke{\|\nabla u\|_{L_x^{\tilde{p}}(B^{+}_{\rho})}+\|\ \omega
\|_{L_x^{\tilde{p}}(B^{+}_{\rho})}}. \] Adding together above estimates, \[
\|\nabla u \|_{L_x^{\tilde{p}}(B^{+}_{r})}\leq \|\nabla v
\|_{L_x^{\tilde{p}}(B^{+}_{r})}+\|\nabla h
\|_{L_x^{\tilde{p}}(B^{+}_{r})}\leq C\|\ \omega
\|_{L_x^{\tilde{p}}(B^{+}_{r})}+C(\frac{r}{\rho}
)^{\frac{3}{\tilde{p}}}\|\nabla u\|_{L_x^{\tilde{p}}(B^{+}_{\rho})}. \] Taking $L^{q}$-norm in time and dividing both sides by $r$, we obtain \eqref{boundary-vorticity-estimate1}. Via the method of iteration, the estimate implies that the scaled norm of gradient of velocity becomes sufficiently small. Since argument is straightforward, we skip its details. We deduce the Theorem via Corollary \ref{cor-boundary-v}. \end{pfthm2}
\section{Appendix} In this Appendix we present the proof of Lemma \ref{mhd-decay} and interior regularity is compared to boundary regularity given in Theorem \ref{main-thm-boundary-v}.
\subsection{Proof of Lemma \ref{mhd-decay}}
As mentioned earlier, the method of proof is quite similar to that of \cite[Lemma 8]{GKT06} and main difference is mostly caused by the presence of magnetic field $b$. Therefore, we give the mainstream of the proof, instead giving all the details.
\begin{pf-mhd-decay} For convenience, we denote $\phi(r):=M_{u}^{\frac{1}{3}}(r)+M_{b}^{\frac{1}{3}}(r)+\tilde{Q}(r)$. Suppose the statement is not true. So for any $\alpha\in(0,1)$ and $C>0$, there exist $z_{n}=(x_{n},t_{n})$, $r_{n}\searrow 0$ and $\epsilon_{n}\searrow 0$ such that \begin{equation*} \phi(r_{n})=\epsilon_{n},\qquad \phi(\theta r_n)>C\theta^{1+\alpha}\phi(r_{n})=C\theta^{1+\alpha}\epsilon_{n}. \end{equation*} Let $w=(y,s)$ where $y=\frac{(x-x_{n})}{r_{n}}$, $s=\frac{(t-t_{n})}{r^{2}_{n}}$ and we define $v_{n}, b_{n}$ and $ \pi_{n}$ as follows: \[ v_{n}(w)=\frac{r_{n}}{\epsilon_{n}}u(z),\quad b_{n}(w)=\frac{r_{n}}{\epsilon_{n}}b(z),\quad \pi_{n}(w)=\frac{r^{2}_{n}}{\epsilon_{n}}(\pi(z)-(\pi)_{B^{+}_{r_n}}(z)). \]
We also introduce some scaling invariant functionals defined by \begin{equation*}
T_{u}(v_{n},\theta):=\frac{1}{\theta^{2}}\int_{Q^{+}_{\theta}}|v_{n}|^{3}dw,\quad T_{b}(b_{n},\theta):=\frac{1}{\theta^{2}}\int_{Q^{+}_{\theta}}|b_{n}|^{3}dw,\quad \end{equation*} \begin{equation*} P_{1}(\pi_{n},\theta):=\frac{1}{\theta}\biggl(\int^{0}_{-\theta^{2}}
\Bigl(\int_{B^{+}_{\theta}}|\nabla
\pi_{n}|^{\kappa}dy\Bigr)^{\frac{\lambda}{\kappa}}ds \biggr)^{\frac{1}{\lambda}}, \end{equation*} \begin{equation*} \tilde{P}(\pi_{n},\theta):=\frac{1}{\theta}\biggl(\int^{0}_{-\theta^{2}}
\Bigl(\int_{B^{+}_{\theta}}|\pi_{n}-(\pi_{n})_{B^{+}_{\theta}}|^{\kappa^{*}}dy \Bigr)^{\frac{\lambda}{\kappa^{*}}}ds\biggr)^{\frac{1}{\lambda}}, \end{equation*} where $\kappa^{*}$, $\kappa$ and $\lambda$ are numbers in \eqref{pq}. Let $\tau_{n}(\theta)=T_u^{\frac{1}{3}}(v_{n},\theta)+T_b^{\frac{1}{3}}(b_{n},\theta) +\tilde{P}(\pi_{n},\theta)$. The change of variables lead to \begin{equation}\label{scaling-1000}
\tau_{n}(1)=\|v_{n}\|_{L_{x,t}^{3,3}(Q^{+}_{1})}+\|b_{n}\|_{L_{x,t}^{3,3}(Q^{+}_{1})}
+\|\pi_{n}\|_{L_{x,t}^{\kappa^{*},\lambda}(Q^{+}_{1})}=1, \quad \tau_{n}(\theta)\geq C\theta^{1+\alpha}. \end{equation} On the other hand, $v_{n}, b_n$ and $ \pi_{n}$ solve the following system in a weak sense: \begin{equation*} \begin{cases} \ \partial_{s}v_{n} -\Delta v_{n}+\epsilon_{n}(v_{n}\cdot\nabla)v_{n}-\epsilon_{n}(b_{n}\cdot\nabla)b_{n} +\nabla\pi_{n}=0,\quad\text{div}\ v_{n}=0&\quad\mbox{ in }\ Q^{+}_{1}, \\ \ \partial_{s}b_{n} -\Delta b_{n}+\epsilon_{n}(v_{n}\cdot\nabla)b_{n}-\epsilon_{n}(b_{n}\cdot\nabla)u_{n}=0,\quad\text{div}\ b_{n}=0&\quad\mbox{ in }\ Q^{+}_{1}, \end{cases} \end{equation*} with boundary data $v_n=0$, $b_n\cdot \nu=0$ and $(\nabla \times b_n)\times \nu=0$ on $B_1\cap\{x_{3}=0\}\times(-1,0)$. Since $\tau_{n}(1)=1$, we have following weak convergence: \begin{equation*} v_{n}\rightharpoonup u \quad\text{in}\ L_{x,t}^{3,3}(Q^{+}_{1}),\qquad\ b_{n}\rightharpoonup b \quad\text{in}\ L_{x,t}^{3,3}(Q^{+}_{1}),\qquad\ \pi_{n}\rightharpoonup \pi\quad\text{in}\ L_{x,t}^{\kappa^*,\lambda}(Q^{+}_{1}), \end{equation*} and $(\pi)_{B^{+}_{1}}(s)=0$. Moreover, we note that $\partial_{s}v_{n}$ and $\partial_{s}b_{n}$ are uniformly bounded in $L^{\lambda}\big((-1,0);(W^{2,2}(B^{+}_{1}))'\big)$, respectively and we also have \begin{equation}\label{timeprimevn} \partial_{s}v_{n}\rightharpoonup\partial_{s}u,\quad\partial_{s}b_{n}\rightharpoonup\partial_{s}b \qquad\text{in}\ L^{\lambda}\big((-1,0);(W^{2,2}(B^{+}_{1}))'\big), \end{equation}
Using the local energy inequality \eqref{local-energy}, $\nabla v_{n}$ and $\nabla b_{n}$ are uniformly bounded in $L_{x,t}^{2,2}(Q^{+}_{3/4})$, which implies
\begin{equation}\label{gradient-l2-weak} v_{n} \rightharpoonup u,\quad b_{n} \rightharpoonup b\qquad\text{in}\ W^{1,2}(Q^{+}_{3/4}). \end{equation} Its verification is rather standard, we skip its details (compare to \cite[Lemma 8]{GKT06}).
\begin{comment}
Indeed, let $\xi$ be a standard cut off function satisfying $\xi$ is smooth, \[ \xi=1 \quad \mbox{on} \ \ Q_{3/4}, \qquad \xi=0 \quad \mbox{on} \ \ ({ \mathbb{R} }^3 \times (-\infty,0)) \backslash Q_1, \ 0\leq \xi \leq 1. \] From the local energy inequality \eqref{local-energy}, we obtain for every $\sigma\in(-1,0)$ \begin{equation*} \begin{split}
\int_{B^{+}_{1}}&(|v_{n}(\cdot,\sigma)|^{2}+|b_{n}(\cdot,\sigma)|^{2})\xi^2(y,\sigma)dy
+\int_{-1}^{\sigma}\int_{B^{+}_{1}}(|\nabla v_{n}(y,s)|^{2}+|\nabla b_{n}(y,s)|^{2})\xi^{2}dyds\\
&\ \ \leq C\bigg(\int_{-1}^{\sigma}\int_{B^{+}_{1}}(|v_{n}|^{2}+|b_{n}|^{2})(|\partial_{s}\xi|
+|\Delta\xi|+|\nabla \xi|)dyds+\epsilon_{n}\int_{-1}^{\sigma}\int_{B^{+}_{1}}|v_{n}|^{3}|\nabla\xi|dyds\\ &\ \ \ \ \
+\epsilon_{n}\int_{-1}^{\sigma}\int_{B^{+}_{1}}|v_{n}||b_{n}|^2|\nabla\xi\xi|dyds
+\int_{-1}^{\sigma}\int_{B^{+}_{1}}|\pi_{n}v_{n}\cdot\nabla\xi\xi|dyds\bigg) \end{split} \end{equation*}
Consider the last two terms in the above inequality. Using H\"{o}lder inequality, we have \begin{equation*}
\epsilon_{n}\int_{-1}^{\sigma}\int_{B^{+}_{1}}|v_{n}||b_{n}|^2|\nabla\xi||\xi|dyds \leq
\||b_n|^2\nabla\xi\|_{L^{\frac{3}{2},\frac{3}{2}}_{x,t}(Q^{+}_{1})} \norm{v_{n}\xi}_{L^{3,3}_{x,t}(Q^{+}_{1})}, \end{equation*} \begin{equation*}
\int_{-1}^{\sigma}\int_{B^{+}_{1}}|\pi_{n}v_{n}\cdot\nabla\xi\xi|dyds
\leq \|\pi_{n}\nabla\xi\|_{L^{\kappa^*,\lambda}_{x,t}(Q^{+}_{1})} \norm{v_{n}\xi}_{L^{p,q}_{x,t}(Q^{+}_{1})}, \end{equation*} where $\kappa^{*},\lambda,p$ and $q$ are numbers in \eqref{pq} and \eqref{pq1}.
In case $q\leq 3$, since $p,q\leq 3$, we have $
\|v_{n}\xi\|_{L^{p,q}\big(Q^{+}_{1}\big)}\leq C\|v_{n}\xi\|_{L^{3,3}_{x,t}(Q^{+}_{1})}$, which implies that $\nabla v_{n},\nabla b_{n}$ is uniformly bounded in $L^{2,2}_{x,t}(Q^{+}_{3/4})$ because of $\tau_{n}(1)=1$.
It remains to consider the case $3<q<\infty$ (equivalently $3/2<p<9/4$). Suppose $2<p<9/4$. In this case, by interpolation, we have \begin{equation*}
\|v_{n}\xi\|_{L^{p,q}(Q^{+}_{1})}\leq C\sup_{-1<s<\sigma}\|v_{n}\xi(\cdot,s)\|_{L^{2,2}_{x,t}(B^{+}_{1})}^{\frac{2(3-p)}{p}}
\|v_{n}\xi\|_{L^{3,3}_{x,t}(Q^{+}_{1})}^{\frac{3(p-2)}{p}}, \end{equation*} where we used that $(p-2)q/p<1$. Since $\sigma$ is arbitrary on $(-1,0)$, we obtain \begin{equation*} \begin{split}
\sup_{-1<\sigma<0}(\|v_{n}&\xi(\cdot,\sigma)\|_{L^{2,2}_{x,t}(B^{+}_{1})}+\|b_{n}\xi(\cdot,\sigma)\|_{L^{2,2}_{x,t}(B^{+}_{1})})+(\|\nabla v_{n}\xi\|_{L^{2}(Q^{+}_{1})}+\|\nabla b_{n}\xi\|_{L^{2}(Q^{+}_{1})})
\\ & \leq C\Big(\|v_{n}\|_{L^{2}(Q^{+}_{1})}+\|b_{n}\|_{L^{2}(Q^{+}_{1})}+\epsilon_{n}\|v_{n}\|_{L^{3,3}_{x,t}(Q^{+}_{1})}
+\epsilon_{n}\|b_{n}\|^2_{L^{3,3}_{x,t}(Q^{+}_{1})}\|v_{n}\|_{L^{3,3}_{x,t}(Q^{+}_{1})}
\\&+\sup_{-1<s<0}\|v_{n}\xi(\cdot,s)\|_{L^{2}(B^{+}_{1})}^{\frac{2(3-p)}{p}}
\|v_{n}\xi\|_{L^{3,3}_{x,t}(Q^{+}_{1})}^{\frac{3(p-2)}{p}}
\|\pi_{n}\|_{L^{\kappa^{*},\lambda}(Q^{+}_{1})}\Big). \end{split} \end{equation*} Using the Young's inequality, we get \begin{equation*} \begin{split}
\sup_{-1<\sigma<0}(\|v_{n}\xi(\cdot,\sigma)&\|_{L^{2}(B^{+}_{1})}+\|b_{n}\xi(\cdot,\sigma)\|_{L^{2}(B^{+}_{1})})
+(\|\nabla v_{n}\xi\|_{L^{2}(Q^{+}_{1})}+\|\nabla b_{n}\xi\|_{L^{2}(Q^{+}_{1})})\\
&\leq C\Bigg((\|v_{n}\|_{L^{2,2}_{x,t}(Q^{+}_{1})}+\|b_{n}\|_{L^{2,2}_{x,t}(Q^{+}_{1})})+\epsilon_{n}\|v_{n}\|_{L^{3}(Q^{+}_{1})}
\\ &+\epsilon_{n}\|b_{n}\|^2_{L^{3,3}_{x,t}(Q^{+}_{1})}\|v_{n}\|_{L^{3,3}_{x,t}(Q^{+}_{1})}
+\|v_{n}\xi\|_{L^{3,3}_{x,t}(Q^{+}_{1})}\|\pi_{n}\|^{\frac{p}{3(p-2)}}_{L^{\kappa^{*}, \lambda}(Q^{+}_{1})}\Bigg). \end{split} \end{equation*} This also implies that $\nabla v_{n}$ and $\nabla b_{n}$ is also uniformly bounded in $L^{2,2}_{x,t}(Q^{+}_{3/4})$.
For the case $3/2<p\leq 2$, we have
$\|v_{n}\xi\|_{L^{p,q}(Q^{+}_{1})}\leq C\sup_{-1<s<\sigma}\norm{v_{n}\xi(\cdot,s)}_{L^{2,2}_{x,t}(B^{+}_{1})}$. By similar procedures as in the previous case, we obtain the uniform bound of $\nabla v_{n}$, $\nabla b_{n}$ in $L^{2,2}_{x,t}(Q^{+}_{3/4})$. So together with \eqref{timeprimevn} and \eqref{timeprimebn} completes the assertion \eqref{gradient-l2-weak}.
\end{comment}
We note that $u$, $b$ and $\pi$ solve the following linear Stokes system \begin{equation*} \begin{cases} \partial_{s}u-\Delta u+\nabla \pi=0,\quad\text{div}\ u=0\quad\text{in}\ Q^{+}_{1}, \\ \partial_{s}b-\Delta b=0,\quad\text{div}\ b=0\quad\text{in}\ Q^{+}_{1}, \end{cases} \end{equation*} with boundary data $u=0$, $b\cdot \nu=0$ and $(\nabla \times b)\times \nu=0$ on $B_1\cap\{x_{3}=0\}\times(-1,0)$. We can show that \begin{equation} \partial_{s}v_n, \ \partial_{s}b_n, \,\Delta v_n, \, \Delta b_n, \ \nabla \pi_n \rightharpoonup \partial_{s} u, \, \partial_{s} b, \ \Delta u, \, \Delta b, \nabla \pi \quad\mbox{ in }\,L_{x,t}^{\kappa,\lambda}(Q^{+}_{5/8}). \end{equation} Indeed, due to H\"older inequality, we see that \[ \norm{\abs{(v_{n}\cdot\nabla) v_{n}}+\abs{(b_{n}\cdot\nabla) b_{n}}+\abs{(v_{n}\cdot\nabla) b_{n}}+\abs{(b_{n}\cdot\nabla) v_{n}}}_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{3/4})} \] \begin{equation}\label{estimate-500}
\leq C(\|\nabla v_{n}\|^{\frac{2}{\lambda}}_{L^{2,2}_{x,t}(Q^{+}_{3/4})}+\|\nabla b_{n}\|^{\frac{2}{\lambda}}_{L^{2,2}_{x,t}(Q^{+}_{3/4})})
(\|v_{n}\|^{\frac{3-2\kappa}{\kappa}}_{L_{x,t}^{2,\infty}(Q^{+}_{3/4})}+
\|b_{n}\|^{\frac{3-2\kappa}{\kappa}}_{L_{x,t}^{2,\infty}(Q^{+}_{3/4})}). \end{equation} Using the local estimates of Stokes system and heat equations near boundary, \[
\|\abs{\partial_{s}v_{n}}+\abs{\partial_{s}b_{n}}+\abs{\nabla^{2}v_{n}} +\abs{\nabla^{2}b_{n}}+\abs{\nabla
\pi_{n}}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{5/8})} \] \[
\leq C\|\abs{v_{n}}+\abs{b_{n}}+\abs{\nabla v_{n}}+\abs{\nabla b_{n}}+\abs{\pi_{n}}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{3/4})} \] \[ +C\epsilon_{n}(\norm{\abs{(v_{n}\cdot\nabla) v_{n}}+\abs{(b_{n}\cdot\nabla) b_{n}}+\abs{(v_{n}\cdot\nabla) b_{n}}+\abs{(b_{n}\cdot\nabla) v_{n}}}_{L_{x,t}^{\kappa,\lambda}(Q^{+}_{3/4})}). \] We note that, due to \eqref{estimate-500}, the righthand side of the above estimate is bounded by $C(1+\epsilon_{n})$.
\begin{comment}
\begin{equation*} \begin{split}
(\|\abs{\partial_{s}&v_{n}}+\abs{\partial_{s}b_{n}}+\abs{\nabla^{2}v_{n}}+\abs{\nabla^{2}b_{n}}+\abs{\nabla
\pi_{n}}\|_{L^{\kappa,\lambda}(Q^{+}_{5/8})}\\
&\leq C\Big((\|v_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})}+\|b_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})})
+(\|\nabla v_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})}+\|\nabla b_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})})\\
&+\epsilon_{n}\|(v_{n}\cdot\nabla)v_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})}+\epsilon_{n}\|(b_{n}\cdot\nabla)b_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})}\\
&+\epsilon_{n}\|(v_{n}\cdot\nabla)b_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})}+\epsilon_{n}\|(b_{n}\cdot\nabla)v_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})}\Big)\\
&+\|\pi_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{3/4})}\leq C(1+\epsilon_{n}). \end{split} \end{equation*}
\end{comment}
According to estimates of the perturbed stokes system near boundary in \cite{SSS06}, $u, b$ are H\"{o}lder continuous in $Q^{+}_{1/2}$ with the exponent $\alpha$ with $0<\alpha<2(1-1/\lambda)$. Here we fix $\alpha_{0}=1-1/\lambda$. Then, by H\"{o}lder continuity of $u, b$ and strong convergence of the $L^{3}-$norm of $v_{n}, b_n$, we obtain \begin{equation}\label{scaling-2000} T_u(v_{n},\theta) \rightarrow T_u(u,\theta),\,\,\,T_b(b_{n},\theta) \rightarrow T_b(b,\theta),\quad T_u^{\frac{1}{3}}(u,\theta)+T_b^{\frac{1}{3}}(b,\theta)\leq C_1\theta^{1+\alpha_{0}}. \end{equation}
Next we need to estimate $\tilde{P}(\pi_n, \theta)$. Let $\tilde{B}^{+}$ be a domain with smooth boundary such that $B^{+}_{11/16}\subset\tilde{B}^{+}\subset B^{+}_{3/4}$, and $\tilde{Q}^{+}:=\tilde{B}^{+}\times(-(3/4)^2,0)$. Now we consider the following initial and boundary problem of $\bar{v}_{n}, \bar{b}_{n}, \bar{\pi}_{n}$ \begin{equation*} \partial_{s}\bar{v}_{n}-\Delta\bar{v}_{n}+\nabla\bar{\pi}_{n} =-\epsilon_{n}(\bar{v}_{n}\cdot\nabla)\bar{v}_{n}+\epsilon_{n}(\bar{b}_{n}\cdot\nabla)\bar{b}_{n},\quad {\rm{div}}\,\bar{v}_{n}=0\qquad \text{in}\ \tilde{Q}^{+}, \end{equation*}
\begin{equation*} (\bar{\pi}_{n})_{\tilde{B}^{+}}(s)=0,\quad s\in (-(\frac{3}{4})^2,0), \end{equation*} \begin{equation*} \bar{v}_{n}=0\quad\text{on}\ \partial\tilde{B}^{+}\times [-(\frac{3}{4})^2,0],\qquad\bar{v}_{n}=0\quad\text{on}\ \tilde{B}^{+}\times\{s=-(\frac{3}{4})^2\}. \end{equation*}
\begin{equation*} \partial_{s}\bar{b}_{n}-\Delta\bar{b}_{n}=-\epsilon_{n}(\bar{v}_{n}\cdot\nabla)\bar{b}_{n} +\epsilon_{n}(\bar{b}_{n}\cdot\nabla)\bar{v}_{n},\quad {\rm{div}}\,\bar{v}_{n}=0\qquad \text{in}\ \tilde{Q}^{+}, \end{equation*}
\begin{equation*} \bar{b}_{n}\cdot \nu=0, \ (\nabla \times \bar{b}_{n})\times \nu=0\quad\text{on}\ \partial\tilde{B}^{+}\times [-(\frac{3}{4})^2,0],\qquad \end{equation*} \[ \bar{b}_{n}=0\quad\text{on}\ \tilde{B}^{+}\times\{s=-(\frac{3}{4})^2\}. \] Using the estimate of Stokes system in Lemma \ref{lem1}, we get \begin{equation}\label{system-10} \begin{split}
\|\partial_{s}&\bar{v}_{n}\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})}
+\|\partial_{s}\bar{b}_{n}\|_{L_{x,t}^{\kappa,\lambda}(\tilde{Q}^{+})}
+\|\bar{v}_{n}\|_{L^{\kappa}((-(3/4)^2,0);W_{0}^{2,\lambda}(\tilde{B}^{+}))}\\
&+\|\bar{b}_{n}\|_{L^{\kappa}((-(3/4)^2,0);W_{0}^{2,\lambda}(\tilde{B}^{+}))}
+\|\bar{\pi}_{n}\|_{L^{\kappa}((-(3/4)^2,0);W^{1,\lambda}(\tilde{B}^{+}))}\\
\leq &\epsilon_{n}(\|(\bar{v}_{n}\cdot
\nabla)\bar{v}_{n}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+})}
+\|(\bar{b}_{n}\cdot
\nabla)\bar{b}_{n}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+})}\\
&+\|(\bar{v}_{n}\cdot
\nabla)\bar{b}_{n}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+})}
+\|(\bar{b}_{n}\cdot
\nabla)\bar{v}_{n}\|_{L_{x,t}^{\kappa,\lambda}(Q^{+})})\leq C\epsilon_{n}. \end{split} \end{equation} Next, we define $\tilde{v}_{n}=v_{n}-\bar{v}_{n}$, $\tilde{b}_{n}=b_{n}-\bar{b}_{n}$ and $\tilde{\pi}_{n}=\pi_{n}-\bar{\pi}_{n}$. Then it is straightforward that $\tilde{v}_{n}$, $\tilde{b}_{n}$ and $\tilde{\pi}_{n}$ solve \begin{equation*} \begin{cases} \partial_{s}\tilde{v}_{n}-\Delta\tilde{v}_{n}+\nabla\tilde{\pi}_{n}=0,\quad\text{div}\ \tilde{v}_{n}=0\quad\text{in}\ \tilde{Q}^{+}, \\ \partial_{s}\tilde{b}_{n}-\Delta\tilde{b}_{n}=0,\quad\text{div}\ \tilde{b}_{n}=0\quad\text{in}\ \tilde{Q}^{+}, \end{cases} \end{equation*} with boundary data $\tilde{v}_{n}=0$, $\tilde{b}_{n}\cdot \nu=0$ and $(\nabla \times \tilde{b}_{n})\times \nu=0$ on $B_1\cap\{x_{3}=0\}\times(-1,0)$.
\begin{comment} \begin{equation*} \tilde{v}_{n}=0\quad\text{on}\ \big(B_{\frac{3}{4}}\cap\{x_{3}=0\}\big)\times(-\frac{3}{4},0), \end{equation*} and \begin{equation*} \tilde{b}_{n}\cdot n=0, \ (\nabla \times \tilde{b}_{n})\times n=0 \quad\text{on}\ \big(B_{\frac{3}{4}}\cap\{x_{3}=0\}\big)\times(-\frac{3}{4},0), \end{equation*} \end{comment}
Using local estimates of Stokes system and heat equation near boundary, we then note that $\tilde{v}_{n}$, $\tilde{b}_{n}$ and $\tilde{\pi}_{n}$ satisfy \[
\|\nabla^{2}\tilde{v}_{n}\|_{L_{x,t}^{\tilde{\kappa},\lambda}(Q^{+}_{9/16})}
+\|\nabla^{2}\tilde{b}_{n}\|_{L_{x,t}^{\tilde{\kappa},\lambda}(Q^{+}_{9/16})}
+\|\nabla\tilde{\pi}_{n}\|_{L_{x,t}^{\tilde{\kappa},\lambda}(Q^{+}_{9/16})}\leq C(1+\epsilon_{n}), \] where $\tilde{\kappa}$ is the number with $3/\tilde{\kappa}+2/\lambda=1$.
\begin{comment} \begin{equation*}
(\|\nabla^{2}\tilde{v}_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{5/8})}+\|\nabla^{2}\tilde{b}_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{5/8})})
+\|\nabla\tilde{\pi}_{n}\|_{L^{\kappa,\lambda}(Q^{+}_{5/8})}\leq C(1+\epsilon_{n}), \end{equation*} and furthermore, for $3/\tilde{\kappa}+2/\lambda=1$, we obtain \[
(\|\nabla^{2}\tilde{v}_{n}\|_{L^{\tilde{\kappa},\lambda}(Q^{+}_{16/9})}+\|\nabla^{2}\tilde{b}_{n}\|_{L^{\tilde{\kappa},\lambda}(Q^{+}_{16/9})})
+\|\nabla\tilde{\pi}_{n}\|_{L^{\tilde{\kappa},\lambda}(Q^{+}_{9/16})}\leq C(1+\epsilon_{n}). \] \end{comment} Now, by the Poincar\'{e} inequality, we have \begin{equation*} \tilde{P}(\pi_{n},\theta)\leq C_{2}\Big(\tilde{P}_{1}(\bar{\pi}_{n},\theta)+\tilde{P}_{1}(\tilde{\pi}_{n},\theta)\Big). \end{equation*} We note that $P_{1}(\bar{\pi}_{n},\theta)$ goes to zero as $n\rightarrow\infty$ because of \eqref{system-10}. On the other hand, using the H\"{o}lder's inequality, we have \begin{equation*} P_{1}(\tilde{\pi}_{n},\theta)
\leq\theta^{2}\biggl(\int^{0}_{-\theta^{2}}\Bigl(\int_{B^{+}_{\theta}}
|\nabla\tilde{\pi}|^{\tilde{\kappa}}dy\Bigr)^{\frac{\lambda}{\tilde{\kappa}}}ds\biggr)^{\frac{1}{\lambda}} \leq C\theta^{2}(1+\epsilon_{n}). \end{equation*} Summing up above observations, we obtain \begin{equation}\label{scaling-3000} \liminf_{n\rightarrow\infty} \tilde{P}(\hat{\pi}_{n},\theta)\leq\lim_{n\rightarrow\infty}C_{2}\theta^{2}(1+\epsilon_{n})\leq C_{2}\theta^{1+\alpha_{0}}. \end{equation} Consequently, if a constant $C$ in \eqref{scaling-1000} is taken bigger than $2(C_{1}+C_{2})$ in \eqref{scaling-2000} and \eqref{scaling-3000}, this leads to a contradiction, since \begin{equation*} 2(C_{1}+C_{2})\theta^{1+\alpha_{0}}\leq C\theta^{1+\alpha_{0}}\leq\liminf_{n\rightarrow\infty}\tau_{n}(\theta) \leq(C_{1}+C_{2})\theta^{1+\alpha_{0}}. \end{equation*} This deduces the Lemma \ref{mhd-decay}. \end{pf-mhd-decay}
Lemma \ref{mhd-decay} is the crucial part of the proof of Proposition \ref{ep-regularity} and since its verification is rather straightforward (compare to \cite[Lemma 7]{GKT06}), the proof of Proposition \ref{ep-regularity} is omitted.
\begin{comment} we present only a brief sketch of the streamline for the proof of Proposition \ref{ep-regularity}.\\ {{\bf{The sketch of the proof of Proposition \ref{ep-regularity}}}\quad We note that due to the Lemma \ref{mhd-decay} there exists a positive constant $\alpha_1<1$ such that \[ M_{u}^{\frac{1}{3}}(r)+M_{b}^{\frac{1}{3}}(r)+\tilde{Q}(r) < C\theta^{1+\alpha_1}\bke{M_{u}^{\frac{1}{3}}(\rho)+M_{b}^{\frac{1}{3}}(\rho)+\tilde{Q}(\rho)},\qquad r<\rho<r_*, \] where $r_*$ is the number in Lemma \ref{mhd-decay}. We consider for any $x\in B^+_{r_*/2}$ and for any $r<r_*/4$ \[ \hat{M_{u}}_a(r)=\frac{1}{r^2}\int_{Q^+_{x,r}}\abs{u(y)-(u)_a}^3dz,\qquad (u)_a=\Xint\diagup_{B^+_{x,r}} u(y)dy, \] \[ \hat{M_{b}}_a(r)=\frac{1}{r^2}\int_{Q^+_{x,r}}\abs{b(y)-(b)_a}^3dz,\qquad (b)_a=\Xint\diagup_{B^+_{x,r}} b(y)dy. \] We can then show that $M_{u}^{\frac{1}{3}}\leq Cr^{1+\alpha_1}$ and $M_{b}^{\frac{1}{3}}\leq Cr^{1+\alpha_1}$, where $C$ is an absolute constant independent of $v$. This can be proved by straightforward computations and thus omit the details. H\"older continuity of $v$ is a direct consequence of this estimate, which immediately implies that $u$ is also H\"older continuous locally near boundary. This completes the proof.
\fbox{}\par
\end{comment}
\subsection{Interior regularity}
In this subsection, we present an interior regularity condition (see Theorem \ref{main-theorem-interior}) and give its proof. As mentioned in Introduction, we very recently became to know that the same result for interior case was obtained in \cite{WZ12}. However, since the proof of ours is different to that of \cite{WZ12}, we give our proof.
\begin{comment}
Here we give a proof of the interior regularity with a little different approach to the paper \cite{WZ12}, in which they are used the backward heat kernel as test function to find the local energy estimate, in particular (Compare to \cite{HD07}). \end{comment}
We first state the main result for interior case.
\begin{theorem}\label{main-theorem-interior}(\textbf{Interior regularity}) Let $(u,b,\pi)$ be a suitable weak solution of the MHD equations \eqref{MHD} in $\mathbb{R}^3 \times I$. Suppose that for every pair $p,q$ satisfying $ 1\leq \frac{3}{p}+\frac{2}{q}\leq 2, \ 1\leq q\leq \infty$, there exists $\epsilon>0$ depending only on $p,q$ such that for some point $z=(x,t)\in \mathbb{R}^3 \times I$ with $u$ is locally in $L^{p,q}_{x,t}$ and \begin{equation}\label{interior-condition1} \limsup_{r\rightarrow 0}r^{-(\frac{3}{p}+\frac{2}{q}-1)} \norm{\norm{u}_{L^p(B_{x,r})}}_{L^q(t-r^2,t)}<\epsilon. \end{equation} Then, $u$ and $b$ are regular at $z=(x,t)$ \end{theorem}
Compared to Theorem \ref{main-thm-boundary-v}, we remark that the range of $q$ in the interior is wider than that of the boundary case. This is mainly due to difference of estimates of the pressure for the interior and boundary cases. Since proof of interior case is simpler than the boundary case, we give the main stream of how the proof goes.
We first observe that the estimate \eqref{boundary-b} is also valid for the interior case for $1\leq q\leq \infty$. Since its verification is rather straightforward, we just state and omit the details (see also \cite[Lemma 3.7]{KL09}).
\begin{lemma}\label{interior-estimate-b} Let $z=(x,t)\in{ \mathbb{R} }^3\times I$. Suppose that $u\in L^{p,q}_{x,t}(Q_{z,r})$ with $\frac{3}{p}+\frac{2}{q}=2$, $\frac{3}{2} \leq p \leq \infty$. Then for $0<r\leq\rho/4$ \begin{equation}\label{interior-b} K_{b}(r)\leq C\bigg(\frac{\rho}{r}\bigg)^3G_{u,p,q}^2\Psi(\rho) +C\bigg(\frac{r}{\rho}\bigg)^2K_{b}(\rho). \end{equation} \end{lemma}
\begin{comment}
\begin{proof} For convenience, we assume that, without loss of generality, $z=(0,0)\in \Gamma\times I$. Let $\xi$ be a cut off supported in $Q_{\rho}$ and $\xi=1$ in $Q_{\frac{\rho}{2}}$. We decompose $b$ as the sum of $\hat{b}, \tilde{b}$ in the following manner: Firstly $\hat{b}$ solves \begin{equation}
\hat{b}_t-\Delta \hat{b}=-\nabla \cdot\bke{ [u\otimes b-b\otimes u]\xi} \qquad \mbox{in} \ \ \ \mathbb{R}^3 \times \mathbb{R}^{+}, \end{equation} and we set $\tilde{b}=b-\hat{b}$ in $Q_{\rho}$. Then we note that \[ \tilde{b}_t-\Delta \tilde{b}=0 \qquad \mbox{in} \ Q_{\frac{\rho}{2}}. \] Moreover, classical regularity theory implies \begin{equation}\label{tilde_b}
\int_{Q_r}|\tilde{b}|^2 dz \leq C\bigg(\frac{r}{\rho}\bigg)^5
\int_{Q_{\frac{\rho}{2}}}|\tilde{b}|^2 dz, \qquad 2r<\rho. \end{equation} Next, we consider $\hat{b}$. Using mixed norm estimates of the heat equation and H\"{o}lder's inequality, we obtain \[
\|\hat{b}\|_{L^{2,2}_{x,t}({ \mathbb{R} }^3 \times I)}\leq C\|\nabla
\hat{b}\|_{L^{\frac{6}{5},2}_{x,t}({ \mathbb{R} }^3\times I)}\leq C\|ub\xi\|_{L^{\frac{6}{5},2}_{x,t}(({ \mathbb{R} }^3\times I)} \] \begin{equation}\label{hat-b}
\leq C\|ub\|_{L^{\frac{6}{5},2}_{x,t}(Q_{\rho})}\leq C\|u\|_{L^{p,q}_{x,t}(Q_{\rho})}\norm{b}_{L^{\alpha,\beta}_{x,t}(Q_{\rho})}, \end{equation} where $\frac{3}{\alpha}+\frac{2}{\beta}=\frac{3}{2}$, $\frac{6}{5}\leq \alpha \leq 6$. First, we deal with the case $2\leq \alpha \leq 6$. As in Lemma \ref{estimate-ub}, we have via \eqref{hat-b} \begin{equation}\label{estimate-tilde_b1}
\frac{1}{\rho^{3}}\|\hat{b}\|^2_{L^{2,2}_{x,t}(Q_{\frac{\rho}{2}})}
\leq
\frac{C}{\rho^2}\|u\|^2_{L^{p,q}_{x,t}(Q_{\rho})}(A_{b}(\rho)+E_{b}(\rho)) \leq CG^2_{u,p,q}\Psi(\rho). \end{equation} Secondly, we consider the case that $\frac{6}{5}\leq \alpha < 2$. In this case, following as like as proof of a reference \cite[Lemma 3.7]{KL09}, \begin{equation}\label{estimate-tilde_b2}
\frac{1}{\rho^{3}}\|\hat{b}\|^2_{L^{2,2}_{x,t}(Q_{\frac{\rho}{2}})}\leq CG^2_{u,p,q}\Psi(\rho). \end{equation} By the estimates \eqref{tilde_b}, \eqref{estimate-tilde_b1} and \eqref{estimate-tilde_b2}, we have \[ K_{b}(r):=\frac{1}{r^3}\norm{b}_{L^{2,2}_{x,t}(Q_r)}^2\leq
\frac{1}{r^3}\|\tilde{b}\|_{L^{2,2}_{x,t}(Q_r)}^2+\frac{1}{r^3}\|\hat{b}\|_{L^{2,2}_{x,t}(Q_r)}^2 \] \[
\leq C\bigg(\frac{r}{\rho}\bigg)^2\frac{1}{\rho^3}\big(\|\hat{b}\|_{L^{2,2}_{x,t}(Q_{\frac{\rho}{2}})}^2
+\|b\|^2_{L^{2,2}_{x,t}(Q_{\frac{\rho}{2}})}\big)
+C\bigg(\frac{\rho}{r}\bigg)^3\frac{1}{\rho^3}\|\hat{b}\|_{L^{2,2}_{x,t}(Q_{\frac{\rho}{2}})}^2 \] \[
\leq C(\Big(\bigg(\frac{r}{\rho}\bigg)^2+\bigg(\frac{\rho}{r}\bigg)^3\Big)\frac{1}{\rho^3}\|\hat{b}\|_{L^{2,2}_{x,t}(Q_{\frac{\rho}{2}})}^2 +C\bigg(\frac{r}{\rho}\bigg)^2K_{b}(\rho). \] \[ \leq C\bigg(\frac{\rho}{r}\bigg)^3G_{u,p,q}^2\Psi(\rho) +C\bigg(\frac{r}{\rho}\bigg)^2K_{b}(\rho). \] This completes the proof. \end{proof} \end{comment}
Next we estimate the pressure in the interior. We first introduce a useful invariant functional in the interior case defined as follows: \[
S(r):=\frac{1}{r^2}\int_{Q_{z,r}}|\pi(y,s)|^{\frac{3}{2}}dyds. \] Now we recall an estimate of the pressure involving the functional above and since its proof is given in \cite[Lemma 3.3]{KL09}, we just state in the following lemma.
\begin{lemma}\label{interior-pressure}
Let $0<r\leq\rho/4$ and $Q_{\rho} \subset { \mathbb{R} }^3 \times I$. Then \begin{equation}\label{estimate-S} S(r)\leq C\bigg(\frac{\rho}{r}\biggr)^2\Psi(\rho)+C\bigg(\frac{r}{\rho}\bigg)S(\rho). \end{equation} \end{lemma}
\begin{comment} \begin{proof} Let $\phi$ be a standard cut off function supported in $B_{\rho}$ such that $\phi=1$ in $B_{\frac{\rho}{2}}$. Note that $-\Delta \pi=\partial_i\partial_j(u_iu_j-b_ib_j)$. We decompose the pressure $\pi$ by the decomposition of $\pi_1$ and $\pi_2$ as follows: \[ \pi_1(x,t):=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{\abs{x-y}} [\partial_i\partial_j(\hat{u}_i\hat{u}_j-\hat{b}_i\hat{b}_j)\phi](y,t)dy \quad \text{in} \ \mathbb{R}^3 \times I, \] where \[ \hat{u}_k=(u_k-(u_k)_{\rho}), \qquad \hat{b}_k=(b_k-(b_k)_{\rho}), \qquad k=1,2,3. \] We set $\pi_2(x,t):=\pi(x,t)-\pi_1(x,t)$ in $Q_{\rho}$.
We note that $\pi_2$ is harmonic in $B_{\frac{\rho}{2}}$ with respect to $x$ and via the mean value property, we have \begin{equation}\label{estimate-pi1} \frac{1}{r^2}\norm{\pi_2}_{L^{\kappa^{*}}_{x}(B_r)}^{\kappa^{*}}\leq C\frac{r}{\rho^3}\norm{\pi_2}_{L^{\kappa^{*}}_{x}(B_{\frac{\rho}{2}})}^{\kappa^{*}}\leq C\frac{r}{\rho^3}\norm{\pi}_{L^{\kappa^{*}}_{x}(B_{\frac{\rho}{2}})}^{\kappa^{*}}+ C\frac{r}{\rho^3}\norm{\pi_1}_{L^{\kappa^{*}}_{x}(B_{\frac{\rho}{2}})}^{\kappa^{*}}. \end{equation} Integrating in time variable, we get \begin{equation}\label{estimate-pi15} \frac{1}{r^2}\norm{\pi_2}_{L^{\kappa^{*},\lambda}_{x,t}(Q_r)}^{\lambda}
\leq C\frac{r}{\rho^3}\norm{\pi}_{L^{\kappa^{*},\lambda}_{x,t}(Q_{\frac{\rho}{2}})}^{\lambda}+ C\frac{r}{\rho^3}\norm{\pi_1}_{L^{\kappa^{*},\lambda}_{x,t}(Q_{\frac{\rho}{2}})}^{\lambda}. \end{equation}
On the other hand, via the Calderon-Zygmund theorem, we have \begin{equation}\label{estimate-pi2} \frac{1}{r^2}\norm{\pi_1}_{L^{\kappa^{*},\lambda}_{x,t}(B_{\frac{\rho}{2}})}^{\lambda}\leq \frac{1}{r^2}\norm{\pi_1}_{L^{\kappa^{*},\lambda}_{x,t}(\mathbb{R}^3)}^{\lambda}\leq \frac{C}{r^2}(\norm{(u-(u)_{\rho})}^{2\lambda}_{L^{2\kappa^{*},2\lambda}_{x,t}(B_{\rho})} +\norm{(b-(b)_{\rho})}^{2\lambda}_{L^{2\kappa^{*},2\lambda}_{x,t}(B_{\rho})}). \end{equation} Combining the above estimate \eqref{estimate-pi1} and \eqref{estimate-pi2} and integrating in time, we get \[ \frac{1}{r^2}\norm{\pi}_{L^{\kappa^{*},\lambda}_{x,t}(Q_r)}^{\lambda}\leq \frac{C}{r^2}\norm{\pi_1}_{L^{\kappa^{*},\lambda}_{x,t}(Q_r)}^{\lambda} +\frac{C}{r^2}\norm{\pi_2}_{L^{\kappa^{*},\lambda}_{x,t}(Q_r)}^{\lambda} \] \[ \leq C(\frac{r}{\rho})\frac{1}{\rho^2}\norm{\pi}_{L^{\kappa^{*},\lambda}_{x,t}(Q_{\frac{\rho}{2}})}^{\lambda}+ C(\frac{\rho}{r})^2\frac{1}{\rho^2}(\norm{(u-(u)_{\rho})}^{2\lambda}_{L^{2\kappa^{*},2\lambda}_{x,t}(B_{\rho})} +\norm{(b-(b)_{\rho})}^{2\lambda}_{L^{2\kappa^{*},2\lambda}_{x,t}(B_{\rho})}). \] Note that \[ \norm{(u-(u)_{\rho})}^{2\lambda}_{L^{2\kappa^{*},2\lambda}_{x,t}(B_{\rho})} \leq C\Psi(\rho) \qquad \mbox{and} \qquad \norm{(b-(b)_{\rho})}^{2\lambda}_{L^{2\kappa^{*},2\lambda}_{x,t}(B_{\rho})} \leq C\Psi(\rho) \]
Therefore, we obtain the estimate \eqref{estimate-Q}. \end{proof} \end{comment}
The other estimates such as $M_u(r)$ and $\int_{Q_{z,r}}|u| |b|^2 dxdt$ in Lemma \ref{estimate-ub} is also valid in the interior by following similar arguments, and thus we skip its details. Now we are ready to give the proof for Theorem \ref{main-theorem-interior}. Here all invariant functionals in this subsection are defined over the interior parabolic balls.
\begin{pfthm3} Under the hypothesis \eqref{interior-condition1}, recalling
Lemma \ref{interior-estimate-b} and Lemma \ref{estimate-ub}, we note first that for $4r<\rho$ \begin{equation}\label{kb-int-est} K_{b}(r)\leq C\epsilon^2(\frac{\rho}{r})^3\Psi(\rho) +C(\frac{r}{\rho})^2K_{b}(\rho)\leq C\bke{\epsilon^2(\frac{\rho}{r})^3+(\frac{r}{\rho})^2}\Psi(\rho), \end{equation} \begin{equation}\label{mu-intmb} M_u(r)+\frac{1}{r^2}\int_{Q_{z,r}}\abs{u}\abs{b}^2dz\leq C\epsilon(\frac{\rho}{r})\Psi(\rho). \end{equation} Next, due to the pressure estimate \eqref{estimate-S}, we obtain \[ \frac{1}{r^2}\int_{Q_{z,r}}\abs{u}\abs{\pi} dz \leq M^{\frac{1}{3}}_u(r)S^{\frac{2}{3}}(r)\leq C\epsilon^{\frac{1}{3}} (\frac{\rho}{r})^{\frac{5}{3}}\Psi(\rho)+C\epsilon^{\frac{1}{3}} (\frac{r}{\rho})^{\frac{1}{3}}\Psi^{\frac{1}{3}}(\rho)S^{\frac{2}{3}}(\rho) \] \begin{equation}\label{int-pre-est} \leq C\epsilon^{\frac{1}{3}} (\frac{\rho}{r})^{\frac{5}{3}}\Psi(\rho)+C\epsilon^{\frac{1}{3}} (\frac{r}{\rho})^{\frac{1}{3}}S(\rho), \end{equation} where Young's inequality is used. Next, combining estimates \eqref{kb-int-est}-\eqref{int-pre-est}, we have via the local energy inequality
\[ \Psi(\frac{r}{2})\leq C\bke{\epsilon^2(\frac{\rho}{r})^3+(\frac{r}{\rho})^2}\Psi(\rho) +C\epsilon^{\frac{2}{3}}(\frac{\rho}{r})^{\frac{2}{3}}\Psi^{\frac{2}{3}}(\rho) \] \begin{equation}\label{est-psi-100} +C\epsilon(\frac{\rho}{r})\Psi(\rho) +C\epsilon^{\frac{1}{3}} (\frac{\rho}{r})^{\frac{5}{3}}\Psi(\rho)+C\epsilon^{\frac{1}{3}} (\frac{r}{\rho})^{\frac{1}{3}}S(\rho) \end{equation} where we used Young's inequality and $\frac{1}{r^3}\int_{Q_{z,r}} \abs{u}^2\leq CM^{\frac{2}{3}}_u(r)$. Let $\epsilon_3$ be a small positive number, which will be specified later. Now via estimates \eqref{estimate-S} and \eqref{est-psi-100} we consider \[ \Psi(\frac{r}{2})+\epsilon_3 S(\frac{r}{2}) \leq \epsilon+ C\bke{\epsilon^2(\frac{\rho}{r})^3+(\frac{r}{\rho})^2 +\epsilon_3(\frac{\rho}{r})^2}\Psi(\rho) +C\bke{\epsilon^{\frac{1}{3}} (\frac{r}{\rho})^{\frac{1}{3}}+\epsilon_3(\frac{r}{\rho})}S(\rho), \] where we used Young's inequality. We fix $\theta \in (0,\frac{1}{4})$ with $\theta<\frac{1}{8^3(C+1)^3}$ and then $\epsilon_3$ and $\epsilon$ are taken to satisfy \[ 0<\epsilon_3<\min\bket{\frac{\theta^2}{16C^2}},\qquad 0<\epsilon<\min\bket{\frac{\epsilon^*}{4}, \,\,\epsilon^3_3,\,\, \frac{\theta^{\frac{3}{2}}}{4C^{\frac{1}{2}}}}, \] where $\epsilon^*$ is the number introduced in \cite[Theorem 1.1]{KL09}. We then obtain \[ \Psi(\theta r)+\epsilon_3 S(\theta r)\leq \frac{\epsilon^*}{4}+\frac{1}{4}\Big(\Psi(r)+\epsilon_3 S(r)\Big). \] Usual method of iteration implies that there exists a sufficiently small $r_0>0$ such that for all $r<r_0$ \[ \Psi(r)+\epsilon_3 S(r)\leq \frac{\epsilon^*}{2}. \] This completes the proof. \end{pfthm3}
\begin{comment}
Recall that \[ \frac{1}{r^2}\int_{Q_{z,r}}\abs{u}\abs{\pi} dz \leq \frac{1}{r}\norm{u}_{L^{p,q}_{x,t}(Q_{z,r})}\frac{1}{r}\norm{\pi}_{L^{\kappa^{*},\lambda}_{x,t}(Q_{z,r})} \] \[ \leq \frac{C}{r}\norm{u}_{L^{p,q}_{x,t}(Q_{z,r})}\Big(\bigg(\frac{\rho}{r}\bigg)^2\Psi(\rho)+\bigg(\frac{r}{\rho}\bigg)Q(\rho)\Big) \leq CG_{u,p,q}\Big(\bigg(\frac{\rho}{r}\bigg)^2\Psi(\rho)+\bigg(\frac{r}{\rho}\bigg)Q(\rho)\Big) \]
Finally, with aid of Lemma \ref{estimate-Mu}, \ref{interior-estimate-b} and \ref{interior-pressure}, we have \begin{equation*} \Psi(\frac{r}{2})\leq C\bigg(\frac{\rho}{r}\bigg)^{\frac{2}{3}}\Psi(\rho)^{\frac{2}{3}}G_{u,p,q}(r)^{\frac{2}{3}} +C\bigg(\frac{\rho}{r}\bigg)\Psi(\rho)G_{u,p,q}(r)+C\bigg(\frac{\rho}{r}\bigg)^3G_{u,p,q}^2\Psi(\rho)+C\bigg(\frac{r}{\rho}\bigg)^2K_{b}(\rho) \end{equation*} \[ +CG_{u,p,q}\Big(\bigg(\frac{\rho}{r}\bigg)^2\Psi(\rho)+\bigg(\frac{r}{\rho}\bigg)Q(\rho)\Big) \]
Hence \[ \Psi(\frac{r}{2}) \leq \epsilon+C\Big((\epsilon^{\frac{1}{2}}+\epsilon)\bigg(\frac{\rho}{r}\bigg)+\epsilon\bigg(\frac{\rho}{r}\bigg)^3+\epsilon\bigg(\frac{\rho}{r}\bigg)^2 \Big)\Psi(\rho)+C\epsilon \bigg(\frac{r}{\rho}\bigg)Q(\rho)+\bigg(\frac{r}{\rho}\bigg)^2K_b(\rho). \]
Since $K_b(\rho)\leq\Psi(\rho)$ and add $\epsilon_2 Q(\frac{r}{2})$ to both side, we have \[ \Psi(\frac{r}{2})+\epsilon_2 Q(\frac{r}{2}) \leq \epsilon+C\Big((\epsilon^{\frac{1}{2}}+\epsilon)\bigg(\frac{\rho}{r}\bigg) +\epsilon\bigg(\frac{\rho}{r}\bigg)^3+(\epsilon+\epsilon_2)\bigg(\frac{\rho}{r}\bigg)^2+\bigg(\frac{r}{\rho}\bigg)^2 \Big)\Psi(\rho)+C(\epsilon+\epsilon_2) \bigg(\frac{r}{\rho}\bigg)Q(\rho) \]
\end{comment}
\begin{comment}
\section{Open Problems} \begin{theorem}\label{main-theorem}(\textbf{Interior regularity}) Let $I=(t_0,t_1]$. Suppose the triple $(u,b,\pi)$ is a suitable weak solution of \eqref{MHD} in $\mathbb{R}^3 \times I$. Let $z=(x,t)\in \mathbb{R}^3 \times I$ and $Q_{z,r} \subset \mathbb{R}^3 \times I.$ There exists a constant $\epsilon(p,q)>0$ such that, if the pair $u,p$ is a suitable weak solution of the MHD equations satisfying Definition \eqref{sws-5dnse} and one of the following conditions holds \\
(a) (\textbf{Velocity gradient condition}) \ $\nabla u \in L^{\tilde{p},q}_{\text{loc}}$ near $z$ and \begin{equation}\label{thm1-condition2} \limsup_{r\rightarrow 0}r^{-(\frac{3}{\tilde{p}}+\frac{2}{q}-2)} \norm{\norm{\nabla u}_{L^{\tilde{p}}(B_{x,r})}}_{L^{q}(t-r^2,t)}<\epsilon, \end{equation} with $ 2\leq \frac{3}{\tilde{p}}+\frac{2}{q}\leq 3, \ 1\leq q\leq \infty,$\\ (b) (\textbf{Vorticity condition}) \ $w:=\nabla \times u \in L^{\tilde{p},q}_{\text{loc}}$ near $z$ and \begin{equation}\label{thm1-condition3} \limsup_{r\rightarrow 0}r^{-(\frac{3}{\tilde{p}}+\frac{2}{q}-2)} \norm{\norm{w}_{L^{\tilde{p}}(B_{x,r})}}_{L^{q}(t-r^2,t)}<\epsilon. \end{equation} with $ 2\leq \frac{3}{\tilde{p}}+\frac{2}{q}\leq 3, \ 1\leq q\leq \infty$ and $(\tilde{p},q)\neq (1,\infty),$\\ (c) (\textbf{Vorticity gradient condition}) \ $\nabla w \in L^{\breve{p},q}_{\text{loc}}$ near $z$ and \begin{equation}\label{thm1-condition4} \limsup_{r\rightarrow 0}r^{-(\frac{3}{\hat{p}}+\frac{2}{q}-2)} \norm{\norm{\nabla w}_{L^{\breve{p}}(B_{x,r})}}_{L^{q}(t-r^2,t)}<\epsilon. \end{equation}with $ 3\leq \frac{3}{\breve{p}}+\frac{2}{q}\leq 4, \ 1\leq q$ and $1 \leq \breve{p}$,
then $(u,b)$ is regular at $z$. \end{theorem}
\end{comment}
\section*{Acknowledgments} K. Kang's work was partially supported by NRF-2012R1A1A2001373. J.-M. Kim's work was partially supported by KRF-2008-331-C00024 and NRF-2009-0088692.
The authors wish to expresses our appreciation to Professor Tai-Peng Tsai for useful comments.
\begin{equation*} \left. \begin{array}{cc} {\mbox{Kyungkeun Kang}}\qquad&\qquad {\mbox{Jae-Myoung Kim}}\\ {\mbox{Department of Mathematics }}\qquad&\qquad
{\mbox{Department of Mathematics}} \\ {\mbox{Yonsei University }}\qquad&\qquad{\mbox{Sungkyunkwan University}}\\ {\mbox{Seoul, Republic of Korea}}\qquad&\qquad{\mbox{Suwon, Republic of Korea}}\\ {\mbox{[email protected] }}\qquad&\qquad {\mbox{[email protected] }} \end{array}\right. \end{equation*}
\end{document} | arXiv |
\begin{document}
\pagebreak
\title{On smoothness of isometries between orbit spaces }
\author{Marcos Alexandrino and Alexander Lytchak}
\keywords{Isometric groups action, singular Riemannian foliations, invariant function, basic form}
\begin{abstract} We discuss the connection between the smooth and metric structure on quotient spaces, prove smoothness of isometries in special cases and discuss an application to a conjecture of Molino.
\end{abstract}
\thanks{The first author was supported by a CNPq-Brazil research fellowship and partially supported by Fapesp. The second author was supported by a Heisenberg grant of the DFG and by the SFB 878 {\it Groups, geometry and actions}}
\maketitle \renewcommand{\arabic{section}.\arabic{equation}}{\arabic{section}.\arabic{equation}} \pagenumbering{arabic}
\section{Smoothness of isometries}
\subsection{The main question} Given an action of a closed group of isometries $G$ on a Riemannian manifold $M$, the quotient $X=M/G$ is equipped with the natural quotient metric and a natural quotient ``smooth structure''. The \emph{smooth structure} on $X$ is given by the sheaf of ``\emph{smooth functions}" $\mathcal C ^{\infty} (U) := (\mathcal C^{\infty } (\pi ^{-1} (U)))^G$, where $U\subset X$ is an arbitrary open set and $\pi:M\to X$ is the canonical projection. One says that a map $F: M/G \to N /H$ between two quotient spaces is \emph{smooth} if the pull-back by $F$ sends smooth functions on $N/H$ to smooth functions on $M/G$. If $F$ is bijective and smooth together with its inverse, it is called a diffeomorpism.
If $X$ is (isometric to) a smooth Riemannian manifold, i.e., if $M \to X$ is a fiber bundle, then the quotient smooth structure on $X$ is the same as the underlying smooth structure of the Riemannian manifold $X$. In this case, the classical theorem of Myers-Steenrod (\cite{msteen}) states that the metric of $X$ determines its smooth structure. This motivates the following natural and simple-minded question:
\begin{quest} \label{firstq} Does the metric of a quotient space $X$ determine its smooth structure? In other words, given two manifolds with isometric actions $(M,G)$ and $(N,H)$ and an isometry $I:M/G \to N /H$, is $I$ always a diffeomorphism? \end{quest}
\subsection{Comments on the smooth structure} \label{subsec2} Before we proceed, we would like to make a few comments about the smooth structure on quotients. First and most important: the terms ``smooth structure", ``smooth function", ``diffeomorphism" may be misleading, since the quotient $X=M/G$ almost never is a smooth manifold. We adopt the notation of \cite{Schwarz3} and hope that the terms are not too ambiguous.
By definition, the projection $\pi:M \to X=M/G$ is smooth; a map $F:X\to Y$ is smooth iff $F\circ \pi $ is smooth. Note that the smooth structure (and the metric on the quotient) does not change if one replaces the action of $G$ by an isometric action of another group, \emph{orbit equivalent} to the action of $G$, i.e., having the same orbits. The question when a smooth map (a diffeomorphism) $F:M/G \to N/H$ (of a space to itself) can be lifted to a smooth map (diffeomorphism) between the total spaces $M$ and $N$ is highly non-trivial (cf. \cite{Strub}, \cite{Schwarz3}, \cite{Schwarz1}).
Clearly, one can always reduce question of smoothness to the the case where the manifolds in question are connected. Below we will always implicitly make this connectedness assumption.
Using distance functions to orbits, one easily constructs arbitrary fine smooth partitions of unity in quotient spaces. In particular, this implies that all questions concerning smoothness are local.
Let $p\in M$ be an arbitrary point and let $V_p ^{\perp}$ be the normal space to the fiber through $p$. The isotropy group $G_p$ acts on $V_p ^{\perp}$ defining the \emph{slice representation} at $p$. The famous slice theorem (cf. \cite{Bredon}) says that the exponential map $\exp:V_p ^{\perp} \to M$ descends to a local diffeomorphism $O\subset V_p ^{\perp}/G_p \to M/G$ from a small neighborhood $O$ of $0$ to a small neighborhood $O'$ of $x=\pi (p) \in X$. The space $V_p ^{\perp}/G_p$ is the tangent space of $X$ at $x$ (in the sense of metric geometry) and the above diffeomorphism will also be denoted by $\exp _x$.
Let now $I:X=M/G \to Y=N/H$ be an isometry and let $x\in X$ a point with $y=I(x)$. Then $I$ sends geodesics
starting at $x$ to geodesics starting in $y$. Thus, on a small neighborhood of $x$ we have $I= \exp _y \circ D_xI \circ \exp_x ^{-1}$, for the induced isometry $D_x I$ between the metric tangent spaces $D_x I: T_x X \to T_y Y$. Since the exponential maps are local diffeomorphisms, $I$ is a local diffeomorphism at $x$ if and only if the isometry $D_x I$ is a diffeomorphism between the quotient spaces $T_x X:= V_p ^{\perp} /G_p \to T_y Y =V_q ^{\perp} /H_q$. Here, $p$ and $q$ are arbitrary points in the orbits corresponding to $x$ and $y$ respectively. This reduces our main Question \ref{firstq} to the case of representations.
Recall that, for a representation of a compact group $G$ on a vector space $V$, the set of $G$-invariant polynomials is finitely generated by some polynomials $f_1,...,f_n$, due to a theorem of Hilbert. By a theorem of Schwarz, any smooth $G$-invariant function on $V$ is a smooth function of the polynomials
$(f_1,...,f_n)$ (\cite{Schwarz2}). Thus to prove that an isometry between quotients of representation is a diffeomorphism, it is sufficient (and necessary) to prove that it preserves invariant polynomials.
The last remark leads to an interesting question in metric geometry. Namely, any $G$-invariant polynomial of degree $n$ on a representation vector space $V$ of $G$ defines a map on the quotient space $V/G$, whose restriction to each geodesic is polynomial of degree at most $n$. It seems interesting to recognize such functions metrically:
\begin{quest} \label{questalex} Let $X$ be an Alexandrov space. Let $f$ be a function whose restriction to each geodesic is a polynomial of degree at most $n$. Does it have some implication on the structure of $X$? \end{quest}
For $n=1$, the answer is yes and a splitting result under the assumption that $X$ does not have boundary has been obtained in \cite{AB}. It corresponds to the easy statement that $G$-invariant linear functions on $V$ can exist only if the set of fixed points of $G$ is non-trivial.
Already for $n=2$, Question \ref{questalex} seems to be much harder.
\subsection{Known results} Probably, the main contribution of this note containing more questions than results, is the observation that Question \ref{firstq} is non-trivial and interesting for applications and for its own sake. We are going to explain now, that our Question \ref{firstq} has been answered in special cases by some famous theorems. Beyond the theorem of Myers-Steenrod, mentioned above, there are two important algebraic results.
The first result is the famous restriction theorem of Chevalley (\cite{Chev}). Let $\mathfrak p$ be the tangent space of a symmetric space $M=G/K$ with the induced isotropy representation of $K$. Let $\mathfrak a$ be a maximal flat in $\mathfrak p$ and let $W$ be the (finite) stabilizer of $\mathfrak a$ in $K$. The isotropy representation is \emph{polar}, meaning that the embedding $\mathfrak a \to \mathfrak p$ induces an isometry $I:\mathfrak a /W \to \mathfrak p /K$; we refer to \cite{Alexgor}, \cite{Michor1}, \cite{LT} for basics facts about polar actions. The theorem of Chevalley says that this isometry induces an isomorphism between the rings of invariants, i.e., that $I$ is a smooth map.
The second result is the theorem of Luna and Richardson (\cite{lr}). Given an isometric action of $G$ on $M$ with non-trivial principal isotropy group $H =G_p$ of a regular point $p\in M$,
let $\bar M$ be the connected component through $p$ of the set of fixed points $M^H$ of $H$. The normalizer $N(H)$ of $H$ in $G$ stabilizes the \emph{generalized section} $\bar M$ (cf. \cite{got}, \cite{Magata}) and the embedding $\bar M \to M$ induces an isometry $I:\bar M/N(H) \to M /G$ (to be precise, if $M^H$ is not connected, one may need to replace $N(H)$ by an open subgroup of it) .
If $M$ is a Euclidean vector space then the main result of \cite{lr} (after applying a complexification and using \cite{Schwarz2}, cf. \cite{str}) says that $I$ is a diffeomorphism. Applying the reduction procedure to slice representations from the previous subsection and using that the tangent space of a generalized section is a generalized section of the slice representation, we deduce that the reduction map $I$ is a diffeomorphism for arbitrary $M$.
\begin{rem} In fact, applying Theorem 2.2 from \cite{lr}, instead of the special case we have applied above, one deduces that the reduction of $(M,G)$ to any generalized section $(\Sigma, N)$ in the sense of \cite{got} induces a diffeomorphism $I:\Sigma /N\to M/G$. \end{rem}
Finally, we mention the extension of Chevalley's restriction theorem to general polar actions by Michor (\cite{Michor1}, \cite{Michor2}).
\subsection{Some new simple observations}
Our first observation is that the analogue of the theorem of Myers-Steenrod is true if the quotient is isometric to a Riemannian orbifold:
\begin{thm} \label{firstthm} Let $X= M/G$ be as above. If there is an isometry $I:X \to B$, where $B$ is a smooth Riemannian orbifold, then $I$ is a diffeomorphism between the quotient smooth structure of $X$ and the underlying smooth orbifold structure of $B$. \end{thm}
A few comments before we proceed. The assumption that $X$ is (isometric to) a Riemannian orbifold is satisfied in many geometric
situations,
for instance, for actions of cohomogeneity at most $2$, for variationally complete or polar actions (\cite{LT}). The quotient $X$ is a Riemannian orbifold if and only if the action is locally diffeomorphic to a polar action, for which reason, such actions are called \emph{infinitesimally polar}. Another equivalent formulation is that all slice representations are polar. However, for polar actions (representations) \tref{firstthm} follows from the proof of Michor (\cite{Michor1}, \cite{Michor2}). The reduction of the global to the infinitesimal problem discussed in Subsection \ref{subsec2} provides now a proof of \tref{firstthm}. (A slightly different proof of the Theorem will be explained in the last section).
From the theorem of Myers-Steenrod one deduces that the smooth structure on a Riemannian orbifold is uniquely determined by the metric. Hence we get
\begin{cor} \label{coro} The answer to Question \ref{firstq} is affirmative if $M/G$ is isometric to a Riemannian orbifold, i.e., if the action of $G$ on $M$ is infinitesimally polar. \end{cor}
We are going to deduce the affirmative answer to our main question in another special case. Again the proof will be an easy consequence of known results. \begin{thm} \label{secondthm} Let $X=M/G$ and $Y=N/H$ be of dimension at most $3$. Then each isometry $I:X\to Y$ is a diffeomorphism. \end{thm}
\begin{proof} If the dimension of $X$ is at most $2$, then the action is infinitesimally polar and the result follows from \cref{coro}. Thus we may assume that $X$ and $Y$ have dimension $3$. Proceeding as in Subsection \ref{subsec2}, we may assume that $M=V$ and $N=W$ are real vector space representations of $G$ and $H$ respectively (we might replace $G$ and $H$ by corresponding isotropy groups). If one of the representations is polar, so is the other, and the result follows again from \cref{coro}. Thus we may and will assume that the representations are not polar.
We may replace $G$ and $H$ by larger groups $G'\subset O(V)$ and $H'\subset O(V)$ having the same orbits as $G$ and $H$ respectively, whenever it is possible. Using the theorem of Luna-Richardson, we may replace $(G',V)$ by the generalized section $(N(G' _p) /G'_p ,V^{G'_p})$ and perform the same operation on $(H',W)$. In this way we get an isometry $\hat I :\hat V /\hat G \to \hat W /\hat H$, where $\hat G$ and $\hat H$ act on $\hat V$ and $\hat W$ respectively, such that the principal isotropy groups are trivial and the groups cannot be enlarged without enlarging the orbits. Due to the theorem of Luna-Richardson, all maps we used producing the ''reduction`` $\hat I$ are diffeomorphism. Hence it suffices to prove that $\hat I$ is a diffeomorphism.
Note that the actions of $\hat G$ (and $\hat H$) are not polar. Now we invoke the classification of all representations of cohomogeneity $3$, as it is discussed in \cite{str}. From the above assumptions on the representations we deduce that $\hat G$ and $\hat H$ are one-dimensional and that $\hat V=\hat W=\mathbb R^4$. In this case, using Section 4 of \cite{str}, one deduces that $\hat G =\hat H$, and that the representations of $\hat H$ and $\hat G$ on $\mathbb R^ 4$ are equivalent. Hence we may assume that $\hat I$ is an isometry of $\mathbb{R} ^4 /\hat G$ to itself. In Section 4 of \cite{str} it is shown that any such isometry is induced by an element $J$ in the normalizer of $\hat G$ in $O(4)$. In particular, $\hat I$ preserves the smooth structure. \end{proof}
\subsection{What to do in general?} The proof of \tref{firstthm} as described above is algebraic. However, at least after the reduction to the case of representations, there is a more geometric proof of \tref{firstthm}, essentially contained in \cite{Terng}, that uses the geometric properties of the Laplacian. We hope that the use of invariant differential operators (cf. \cite{mendes}) may help to understand the structure of smooth functions on quotients and to answer Question \ref{firstq}.
Another algebraic way, chosen in the proof of \tref{secondthm} consists in understanding (classification) of representations having isometric quotients. The description of such \emph{quotient equivalence classes} of representations is probably possible in each concrete case, but seems to be difficult in general. At least in small codimensions, it should be possible to prove the analogue of \tref{secondthm} along the same lines. Some ideas and results concerning quotient-equivalence classes can be found in \cite{GL}. These results and the proof of \tref{secondthm} motivate the following question. An affirmative answer to it would provide an answer to Question \ref{firstq} as well.
\begin{quest} Let $(V,G)$ and $(W,H)$ be two representations and let $I:V/G \to W/H$ be an isometry. Can $I$ be obtained as a compositions of following three kinds of isometries between quotients: The isometry induced by an orbit equivalence (i.e., replacing a group by a larger group having the same orbit); the isometry induced by the reduction as in the theorem of Luna-Richardson; the isometry $\hat I$ of a quotient $U/K$ to itself induced by an element $g$ in the normalizer of $K$ in the orthogonal group $O(U)$? \end{quest}
\section{Basic forms} The quotient space $X=M/G$ contains an open dense Riemannian manifold $X_0$ that consists of the set of principal orbits. A smooth function on $X$ as defined in the previous section, is just a smooth function on the manifold $X_0$, whose pull-back to $M$ extends to a smooth function on $M$. It is natural to define generalizations of smooth forms in the same way.
A \emph{basic $p$-form} on an open subset $U$ of $X$ is a smooth $p$-form on $X_0 \cap U$, whose pull-back to $M$ extends to a smooth $p$-form on $\pi ^{-1} (U)$. The basic forms constitute a complex (of sheaves on $X$) and its cohomology coincides with the singular cohomology of $X$ (The sheaves are fine and the usual lemma of Poincare holds with the usual proof, cf. \cite{Kosz}). Despite this fact and the very natural definition of the complex, the following question seems to be difficult in general:
\begin{quest} \label{secondq} Does the smooth structure determine the basic forms on a quotient? In other words, given a diffeomorphism $F:M_1/G_1 \to M_2/G_2$, does $F$ induce a bijection between basic forms? \end{quest}
The following question generalizes Question \ref{firstq}: \begin{quest} \label{thirdq} Does an isometry $I:M/G \to N/H$ always preserve the sheaves of basic forms? \end{quest}
The arguments from Subsection \ref{subsec2} apply to this situation as well and reduce both question to the case of representations. Again, in the case of infinitesimally polar actions the answer to both question is affirmative and an easy consequence of known results:
\begin{thm} If $X=M/G$ is isometric to a smooth Riemannian orbifold then the basic forms of $X$ are precisely the smooth forms of the underlying smooth orbifold. \end{thm}
\begin{proof} The reduction explained in Subsection \ref{subsec2} reduces the question to isotropy representations. For infinitesimally polar actions, the isotropy representations are polar, in which case the result is known (\cite{Michor1}, \cite{Michor2}). \end{proof}
Thus we deduce:
\begin{cor} The answer to Question \ref{secondq} and Question \ref{thirdq} is affirmative for infinitesimally polar actions. \end{cor}
\section{Singular Riemannian foliations} \subsection{Definition} The definition of smooth structures, in particular of basic forms, on a quotient space $X=M/G$ does not depend on the group action of $G$ on $M$, but only on the decomposition of $M$ into $G$-orbits.
Such a decomposition is a special case of a \emph{singular Riemannian foliation}, a notion that generalizes Riemannian foliations and decomposition in orbits of isometric group actions. We just recall the definition here and refer the reader to \cite{Molino}, \cite{Alexgor}, \cite{LT} for more on details.
A \emph{transnormal system} on a Riemannian manifold $M$ is a decomposition of $M$ into pairwise disjoint isometrically immersed submanifolds $M=\cup_{x} L(x)$, called \emph{leaves}, such that a geodesic starting orthogonally to a leaf remains orthogonal to all leaves it intersects. A\emph{transnormal system} is a called a \emph{singular Riemannian foliation} if for each leaf $L$ and each $v\in TL$ with footpoint $p,$ there is a vector field $X$ tangent to the leaves so that $X(p)=v.$
While there is some evidence that the answer to the following question is affirmative (see the final remarks of
\cite{Wilk}), the question seems to be highly non-trivial:
\begin{quest} \label{trans}
Is any transnormal system automatically a singular Riemannian foliation? \end{quest}
\subsection{Smooth structure}
From now on let $\mathcal F$ be a singular Riemannian foliation on a Riemannian manifold $M$. A smooth $p$-form on an open subset $V\subset M$ is called \emph{basic} if the restriction of the form to the set of regular leaves $V_{0} \subset V$ is the pull-back of a smooth $p$-form on a local quotient, locally around each point $x\in V_0$. Again, the basic forms constitute a complex of sheaves, whose cohomology, the \emph{basic cohomology} is a very important invariant of the foliation.
Again locally around each point $p\in M$ the foliation is locally diffeomorph to a (uniquely defined) singular Riemannian
foliation $T_p \mathcal F$ on the Euclidean space $T_p M$, which is invariant with respect to scalar multiplications (cf. \cite{LT}).
This can be used to reduce questions to the case of singular Riemannian foliations on Euclidean spaces. However, most fundamental results known in case of representations (for instance the main theorems from \cite{Schwarz2} and \cite{Schwarz3}) have not been answered in this more general situation until now. We would like to formulate:
\begin{quest} Let $\mathcal F$ be a singular Riemannian foliation on a Euclidean space $\mathbb R^n$. Assume that the leaf through the origin $0$ consists of only one point. Is the space of basic functions finitely generated? \end{quest}
\subsection{Smoothness of isometries} The singular Riemannian foliation $\mathcal F$ is called \emph{closed} if all of its leaves are closed. Assume now that $M$ is complete and that $\mathcal F$ is closed. Then the set of leaves $X=M/\mathcal F$ carries a natural quotient metric and the sheaf of basic forms should be considered as a sheaf on $X$. Again there are arbitrary fine partitions of unity consisting of basic functions and the usual Lemma of Poincare holds with the usual proof, thus showing that the basic cohomology coincides in this case the singular cohomology (with real coefficients) of the metric space $X$.
The following generalization of Question \ref{firstq} and Question \ref{thirdq} is further remote from algebra and representation theory, and one can hope, that the right geometric answer to Question \ref{firstq} would answer the following question as well.
\begin{quest} \label{fourthq} Let $M,N$ be complete Riemannian manifolds with closed singular Riemannian foliations $\mathcal F$ and $\mathcal G$ respectively. Does an isometry $I:M/\mathcal F \to N/\mathcal G$ induce a bijection between the basic forms? \end{quest}
While we believe that the answer to this question is affirmative in general, we only know a proof in a very special situation. Recall that a singular Riemannian foliation $\mathcal F$ is called \emph{polar} (also known as a singular Riemannian foliation with section, cf. \cite{Alexandrino}) if any point $p \in M$ is contained in a submanifold $\Sigma$ (called \emph{section}) that intersects all leaves orthogonally and the regular leaves transversally. A singular Riemannian foliation $\mathcal F$ is called \emph{infinitesimally polar} if and only if at all points $p\in M$ the infinitesimal foliation $T_p \mathcal F$ is polar (\cite{LT}). Again, all singular Riemannian foliations of codimension at most $2$ are infinitesimally polar.
If $M$ is complete and $\mathcal F$ is closed, infinitesimal polarity is equivalent to the assumption that $M/\mathcal F$ is isometric to a smooth Riemannian orbifold. We have:
\begin{thm} \label{sing} Let $\mathcal F$ be a closed singular Riemannian foliation on a complete Riemannian manifold $M$. If there is an isometry $I:M/\mathcal F \to B$ to a smooth Riemannian orbifold $B$ then $I$ induces an isomorphism between the sheaves of smooth forms on $B$ and the sheaves of basic forms on $M/\mathcal F$. In particular, the answer to Question \ref{fourthq} is affirmative for infinitesimally polar foliations. \end{thm}
A short proof of the theorem above can be obtained along the same lines as the proof of \tref{firstthm} relying on \cite{Alexgor} instead of \cite{Michor1}. We are going to explain another only slightly different proof that is also valid for non-closed foliations. To do this we recall from \cite{Lyt}
that for any infinitesimally polar singular Riemannian foliation $\mathcal F$ on a Riemannian manifold $M$ there is a Riemannian manifold $\hat M$ (called the \emph{geometric resolution} of $M$) with a regular (!) Riemannian foliation $\hat {\mathcal F}$ and a smooth surjective map $F:\hat M \to M$, such that the following holds true. The preimages of leaves of $\mathcal F$ are leaves of $\hat {\mathcal F}$. The map $F$ is a diffeomorphism, when restricted to $\hat M _0$, the preimage of the regular part of $M$. The map $F$ preserves the lengths of all horizontal curves. If $M$ is complete and $\mathcal F$ is closed then so are $\hat M$ and $ \hat {\mathcal F}$, and $F$ induces an isometry $\bar F$ between the quotients $\hat M / \hat {\mathcal F}$ and $M/\mathcal F$. Note that since $\hat {\mathcal F}$ is a regular foliation, the $\hat{\mathcal{ F}}$-basic forms are the smooth forms on the quotient orbifold $\hat M /\hat {\mathcal F}$ (cf. \cite{Molino}).
Thus the following observation generalizes and proves \tref{sing}.
\begin{prop} \label{help} Let $\mathcal F$ be an infinitesimally polar singular Riemannian foliation on a Riemannian manifold $M$. Let $ F:(\hat M, \hat {\mathcal F}) \to (M,\mathcal F)$ be the geometric resolution of $\mathcal F$. Then $F$ induces a bijection between the sheaves of basic forms. \end{prop}
\begin{proof} Since $F$ maps leaves to leaves, the pull-back by $F$ defines a map from basic forms on $V\subset M$ to basic forms on $F^{-1} (V) =\hat V$. Since the restriction to $\hat M _0$ is a diffeomorphism, the pull-back by $F$ is injective. And it remains to prove, that any basic form on $\hat V$ is the pull-back of some form on $V$. By injectivety, this question is local and it is enough to prove it for a \emph{distinguished neighborhood} $U$ of a given point $x\in M$. Identifying the restriction of $\mathcal F$ to $U$ with a polar foliation and using the fact that the resolution map $F$ is constructed in a canonical and local way, we reduce the question to the case of polar foliations on Euclidean spaces. Here we apply \cite{Alexgor} to finish the proof. \end{proof}
\subsection{Conjecture of Molino} We would like to discuss an application of Question \ref{fourthq} to the so called Conjecture of Molino. Given a complete Riemannian manifold $M$, Molino has shown that for any regular Riemannian foliation $\mathcal F$ on $M$, the closure $\bar {\mathcal F}$ of $\mathcal F$ consisting of leaf closures of $\mathcal F$ is a singular Riemannian foliation (cf \cite{Molino}). Molino has conjectured that the closure $\mathcal F$ of a \emph{singular} Riemannian foliation is a singular Riemannian foliation as well.
For a singular Riemannain foliation $\mathcal F$ on a complete Riemannian manifold $M$ any pair of leaves are equidistant. Thus so are any pair of closures of two leaves. Therefore, the leaf closure $\bar {\mathcal F}$ is a \emph{transnormal system}. Thus the problem whether $\bar { \mathcal F}$ is a singular Riemannian foliation amounts to finding smooth vector fields tangent to $\bar {\mathcal F}$ and generating this transnormal system. In particular, the conjecture of Molino is a special case of Question \ref{trans}.
Using \cite{Schwarz3}, it is possible to prove that an affirmative answer to Question \ref{firstq} would prove the conjecture of Molino for all singular Riemannian foliations, all of whose infinitesimal foliations are given by isometric group actions. Similarly, an affirmative answer to Question \ref{fourthq} together with a generalization of \cite{Schwarz3} to singular Riemannian foliations would prove the conjecture of Molino in general. Our interests in the smoothness issues discussed in this note originated from these observations. We would like to finish our exposition by sketching the proof of the conjecture of Molino for infinitesimally polar foliations,
a result previously shown in \cite{ale} for polar foliations.
\begin{thm} \label{lastthm} Let $ \mathcal F$ be an infinitesimally polar foliation on a complete Riemannian manifold $M$. Then the closure $\bar {\mathcal F}$ is a singular Riemannian foliation. \end{thm}
\begin{proof} Consider the resolution $\hat M$ of $M$ discussed in the previous subsection. The map $F: \hat M \to M$ is proper, thus the leaf closures $\hat {\mathcal F}$ are exactly the preimages of the ``leaves" of $\bar {\mathcal F}$. Due to the theorem of Molino, the closure $\mathcal G$ of $\hat {\mathcal F}$ is a singular Riemannian foliation. Moreover, locally, the generating smooth vector field of $\mathcal G$ are given by the vector fields generating $\hat {\mathcal F}$ and a family of smooth horizontal fields (``the horizontal lifts of basic Killing fields", cf. \cite{Molino}).
However, by duality with respect to our Riemannian metric, smooth horizontal fields are in one-to-one correspondence with basic 1-forms. Due to \pref{help} these basic one forms descend to basic 1-forms on $M$. Dualizing again, one obtains the smooth horizontal vector fields on $M$ that together with the generating vector fields of $\mathcal F$ generate $\bar {\mathcal F}$. \end{proof}
\begin{rem} The above proof shows that the vector fields generating the closure $\bar {\mathcal F}$ in \tref{lastthm} can locally be obtained by adding to the vector fields generating $\mathcal F$ the horizontal lifts of ``transversal Killing fields", as expected by Molino. In the case of infinitesimally polar foliations, these horizontal vector fields are uniquely determined by their restrictions to any minimal stratum, cf. \cite{ale}. \end{rem}
\noindent\textbf{Acknowledgements} For helpful discussions and comments we would like to express our gratitude to Claudio Gorodski, Gerald Schwarz and Stephan Wiesendorf.
\end{document} | arXiv |
\begin{document}
\setlength{\baselineskip}{1\baselineskip} \thispagestyle{empty} \title{Between classical and quantum\footnote{To appear in Elsevier's forthcoming {\it Handbook of the Philosophy of Science}, Vol.\ 2: {\it Philosophy of Physics} (eds.\ John Earman \&\ Jeremy Butterfield). The author is indebted to Stephan de Bi\`{e}vre, Jeremy Butterfield, Dennis Dieks, Jim Hartle, Gijs Tuynman, Steven Zelditch, and Wojciech Zurek for detailed comments on various drafts of this paper. The final version has greatly benefited from the 7 Pines Meeting on `The Classical-Quantum Borderland' (May, 2005); the author wishes to express his gratitude to Lee Gohlike and the Board of the 7 Pines Meetings for the invitation, and to the other speakers (M. Devoret, J. Hartle, E. Heller, G. `t Hooft, D. Howard, M. Gutzwiller, M. Janssen, A. Leggett, R. Penrose, P. Stamp, and W. Zurek) for sharing their insights with him. }} \author{\textbf{N.P. Landsman} \\ \mbox{}
\\ Radboud Universiteit Nijmegen\\ Institute for Mathematics, Astrophysics, and Particle Physics\\ Toernooiveld 1, 6525 ED NIJMEGEN\\ THE NETHERLANDS\\ \mbox{}
\\ email \texttt{[email protected]}} \date{\today} \maketitle \begin{abstract} The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg's `quantum-theoretical \textit{Umdeutung} (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schr\"{o}dinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from \qm. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail.
On the assumption that \qm\ is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit $\hbar\rightarrow 0$ of small Planck's constant (in a finite system), in the limit $N\raw\infty$ of a large system with $N$ degrees of freedom (at fixed $\hbar$), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of \ca s and the concepts of superselection sectors and macroscopic observables. In these limits,
the classical world does not emerge as a sharply defined objective reality, but rather as an approximate {\it appearance} relative to certain ``classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are ``einselected", i.e.\ robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories.
This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the {\it elimination} of certain states and observables from quantum theory.
Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it.
\end{abstract}
\tableofcontents
\begin{quote} `But the worst thing is that I am quite unable to clarify the transition [of matrix mechanics] to the classical theory.'
(Heisenberg to Pauli, October 23th, 1925)\footnote{`Aber das Schlimmste ist, da\ss\ ich \"{u}ber den \"{U}bergang in die klassische Theorie nie Klarheit bekommen kann.' See Pauli (1979), p.\ 251.} \end{quote} \begin{quote} `Hendrik Lorentz considered the establishment of the correct relation between the classical and the quantum theory as the most fundamental problem of future research. This problem bothered him as much as it did Planck.' (Mehra \&\ Rechenberg, 2000, p.\ 721) \end{quote} \begin{quote} `Thus \qm\ occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.' (Landau \&\ Lifshitz, 1977, p.\ 3) \end{quote} \section{Introduction}\label{S1} Most modern physicists and philosophers would agree that a decent interpretation of quantum mechanics should fullfil at least two criteria. Firstly, it has to elucidate the physical meaning of its mathematical formalism and thereby secure the empirical content of the theory. This point (which we address only in a derivative way) was clearly recognized by all the founders of quantum theory.\footnote{The history of quantum theory has been described in a large number of books. The most detailed presentation is in Mehra \&\ Rechenberg (1982--2001), but this multi-volume series has by no means superseded smaller works such as Jammer (1966), vander Waerden (1967), Hendry (1984), Darrigol (1992), and Beller (1999). Much information may also be found in biographies such as Heisenberg (1969), Pais (1982), Moore (1989), Pais (1991), Cassidy (1992), Heilbron (2000), Enz (2002), etc. See also Pauli (1979). A new project on the history of matrix mechanics led by J\"{u}rgen Renn is on its way.\label{historybooks}} Secondly (and this {\it is} the subject of this paper), it has to explain at least the {\it appearance} of the classical world.\footnote{That these point are quite distinct is shown by the Copenhagen Interpretation, which exclusively addresses the first at utter neglect of the second. Nonetheless, in most other approaches to \qm\ there is substantial overlap between the various mechanisms that are proposed to fullfil the two criteria in question.}
As shown by our second quotation above, Planck saw the difficulty this poses, and as a first contribution he noted that the high-temperature limit of his formula for black-body radiation converged to the classical expression.
Although Bohr believed that {\it \qm\ should be interpreted through classical physics}, among the founders of the theory he seems to have been unique in his lack of appreciation of the problem of {\it deriving classical physics from quantum theory}. Nonetheless, through his correspondence principle (which he proposed in order to address the {\it first} problem above rather than the second) Bohr made one of the most profound contributions to the issue. Heisenberg initially recognized the problem, but quite erroneously came to believe he had solved it in his renowned paper on the uncertainty relations.\footnote{`One can see that the transition from micro- to macro-mechanics is now very easy to understand: classical mechanics is altogether part of \qm.' (Heisenberg to Bohr, 19 March 1927, just before the submission on 23 March of Heisenberg (1927). See {\it Bohr's Scientific Correspondence} in the {\it Archives for the History of Quantum Physics}).} Einstein famously did not believe in the fundamental nature of quantum theory, whereas Schr\"{o}dinger was well aware of the problem from the beginning, later highlighted the issue with his legendary cat, and at various stages in his career made important technical contributions towards its resolution. Ehrenfest stated the well-known theorem named after him. Von Neumann saw the difficulty, too, and addressed it by means of his well-known analysis of the measurement procedure in \qm.
The problem is actually even more acute than the founders of quantum theory foresaw. The experimental realization of Schr\"{o}dinger's cat is nearer than most physicists would feel comfortable with (Leggett, 2002; Brezger et al., 2002; Chiorescu et al., 2003; Marshall et al., 2003; Devoret et al., 2004). Moreover, awkward superpositions are by no means confined to physics
laboratories: due to its chaotic motion, Saturn's moon Hyperion (which is about the size of New York) has been estimated to spread out all over its orbit within 20 years if treated as an isolated quantum-mechanical wave packet (Zurek \&\ Paz, 1995). Furthermore, decoherence theorists have made the point that ``measurement" is not only a procedure carried out by experimental physicists in their labs, but takes place in Nature all the time without any human intervention. On the conceptual side, parties as diverse as Bohm \&\ Bell and their followers on the one hand and the quantum cosmologists on the other have argued that a ``Heisenberg cut" between object and observer cannot possibly lie at the basis of a fundamental theory of physics.\footnote{Not to speak of the problem, also raised by quantum cosmologists, of deriving classical space-time from some theory of quantum gravity. This is certainly part of the general program of deriving classical physics from quantum theory, but unfortunately it cannot be discussed in this paper. }
These and other remarkable insights of the past few decades have drawn wide attention to the importance of the problem of interpreting \qm, and in particular of explaining classical physics from it.
We will discuss these ideas in more detail below, and indeed our discussion of the relationship between classical and \qm\ will be partly historical. However, other than that it will be technical and mathematically rigorous. For the problem at hand is so delicate that in this area sloppy mathematics is almost guaranteed to lead to unreliable physics and conceptual confusion (notwithstanding the undeniable success of poor man's math elsewhere in theoretical physics). Except for von Neumann, this was not the attitude of the pioneers of \qm; but while it has to be acknowledged that many of their ideas are still central to the current discussion, these ideas {\it per se} have {\it not} solved the problem. Thus we assume the reader to be familiar with the Hilbert space formalism of \qm,\footnote{ Apart from seasoned classics such as Mackey (1963), Jauch (1968), Prugovecki (1971), Reed \&\ Simon (1972), or Thirring (1981), the reader might consult more recent books such as Gustafson \&\ Sigal (2003) or Williams (2003). See also Dickson (2005).} and for some parts of this paper (notably Section \ref{S6} and parts of Section \ref{S4}) also with the basic theory of \ca s and its applications to quantum theory.\footnote{For physics-oriented introductions to \ca s see Davies (1976), Roberts \&\ Roepstorff (1969), Primas (1983), Thirring (1983), Emch (1984), Strocchi (1985), Sewell (1986), Roberts (1990), Haag (1992), Landsman (1998), Araki (1999), and Sewell (2002). Authoratitive mathematical texts include Kadison \&\ Ringrose (1983, 1986) and Takesaki (2003). \label{Cstarlit}} In addition, some previous encounter with the conceptual problems of quantum theory would be helpful.\footnote{Trustworthy books include, for example, Scheibe (1973), Jammer (1974), van Fraassen (1991), dÕEspagnat (1995), Peres (1995), Omn\`{e}s (1994, 1999), Bub (1997), and Mittelstaedt (2004).\label{QMtexts}}
Which ideas {\it have} solved the problem of explaining the appearance of the classical world from quantum theory? In our opinion, none have, although since the founding days of \qm\ a number of new ideas have been proposed that almost certainly will play a role in the eventual resolution, should it ever be found. These ideas surely include: \begin{itemize} \item The limit $\hbar\rightarrow 0$ of small Planck's constant (coming of age with the mathematical field of microlocal analysis); \item The limit $N\raw\infty$ of a large system with $N$ degrees of freedom (studied in a serious only way after the emergence of \ca ic methods); \item Decoherence and consistent histories. \end{itemize} Mathematically, the second limit may be seen as a special case of the first, though the underlying physical situation is of course quite different. In any case, after a detailed analysis our conclusion will be that none of these ideas in isolation is capable of explaining the classical world, but that there is some hope that by combining all three of them, one might do so in the future.
Because of the fact that the subject matter of this review is unfinished business, to date one may adopt a number of internally consistent but mutually incompatible philosophical stances on the relationship between classical and quantum theory. Two extreme ones, which are always useful to keep in mind whether one holds one of them or not, are: \begin{enumerate} \item Quantum theory is fundamental and universally valid, and the classical world has only ``relative" or ``perspectival" existence. \item Quantum theory is an approximate and derived theory, possibly false, and the classical world exists absolutely. \end{enumerate} An example of a position that our modern understanding of the measurement problem\footnote{See the books cited in footnote \ref{QMtexts}, especially Mittelstaedt (2004).} has rendered internally inconsistent is: \begin{quote} 3. Quantum theory is fundamental and universally valid, and (yet) the classical world exists absolutely. \end{quote}
In some sense stance 1 originates with Heisenberg (1927), but the modern era started with Everett (1957).\footnote{\label{MWM} Note, though, that stance 1 by no means implies the so-called Many-Worlds Interpretation, which also in our opinion is `simply a meaningless collage of words' (Leggett, 2002).} These days, most decoherence theorists, consistent historians, and modal interpreters seem to support it. Stance 2 has a long and respectable pedigree unequivocally, including among others Einstein, Schr\"{o}dinger, and Bell. More recent backing has come from Leggett as well as from ``spontaneous collapse" theorists such as Pearle, Ghirardi, Rimini, Weber, and others. As we shall see in Section \ref{S3}, Bohr's position eludes classification according to these terms; our three stances being of an ontological nature, he probably would have found each of them unattractive.\footnote{To the extent that it was inconclusive, Bohr's debate with Einstein certainly suffered from the fact that the latter attacked strawman 3 (Landsman, 2006). The fruitlessness of discussions such as those between Bohm and Copenhagen (Cushing, 1994) or between Bell (1987, 2001) and Hepp (1972) has the same origin.}
Of course, one has to specify what the terminology involved means. By quantum theory we mean standard \qm\ including the eigenvector-eigenvalue link.\footnote{Let $A$ be a selfadjoint operator on a \Hs\ ${\mathcal H}$, with associated projection-valued measure $P(\Delta)$, $\Delta\subset \R$, so that $A=\int dP(\lm)\, \lm$ (see also footnote \ref{PVM} below). The eigenvector-eigenvalue link states that a state $\Psi$ of the system lies in $P(\Delta){\mathcal H}$ if and only if $A$ takes some value in $\Dl$ for sure. In particular, if $\Psi$ is an eigenvector of $A$ with eigenvalue $\lm$ (so that $P(\{\lm\})\neq 0$ and $\Psi\in P(\{\lm\}){\mathcal H}$), then $A$ takes the value $\lm$ in the state $\Psi$ with probability one. In general, the probability $p_{\Ps}(\Delta)$ that in a state $\Psi$ the observable $a$ takes some value in $\Dl$ (``upon measurement") is given by the Born--von Neumann rule $p_{\Ps}(\Delta)=(\Psi, P(\Delta)\Psi)$.} Modal interpretations of \qm\ (Dieks (1989a,b; van Fraassen, 1991; Bub, 1999; Vermaas, 2000; Bene \& Dieks, 2002; Dickson, 2005) deny this link, and lead to positions close to or identical to stance 1.
The projection postulate is neither endorsed nor denied when we generically speak of quantum theory.
It is a bit harder to say what ``the classical world" means. In the present discussion we evidently can {\it not} define the classical world as the world that exists independently of observation - as Bohr did, see Subsection \ref{Pcl} - but neither can it be taken to mean the part of the world that is described by the laws of classical physics full stop; for if stance 1 is correct, then these laws are only approximately valid, if at all. Thus we simply put it like this: \begin{quote}{\it The classical world is what observation shows us to behave - with appropriate accuracy - according to the laws of classical physics}. \end{quote} There should be little room for doubt as to what `with appropriate accuracy' means: the existence of the colour grey does not imply the nonexistence of black and white!
We {\it can} define the {\it absolute existence of} the classical world \`{a} la Bohr as its existence independently of observers or measuring devices. Compare with Moore's (1939) proof of the existence of the external world: \begin{quote} How? By holding up my two hands, and saying, as I make a certain gesture with the right hand, `Here is one hand', and adding, as I make a certain gesture with the left, `and here is another'.\end{quote}
Those holding position 1, then, maintain that {\it the classical world exists only as an appearance relative to a certain specification}, where the specification in question could be an observer (Heisenberg), a certain class of observers and states (as in decoherence theory), or some coarse-graining of the Universe defined by a particular consistent set of histories, etc. If the notion of an observer is construed in a sufficiently abstract and general sense, one might also formulate stance 1 as claiming that the classical world merely exists {\it from the perspective of the observer} (or the corresponding class of observables).\footnote{The terminology ``perspectival" was suggested to the author by Richard Healey.} For example,
Schr\"{o}dinger's cat ``paradox" dissolves at once when the appropriate perspective is introduced; cf.\ Subsection \ref{hepps}.
Those holding stance 2, on the other hand, believe that the classical world exists in an absolute sense (as Moore did). Thus stance 2 is akin to common-sense realism, though the distinction between 1 and 2 is largely independent of the issue of scientific realism.\footnote{See Landsman (1995) for a more elaborate discussion of realism in this context. Words like ``objective" or ``subjective" are not likely to be helpful in drawing the distinction either: the claim that `my children are the loveliest creatures in the world' is at first glance subjective, but it can trivially be turned into an objective one through the reformulation that `Klaas Landsman finds his children the loveliest creatures in the world'. Similarly, the proposition that (perhaps due to decoherence) `local observers find that the world is classical' is perfectly objective, although it describes a subjective experience. See also Davidson (2001). } For defendants of stance 1 usually still believe in the existence of some observer-independent reality (namely somewhere in the quantum realm), but deny that this reality incorporates the world observed around us. This justifies a pretty vague specification of such an important notion as the classical world: one of the interesting outcomes of the otherwise futile discussions surrounding the Many Worlds Interpretation has been the insight that {\it if \qm\ is fundamental, then the notion of a classical world is intrinsically vague and approximate}. Hence it would be self-defeating to be too precise at this point.\footnote{See Wallace (2002, 2003); also cf.\ Butterfield (2002). This point was not lost on Bohr and Heisenberg either; see Scheibe (1973).}
Although stance 1 is considered defensive if not cowardly by adherents of stance 2, it is a highly nontrivial mathematical fact that so far it seems supported by the formalism of \qm. In his derision of what he called `FAPP' (= For All Practical Purposes) solutions to the measurement problem (and more general attempts to explain the appearance of the classical world from quantum theory), Bell (1987, 2001) and others in his wake mistook a profound epistemological stance for a poor defensive move.\footnote{The insistence on ``precision" in such literature is reminiscent of Planck's long-held belief in the absolute nature of irreversibility (Darrigol, 1992; Heilbron, 2002). It should be mentioned that although Planck's stubbornness by historical accident led him to take the first steps towards quantum theory, he eventually gave it up to side with Boltzmann.} It is, in fact, stance 2 that we would recommend to the cowardly: for
proving or disproving stance 1 seems the real challenge of the entire debate, and we regard the technical content of this paper as a survey of progress towards actually proving it. Indeed, to sum up our conclusions, we claim that there is good evidence that: \begin{enumerate} \item Classical physics emerges from quantum theory in the limit $\hbar\rightarrow 0$ or $N\raw\infty$ {\it provided that the system is in certain ``classical" states and is monitored with ``classical" observables only}; \item Decoherence and consistent histories will probably explain {\it why} the system happens to be in such states and has to be observed in such a way. \end{enumerate} However, even if one fine day this scheme will be made to work, the explanation of the appearance of the classical world from quantum theory will be predicated on an external solution of the notorious `from ``and" to ``or" problem': If \qm\ predicts various possible outcomes with certain probabilities, why does only {\it one} of these appear to us?\footnote{It has to be acknowledged that we owe the insistence on this question to the defendants of stance 2. See also footnote \ref{MWM}.}
For a more detailed outline of this paper we refer to the table of contents above. Most philosophical discussion will be found in Section \ref{S3} on the Copenhagen interpretation, since whatever its merits, it undeniably set the stage for the entire discussion on the relationship between classical and quantum.\footnote{We do not discuss the classical limit of \qm\ in the philosophical setting of theory reduction and intertheoretic relations; see, e.g., Scheibe (1999) and Batterman (2002).}
The remainder of the paper will be of an almost purely technical nature. Beyond this point we will try to avoid controversy, but when unavoidable it will be confined to the Epilogues appended to Sections \ref{S3}-\ref{S6}. The final Epilogue (Section \ref{S8}) expresses our deepest thoughts on the subject. \section{Early history}\label{S2} This section is a recapitulation of the opinions and contributions of the founders of \qm\ regarding the relationship between classical and quantum. More detail may be found in the books cited in footnote \ref{historybooks} and in specific literature to be cited; for an impressive bibliography see also Gutzwiller (1998). The early history of quantum theory is of interest in its own right, concerned as it is with one of the most significant scientific revolutions in history. Although this history is not a main focus of this paper, it is of special significance for our theme. For the usual and mistaken interpretation of Planck's work (i.e.\ the idea that he introduced something like a ``quantum postulate", see Subsection \ref{HC} below) appears to have triggered the belief that quantum theory and Planck's constant are related to a universal discontinuity in Nature. Indeed, this discontinuity is sometimes even felt to mark the basic difference between classical and quantum physics. This belief is particularly evident in the writings of Bohr, but still resonates even today. \subsection{Planck and Einstein} The relationship between classical physics and quantum theory is so subtle and confusing that historians and physicists cannot even agree about the precise way the classical gave way to the quantum!
As Darrigol (2001) puts it: `During the past twenty years, historians [and physicists] have disagreed over the meaning of the quanta which Max Planck introduced in his black-body theory of 1900. The source of this confusion is the publication (\ldots) of Thomas Kuhn's [(1978)] iconoclastic thesis that Planck did not mean his energy quanta to express a quantum discontinuity.'
As is well known (cf.\ Mehra \&\ Rechenberg, 1982a, etc.), Planck initially derived Wien's law for blackbody radiation in the context of his (i.e.\ Planck's) program of establishing the absolute nature of irreversibility (competing with Boltzmann's probabilistic approach, which eventually triumphed). When new high-precision measurements in October 1900 turned out to refute Wien's law, Planck first guessed his expression \begin{equation} E_{\nu}/N_{\nu}=h\nu/(e^{h\nu/kT}-1) \label{Planck}\end{equation} for the correct law, \textit{en passant} introducing two new constants of nature $h$ and $k$,\footnote{Hence Boltzmann's constant $k$ was introduced by Planck, who was the first to write down the formula $S= k\log W$.} and subsequently, on December 14, 1900, presented a theoretical derivation of his law in which he allegedly introduced the idea that the energy of the resonators making up his black body was quantized in units of $\varep_{\nu}=h\nu$ (where $\nu$ is the frequency of a given resonator). This derivation is generally seen as the birth of quantum theory, with the associated date of birth just mentioned.
However, it is clear by now (Kuhn, 1978; Darrigol, 1992, 2001; Carson, 2000; Brush, 2002) that Planck was at best agnostic about the energy of his resonators, and at worst assigned them a continuous energy spectrum. Technically, in the particular derivation of his empirical law that eventually turned out to lead to the desired result (which relied on Boltzmann's concept of entropy),\footnote{Despite the fact that Planck only converted to Boltzmann's approach to irreversibility around 1914.} Planck had to count the number of ways a given amount of energy $E_{\nu}$ could be distributed over a given number of resonators $N_{\nu}$ at frequency $\nu$. This number is, of course, infinite, hence in order to find a finite answer Planck followed Boltzmann in breaking up $E_{\nu}$ into a large number $A_{\nu}$ of portions of identical size $\varep_{\nu}$, so that $A_{\nu}\varep_{\nu}=E_{\nu}$.\footnote{The number in question is then given by $(N+A-1)!/(N-1)!A!$, dropping the dependence on $\nu$ in the notation.} Now, as we all know, whereas Boltzmann let $\varep_{\nu}\raw 0$ at the end of his corresponding calculation for a gas, Planck discovered that his empirical blackbody law emerged if he assumed the relation $\varep_{\nu}=h\nu$.
However, this postulate did \textit{not} imply that Planck quantized the energy of his resonators. In fact, in his definition of a given distribution he counted the number of resonators with energy \textit{between} say $(k-1)\varep_{\nu}$ and $k\varep_{\nu}$ (for some $k\in\mathbb{N}$), as Boltzmann did in an analogous way for a gas, rather than the number of resonators \text{with} energy $k\varep_{\nu}$, as most physicists came to interpret his procedure. More generally, there is overwhelming textual evidence that Planck himself by no means believed or implied that he had quantized energy; for one thing, in his Nobel Prize Lecture in 1920 he attributed the correct interpretation of the energy-quanta $\varep_{\nu}$ to Einstein. Indeed, the modern understanding of the earliest phase of quantum theory is that it was Einstein rather than Planck who, during the period 1900--1905, clearly realized that Planck's radiation law marked a break with classical physics (B\"{u}ttner, Renn, \&\ Schemmel, 2003). This insight, then, led Einstein to the quantization of energy. This he did in a twofold way, both in connection with Planck's resonators - interpreted by Einstein as harmonic oscillators in the modern way - and, in a closely related move, through his concept of a photon. Although Planck of course introduced the constant named after him, and as such is the founding {\it father} of the theory characterized by $\hbar$, it is the introduction of the photon that made Einstein at least the {\it mother} of quantum theory. Einstein himself may well have regarded the photon as his
most revolutionary discovery, for what he wrote about his pertinent paper is not matched in self-confidence by anything he said about relativity:
`Sie handelt \"{u}ber die Strahlung und die energetischen Eigenschaften des Lichtes und ist sehr revolution\"{a}r.'\footnote{`[This paper] is about radiation and the energetic properties of light, and is very revolutionary.' See also the Preface to Pais (1982).}
Finally, in the light of the present paper, it deserves to be mentioned that Einstein (1905) and Planck (1906) were the first to comment on the classical limit of quantum theory; see the preamble to Section \ref{S5} below. \subsection{Bohr}\label{Bohr1}
Bohr's brilliant model of the atom reinforced his idea that quantum theory was a theory of quanta.\footnote{Although at the time Bohr followed practically all physicists in their rejection of Einstein's photon, since he believed that during a quantum jump the atom emits electromagnetic radiation in the form of a spherical wave. His model probably would have gained in consistency by adopting the photon picture of radiation, but in fact Bohr was to be the last prominent opponent of the photon, resisting the idea until 1925. See also Blair Bolles (2004) and footnote \ref{Bohropp} below.\label{BFN1}}
Since this model simultaneously highlighted the clash between classical and quantum physics {\it and} carried the germ of a resolution of this conflict through Bohr's equally brilliant correspondence principle, it is worth saying a few words about it here.\footnote{Cf.\ Darrigol (1992) for a detailed treatment; also see Liboff (1984) and Steiner (1998).} Bohr's atomic model addressed the radiative instability of Rutherford's solar-system-style atom:\footnote{The solar system provides the popular visualization of Rutherford's atom, but his own picture was more akin to Saturn' rings than to a planet orbiting the Sun.} according to the electrodynamics of Lorentz, an accelerating electron should radiate, and since the envisaged circular or elliptical motion of an electron around the nucleus is a special case of an accelerated motion, the electron should continuously lose energy and spiral towards the nucleus.\footnote{In addition, any Rutherford style atom with more than one electron is mechanically unstable, since the electrons repel each other, as opposed to planets, which attract each other.} Bohr countered this instability by three simultaneous moves, each of striking originality: \begin{enumerate} \item He introduced a quantization condition that singled out only a discrete number of allowed electronic orbits (which subsequently were to be described using classical mechanics, for example, in Bohr's calculation of the Rydberg constant $R$). \item He replaced the emission of continuous radiation called for by Lorentz by quantum jumps with unpredictable destinations taking place at unpredictable moments, during which the atom emits light with energy equal to the energy difference of the orbits between which the electron jumps. \item He prevented the collapse of the atom through such quantum jumps by introducing the notion of ground state, below which no electron could fall. \end{enumerate} With these postulates, for which at the time there existed no foundation whatsoever,\footnote{\label{SigalF}What has hitherto been mathematically proved of Bohr's atomic model is the existence of a ground state (see Griesemer, Lieb, \&\ Loss, 2001, and references therein for the greatest generality available to date) and the metastability of the excited states of the atom after coupling to the electromagnetic field (cf.\ Bach, Fr\"{o}hlich, \&\ Sigal, 1998, 1999 and Gustafson \&\ Sigal, 2003). The energy spectrum is discrete only if the radiation field is decoupled, leading to the usual computation of the spectrum of the hydrogen atom first performed by Schr\"{o}dinger and Weyl. See also the end of Subsection \ref{PSL}.}
Bohr explained the spectrum of the hydrogen atom, including an amazingly accurate calculation of $R$. Moreover, he proposed what was destined to be the key guiding principle in the search for quantum mechanics in the coming decade, viz.\ the correspondence principle (cf.\ Darrigol, 1992, {\it passim}, and Mehra \&\ Rechenberg, 1982a, pp.\ 249--257).
In general, there is no relation between the energy that an electron loses during a particular quantum jump and the energy it would have radiated classically (i.e.\ according to Lorentz) in the orbit it revolves around preceding this jump. Indeed, in the ground state it cannot radiate through quantum jumps at all, whereas according to classical electrodynamics it should radiate all the time. However, Bohr saw that in the opposite case of very wide orbits (i.e.\ those having very large principal quantum numbers $n$), the frequency $\nu=(E_n-E_{n-1})/h$ (with $E_n=-R/n^2$) of the emitted radiation approximately corresponds to the frequency of the lowest harmonic of the classical theory, applied to electron motion in the initial orbit.\footnote{Similarly, higher harmonics correspond to quantum jumps $n\raw n-k$ for $k>1$.} Moreover, the measured intensity of the associated spectral line (which theoretically should be related to the probability of the quantum jump, a quantity out of the reach of early quantum theory), similarly turned out to be given by classical electrodynamics. This property, which in simple cases could be verified either by explicit computation or by experiment, became a guiding principle in situations where it could not be verified, and was sometimes even extended to low quantum numbers, especially when the classical theory predicted selection rules.
It should be emphasized that {\it Bohr's correspondence principle was concerned with the properties of radiation, rather than with the mechanical orbits themselves}.\footnote{As such, it remains to be verified in a rigorous way.} This is not quite the same as what is usually called the correspondence principle in the modern literature.\footnote{A typical example of the modern version is: `Non-relativistic \qm\ was founded on the correspondence principle of Bohr: ``When the Planck constant $\hbar$ can be considered small with respect to the other parameters such as masses and distances, quantum theory approaches classical Newton theory."' (Robert, 1998, p.\ 44). The reference to Bohr is historically inaccurate!} In fact, although also this modern correspondence principle has a certain range of validity (as we shall see in detail in Section \ref{S5}), Bohr never endorsed anything like that, and is even on record as opposing such a principle:\footnote{Quoted from Miller (1984), p.\ 313.} \begin{quote} `The place was Purcell's office where Purcell and others had taken Bohr for a few minutes of rest [during a visit to the Physics Department at Harvard University in 1961]. They were in the midst of a general discussion when Bohr commented: ``People say that classical mechanics is the limit of \qm\ when $h$ goes to zero." Then, Purcell recalled, Bohr shook his finger and walked to the blackboard on which he wrote $e^2/hc$. As he made three strokes under $h$, Bohr turned around and said, ``you see $h$ is in the denominator."' \end{quote} \subsection{Heisenberg}\label{heis} Heisenberg's (1925) paper \textit{\"{U}ber die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen}\footnote{\textit{On the quantum theoretical reinterpretation of kinematical and mechanical relations}. English translation in vander Waerden, 1967.} is generally seen as a turning point in the development of \qm. Even A. Pais, no friend of Heisenberg's,\footnote{For example, in Pais (2000), claiming to portray the `genius of science', Heisenberg is conspicously absent.} conceded that Heisenberg's paper marked 'one of the great jumps - perhaps the greatest - in the development of twentieth century physics.' What did Heisenberg actually accomplish? This question is particularly interesting from the perspective of our theme.
At the time, atomic physics was in a state of crisis, to which various camps responded in different ways. Bohr's approach might best be described as \textit{damage control}: his quantum theory was a hybrid of classical mechanics adjusted by means of \textit{ad hoc} quantization rules, whilst keeping electrodynamics classical at all cost.\footnote{\label{Bohropp} Continuing footnote \ref{BFN1}, we quote from
Mehra \&\ Rechenberg, 1982a, pp 256--257: `Thus, in the early 1920s, Niels Bohr arrived at a definite point of view how to proceed forward in atomic theory. He wanted to make maximum use of what he called the ``more dualistic prescription" (\ldots) In it the atom was regarded as a mechanical system having discrete states and emitting radiation of discrete frequencies, determined (in a nonclassical way) by the energy differences between stationary states; radiation, on the other hand, had to be described by the classical electrodynamic theory.'} Einstein, who had been the first physicist to recognize the need to quantize classical electrodynamics, in the light of his triumph with General Relativity nonetheless dreamt of a classical field theory with singular solutions as the ultimate explanation of quantum phenomena. Born led the radical camp, which included Pauli: he saw the need for an entirely new mechanics replacing classical mechanics,\footnote{It was Born who coined the name \textit{quantum mechanics} even before Heisenberg's paper.} which was to be based on discrete quantities satisfying difference equations.\footnote{This idea had earlier occurred to Kramers.} This was a leap in the dark, especially because of Pauli's frowning upon the correspondence principle (Hendry, 1984; Beller, 1999).
It was Heisenberg's genius to {\it interpolate} between Bohr and Born.\footnote{Also literally! Heisenberg's traveled between Copenhagen and G\"{o}ttingen most of the time.} The meaning of his \textit{Umdeutung} was to keep the classical equations of motion,\footnote{This crucial aspect of \textit{Umdeutung} was appreciated at once by Dirac (1926): `In a recent paper Heisenberg puts forward a new theory which suggests that it is not the equations of classical mechanics that are in any way at fault, but that the mathematical operations by which physical results are deduced from them require modification. (\ldots) The correspondence between the quantum and classical theories lies not so much in the limiting agreement when $\hbar\raw 0$ as in the fact that the mathematical operations on the two theories obey in many cases the same laws.'} whilst reinterpreting the mathematical symbols occurring therein as (what were later recognized to be) matrices. Thus his \textit{Umdeutung} $x\mapsto a(n,m)$ was a precursor of what now would be called a quantization map $f\mapsto Q_{\hbar}(f)$, where $f$ is a classical observable, i.e.\ a function on phase space, and $Q_{\hbar}(f)$ is a quantum mechanical observable, in the sense of an operator on a \Hs\ or, more abstractly, an element of some \ca. See Section \ref{S4} below. As Heisenberg recognized, this move implies the noncommutativity of the quantum mechanical observables; it is this, rather than something like a ``quantum postulate" (see Subsection \ref{HC} below), that is the defining characteristic of quantum mechanics. Indeed, most later work on quantum physics and practically all considerations on the connection between classical and quantum rely on Heisenberg's idea of \textit{Umdeutung}. This even applies to the mathematical formalism as a whole; see Subsection \ref{vNs}.
We here use the term ``observable" in a loose way. It is now well recognized (Mehra \&\ Rechenberg, 1982b; Beller, 1999; Camilleri, 2005) that Heisenberg's claim that his formalism could be physically interpreted as the replacement of atomic orbits by observable quantities was a red herring, inspired by his discussions with Pauli.
In fact, in quantum mechanics \textit{any} mechanical quantity has to be ``reinterpreted", whether or not it is observable. As Heisenberg (1969) recalls, Einstein reprimanded him for the illusion that physics admits an \textit{a priori} notion of an observable, and explained that a theory determines what can be observed. Rethinking the issue of observability then led Heisenberg to his second major contribution to \qm, namely his uncertainty relations.
These relations were Heisenberg's own answer to the quote opening this paper. Indeed, matrix mechanics was initially an extremely abstract and formal scheme, which lacked not only any visualization but also the concept of a state (see below). Although these features were initially quite to the liking of Born, Heisenberg, Pauli, and Jordan, the success of Schr\"{o}dinger's work forced them to renege on their radical stance, and look for a semiclassical picture supporting their mathematics; this was a considerable U-turn (Beller, 1999; Camilleri, 2005). Heisenberg (1927) found such a picture, claiming that his uncertainty relations provided the `intuitive content of the quantum theoretical kinematics and mechanics' (as his paper was called). His idea was that {\it the classical world emerged from \qm\ through observation}: `The trajectory only comes into existence because we observe it.' \footnote{`Die Bahn entsteht erst dadurch, da\ss\ wir sie beobachten.'} This idea was to become extremely influential, and could be regarded as the origin of stance 1 in the Introduction. \subsection{Schr\"{o}dinger} \label{Ssection} The history of \qm\ is considerably clarified by the insight that Heisenberg and Schr\"{o}dinger did not, as is generally believed, discover two equivalent formulations of the theory, but rather that Heisenberg (1925) identified the mathematical nature of the observables, whereas Schr\"{o}dinger (1926a) found the description of states.\footnote{See also Muller (1997).} Matrix mechanics lacked the notion of a state, but by the same token wave mechanics initially had no observables; it was only in his attempts to relate wave mechanics to matrix mechanics that Schr\"{o}dinger (1926c) introduced the position and momentum operators\footnote{Here $j=1,2,3$. In modern terms, the expressions on the right-hand side are unbounded operators on the \Hs\ ${\mathcal H}=L^2(\R^n)$. See Section \ref{S4} for more details. The expression $x^i$ is a multiplication operator, i.e.\ $(x^j\Psi)(x)=x^j\Psi(x)$, whereas, obviously, $(\partial/\partial x^j \Psi)(x)= (\partial\Psi/\partial x^j)(x)$.} \begin{eqnarray} \CQ_{\hbar}(q^j) & =& x^j; \nn \\ \CQ_{\hbar}(p_j) & =& -i\hbar \frac{\partial}{\partial x^j}.\label{SOP} \end{eqnarray} This provided a new basis for Schr\"{o}dinger's equation\footnote{Or the corresponding time-independent one, with $E\Psi$ on the right-hand side.} \begin{equation} \left(-\frac{\hbar^2}{2m}\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2} +V(x)\right)\Psi=i\hbar \frac{\partial\Psi}{\partial t},\label{Schreq}\end{equation} by interpreting the left-hand side as $H\Psi$, with $H=\CQ_{\hbar}(h)$ in terms of the classical Hamiltonian $h(p,q)=\sum_j p_j^2/2m +V(q)$. Thus Schr\"{o}dinger founded the theory of the operators now named after him,\footnote{\label{SOPrefs} See Reed \& Simon (1972, 1975, 1987, 1979), Cycon et al. (1987), Hislop \& Sigal (1996), Hunziker \&\ Sigal (2000), Simon (2000), Gustafson \&\ Sigal (2003). For the mathematical origin of the Schr\"{o}dinger equation also cf.\ Simon (1976). } and in doing so gave what is still the most important example of Heisenberg's idea of {\it Umdeutung} of classical observables.
Subsequently, correcting and expanding on certain ideas of Dirac, Pauli, and Schr\"{o}dinger, von Neumann (1932) brilliantly glued these two parts together through the concept of a \Hs. He also gave an abstract form of the formulae of Born, Pauli, Dirac, and Jordan for the transition probabilities, thus completing the mathematical formulation of \qm.
However, this is not how Schr\"{o}dinger saw his contribution. He intended wave mechanics as a full-fledged classical field theory of reality, rather than merely as one half (namely in modern parlance the state space half) of a probabilistic description of the world that still incorporated the quantum jumps he so detested (Mehra \&\ and Rechenberg, 1987; G\"{o}tsch, 1992; Bitbol \&\ Darrigol, 1992; Bitbol, 1996; Beller, 1999). Particles were supposed to emerge in the form of wave packets, but it was immediately pointed out by Heisenberg, Lorentz, and others that in realistic situations such wave packets tend to spread in the course of time. This had initially been overlooked by Schr\"{o}dinger (1926b), who had based his intuition on the special case of the harmonic oscillator. On the positive side, in the course of his unsuccessful attempts to derive classical particle mechanics from wave mechanics through the use of wave packets, Schr\"{o}dinger (1926b) gave the first example of what is now called a {\it coherent state}. Here a quantum wave function $\Psi_z$ is labeled by a `classical' parameter $z$, in such a way that the quantum-mechanical time-evolution $\Psi_z(t)$ is approximately given by $\Psi_{z(t)}$, where $z(t)$ stands for some associated classical time-evolution; see Subsections \ref{PSQ} and \ref{CEOM} below. This has turned out to be a very important idea in understanding the transition from quantum to classical mechanics.
Furthermore, in the same paper Schr\"{o}dinger (1926b) proposed the following wave-mechanical version of Bohr's correspondence principle: classical atomic states should come from superpositions of a very large number (say at least 10,000) of highly excited states (i.e.\ energy eigenfunctions with very large quantum numbers). After decades of limited theoretical interest in this idea, interest in wave packets in atomic physics was revived in the late 1980s due to the development of modern experimental techniques based on lasers (such as pump-probing and phase-modulation). See Robinett (2004) for a recent technical review, or Nauenberg, Stroud, \&\ Yeazell (1994) for an earlier popular account. Roughly speaking, the picture that has emerged is this: a localized wave packet of the said type initially follows a time-evolution with almost classical periodicity, as Schr\"{o}dinger hoped, but subsequently spreads out after a number of orbits. Consequently, during this second phase the probability distribution approximately fills the classical orbit (though not uniformly). Even more surprisingly, on a much longer time scale there is a phenomenon of {\it wave packet revival}, in which the wave packet recovers its initial localization. Then the whole cycle starts once again, so that one does see periodic behaviour, but not of the expected classical type. Hence even in what naively would be thought of as the thoroughly classical regime, wave phenomena continue to play a role, leading to quite unusual and unexpected behaviour. Although a rigorous mathematical description of wave packet revival has not yet been forthcoming, the overall picture (based on both ``theoretical physics" style mathematics and experiments) is clear enough.
It is debatable (and irrelevant) whether the story of wave packets has evolved
according to Schr\"{o}dinger's intentions (cf.\ Littlejohn, 1986); what is certain is that his other main idea on the relationship between classical and quantum has been extremely influential. This was, of course, Schr\"{o}dinger's (1926a) ``derivation" of his wave equation from the Hamilton--Jacobi formalism of classical mechanics. This gave rise to the WKB approximation and related methods; see Subsection \ref{WKBS}.
In any case, where Schr\"{o}dinger hoped for a classical interpretation of his wave function, and Heisenberg wanted to have nothing to do with it whatsoever (Beller, 1999), Born and Pauli were quick to realize its correct, probabilistic significance. Thus they deprived the wave function of its naive physical nature, and effectively
degraded it to the purely mathematical status of a probability amplitude. And in doing so, Born and Pauli rendered the connection between \qm\ and classical mechanics almost incomprehensible once again! It was this incomprehensibility that Heisenberg addressed with his uncertainty relations.
\subsection{von Neumann}\label{vNs} Through its creation of the Hilbert space formalism of \qm, von Neumann's book (1932) can be seen as a mathematical implementation of Heisenberg's idea of {\it Umdeutung}. Von Neumann in effect proposed the following quantum-theoretical reinterpretations: \begin{trivlist} \item Phase space $M$ $\mathbf{\mapsto}$ Hilbert space ${\mathcal H}$; \item Classical observable (i.e.\ real-valued measurable function on $M$) $\mathbf{\mapsto}$ self-adjoint operator on ${\mathcal H}$; \item Pure state (seen as point in $M$) $\mathbf{\mapsto}$ unit vector (actually ray) in ${\mathcal H}$; \item Mixed state (i.e.\ probability measure on $M$) $\mathbf{\mapsto}$ density matrix on ${\mathcal H}$; \item Measurable subset of $M$ $\mathbf{\mapsto}$ closed linear subspace of ${\mathcal H}$; \item Set complement $\mathbf{\mapsto}$ orthogonal complement; \item Union of subsets $\mathbf{\mapsto}$ closed linear span of subspaces; \item Intersection of subsets $\mathbf{\mapsto}$ intersection of subspaces; \item Yes-no question (i.e.\ characteristic function on $M$) $\mathbf{\mapsto}$ projection operator.\footnote{Later on, he of course added the {\it Umdeutung} of a Boolean lattice by a modular lattice, and the ensuing {\it Umdeutung} of classical logic by quantum logic (Birkhoff \&\ von Neumann, 1936). }
\end{trivlist}
Here we assume for simplicity that
quantum observables $R$ on a \Hs\ ${\mathcal H}$ are bounded operators, i.e.\ $R\in\BH$. Von Neumann actually {\it derived} his {\it Umdeutung} of classical mixed states as density matrices from his axiomatic characterization of quantum-mechanical states as linear maps ${\rm Exp}: \BH\raw \C$ that satisfy ${\rm Exp}(R)\geq 0$ when $R\geq 0$,\footnote{I.e., when $R$ is self-adjoint with positive spectrum, or, equivalently, when $R=S^*S$ for some $S\in \BH$.} ${\rm Exp}(1)=1$,\footnote{Where the $1$ in ${\rm Exp}(1)$ is the unit operator on ${\mathcal H}$.}, and countable additivity on a commuting set of operators. For he proved that such a map ${\rm Exp}$ is necessarily given by a density matrix $\rh$ according to ${\rm Exp}(R)=\Tr(\rh R)$.\footnote{This result has been widely misinterpreted (apparently also by von Neumann himself) as a theorem excluding hidden variables in \qm. See Scheibe (1991). However, Bell's characterization of von Neumann's linearity assumption in the definition of a state as ``silly'' is far off the mark, since it holds both in classical mechanics and in \qm. Indeed, von Neumann's theorem {\it does} exclude all hidden variable extensions of \qm\ that are classical in nature, and it is precisely such extensions that many physicists were originally looking for. See R\'{e}dei \&\ St\"{o}ltzner (2001) and Scheibe (2001) for recent discussions of this issue.} A unit vector $\Psi\in{\mathcal H}$ defines a pure state in the sense of von Neumann, which we call $\ps$, by $\ps(R)=(\Psi,R\Psi)$ for $R\in\BH$. Similarly, a density matrix $\rh$ on ${\mathcal H}$ defines a (generally mixed) state, called $\rh$ as well, by $\rh(R)=\Tr(\rh R)$. In modern terminology, a state on $\BH$ as defined by von Neumann would be called a {\it normal} state.
In the \ca ic formulation of quantum physics (cf.\ footnote \ref{Cstarlit}), this axiomatization has been maintained until the present day; here $\BH$ is replaced by more general algebras of observables in order to accommodate possible superselection rules (Haag, 1992).
Beyond his mathematical axiomatization of \qm, which (along with its subsequent extension by the \ca ic formulation) lies at the basis of all serious efforts to relate classical and \qm, von Neumann contributed to this relationship through his analysis of the measurement problem.\footnote{Von Neumann (1932) refrained from discussing either the classical limit of \qm\ or (probably) the notion of quantization. In the latter direction, he declares that `If the quantity $\mathfrak{R}$ has the operator $R$, then the quantity $f(\mathfrak{R})$ has the operator $f(R)$', and that `If the quantities $\mathfrak{R}$, $\mathfrak{S}$, $\cdots$ have the operators $R$, $S$, $\cdots$, then the quantity $\mathfrak{R}+\mathfrak{S}+\cdots$ has the operator $R+S+\cdots$'. However, despite his legendary clarity and precision, von Neumann is rather vague about the meaning of the transition $\mathfrak{R}\mapsto R$. It is tempting to construe
$\mathfrak{R}$ as a classical observable whose quantum-mechanical counterpart is $R$, so that the above quotations might be taken as axioms for quantization. However, such an interpretation is neither supported by the surrounding text, nor by our current understanding of quantization (cf.\ Section \ref{S4}). For example, a quantization map $\mathfrak{R}\mapsto \CQ_{\hbar}(\mathfrak{R})$ cannot satisfy $f(\mathfrak{R})\mapsto f(\CQ_{\hbar}(\mathfrak{R}))$ even for very reasonable functions such as $f(x)=x^2$.} Since here the apparent clash between classical and quantum physics comes to a head, it is worth summarizing von Neumann's analysis of this problem here. See also Wheeler \&\ Zurek (1983), Busch, Lahti \&\ Mittelstaedt (1991), Auletta (2001) and Mittelstaedt (2004) for general discussions of the measurement problem.
The essence of the measurement problem is that certain states are never seen in nature, although they are not merely allowed by \qm\ (on the assumption of its universal validity), but are even predicted to arise in typical measurement situations. Consider a system $S$, whose pure states are mathematically described by normalized vectors (more precisely, rays) in a Hilbert space ${\mathcal H}_S$. One wants to measure an observable $\mathcal{O}$, which is mathematically represented by a self-adjoint operator $O$ on ${\mathcal H}_S$. Von Neumann assumes that $O$ has discrete spectrum, a simplification which does not hide the basic issues in the measurement problem. Hence $O$ has unit eigenvectors $\Psi_n$ with real eigenvalues $o_n$. To measure $\mathcal{O}$, one couples the system to an apparatus $A$ with Hilbert space ${\mathcal H}_A$ and ``pointer" observable $\mathcal{P}$, represented by a self-adjoint operator $P$ on ${\mathcal H}_A$, with discrete eigenvalues $p_n$ and unit eigenvectors $\Phi_n$. The pure states of the total system $S+A$ then correspond to unit vectors in the tensor product ${\mathcal H}_S\otimes {\mathcal H}_A$. A good (``first kind") measurement is then such that after the measurement, $\Psi_n$ is correlated to $\Phi_n$, that is, for a suitably chosen initial state $I \in{{\mathcal H}}_A$, a state
$\Psi_n\otimes I$ (at $t=0$) almost immediately evolves into $\Psi_n\otimes\Phi_n$. This can indeed be achieved by a suitable Hamiltonian.
The problem, highlighted by Schr\"{o}dinger's cat, now arises if one selects the initial state of $S$ to be $\sum_n c_n \Psi_n$ (with $\sum |c_n|^2=1$), for then the superposition principle leads to the conclusion that the final state of the coupled system is $\sum_n c_n \Psi_n \otimes\Phi_n$.
Now, basically all von Neumann said was that if one restricts the final state to the system $S$, then the resulting density matrix is the mixture $\sum_n |c_n|^2
[\Psi_n]$ (where $[\Psi]$ is the orthogonal projection onto a unit vector $\Psi$),\footnote{I.e., $[\Psi]f=(\Psi,f)\Psi$; in Dirac notation one would have $[\Psi]=|\Psi\rangle\langle\Psi|$.} so that, {\it from the perspective of the system alone}, the measurement appears to have caused a transition from the pure state
$\sum_{n,m} c_n\ovl{c_m} \Psi_n\Psi_m^*$
to the mixed state
$\sum_n |c_n|^2 [\Psi_n]$, in which interference terms $\Psi_n\Psi_m^*$ for $n\neq m$
are absent. Here the operator $\Psi_n\Psi_m^*$ is defined by $\Psi_n\Psi_m^*f=(\Psi_m,f)\Psi_n$; in particular, $\Psi\Psi^*=[\Psi]$.\footnote{In Dirac notation one would have $\Psi_n\Psi_m^*=|\Psi_n\rangle\langle\Psi_m|$. }
Similarly, the apparatus, taken by itself, has evolved from the pure state $\sum_{n,m} c_n\ovl{c_m} \Phi_n\Phi^*_m$ to the mixed state
$\sum_n |c_n|^2 [\Phi_n]$. This is simply a mathematical theorem (granted the possibility of coupling the system to the apparatus in the desired way), rather than a proposal that there exist two different time-evolutions in Nature, viz.\ the unitary propagation according to the Schr\"{o}dinger equation side by side with the above ``collapse" process.
In any case, by itself this move by no means solves the measurement problem.\footnote{Not even in an ensemble-interpretation of \qm, which was the interpretation von Neumann unfortunately adhered to when he wrote his book.} Firstly, in the given circumstances one is not allowed to adopt the ignorance interpretation of mixed states (i.e.\ assume that the system really is in one of the states $\Psi_n$); cf., e.g., Mittelstaedt (2004). Secondly, even if one were allowed to do so, one could restore the problem (i.e.\ the original superposition $\sum_n c_n \Psi_n\otimes \Phi_n$) by once again taking the other component of the system into account.
Von Neumann was well aware of at least this second point, to which he responded by his construction of a {\it chain}: one redefines $S+A$ as the system, and couples it to a new apparatus $B$, etc. This eventually leads to a post-measurement state $\sum_n c_n \Psi_n \otimes\Phi_n\otimes \ch_n$ (in hopefully self-explanatory notation, assuming the vectors $\ch_n$ form an orthonormal set), whose restriction to $S+A$ is the mixed state $\sum_n |c_n|^2 [\Psi_n]\otimes [\Phi_n]$. The restriction of the latter state to $S$ is, once again, $\sum_n |c_n|^2 [\Psi_n]$. This procedure may evidently be iterated; the point of the construction is evidently to pass on superpositions in some given system to arbitrary systems higher up in the chain. It follows that for the final state of the original system it does not matter where one ``cuts the chain" (that is, which part of the chain one leaves out of consideration), as long as it is done {\it somewhere}. Von Neumann (1932, in beautiful prose) and others suggested identifying the cutting with the act of observation, but it is preferable and much more general to simply say that {\it some} end of the chain is omitted in the description.
The burden of the measurement problem, then, is to \begin{enumerate} \item Construct a suitable chain along with an appropriate cut thereof; it doesn't matter where the cut is made, as long as it is done. \item Construct a suitable time-evolution accomplishing the measurement.
\item Justify the ignorance interpretation of mixed states. \end{enumerate} As we shall see, these problems are addressed, in a conceptually different but mathematically analogous way, in the Copenhagen interpretation as well as in the decoherence approach. (The main conceptual difference will be that the latter aims to solve also the more ambitious problem of explaining the appearance of the classical world, which in the former seems to be taken for granted).
We conclude this section by saying that despite some brilliant ideas, the founders of \qm\ left wide open the problem of deriving classical mechanics as a certain regime of their theory.
\section{Copenhagen: a reappraisal}\label{S3}\setcounter{equation}{0} The so-called ``Copenhagen interpretation" of \qm\ goes back to ideas first discussed and formulated by Bohr, Heisenberg, and Pauli around 1927. Against the idea that there has been a ``party line" from the very beginning, it has frequently been pointed out that in the late 1920s there were actually sharp differences of opinion between Bohr and Heisenberg on the interpretation of \qm\ and that they never really arrived at a joint doctrine (Hooker, 1972; Stapp, 1972; Hendry, 1984; Beller, 1999; Howard, 2004; Camilleri, 2005). For example, they never came to agree about the notion of complementarity (see Subsection \ref{compl}). More generally, Heisenberg usually based his ideas on the mathematical formalism of quantum theory, whereas Bohr's position was primarily philosophically oriented. Nonetheless, there is a clearly identifiable core of ideas on which they {\it did} agree, and since this core has everything to do with the relationship between classical and quantum, we are going to discuss it in some detail.
The principal primary sources are Bohr's Como Lecture, his reply to {\sc epr}, and his essay dedicated to Einstein (Bohr, 1927, 1935, 1949).\footnote{ These papers were actually written in collaboration with Pauli (after first attempts with Klein), Rosenfeld, and Pais, respectively.} Historical discussions of the emergence and reception of these papers are given in Bohr (1985, 1996) and in Mehra \&\ Rechenberg (2001). As a selection of the enormous literature these papers have given rise to, we mention among relatively recent works Hooker (1972), Scheibe (1973), Folse (1985), Murdoch (1987), Lahti \&\ Mittelstaedt (1987), Honner (1987), Chevalley (1991, 1999), Faye (1991), Faye \&\ Folse (1994), Held (1994), Howard (1994), Beller (1999), Faye (2002), and Saunders (2004). For Bohr's sparring partners see Heisenberg (1930, 1942, 1958, 1984a,b, 1985) with associated secondary literature (Heelan, 1965; H\"{o}rz, 1968; Geyer et al., 1993; Camilleri, 2005), and Pauli (1933, 1949, 1979, 1985, 1994), along with Laurikainen (1988) and Enz (2002).
As with Wittgenstein (and many other thinkers), it helps to understand Bohr if one makes a distinction between an ``early" Bohr and a ``later" Bohr.\footnote{Here we side with Held (1994) and Beller (1999) against Howard (1994) and Suanders (2004). See also Pais (2000), p.\ 22: `Bohr's Como Lecture did not bring the house down, however. He himself would later frown on expressions he used there, such as ``disturbing the phenomena by observation". Such language may have contributed to the considerable confusion that for so long has reigned around this subject.'\label{paisnote}} Despite a good deal of continuity in his thought (see below), the demarcation point is his response to {\sc epr}\ (Bohr, 1935),\footnote{This response is problematic, as is {\sc epr}\ itself. Consequently, there exists a considerable exegetical literature on both, marked by the fact that equally competent and well-informed pairs of commentators manage to flatly contradict each other while at the same time both claiming to explain or reconstruct what Bohr ``really" meant. } and the main shift he made afterwards lies in his sharp insistence on the indivisible unity of object and observer after 1935, focusing on the concept of a {\it phenomenon}.
Before {\sc epr}, Bohr equally well believed that object and observer were both necessary ingredients of a complete description of quantum theory, but he then thought that although their interaction could never be neglected, they might at least logically be considered separately. After 1935, Bohr gradually began to claim that object and observer no longer even had separate identities, together forming a ``phenomenon". Accordingly, also his notion of complementarity changed, increasingly focusing on the idea that the specification of the experimental conditions is crucial for the unambiguous use of (necessarily) classical concepts in quantum theory (Scheibe, 1973; Held, 1994). See also Subsection \ref{compl} below. This development culminated in Bohr's eventual denial of the existence of the quantum world: \begin{quote} `There is no quantum world. There is only an abstract quantum-physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature. (\ldots) What is it that we humans depend on? We depend on our words. Our task is to communicate experience and ideas to others. We are suspended in language.' (quoted by Petersen (1963), p.\ 8.)\footnote{See Mermin (2004) for a witty discussion of this controversial quotation.} \end{quote} \subsection{The doctrine of classical concepts}\label{Pcl} Despite this shift, it seems that Bohr stuck to one key thought throughout his career: \begin{quote}
`However far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. (\ldots) The argument is simply that by the word {\it experiment} we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangements and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.' (Bohr, 1949, p.\ 209). \end{quote}
This is, in a nutshell, Bohr's {\it doctrine of classical concepts}. Although his many drawings and stories may suggest otherwise, Bohr does not quite express the idea here that the goal of physics lies in the description of experiments.\footnote{Which often but misleadingly has been contrasted with Einstein's belief that the goal of physics is rather to describe reality. See Landsman (2006) for a recent discussion. } In fact, he merely points out the need for ``unambiguous" communication, which he evidently felt threatened by \qm.\footnote{Here ``unambiguous" means ``objective" (Scheibe, 1973; Chevalley, 1991).} The controversial part of the quote lies in his identification of the means of unambiguous communication with the language of classical physics, involving particles and waves and the like. We will study Bohr's specific argument in favour of this identification shortly, but it has to be said that, like practically all his foundational remarks on quantum mechanics, Bohr presents his reasoning as self-evident, necessary, and not in need of any further analysis (Scheibe, 1973; Beller, 1999). Nonetheless, young Heisenberg clashed with Bohr on precisely this point, for Heisenberg felt that the abstract mathematical formalism of quantum theory (rather than Bohr's world of words and pictures) provided those means of unambiguous communication.\footnote{It is hard to disagree with Beller's (1999) conclusion that Bohr was simply not capable of understanding the formalism of post-1925 quantum mechanics, turning his own need of understanding this theory in terms of words and pictures into a deep philosophical necessity.}
By classical physics Bohr undoubtedly meant the theories of Newton, Maxwell, and Lorentz, but that is not the main point.\footnote{Otherwise, one should wonder why one shouldn't use the physics of Aristotle and the scholastics for this purpose, which is a much more effective way of communicating our naive impressions of the world. In contrast, the essence of physics since Newton has been to unmask a reality behind the phenomena. Indeed, Newton himself empasized that his physics was intended for those capable of natural philosophy, in contrast to \textit{ye vulgar} who believed naive appearances. The fact that Aristotle's physics is now known to be wrong should not suffice to disqualify its use for Bohr's purposes, since the very same comment may be made about the physics of Newton etc.} For Bohr, the \textit{defining} property of classical physics was the property that it was \textit{objective}, i.e.\ that it could be studied in an observer-independent way:
\begin{quote} `All description of experiences so far has been based on the assumption, already inherent in ordinary conventions of language, that it is possible to distinguish sharply between the behaviour of objects and the means of observation. This assumption is not only fully justified by everyday experience, {\it but even constitutes the whole basis of classical physics}' (Bohr, 1958, p.\ 25; italics added).\footnote{Despite the typical imperative tone of this quotation, Bohr often regarded certain other properties as essential to classical physics, such as determinism, the combined use of space-time concepts and dynamical conservation laws, and the possibility of pictorial descriptions. However, these properties were in some sense secondary, as Bohr considered them to be {\it consequences} of the possibility of isolating an object in classical physics. For example: `The assumption underlying the ideal of causality [is] that the behaviour of the object is uniquely determined, quite independently of whether it is observed or not' (Bohr, 1937), and then again, now negatively: `the renunciation of the ideal of causality [in \qm] is founded logically only on our not being any longer in a position to speak of the autonomous behaviour of a physical object' (Bohr, 1937). See Scheibe (1973).} \end{quote}
See also Hooker (1972), Scheibe (1973) and Howard (1994). Heisenberg (1958, p.\ 55) shared this view:\footnote{As Camilleri (2005, p.\ 161) states: `For Heisenberg, classical physics is the fullest expression of the ideal of objectivity.'}
\begin{quote}
`In classical physics science started from the belief - or should one say from the illusion? - that we could describe the world or at least part of the world without any reference to ourselves. This is actually possible to a large extent. We know that the city of London exists whether we see it or not. It may be said that classical physics is just that idealization in which we can speak about parts of the world without any reference to ourselves. Its success has led to the general idea of an objective description of the world.' \end{quote}
On the basis of his ``quantum postulate" (see Subsection \ref{HC}), Bohr came to believe that, similarly, the
\textit{defining} property of quantum physics was precisely the opposite, i.e.\ the necessity of the role of the observer (or apparatus - Bohr did not distinguish between the two and never assigned a special role to the mind of the observer or endorsed a subjective view of physics). Identifying unambiguous communication with an objective description, in turn claimed to be the essence of classical physics,
Bohr concluded that despite itself quantum physics had to be described entirely in terms of classical physics. Thus his doctrine of classical concepts has an epistemological origin, arising from an analysis of the conditions for human knowledge.\footnote{See, for example, the very {\it title} of Bohr (1958)!} In that sense it may be said to be Kantian in spirit (Hooker, 1972; Murdoch, 1987; Chevalley, 1991, 1999).
Now, Bohr himself is on record as saying: `They do it smartly, but what counts is to do it right' (Rosenfeld, p.\ 129).\footnote{`They' refers to {\sc epr}.} The doctrine of classical concepts is certainly smart, but is it right? As we have seen, Bohr's argument starts from the claim that classical physics is objective (or `unambiguous') in being independent of the observer. In fact, nowadays it is widely believed that \qm\ leads to the {\it opposite} conclusion that ``quantum reality" (whatever that may be) is objective (though ``veiled" in the terminology of dÕEspagnat (1995)), while ``classical reality" only comes into existence relative to a certain specification: this is stance 1 discussed in the Introduction.\footnote{ Indeed, interesting recent attempts to make Bohr's philosophy of \qm\ precise accommodate the a priori status of classical observables into some version of the modal interpretation; see Dieks (1989b), Bub (1999), Halvorson \&\ Clifton (1999, 2002), and Dickson (2005). It should give one some confidence in the possibility of world peace that the two most hostile interpretations of \qm, viz.\ Copenhagen and Bohm (Cushing, 1994) have now found a common home in the modal interpretation in the sense of the authors just cited! Whether or not one agrees with Bub's (2004) criticism of the modal interpretation, Bohr's insistence on the necessity of classical concepts is not vindicated by any current version of it.}
Those who disagree with stance 1 cannot use stance 2 (of denying the fundamental nature of quantum theory) at this point either, as that is certainly not what Bohr had in mind.
Unfortunately, in his most outspoken defence of Bohr, even Heisenberg (1958, p.\ 55) was unable to find a better argument for Bohr's doctrine than the lame remark that `the use of classical concepts is finally a consequence of the general human way of thinking.'\footnote{And similarly: `We are forced to use the language of classical physics, simply because we have no other language in which to express the results.' (Heisenberg, 1971, p.\ 130). This in spite of the fact that the later Heisenberg thought about this matter very deeply; see, e.g., his (1942), as well as Camilleri (2005). Murdoch (1987, pp.\ 207--210) desperately tries to boost the doctrine of classical concepts into a profound philosophical argument by appealing to Strawson (1959).}
In our opinion, Bohr's {\it motivation} for his doctrine has to be revised in the light of our current understanding of quantum theory; we will do so in Subsection \ref{primas}. In any case, whatever its motivation, the doctrine {\it itself} seems worth keeping: apart from the fact that it evidently describes experimental practice, it provides a convincing explanation for the probabilistic nature of \qm\ (cf.\ the next subsection).
\subsection{Object and apparatus: the Heisenberg cut}\label{HC} Describing quantum physics in terms of classical concepts
sounds like an impossible and even self-contradictory task (cf.\ Heisenberg, 1958). For one, it precludes a completely quantum-mechanical description of the world: `However far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms.' But at the same time it precludes a purely classical description of the world, for underneath classical physics one has quantum theory.\footnote{This peculiar situation makes it very hard to give a realist account of the Copenhagen interpretation, since quantum reality is denied whereas classical reality is neither fundamental nor real.} The fascination of Bohr's philosophy of \qm\ lies precisely in his brilliant resolution of this apparently paradoxical situation.
The first step of this resolution that he and Heisenberg proposed is to divide the system whose description is sought into two parts: one, the object, is to be described quantum-mechanically, whereas the other, the apparatus, is treated \textit{as if it were classical}. Despite innumerable claims to the contrary in the literature (i.e.\ to the effect that Bohr held that a separate realm of Nature was intrinsically classical), there is no doubt that both Bohr and Heisenberg believed in the fundamental and universal nature of \qm, and saw the classical description of the apparatus as a {\it purely epistemological move without any counterpart in ontology}, expressing the fact that a given {\it quantum} system is {\it being used} as a measuring device.\footnote{See especially Scheibe (1973) on Bohr, and Heisenberg (1958). The point in question has also been made by R. Haag (who knew both Bohr and Heisenberg) in most of his talks on \qm\ in the 1990s. In this respect we disagree with Howard (1994), who claims that according to Bohr a classical description of an apparatus amounts to picking a particular (maximally) abelian subalgebra of its quantum-mechanical algebra of `beables', which choice is dictated by the measurement context. But having a commutative algebra falls far short of a classical description, since in typical examples one obtains only half of the canonical classical degrees of freedom in this way. Finding a classical description of a
quantum-mechanical system is a much deeper problem, to which we shall return throughout this paper.} For example: `The construction and the functioning of all apparatus like diaphragms and shutters, serving to define geometry and timing of the experimental arrangements, or photographic plates used for recording the localization of atomic objects, will depend on properties of materials which are themselves essentially determined by the quantum of action' (Bohr, 1948), as well as: `We are free to make the cut only within a region where the quantum mechanical description of the process concerned is effectively equivalent with the classical description' (Bohr, 1935).\footnote{This last point suggests that the cut has something to do with the division between a microscopic and a macroscopic realm in Nature, but although this division often facilitates making the cut when it is well defined, this is by no means a matter of principle. Cf.\ Howard (1994). In particular, all objections to the Copenhagen interpretation to the effect that the interpretation is ill-defined because the micro-macro distinction is blurred are unfounded. }
The separation between object and apparatus called for here is usually called the \textit{Heisenberg cut}, and it plays an absolutely central role in the Copenhagen interpretation of \qm.\footnote{Pauli (1949) went as far as saying that the Heisenberg cut provides the appropriate generalization modern physics offers of the old Kantian opposition between a knowable object and a knowing subject: 'Auf diese Weise verallgemeinert die moderne Physik die alte Gegen\"{u}berstellung von erkennenden Subjekt auf der einen Seite und des erkannten Objektes auf der anderen Seite zu der Idee des Schnittes zwischen Beobachter oder Beobachtungsmittel und dem beobachten System.' (`In this way, modern physics generalizes the old opposition between the knowing subject on the one hand and the known object on the other to the idea of the cut between observer or means of observation and the observed system.') He then continued calling the cut a necessary condition for human knowledge: see footnote \ref{PFN2}. } The idea, then, is that {\it a quantum-mechanical object is studied exclusively through its influence on an apparatus that is described classically}. Although {\it described} classically, the apparatus {\it is} a quantum system, and is supposed to be influenced by its {\it quantum-mechanical} coupling to the underlying (quantum) object.
The alleged necessity of including both object and apparatus in the description was initially claimed to be a consequence of the so-called ``quantum postulate". This notion played a key role in Bohr's thinking: his Como Lecture (Bohr, 1927) was even entitled `The quantum postulate and the recent development of atomic theory'. There he stated its contents as follows: `The essence of quantum theory is the quantum postulate: every atomic process has an essential discreteness - completely foreign to classical theories - characterized by PlanckÕs quantum of action.'\footnote{Instead of `discreteness', Bohr alternatively used the words `discontinuity' or `individuality' as well. He rarely omitted amplifications like `essential'.} Even more emphatically, in his reply to {\sc epr}\ (Bohr, 1935): `Indeed the finite interaction between object and measuring agencies conditioned by the very existence of the quantum of action entails - because of the impossibility of controlling the reaction of the object on the measurement instruments if these are to serve their purpose - the necessity of a final renunication of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality.'
Also, Heisenberg's uncertainty relations were originally motivated by the quantum postulate in the above form. According to Bohr and Heisenberg around 1927, this `essential discreteness' causes an `uncontrollable disturbance' of the object by the apparatus during their interaction. Although the ``quantum postulate" is not supported by the mature mathematical formalism of \qm\ and is basically obsolete, the intuition of Bohr and Heisenberg that a measurement of a quantum-mechanical object causes an `uncontrollable disturbance' of the latter is actually quite right.\footnote{Despite the fact that Bohr later distanced himself from it; cf.\ Beller (1999) and footnote \ref{paisnote} above. In a correct analysis, what is disturbed upon coupling to a classical apparatus is the quantum-mechanical state of the object (rather than certain sharp values of classical observables such as position and momentum, as the early writings of Bohr and Heisenberg suggest). }
In actual fact, the reason for this disturbance does not lie in the ``quantum postulate", but in the phenomenon of entanglement, as further discussed in Subsection \ref{primas}. Namely, from the point of view of von Neumann's measurement theory (see Subsection \ref{vNs})
the Heisenberg cut is just a two-step example of a von Neumann chain, with the special feature that after the quantum-mechanical interaction has taken place, the second link (i.e.\ the apparatus) is {\it described} classically. The latter feature not only supports Bohr's philosophical agenda, but, more importantly, also suffices to guarantee the applicability of the ignorance interpretation of the mixed state that arises after completion of the measurement.\footnote{In a purely quantum-mechanical von Neumann chain the final state of system plus apparatus is pure, but if the apparatus is classical, then the post-measurement state is mixed.}
All of von Neumann's analysis of the arbitrariness of the location of the cut applies here, for one may always extend the definition of the quantum-mechanical object by coupling the original choice to any other purely quantum-mechanical system one likes, and analogously for the classical part. Thus the two-step nature of the Heisenberg cut includes the possibility that the first link or object is in fact a lengthy chain in itself (as long as it is quantum-mechanical), and similarly for the second link (as long as it is classical).\footnote{\label{PFN2} Pauli (1949) once more: 'W\"{a}hrend die {\sc Existenz} eines solchen Schnittes eine notwendige Bedingung menschlicher Erkenntnis ist, fa\ss t sie die {\sc Lage} des Schnittes als bis zu einem gewissen Grade willk\"{u}rlich und als Resultat einer durch Zweckm\"{a}\ss igkeitserw\"{a}gungen mitbestimmten, also teilweise freien Wahl auf.' (`While the {\sc existence} of such a [Heisenberg] cut is a necessary condition for human knowledge, its {\sc location} is to some extent arbitrary as a result of a pragmatic and thereby partly free choice.')} This arbitrariness, subject to the limitation expressed by the second (1935) Bohr quote in this subsection, was well recognized by Bohr and Heisenberg, and was found at least by Bohr to be of great philosophical importance.
It is the interaction between object and apparatus that causes the measurement to `disturb' the former, but it is only and precisely the classical description of the latter that (through the ignorance interpretation of the final state) makes the disturbance `uncontrollable'.\footnote{These points were not clearly separated by Heisenberg (1927) in his paper on the uncertainty relations, but were later clarified by Bohr. See Scheibe (1973).} In the Copenhagen interpretation, {\it probabilities arise solely because we look at the quantum world through classical glasses}. \begin{quote} `Just the necessity of accounting for the function of the measuring agencies on classical lines excludes in principle in proper quantum phenomena an accurate control of the reaction of the measuring instruments on the atomic objects.' (Bohr, 1956, p.\ 87) \end{quote} \begin{quote} `One may call these uncertainties objective, in that they are simply a consequence of the fact that we describe the experiment in terms of classical physics; they do not depend in detail on the observer. One may call them subjective, in that they reflect our incomplete knowledge of the world.' (Heisenberg, 1958, pp.\ 53--54) \end{quote}
Thus the picture that arises is this: Although the quantum-mechanical side of the Heisenberg cut is described by the Schr\"{o}dinger equation (which is deterministic), while the classical side is subject to Newton's laws (which are equally well deterministic),\footnote{But see Earman (1986, 2005).} unpredictability arises because the quantum system serving as an apparatus is approximated by a classical system.
The ensuing probabilities reflect the ignorance arising from the decision (or need) to ignore the quantum-mechanical degrees of freedom of the apparatus. Hence the probabilistic nature of quantum theory is not intrinsic but extrinsic, and as such is entirely a consequence of
the doctrine of classical concepts, which by the same token {\it explains} this nature.
Mathematically, the simplest illustration of this idea is as follows. Take a finite-dimensional Hilbert space ${\mathcal H}=\C^n$ with the ensuing algebra of observables $\CA=M_n(\C)$ (i.e.\ the $n\times n$ matrices). A unit vector $\Ps\in\C^n$ determines a quantum-mechanical state in the usual way. Now describe this quantum system as if it were classical by ignoring all observables except the diagonal matrices. The state then immediately collapses to a probability measure on the set of $n$ points, with probabilities given by the Born rule
$p(i)=|(e_i,\Ps)|^2$, where $(e_i)_{i=1,\ldots,n}$ is the standard basis of $\C^n$.
Despite the appeal of this entire picture, it is not at all clear that it actually applies! There is no a priori guarantee whatsoever that one may indeed describe a quantum system ``as if it were classical". Bohr and Heisenberg apparently took this possibility for granted, probably on empirical grounds, blind to the extremely delicate theoretical nature of their assumption. It is equally astounding that they never reflected in print on the question if and how the classical worlds of mountains and creeks they loved so much emerges from a quantum-mechanical world. In our opinion, the main difficulty in making sense of the Copenhagen interpretation is therefore not of a philosophical nature, but is a mathematical one. This difficulty is the main topic of this paper, of which Section \ref{S6} is of particular relevance in the present context. \subsection{Complementarity}\label{compl} The notion of a Heisenberg cut is subject to a certain arbitrariness even apart from the precise location of the cut within a given chain, for one might in principle construct the chain in various different and incompatible ways. This arbitrariness was analyzed by Bohr in terms of what he called \textit{complementarity}.\footnote{Unfortunately and typically, Bohr once again presented complementarity as a necessity of thought rather than as the truly amazing possible mode of description it really is.}
Bohr never gave a precise definition of complementarity,\footnote{Perhaps he preferred this approach because he felt a definition could only reveal part of what was supposed to be defined: one of his favourite examples of complementarity was that between definition and observation.} but restricted himself to the analysis of a number of examples.\footnote{We refrain from discussing the complementarity between truth and clarity, science and religion, thoughts and feelings, and objectivity and introspection here, despite the fact that on this basis Bohr's biographer Pais (1997) came to regard his subject as the greatest philosopher since Kant.} A prominent such example is the complementarity between a ``causal" \footnote{\label{caudet} Bohr's use the word ``causal" is quite confusing in view of the fact that in the British empiricist tradition causality is often interpreted in the sense of a space-time description. But Bohr's ``causal" is meant to be {\it complementary} to a space-time description!} description of a quantum system in which conservation laws hold, and a space-time description that is necessarily statistical in character. Here Bohr's idea seems to have been that a stationary state (i.e.\ an energy eigenstate) of an atom is incompatible with an electron moving in its orbit in space and time - see Subsection \ref{PSL} for a discussion of this issue. Heisenberg (1958), however, took this example of complementarity to mean that a system on which no measurement is performed evolves deterministically according to the Schr\"{o}dinger equation, whereas a rapid succession of measurements produces a space-time path whose precise form quantum theory is only able to predict statistically (Camilleri, 2005). In other words, this example reproduces precisely the picture through which Heisenberg (1927) believed he had established the connection between classical and quantum mechanics; cf.\ Subsection \ref{heis}.
Bohr's other key example was the complementarity between particles and waves. Here his principal aim was to make sense of Young's double-slit experiment. The well-known difficulty with a classical visualization of this experiment is that a particle description appears impossible because a particle has to go through a single slit, ruining the interference pattern gradually built up on the detection screen, whereas a wave description seems incompatible with the point-like localization on the screen once the wave hits it. Thus Bohr suggested that whilst each of these classical descriptions is incomplete, the union of them is necessary for a complete description of the experiment.
The deeper epistemological point appears to be that although the {\it completeness} of the quantum-mechanical description of the microworld systems seems to be endangered by the doctrine of classical concepts, it is actually restored by the inclusion of {\it two} ``complementary" descriptions (i.e.\ of a given quantum system plus a measuring device that is necessarily described classicaly, `if it is to serve its purpose'). Unfortunately, despite this attractive general idea it is unclear to what precise definition of complementarity Bohr's examples should lead. In the first, the complementary notions of determinism and a space-time description are in mutual harmony as far as classical physics is concerned, but are apparently in conflict with each other in \qm. In the second, however, the wave description of some entity contradicts a particle description of the same entity precisely in classical physics, whereas in \qm\ these descriptions somehow coexist.\footnote{On top of this, Bohr mixed these examples in conflicting ways. In discussing bound states of electrons in an atom he jointly made determinism and particles one half of a complementary pair, waves and space-time being the other. In his description of electron-photon scattering he did it the other way round: this time determinism and waves formed one side, particles and space-time the other (cf.\ Beller, 1999).}
Scheibe (1973, p.\ 32) notes a `clear convergence [in the writings of Bohr] towards a preferred expression of a complementarity between phenomena', where a Bohrian {\it phenomenon} is an indivisible union (or ``whole") of a quantum system and a classically described experimental arrangement used to study it; see item \ref{item2list} below. Some of Bohr's early examples of complementarity can be brought under this heading, others cannot (Held, 1994). For many students of Bohr (including the present author), the fog has yet to clear up.\footnote{Even Einstein (1949, p.\ 674) conceded that throughout his debate with Bohr he had never understood the notion of complementarity, `the sharp formulation of which, moreover,
I have been unable to achieve despite much effort which I have expended on it.' See Landsman (2006) for the author's view on the Bohr--Einstein debate. }
Nonetheless, the following mathematical interpretations might assign some meaning to the idea of complementarity in the framework of von Neumann's formalism of \qm.\footnote{This exercise is quite against the spirit of Bohr, who is on record as saying that `von Neumann's approach (\ldots) did not {\it solve} problems but created {\it imaginary difficulties} (Scheibe, 1973, p.\ 11, quoting Feyerabend; italics in original).} \begin{enumerate} \item Heisenberg (1958) {\it identified complementary pictures of a quantum-mechanical system with equivalent mathematical representations thereof}. For example, he thought of the complementarity of $x$ and $p$ as the existence of what we now call the Schr\"{o}dinger representations of the canonical commutation relations (CCR) on $L^2(\R^n)$ and its Fourier transform to momentum space. Furthermore, he felt that in quantum field theory particles and waves gave two {\it equivalent} modes of description of quantum theory because of second quantization. Thus for Heisenberg complementary pictures are classical because there is an underlying classical variable, with no apparatus in sight, and such pictures are not mutually contradictory but (unitarily) equivalent. See also Camilleri (2005, p.\ 88), according to whom `Heisenberg never accepted Bohr's complementarity arguments'. \item \label{item2list} Pauli (1933) simply stated that {\it two observables are complementary when the corresponding operators fail to commute}.\footnote{ More precisely, one should probably require that the two operators in question generate the ambient algebra of observables, so that complementarity in Pauli's sense is really defined between two commutative subalgebras of a given algebra of observables (again, provided they jointly generate the latter).} Consequently, it then follows from Heisenberg's uncertainty relations that complementary observables cannot be measured simultaneously with arbitrary precision. This suggests (but by no means proves) that they should be measured independently, using mutually exclusive experimental arrangements. The latter feature of complementarity was emphasized by Bohr in his later writings.\footnote{We follow Held (1994) and others. Bohr's earlier writings do not quite conform to Pauli's approach. In Bohr's discussions of the double-slit experiment particle and wave form a complementary pair, whereas Pauli's complementary observables are position and momentum, which refer to a single side of Bohr's pair.} This approach makes the notion
of complementarity unambiguous and mathematically precise, and perhaps for this reason the few physicists who actually use the idea of complementarity in their work tend to follow Pauli and the later Bohr.\footnote{Adopting this point of view, it is tempting to capture the complementarity between position and momentum by means of the following conjecture: \textit{Any normal pure state $\om$ on $\CB(L^2(\R^n))$} (that is, any wave function seen as a state in the sense of \ca s) {\it is determined by the pair $\{\om| L^{\infty}(\R^n), \om| FL^{\infty}(\R^n)F\inv\}$} (in other words, by its restrictions to position and momentum). Here $L^{\infty}(\R^n)$ is the \vna\ of multiplication operators on $L^2(\R^n)$, i.e.\
the \vna\ generated by the position operator, whereas $FL^{\infty}(\R^n)F\inv$ is its Fourier transform, i.e.\ the \vna\ generated by the momentum operator. The idea is that each of its restrictions $\om| L^{\infty}(\R^n)$ and $\om| FL^{\infty}(\R^n)F\inv$ gives a classical picture of $\om$. These restrictions are a measure on $\R^n$ interpreted as position space, and another measure on $\R^n$ interpreted as momentum space. Unfortunately, this conjecture is false. The following counterexample was provided by D. Buchholz (private communication): take $\om$ as the state defined by the wave function
$\Psi(x) \sim \exp(- a x^2 / 2)$ with $\Re (a) > 0$, $\Im(a)\neq 0$, and $|a|^2=1$. Then
$\om$ depends on $\Im(a)$, whereas neither $\om| L^{\infty}(\R^n)$ nor $\om| FL^{\infty}(\R^n)F\inv$ does.
There is even a counterexample to the analogous conjecture for the \ca\ of $2\x 2$ matrices, found by H. Halvorson: if $A$ is the commutative \ca\ generated by $\sg_x$, and $B$ the one generated by $\sg_y$, then the two different eigenstates of $\sg_z$ coincide on $A$ and on $B$. One way to improve our conjecture might be to hope that if, in the Schr\"{o}dinger picture, two states coincide on the two given commutative \vna s for all times, then they must be equal. But this can only be true for certain ``realistic" time-evolutions, for the trivial Hamiltonian $H=0$ yields the above counterexample. We leave this as a problem for future research. At the time of writing, Halvorson (2004) contains the only sound mathematical interpretation of the complementarity between position and momentum, by relating it to the representation theory of the CCR. He shows that in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa.} \item The present author proposes that {\it observables and pure states are complementary}. For in the Schr\"{o}dinger representation of elementary \qm, the former are, roughly speaking, generated by the position and momentum operators, whereas the latter are given by wave functions. Some of Bohr's other examples of complementarity also square with this interpretation (at least if one overlooks the collapse of the wavefunction upon a measurement). Here one captures the idea that both ingredients of a complementary pair are necessary for a complete description, though the alleged mutual contradiction between observables and states is vague. Also, this reading of complementarity relies on a specific representation of the canonical commutation relations. It is not quite clear what one gains with this ideology, but perhaps it deserves to be developed in some more detail. For example, in quantum field theory it is once more the observables that carry the space-time description, especially in the algebraic description of Haag (1992). \end{enumerate} \subsection{Epilogue: entanglement to the rescue?}\label{primas} Bohr's ``quantum postulate" being obscure and obsolete, it is interesting to consider Howard's (1994) `reconstruction' of Bohr's philosophy of physics on the basis of entanglement.\footnote{We find little evidence that Bohr himself ever thought along those lines. With approval we quote Zeh, who, following a statement of the quantum postulate by Bohr similar to the one in Subsection \ref{HC} above, writes: `The later revision of these early interpretations of quantum theory (required by the important role of entangled quantum states for much larger systems) seems to have gone unnoticed by many physicists.' (Joos et al., 2003, p.\ 23.) See also Howard (1990) for an interesting historical perspective on entanglement, and cf.\ Raimond, Brune, \&\ Haroche (2001) for the experimental situation. } His case can perhaps be strengthened by an appeal to the analysis Primas (1983) has given of the need for classical concepts in quantum physics.\footnote{See also Amann \&\ Primas (1997) and Primas (1997).} Primas proposes to define a ``quantum object" as a physical system $\CS$ that is free from what he calls ``{\sc epr}-correlations" with its environment. Here the ``environment" is meant to include apparatus, observer, the rest of the universe if necessary, and what not. In elementary \qm, quantum objects in this sense exist only in very special states: if ${\mathcal H}_S$ is the \Hs\ of the system $S$, and ${\mathcal H}_E$ that of the environment $E$, any pure state of the form $\sum_i c_i \Psi_i\otimes\Phi_i$ (with more than one term) by definition correlates $S$ with $E$; the only uncorrelated pure states are those of the form $\Psi\otimes\Phi$ for unit vectors $\Psi\in {\mathcal H}_S$, $\Phi\in{\mathcal H}_E$. The restriction of an {\sc epr}-correlated state on $S+E$ to $S$ is mixed, so that the (would-be) quantum object `does not have its own pure state'; equivalently, the restriction of an {\sc epr}-correlated state $\om$ to $S$ together with its restriction to $E$ do not jointly determine $\om$. Again in other words, if the state of the total $S+E$ is {\sc epr}-correlated, a complete characterization of the state of $S$ requires $E$ (and vice versa). But (against Bohr!) mathematics defeats words: the sharpest characterization of the notion of {\sc epr}-correlations can be given in terms of operator algebras, as follows. In the spirit of the remainder of the paper we proceed in a rather general and abstract way.\footnote{Though Summers \&\ Werner (1987) give even more general results, where the tensor product $\CA\hat{\otimes}\CB$ below is replaced by an arbitrary \ca\ $\CC$ containing $\CA$ and $\CB$ as $C^*$-subalgebras. }
Let $\CA$ and $\CB$ be $C^*$-algebras,\footnote{\label{CSQP} Recall that a $C^*$-algebra is a complex algebra $\CA$ that is complete in a norm $\|\cdot\|$ that satisfies $\| AB\|\,\leq\, \| A\|\,\|
B\|$ for all $A,B\in\CA$, and has an involution $A\raw A^*$ such that
$\| A^*A\|=\| A\|^2$. A basic examples is
$\CA=\BH$, the algebra of all bounded operators on a \Hs\ ${\mathcal H}$, equipped with the usual operator norm and adjoint. By the Gelfand--Naimark theorem, any \ca\ is isomorphic to a norm-closed self-adjoint subalgebra of $\BH$, for some \Hs\ ${\mathcal H}$. Another key example is $\CA=C_0(X)$, the space of all continuous complex-valued functions on a (locally compact Hausdorff) space $X$ that vanish at infinity (in the sense that for every $\varep>0$ there is a {\it compact} subset $K\subset X$ such that $|f(x)|<\varep$ for all $x\notin K$), equipped with the supremum norm
$\| f\|_{\infty}:=\sup_{x\in X} |f(x)|$, and involution given by (pointwise) complex conjugation. By the Gelfand--Naimark lemma, any commutative \ca\ is isomorphic to $C_0(X)$ for some locally compact Hausdorff space $X$.
\label{Cstar}}
with tensor product $\CA\hat{\otimes}\CB$.\footnote{\label{tensorproducts} The tensor product of two (or more) \ca s is not unique, and we here need the so-called {\it projective} tensor product $\CA\hat{\otimes}\CB$, defined as the completion of the algebraic tensor product $\CA\otimes\CB$ in the {\it maximal} $C^*$-cross-norm. The choice of the projective tensor product guarantees that each state on $\CA\otimes\CB$ extends to a state on $\CA\hat{\otimes}\CB$ by continuity; conversely, since $\CA\otimes\CB$ is dense in $\CA\hat{\otimes}\CB$, each state on the latter is uniquely determined by its values on the former. See Wegge-Olsen (1993), Appendix T, or Takesaki (2003), Vol.\ {\sc i}, Ch.\ {\sc iv}. In particular, product states $\rh\otimes\sg$ and mixtures $\om=\sum_i p_i \rh_i\otimes\sg_i$ thereof as considered below are well defined on $\CA\hat{\otimes}\CB$. If $\CA\subset \CB({\mathcal H}_S)$ and $\CB\subset \CB({\mathcal H}_E)$ are von Neumann algebras, as in the analysis of Raggio (1981, 1988), it is easier (and sufficient) to work with the {\it spatial} tensor product $\CA\ovl{\otimes}\CB$, defined as the double commutant (or weak completion) of $\CA\otimes\CB$ in $\CB({\mathcal H}_S\otimes {\mathcal H}_E)$. For any {\it normal} state on $\CA\otimes\CB$ extends to a normal state on $\CA\ovl{\otimes}\CB$ by continuity.} Less abstractly, just think of two \Hs s ${\mathcal H}_S$ and ${\mathcal H}_E$ as above, with tensor product ${\mathcal H}_S\otimes{\mathcal H}_E$, and assume that $\CA=\CB({\mathcal H}_S)$ while $\CB$ is either $\CB({\mathcal H}_E)$ itself or some (norm-closed and involutive) commutative subalgebra thereof. The tensor product $\CA\hat{\otimes}\CB$ is then a (norm-closed and involutive) subalgebra of $\CB({\mathcal H}_S\otimes{\mathcal H}_E)$, the algebra of all bounded operators on ${\mathcal H}_S\otimes{\mathcal H}_E$.
A {\it product state} on $\CA\hat{\otimes}\CB$ is a state of the form $\om=\rh\otimes\sg$, where the states $\rh$ on $\CA$ and $\sg$ on $\CB$ may be either pure or mixed.\footnote{ We use the notion of a state that is usual in the algebraic framework. Hence a {\it state} on a \ca\ $\CA$ is a linear functional $\rh:\CA\raw\C$ that is {\it positive} in that $\rh(A^*A)\geq 0$ for all $A\in\CA$ and {\it normalized} in that $\rh(1)=1$, where $1$ is the unit element of $\CA$. If $\CA$ is a von Neumann algebra, one has the notion of a
{\it normal} state, which satisfies an additional continuity condition. If $\CA=\CB({\mathcal H})$, then a fundamental theorem of von Neumann states that each normal state $\rh$ on $\CA$ is given by a density matrix $\hat{\rh}$ on ${\mathcal H}$, so that $\rh(A)=\Tr (\hat{\rh} A)$ for each $A\in\CA$. In particular, a normal pure state on $\CB({\mathcal H})$ (seen as a von Neumann algebra) is necessarily of the form $\ps(A)=(\Ps,A\Ps)$ for some unit vector $\Ps\in{\mathcal H}$.} We say that a state $\om$ on $\CA\hat{\otimes}\CB$ is {\it decomposable} when it is a mixture of product states, i.e.\ when
$\om=\sum_i p_i \rh_i\otimes\sg_i$, where the coefficients $p_i>0$ satisfy $\sum_i p_i=1$.\footnote{Infinite sums are allowed here. More precisely, $\om$ is decomposable if it is in the $w^*$-closure of the convex hull of the product states on $\CA\hat{\otimes}\CB$.} A decomposable state $\om$ is pure precisely when it is a product of pure states. This has the important consequence that both its restrictions $\om_{|\CA}$ and $\om_{|\CB}$ to $\CA$ and $\CB$, respectively, are pure as well.\footnote{The restriction $\om_{|\CA}$ of a state $\om$ on $\CA\hat{\otimes}\CB$ to, say, $\CA$ is given by $\om_{|\CA}(A)=\om(A\otimes 1)$, where $1$ is the unit element of $\CB$, etc.} On the other hand, a state on $\CA\hat{\otimes}\CB$ may be said to be {\it {\sc epr}-correlated} (Primas, 1983) when it is {\it not} decomposable. An {\sc epr}-correlated {\it pure} state has the property that its
restriction to $\CA$ or $\CB$ is {\it mixed}.
Raggio (1981) proved that {\it each normal state on $\CA\hat{\otimes}\CB$ is decomposable
if and only if $\CA$ or $\CB$ is commutative.} In other words, {\it {\sc epr}-correlated states exist if and only if $\CA$ and $\CB$ are both noncommutative.}\footnote{Raggio (1981) proved this for von Neumann algebras and normal states. His proof was adapted to \ca s by Bacciagaluppi (1993).} As one might expect, this result is closely related to the Bell inequalities. Namely,
the Bell-type (or Clauser--Horne) inequality \begin{equation} \sup\{\om(A_1(B_1+B_2)+A_2(B_1-B_2))\} \leq 2, \label{bell} \end{equation}
where {\it for a fixed state $\om$} the supremum is taken over all self-adjoint operators $A_1,A_2\in \CA$, $B_1,B_2\in\CB$, each of norm $\leq 1$, {\it holds if and only if $\om$ is decomposable} (Baez, 1987; Raggio, 1988). Consequently, the inequality \eqref{bell} can only be violated in some state $\om$ when
the algebras $\CA$ and $\CB$ are both noncommutative. If, on the other hand, \eqref{bell} is satisfied, then one knows that there exists a classical probability space and probability measure (and hence a ``hidden variables" theory) reproducing the given correlations (Pitowsky, 1989). As stressed by Bacciagaluppi (1993), such a description does {\it not} require the entire setting to be classical; as we have seen, only one of the algebras $\CA$ and $\CB$ has to be commutative for the Bell inequalities to hold.
Where does this leave us with respect to Bohr? If we follow Primas (1983) in describing a (quantum) object as a system free from {\sc epr}-correlations with its environment, then the mathematical results just reviewed leave us with two possibilities. Firstly, we may pay lip-service to Bohr in taking the algebra $\CB$ (interpreted as the algebra of observables of the environment in the widest possible sense, as above) to be commutative {\it as a matter of description}. In that case, our object is really an ``object" in {\it any} of its states.
But clarly this is not the only possibility. For even in the case of elementary \qm\ - where $\CA=\CB({\mathcal H}_S)$ and $\CB=\CB({\mathcal H}_E)$ - the system is still an ``object" in the sense of Primas as long as the total state $\om$ of $S+E$ is decomposable. In general, for pure states this just means that $\om=\psi\otimes\phi$, i.e.\ that the total state is a product of pure states. To accomplish this, one has to define the Heisenberg cut in an appropriate way, and subsequently hope that the given product state remains so under time-evolution (see Amann \&\ Primas (1997) and Atmanspacher, Amann \&\ M\"{u}ller-Herold, 1999, and references therein). This selects certain states on $\CA$ as ``robust" or ``stable", in much the same way as in the decoherence approach. We therefore continue this discussion in Section \ref{S7} (see especially point \ref{CHD} in Subsection \ref{DSS}). \section{Quantization}\label{S4}\setcounter{equation}{0} Heisenberg's (1925) idea of {\it Umdeutung} ({\it reinterpretation}) suggests that it is possible to construct a quantum-mechanical description of a physical system whose classical description is known. As we have seen, this possibility was realized by Schr\"{o}dinger (1925c), who found the simplest example \eqref{SOP} and \eqref{Schreq} of {\it Umdeutung} in the context of atomic physics. This early example was phenomenally successful, as almost all of atomic and molecular physics is still based on it.
Quantization theory is an attempt to understand this example, make it mathematically precise, and generalize it to more complicated systems. It has to be stated from the outset that, like the entire classical-quantum interface, the nature of quantization is not yet well understood. This fact is reflected by the existence of a fair number of competing quantization procedures, the most transparent of which we will review below.\footnote{The path integral approach to quantization is still under development and so far has had no impact on foundational debates, so we will not discuss it here. See Albeverio \&\ H\o egh-Krohn (1976) and Glimm \&\ Jaffe (1987).} Among the first mathematically serious discussions of quantization are Mackey (1968) and Souriau (1969); more recent and comprehensive treatments are, for example, Woodhouse (1992), Landsman (1998), and Ali \&\ Englis (2004). \subsection{Canonical quantization and systems of imprimitivity}\label{Mackey} The approach based on \eqref{SOP} is often called {\it canonical quantization}. Even apart from the issue of mathematical rigour, one can only side with Mackey (1992, p.\ 283), who wrote: `Simple and elegant as this model is, it appears at first sight to be quite arbitrary and ad hoc. It is difficult to understand how anyone could have guessed it and by no means obvious how to modify it to fit a model for space different from $\R^r$.'
One veil of the mystery of quantization was lifted by von Neumann (1931), who (following earlier heuristic proposals by Heisenberg, Schr\"{o}dinger, Dirac, and Pauli) recognized that \eqref{SOP} does not merely provide {\it a} \rep\ of the canonical commutation relations \begin{equation} [\CQ_{\hbar}(p_j),\CQ_{\hbar}(q^k)]=-i\hbar\dl^k_j, \label{ccr}\end{equation} but (subject to a regularity condition)\footnote{It is required that the unbounded operators $\CQ_{\hbar}(p_j)$ and $\CQ_{\hbar}(q^k)$ integrate to a unitary \rep\ of the $2n+1$-dimensional Heisenberg group $H_n$, i.e.\ the unique connected and simply connected Lie group with $2n+1$-dimensional Lie algebra with generators $X_i,Y_i,Z$ ($i=1,\ldots, n$) subject to the Lie brackets $[X_i,X_j]=[Y_i,Y_j]=0$, $[X_i,Y_j]=\dl_{ij}Z$, $[X_i,Z]=[Y_i,Z]=0$. Thus von Neumann's uniqueness theorem for \rep s of the canonical commutation relations is (as he indeed recognized himself) really a uniqueness theorem for unitary \rep s of $H_n$ for which the central element $Z$ is mapped to $-i\hbar\inv 1$, where $\hbar\neq 0$ is a {\it fixed} constant. See, for example, Corwin \&\ Greenleaf (1989) or Landsman (1998).} is {\it the only} such \rep\ that is irreducible (up to unitary equivalence). In particular, the seemingly different formulations of quantum theory by Heisenberg and Schr\"{o}dinger (amended by the inclusion of states and of observables, respectively - cf.\ Section \ref{S2}) simply involved superficially different but unitarily equivalent \rep s of \eqref{ccr}: the difference between matrices and waves was just one between coordinate systems in \Hs, so to speak. Moreover, any other conceivable formulation of \qm\ - now simply {\it defined} as a (regular) \Hs\ \rep\ of \eqref{ccr} - has to be equivalent to the one of Heisenberg and Schr\"{o}dinger.\footnote{This is unrelated to the issue of the Heisenberg picture versus the Schr\"{o}dinger picture, which is about the time-evolution of observables versus that of states.}
This, then, transfers the quantization problem of a particle moving on $\R^n$ to the canonical commutation relations \eqref{ccr}. Although a mathematically rigorous theory of these commutation relations (as they stand) exists (J\o rgensen,\&\ Moore, 1984; Schm\"{u}dgen, 1990), they are problematic nonetheless. Firstly, technically speaking the operators involved are unbounded, and in order to represent physical observables they have to be self-adjoint; yet on their respective domains of self-adjointness the commutator on the left-hand side is undefined. Secondly, and more importantly, \eqref{ccr} relies on the possibility of choosing global coordinates on $\R^n$, which precludes a naive generalization to arbitrary configuration spaces. And thirdly, even if one has managed to quantize $p$ and $q$ by finding a \rep\ of \eqref{ccr}, the problem of quantizing other observables remains - think of the Hamiltonian and the Schr\"{o}dinger equation.
About 50 years ago, Mackey set himself the task of making good sense of canonical quantization; see Mackey (1968, 1978, 1992) and the brief exposition below for the result. Although the author now regards Mackey's reformulation of quantization in terms of induced \rep s and systems of imprimitivity merely as a stepping stone towards our current understanding based on deformation theory and groupoids (cf.\ Subsection \ref{DQsection} below), Mackey's approach is (quite rightly) often used in the foundations of physics, and one is well advised to be familiar with it. In any case, Mackey (1992, p.\ 283 - continuing the previous quotation) claims with some justification that his approach to quantization `removes much of the mystery.'
Like most approaches to quantization, Mackey assigns momentum and position a quite different role in \qm, despite the fact that in classical mechanics $p$ and $q$ can be interchanged by a canonical transformation:\footnote{Up to a minus sign, that is. This is true globally on $\R^n$ and locally on any symplectic manifold, where local Darboux coordinates do not distinguish between position and momentum.} \begin{enumerate} \item The position operators $\CQ_{\hbar}(q^j)$ are collectively replaced by a single projection-valued measure $P$ on $\R^n$,\footnote{\label{PVM} A projection-valued measure $P$ on a space $\Om$ with Borel structure (i.e.\ equipped with a $\sg$-algebra of measurable sets defined by the topology) with values in a \Hs\ ${\mathcal H}$ is a map $E\mapsto P(E)$ from the Borel subsets $E\subset\Om$ to the projections on ${\mathcal H}$ that satisfies $P(\emptyset)=0$, $P(\Om)=1$, $P(E) P(F)=P(F)P(E)=P(E\cap F)$ for all measurable $E,F\subset\Om$, and $P(\cup_{i=1}^{\infty} E_i)=\sum_{i=1}^{\infty} P(E_i)$ for all countable collections of mutually disjoint $E_i\subset\Om$. } which on $L^2(\R^n)$ is given by $P(E)=\ch_E$ as a multiplication operator. Given this $P$, any multiplication operator defined by a (measurable) function $f:\R^n\raw\R$ can be represented as $\int_{\R^n} dP(x)\, f(x)$, which is defined and self-adjoint on a suitable domain.\footnote{\label{domain} This domain consists of all
$\Psi\in{\mathcal H}$ for which $\int_{\R^n} d(\Psi,P(x)\Psi)\, |f(x)|^2<\infty$.} In particular, the position operators $\CQ_{\hbar}(q^j)$ can be reconstructed from $P$ by choosing $f(x)=x^j$, i.e.\ \begin{equation} \CQ_{\hbar}(q^j)=\int_{\R^n} dP(x)\, x^j.\end{equation} \item The momentum operators $\CQ_{\hbar}(p_j)$ are collectively replaced by a single unitary group \rep\ $U(\R^n)$, defined on $L^2(\R^n)$ by $$U(y)\Psi(x):=\Psi(x-y).$$ Each $\CQ_{\hbar}(p_j)$ can be reconstructed from $U$ by means of \begin{equation} \CQ_{\hbar}(p_j)\Psi:=i\hbar \lim_{t_j\raw 0} t_j\inv(U(t_j)-1)\Psi,\end{equation}
where $U(t_j)$ is $U$ at $x^j=t_j$ and $x^k=0$ for $k\neq j$.\footnote{By Stone's theorem (cf.\ Reed \&\ Simon, 1972), this operator is defined and self-adjoint on the set of all $\Psi\in H$ for which the limit exists.} \end{enumerate}
Consequently, it entails no loss of generality to work with the pair $(P,U)$ instead of the pair $(\CQ_{\hbar}(q^k),\CQ_{\hbar}(p_j))$. The commutation relations \eqref{ccr} are now replaced by \begin{equation} U(x)P(E)U(x)\inv=P(xE), \label{impr}\end{equation} where $E$ is a (Borel) subset of $\R^n$ and $xE=\{x\om\mid\om\in E\}$. On the basis of this reformulation, Mackey proposed the following sweeping generalization of the the canonical commutation relations:\footnote{All groups and spaces are supposed to be locally compact, and actions and \rep s are assumed continuous.} \begin{quote}A {\it system of imprimitivity} $({\mathcal H},U,P)$ for a given action of a group $G$ on a space $Q$ consists of a \Hs\ ${\mathcal H}$, a unitary \rep\ $U$ of $G$ on ${\mathcal H}$, and a projection-valued measure $E\mapsto P(E)$ on $Q$ with values in ${\mathcal H}$, such that \eqref{impr} holds for all $x\in G$ and all Borel sets $E\subset Q$. \end{quote}
In physics such a system describes the \qm\ of a particle moving on a configuration space $Q$ on which $G$ acts by symmetry transformations; see Subsection \ref{DQsection} for a more detailed discussion. When everything is smooth,\footnote{I.e.\ $G$ is a Lie group, $Q$ is a manifold, and the $G$-action is smooth.} each element $X$ of the Lie algebra $\g$ of $G$ defines a generalized momentum operator \begin{equation} \CQ_{\hbar}(X)=i\hbar dU(X)\label{mom}\end{equation} on ${\mathcal H}$.\footnote{This operator is defined and self-adjoint on the domain of vectors $\Psi\in {\mathcal H}$ for which $dU(X)\Psi:=\lim_{t\raw 0} t\inv(U(\exp(tX))-1)\Psi$ exists.\label{dUFN}} These operators satisfy the generalized canonical commutation relations\footnote{As noted before in the context of \eqref{ccr}, the commutation relations \eqref{Gccr}, \eqref{Gccr2} and \eqref{Gccr3} do not hold on the domain of self-adjointness of the operators involved, but on a smaller common core.} \begin{equation} [\CQ_{\hbar}(X),\CQ_{\hbar}(Y)] = i\hbar \CQ_{\hbar}([X,Y]). \label{Gccr}\end{equation} Furthermore, in terms of the operators\footnote{For the domain of $\CQ_{\hbar}(f)$ see footnote \ref{domain}.} \begin{equation} \CQ_{\hbar}(f)=\int_Q dP(x)\, f(x), \label{pifP}\end{equation} where $f$ is a smooth function on $Q$ and $X\in\g$, one in addition has
\begin{equation} [\CQ_{\hbar}(X),\CQ_{\hbar}(f)] =i\hbar\CQ_{\hbar}(\xi^Q_X f), \label{Gccr2}\end{equation} where $\xi^Q_X$ is the canonical vector field on $Q$ defined by the $G$-action,\footnote{I.e.\ $\xi^Q_X f(y)=d/dt|_{t=0} [f(\exp(-tX)y)]$.\label{VFFN}}
and \begin{equation} [\CQ_{\hbar}(f_1),\CQ_{\hbar}(f_2)]=0. \label{Gccr3}\end{equation}
Elementary \qm\ on $\R^n$ corresponds to the special case $Q=\R^n$ and $G=\R^n$ with the usual additive group structure. To see this, we denote the standard basis of $\R^3$ (in its guise as the Lie algebra of $\R^3$) by the name $(p_j)$, and furthermore take $f_1(q)=q^j$, $f_2(q)=f(q)=q^k$. Eq.\ \eqref{Gccr} for $X=p_j$ and $Y=p_k$ then reads $[\CQ_{\hbar}(p_j),\CQ_{\hbar}(p_k)]=0$, eq.\ \eqref{Gccr2} yields the canonical commutation relations \eqref{ccr}, and \eqref{Gccr3} states the commutativity of the position operators, i.e.\ $[\CQ_{\hbar}(q^j),\CQ_{\hbar}(q^k)]=0$.
In order to incorporate spin, one picks $G=E(3)=SO(3)\ltimes\R^3$ (i.e.\ the Euclidean motion group), acting on $Q=\R^3$ in the obvious (defining) way. The Lie algebra of $E(3)$ is $\R^6=\R^3\x\R^3$ as a vector space; we extend the basis $(p_j)$ of the second copy of $\R^3$ (i.e.\ the Lie algebra of $\R^3$) by a basis $(J_i)$ of the first copy of $\R^3$ (in its guise as the Lie algebra of $SO(3)$) , and find that the $\CQ_{\hbar}(J_i)$ are just the usual angular momentum operators.\footnote{The commutation relations in the previous paragraph are now extended by the familiar relations $[\CQ_{\hbar}(J_i),\CQ_{\hbar}(J_j)] = i\hbar\ep_{ijk}\CQ_{\hbar}(J_k)$,
$ [\CQ_{\hbar}(J_i),\CQ_{\hbar}(p_j)] = i\hbar\ep_{ijk}\CQ_{\hbar}(p_k)$, and
$[\CQ_{\hbar}(J_i),\CQ_{\hbar}(q^j)] = i\hbar\ep_{ijk}\CQ_{\hbar}(q^k)$. \label{AMCCR}}
Mackey's generalization of von Neumann's (1931) uniqueness theorem for the irreducible representation s of the canonical commutation relations \eqref{ccr} is his {\it imprimitivity theorem}. This theorem applies to the special case where $Q=G/H$ for some (closed) subgroup $H\subset G$, and states that (up to unitary equivalence) there is a bijective correspondence between: \begin{enumerate} \item Systems of imprimitivity $({\mathcal H},U,P)$ for the left-translation of $G$ on $G/H$; \item Unitary \rep s $U_{\ch}$ of $H$. \end{enumerate} This correspondence preserves irreducibility.\footnote{Specifically, given $U_{\ch}$ the triple $({\mathcal H}^{\ch},U^{\ch},P^{\ch})$ is a system of imprimitivity, where ${\mathcal H}^{\ch}=L^2(G/H)\otimes {\mathcal H}_{\ch}$ carries the \rep\ $U^{\ch}(G)$ induced by $U_{\ch}(H)$, and the $P^{\ch}$ act like multiplication operators. Conversely, if $({\mathcal H},U,P)$ is a system of imprimitivity, then there exists a unitary \rep\ $U_{\ch}(H)$ such that the triple $({\mathcal H},U,P)$ is unitarily equivalent to the triple $({\mathcal H}^{\ch},U^{\ch},P^{\ch})$ just described. For example, for $G=E(3)$ and $H=SO(3)$ one has $\ch=j=0,1,2,\ldots$ and ${\mathcal H}^j=L^2(\R^3)\otimes {\mathcal H}_j$ (where ${\mathcal H}_j=\C^{2j+1}$ carries the given \rep\ $U_j(SO(3))$). \label{MIT}}
For example, von Neumann's theorem is recovered as a special case of Mackey's by making the choice $G=\R^3$ and $H=\{e\}$ (so that $Q=\R^3$, as above): the uniqueness of the (regular) irreducible representation\ of the canonical commutation relations here follows from the uniqueness of the irreducible representation\ of the trivial group.
A more illustrative example is $G=E(3)$ and $H=SO(3)$ (so that $Q=\R^3$), in which case the irreducible representation s of the associated system of imprimitivity are classified by spin $j=0,1,\ldots$.\footnote{By the usual arguments (Wigner's theorem), one may replace $SO(3)$ by $SU(2)$, so as to obtain $j=0,1/2,\ldots$.} Mackey saw this as an explanation for the emergence of spin as a purely quantum-mechanical degree of freedom. Although the opinion that spin has no classical analogue was widely shared also among the pioneers of quantum theory,\footnote{This opinion goes back to Pauli (1925), who talked about a `klassisch nicht beschreibbare Zweideutigkeit in den quantentheoretischen Eigenschaften des Elektrons,' (i.e.\ an `ambivalence in the quantum theoretical properties of the electron that has no classical description') which was later identified as spin by Goudsmit and Uhlenbeck. Probably the first person to draw attention to the classical counterpart of spin was Souriau (1969).
Another misunderstanding about spin is that its ultimate explanation must be found in relativistic \qm.} it is now obsolete (see Subsection \ref{DQsection} below). Despite this unfortunate misinterpretation, Mackey's approach to canonical quantization is hard to surpass in power and clarity, and has many interesting applications.\footnote{This begs the question about the `best' possible proof of Mackey's imprimitivity theorem. Mackey's own proof was rather measure-theoretic in flavour, and did not shed much light on the origin of his result. Probably the shortest proof has been given by \O rsted (1979), but the insight brevity gives is still rather limited. Quite to the contrary, truly transparent proofs reduce a mathematical claim to a tautology. Such proofs, however, tend to require a formidable machinery to make this reduction work; see Echterhoff et al. (2002) and Landsman (2005b) for two different approaches to the imprimitivity theorem in this style.}
We mention one of specific interest to the philosophy of physics, namely the {\it Newton--Wigner position operator} (as analyzed by Wightman, 1962).\footnote{Fleming \&\ Butterfield (2000) give an up-to-date introduction to particle localization in relativistic quantum theory. See also De Bi\`{e}vre (2003).} Here the general question is whether a given unitary \rep\ $U$ of $G=E(3)$ on some \Hs\ ${\mathcal H}$ may be extended to a system of imprimitivity with respect to $H=SO(3)$ (and hence $Q=\R^3$, as above); in that case, $U$ (or rather the associated quantum system) is said to be {\it localizable} in $\R^3$. Following Wigner's (1939) suggestion that a relativistic particle is described by an irreducible \rep\ $U$ of the Poincar\'{e} group $P$, one obtains a \rep\ $U(E(3))$ by restricting $U(P)$ to the subgroup $E(3)\subset P$.\footnote{Strictly speaking, this hinges on the choice of an inertial frame in Minkowski space, with associated adapted co-ordinates such that the configuration space $\R^3$ in question is given by $x^0=0$.} It then follows from the previous analysis that the particle described by $U(P)$ is localizable if and only if $U(E(3))$ is induced by some \rep\ of $SO(3)$. This can, of course, be settled, with the result that massive particles of arbitrary spin can be localized in $\R^3$ (the corresponding position operator being precisely the one of Newton and Wigner), whereas
massless particles may be localized in $\R^3$ if and only if their helicity is less than one. In particular, the photon (and the graviton) cannot be localized in $\R^3$ in the stated sense.\footnote{ Seeing photons as quantized light waves with two possible polarizations transverse to the direction of propagation, this last result is physically perfectly reasonable. }
To appreciate our later material on both phase space quantization and deformation quantization, it is helpful to give a \ca ic reformulation of Mackey's approach. Firstly, by the spectral theorem (Reed \&\ Simon, 1972; Pedersen, 1989), a projection-valued measure $E\mapsto P(E)$ on a space $Q$ taking values in a \Hs\ ${\mathcal H}$ is equivalent to a nondegenerate \rep\ $\pi$ of the commutative \ca\ $C_0(Q)$ on ${\mathcal H}$ through the correspondence \eqref{pifP}.\footnote{A {\it representation} of a \ca\ $\CA$ on a \Hs\ ${\mathcal H}$ is a linear map $\pi:\CA\raw\BH$ such that $\pi(AB)=\pi(A)\pi(B)$ and $\pi(A^*)=\pi(A)^*$ for all $A,B\in\CA$. Such a \rep\ is called {\it nondegenerate} when $\pi(A)\Psi =0$ for all $A\in\CA$ implies $\Psi=0$.} Secondly, if ${\mathcal H}$ in addition carries a unitary \rep\ $U$ of $G$, the defining condition \eqref{impr} of a system of imprimitivity (given a $G$-action on $Q$) is equivalent to the covariance condition \begin{equation} U(x)\CQ_{\hbar}(f)U(x)\inv=\CQ_{\hbar}(L_xf) \label{Gcov}\end{equation}
for all $x\in G$ and $f\in C_0(Q)$, where $L_xf (m)=f(x\inv m)$. Thus a system of imprimitivity for a given $G$-action on $Q$ is ``the same" as a covariant nondegenerate \rep\ of $C_0(Q)$. Thirdly, from a $G$-action on $Q$ one can construct a certain \ca\ $C^*(G,Q)$, the so-called {\it transformation group \ca} defined by the action, which has the property that its nondegenerate \rep s correspond bijectively (and ``naturally") to covariant nondegenerate \rep s of $C_0(Q)$, and therefore to systems of imprimitivity for the given $G$-action (Effros \&\ Hahn, 1967; Pedersen, 1979; Landsman, 1998). In the \ca ic approach to quantum physics, $C^*(G,Q)$ is the algebra of observables of a particle moving on $Q$ subject to the symmetries defined by the $G$-action; its inequivalent irreducible representation s correspond to the possible superselection sectors of the system (Doebner \&\ Tolar, 1975; Majid, 1988, 1990; Landsman, 1990a, 1990b, 1992).\footnote{Another reformulation of Mackey's approach, or rather an extension of it, has been given by Isham (1984). In an attempt to reduce the whole theory to a problem in group \rep s, he proposed that the possible quantizations of a particle with configuration space $G/H$ are given by the inequivalent irreducible representation s of a ``canonical group" $G_c=G\ltimes V$, where $V$ is the lowest-dimensional vector space that carries a \rep\ of $G$ under which $G/H$ is an orbit in the dual vector space $V^*$. All pertinent systems of imprimitivity then indeed correspond to unitary \rep s of $G_c$, but this group has many other \rep s whose physical interpretation is obscure. See also footnote \ref{Isham2}.\label{Isham1}} \subsection{Phase space quantization and coherent states}\label{PSQ} In Mackey's approach to quantization, $Q$ is the {\it configuration space} of the system; the associated position coordinates {\it commute} (cf.\ \eqref{Gccr3}). This is reflected by the correspondence just discussed between projection-valued measures on $Q$ and \rep s of the {\it commutative} \ca\ $C_0(Q)$. The noncommutativity of observables (and the associated uncertainty relations) typical of \qm\ is incorporated by adding the symmetry group $G$ to the picture and imposing the relations \eqref{impr} (or, equivalently, \eqref{Gccr2} or \eqref{Gcov}). As we have pointed out, this procedure upsets the symmetry between the phase space variables position and momentum in classical mechanics.
This somewhat unsatisfactory feature of Mackey's approach may be avoided by replacing $Q$ by the {\it phase space} of the system, henceforth called $M$.\footnote{Here the reader may think of the simplest case $M=\R^6$, the space of $p$'s and $q$'s of a particle moving on $\R^3$. More generally, if $Q$ is the configuration space, the associated phase space is the cotangent bundle $M=T^*Q$. Even more general phase spaces, namely arbitrary symplectic manifolds, may be included in the theory as well. References for what follows include Busch, Grabowski, \&\ Lahti, 1998, Schroeck, 1996, and Landsman, 1998, 1999a. } In this approach, noncommutativity is incorporated by a treacherously tiny modification to Mackey's setup. Namely, the projection-valued measure $E\mapsto P(M)$ on $M$ with which he starts is now replaced by a {\it positive-operator-valued measure} or {\it POVM} on $M$, still taking values in some \Hs\ $\CK$. This is a map $E\mapsto A(E)$ from the (Borel) subsets $E$ of $M$ to the collection of {\it positive} bounded operators on $\CK$,\footnote{A bounded operator $A$ on $\CK$ is called positive when $(\Psi,A\Psi)\geq 0$ for all $\Psi\in\CK$. Consequently, it is self-adjoint with spectrum contained in $\R^+$.} satisfying $A(\emptyset)=0$, $A(M)=1$, and $A(\cup_i E_i)=\sum_i A(E_i)$ for any countable collection of disjoint Borel sets $E_i$.\footnote{Here the infinite sum is taken in the weak operator topology.
Note that the above conditions force $0\leq A(E)\leq 1$, in the sense that $0\leq (\Psi, A(E)\Psi)\leq (\Psi,\Psi)$ for all $\Psi\in\CK$.} A POVM that satisfies $A(E\cap F) = A(E)A(F)$ for all (Borel) $E,F\subset M$ is precisely a projection-valued measure, so that a POVM is a generalization of the latter.\footnote{This has given rise to the so-called {\it operational approach} to quantum theory, in which observables are not represented by self-adjoint operators (or, equivalently, by their associated projection-valued measures), but by POVM's. The space $M$ on which the POVM is defined is the space of outcomes of the measuring instrument; the POVM is determined by both $A$ and a calibration procedure for this instrument. The probability that in a state $\rh$ the outcome of the experiment lies in $E\subset M$ is taken to be $\Tr (\rh A(E))$. See Davies (1976), Holevo (1982), Ludwig (1985), Schroeck (1996), Busch, Grabowski, \&\ Lahti (1998), and De Muynck (2002).} The point, then, is that {\it a given POVM defines a quantization procedure} by the stipulation that a classical observable $f$ (i.e. a measurable function on the phase space $M$, for simplicity assumed bounded) is quantized by the operator\footnote{The easiest way to define the right-hand side of \eqref{berezin} is to fix $\Psi\in\CK$ and define a probability measure $p_{\Psi}$ on $M$ by means of $p_{\Psi}(E)=(\Psi,A(E)\Psi)$. One then {\it defines} $\CQ(f)$ as an operator through its expectation values $(\Psi, \CQ(f)\Psi)=\int_M dp_{\Psi}(x)\, f(x)$. The expression \eqref{berezin} generalizes \eqref{pifP}, and also generalizes the spectral resolution of the operator $f(A)= \int_{\R} dP(\lm) f(\lm)$, where $P$ is the projection-valued measure defined by a self-adjoint operator $A$. } \begin{equation} \CQ(f)=\int_M dA(x) f(x). \label{berezin} \end{equation} Thus the seemingly slight move from projection-valued measures on configuration space to positive-operator valued measures on phase space gives a wholly new perspective on quantization, actually reducing this task to the problem of finding such POVM's.\footnote{An important feature of $\CQ$ is that it is {\it positive} in the sense that if $f(x)\geq 0$ for all $x\in M$, then $(\Psi,\CQ(f)\Psi)\geq 0$ for all $\Psi\in\CK$. In other words, $\CQ$ is positive as a map from the \ca\ $C_0(M)$ to the \ca\ $\BH$.\label{BQP}}
The solution to this problem is greatly facilitated by {\it Naimark's dilation theorem}.\footnote{See, for example, Riesz and Sz.-Nagy (1990). It is better, however, to see Naimark's theorem as a special case of Stinesprings's, as explained e.g.\ in Landsman, 1998, and below.} This states that, given a POVM $E \mapsto A(E)$ on $M$ in a \Hs\ $\CK$, there exists a \Hs\ ${\mathcal H}$ carrying a projection-valued measure $P$ on $M$ and an isometric injection $\CK\hraw{\mathcal H}$, such that \begin{equation} A(E)=[\CK]P(E)[\CK] \label{AEP} \end{equation} for all $E\subset M$ (where $[\CK]$ is the orthogonal projection from ${\mathcal H}$ onto $\CK$). Combining this with Mackey's imprimitivity theorem yields a powerful generalization of the latter (Poulsen, 1970; Neumann, 1972; Scutaru, 1977; Cattaneo, 1979; Castrigiano \&\ Henrichs, 1980).
First, define a {\it generalized system of imprimitivity} $(\CK,U,A)$ for a given action of a group $G$ on a space $M$ as a POVM $A$ on $M$ taking values in a \Hs\ $\CK$, along with a unitary \rep\ $V$ of $G$ on $\CK$ such that \begin{equation} V(x)A(E)V(x)\inv=A(xE) \label{gimpr}\end{equation} for all $x\in G$ and $E\subset M$; cf.\ \eqref{impr}. Now assume $M=G/H$ (and the associated canonical left-action on $M$). The {\it generalized imprimitivity theorem} states that a generalized system of imprimitivity $(\CK,V,A)$ for this action is necessarily (unitarily equivalent to) a reduction of a system of imprimitivity $({\mathcal H}, U,P)$ for the same action. In other words, the \Hs\ ${\mathcal H}$ in Naimark's theorem carries a unitary \rep\ $U(G)$ that commutes with the projection $[\CK]$, and the \rep\ $V(G)$ is simply the restriction of $U$ to $\CK$. Furthermore, the POVM $A$ has the form \eqref{AEP}. The structure of $({\mathcal H}, U,P)$ is fully described by Mackey's imprimitivity theorem, so that one has a complete classification of generalized systems of imprimitivity.\footnote{Continuing footnote \ref{MIT}: $V(G)$ is necessarily a sub\rep\ of some \rep\ $U^{\ch}(G)$ induced by $U_{\ch}(H)$.} One has \begin{equation} \CK=p{\mathcal H}; \:\:\: {\mathcal H}= L^2(M)\otimes{\mathcal H}_{\ch}, \label{pH}\end{equation} where $L^2$ is defined with respect to a suitable measure on $M=G/H$,\footnote{In the physically relevant case that $G/H$ is symplectic (so that it typically is a coadjoint orbit for $G$) one should take a multiple of the Liouville measure.} the \Hs\ ${\mathcal H}_{\ch}$ carries a unitary \rep\ of $H$, and $p$ is a projection in the commutant of the \rep\ $U^{\ch}(G)$ induced by $U_{\ch}(G)$.\footnote{The explicit form of $U^{\ch}(g)$, $g\in G$, depends on the choice of a cross-section $\sg:G/H\raw G$ of the projection $\pi: G\raw G/H$ (i.e.\ $\pi\circ\sg=\mbox{\rm id}$). If the measure on $G/H$ defining $L^2(G/H)$ is $G$-invariant, the explicit formula is $U^{\ch}(g)\Psi(x)=U_{\ch}(s(x)\inv g s(g\inv x))\Psi(g\inv x)$.} The quantization \eqref{berezin} is given by \begin{equation} \CQ(f)=pfp, \label{PSQE}\end{equation} where $f$ acts on $L^2(M)\otimes{\mathcal H}_{\ch}$ as a multiplication operator, i.e.\ $(f\Psi)(x)=f(x)\Psi(x)$. In particular, one has $P(E)=\ch_E$ (as a multiplication operator) for
a region $E\subset M$ of phase space, so that $\CQ(\ch_E)=A(E)$. Consequently, the probability that in a state $\rh$ (i.e.\ a density matrix on $\CK$) the system is localized in $E$ is given by $\Tr (\rh A(E))$.
In a more natural way than in Mackey's approach, the covariant POVM quantization method allows one to incorporate space-time symmetries {\it ab initio} by taking $G$ to be
the Galilei group or the Poincar\'{e} group, and choosing $H$ such that $G/H$ is a physical phase space (on which $G$, then, canonically acts).
See Ali et al.\ (1995) and Schroeck (1996).
Another powerful method of constructing POVM's on phase space (which in the presence of symmetries overlaps with the preceding one)\footnote{ Suppose there is a vector $\Om\in\CK$ such that
$\int_{G/H}d\mu(x) |(\Om, V(\sg(x))\Om)|^2 <\infty$ with respect to some cross-section $\sg: G/H\raw G$ and a $G$-invariant measure $\mu$, as well as $V(h)\Om=U_{\ch}(h)\Om$ for all $h\in H$, where $U_{\ch}:H\raw \C$ is {\it one-dimensional}. Then (taking $\hbar=1$) the vectors $V(\sg(x))\Om$ (suitably normalized) form a family of coherent states on $G/H$ (Ali et al., 1995; Schroeck, 1996; Ali, Antoine, \&\ Gazeau, 2000). For example, the coherent states \eqref{pqcohst} are of this form for the Heisenberg group.}
is based on {\it coherent states}.\footnote{\label{CSFNO}See Klauder \&\ Skagerstam, 1985, Perelomov, 1986, Odzijewicz, 1992, Paul \&\ Uribe, 1995, 1996, Ali et al., 1995, and Ali, Antoine, \&\ Gazeau, 2000, for general discussions of coherent states. } The minimal definition of coherent states in a \Hs\ ${\mathcal H}$ for a phase space $M$ is that (for some fixed value of Planck's constant $\hbar$, for the moment) one has an injection\footnote{This injection must be continuous as a map from $M$ to $\mathbb{P}{\mathcal H}$, the projective \Hs\ of ${\mathcal H}$.}
$M\hraw {\mathcal H}$, $z\mapsto\Psi_z^{\hbar}$, such that
\begin{equation} \|\Psi_z^{\hbar}\|=1 \label{normcs} \end{equation}
for all $z\in M$, and \begin{equation}
c_{\hbar} \int_M d\mu_L(z)\,|(\Psi_z^{\hbar},\Phi)|^2 =1, \label{qhnorm} \end{equation} for each $\Phi\in{\mathcal H}$ of unit norm (here $\mu_L$ is the Liouville measure on $M$ and $c_{\hbar}>0$ is a suitable constant).\footnote{Other measures might occur here; see, for example, Bonechi \&\ De Bi\`{e}vre (2000).} Condition (\ref{qhnorm}) guarantees that we may define a POVM on $M$ in $\CK$ by\footnote{Recall that $[\Psi]$ is the orthogonal projection onto a unit vector $\Psi$.} \begin{equation} A(E)= c_{\hbar} \int_E d\mu_L(z)\,[\Psi_z^{\hbar}]. \label{pov} \end{equation} Eq.\ (\ref{berezin}) then simply reads (inserting the $\hbar$-dependence of $\CQ$ and a suffix $B$ for later use) \begin{equation} \CQ_{\hbar}^B(f)= c_{\hbar}\int_{M} d\mu_L(z)\, f(z) [\Psi^{\hbar}_z]. \label{b2} \end{equation}
The time-honoured example, due to Schr\"{o}dinger (1926b), is $M=\R^{2n}$, ${\mathcal H}=L^2(\R^n)$, and \begin{equation} \Psi_{(p,q)}^{\hbar}(x)=(\pi\hbar)^{-n/4}e^{- ipq/2\hbar}e^{ipx/\hbar}e^{-(x-q)^2/2\hbar}.\label{pqcohst} \end{equation} Eq.\ (\ref{qhnorm}) then holds with $d\mu_L(p,q)=(2\pi)^{-n}d^npd^nq$ and $c_{\hbar}=\hbar^{-n}$. One may verify that $\CQ_{\hbar}^B(p_j)$ and $\CQ_{\hbar}^B(q^j)$ coincide with Schr\"{o}dinger's operators \eqref{SOP}. This example illustrates that coherent states need {\it not} be mutually orthogonal; in fact, in terms of $z=p+iq$ one has for the states in \eqref{pqcohst} \begin{equation}
|(\Psi^{\hbar}_{z},\Psi^{\hbar}_{w})|^2=e^{-|z-w|^2/2\hbar}; \label{tpbt} \end{equation} the significance of this result will emerge later on.
In the general case, it is an easy matter to verify Naimark's dilation theorem for the POVM \eqref{pov}:
changing notation so that the vectors $\Psi_z^{\hbar}$ now lie in $\CK$,
one finds
\begin{equation} {\mathcal H}=L^2(M, c_{\hbar}\mu_L), \label{HSPSQ}
\end{equation}
the embedding $W:\CK\hraw{\mathcal H}$ being given by $(W\Phi)(z)=(\Psi^{\hbar}_z,\Phi)$. The projection-valued measure $P$ on ${\mathcal H}$ is just $P(E)=\ch_E$ (as a multiplication operator), and the projection $p$ onto $W\CK$ is given by
\begin{equation} p\Psi(z)=c_{\hbar} \int_M d\mu_L(w) (\Psi^{\hbar}_{z},\Psi^{\hbar}_{w})\Psi(w).\end{equation}
Consequently, \eqref{b2} is unitarily equivalent
to \eqref{PSQE}, where $f$ acts on $L^2(M)$ as a multiplication operator.\footnote{This leads to a close relationship between coherent states and \Hs s with a reproducing kernel; see Landsman (1998) or Ali, Antoine, \&\ Gazeau (2000).} \begin{quote}{\it Thus \eqref{PSQE} and \eqref{HSPSQ} (or its possible extension \eqref{pH}) form the essence of phase space quantization.}\footnote{See also footnote \ref{Koopman} below.} \end{quote}
We close this subsection in the same fashion as the previous one, namely by pointing out the \ca ic significance of POVM's. This is extremely easy: whereas
a projection-valued measure on $M$ in ${\mathcal H}$ is the same as a nondegenerate {\it \rep} of $C_0(M)$ in ${\mathcal H}$, a POVM on $M$ in a \Hs\ $\CK$ is nothing but a nondegenerate {\it completely positive map} $\phv:C_0(M)\raw \mathcal{B}(\CK)$.\footnote{A map $\phv:\CA\raw\CB$ between \ca s is called positive when $\phv(A)\geq 0$
whenever $A\geq 0$; such a map is called {\it completely positive} if for all $n\in \N$ the map $\phv_n: \CA\otimes M_n(\C)\raw \CB\otimes M_n(\C)$, defined by linear extension of $\phv\otimes\mbox{\rm id}$ on elementary tensors, is positive (here $M_n(\C)$ is the \ca\ of $n\x n$ complex matrices). When $\CA$ is commutative
a nondegenerate positive map $\CA\raw \CB$ is automatically completely positive for any $\CB$.}
Consequently, Naimark's dilation theorem becomes a special case of Stinespring's (1955) theorem: if $\CQ:\CA\raw \CB(\CK)$ is a completely positive map, there exists a \Hs\ ${\mathcal H}$ carrying a \rep\ $\pi$ of $C_0(M)$
and an isometric injection $\CK\hraw{\mathcal H}$, such that $\CQ(f)=[\CK]\pi(f)[\CK]$ for all $f\in C_0(M)$. In terms of $\CQ(C_0(M))$, the covariance condition
\eqref{gimpr} becomes $U(x)\CQ(f)U(x)\inv=\CQ(L_xf)$, just like
\eqref{Gcov}.
\subsection{Deformation quantization}\label{DQsection}
So far, we have used the word `quantization' in a heuristic way, basing our account on historical continuity rather than on axiomatic foundations. In this subsection and the next we set the record straight by introducing two alternative ways of looking at quantization in an axiomatic way. We start with the approach that historically came last, but which conceptually is closer to the material just discussed. This is {\it deformation quantization}, originating in the work of Berezin (1974, 1975a, 1975b), Vey (1975), and Bayen et al.\ (1977). We here follow the \ca ic approach to deformation quantization proposed by Rieffel (1989a, 1994), since it is not only mathematically transparent and rigorous, but also reasonable close to physical practice.\footnote{See also Landsman (1998) for an extensive discussion of the \ca ic approach to deformation quantization. In other approaches to deformation quantization, such as the theory of star products, $\hbar$ is a formal parameter rather than a real number. In particular, the meaning of the limit $\hbar\raw 0$ is obscure.} Due to the mathematical language used, this method of course naturally fits into the general $C^*$-algebraic approach to quantum physics.
The key idea of deformation quantization is that quantization should be defined through the property of having the correct classical limit. Consequently, Planck's ``constant" $\hbar$ is treated as a variable, so that for each of its values one should have a quantum theory. The key requirement is that this family of quantum theories converges to the underlying classical theory as $\hbar\raw 0$.\footnote{Cf.\ the preamble to Section \ref{S5} for further comments on this limit.} The mathematical implementation of this idea is quite beautiful, in that the classical algebra of observables is ``glued" to the family of quantum algebras of observables in such a way that the classical theory literally forms the boundary of the space containing the pertinent quantum theories (one for each value of $\hbar>0$). Technically, this is done through the concept of a {\it continuous field of
\ca s}.\footnote{See Dixmier (1977), Fell \&\ Doran (1988), and Kirchberg \&\ Wassermann (1995) for three different approaches to the same concept. Our definition follows the latter; replacing $I$ by an arbitrary locally compact Hausdorff space one finds the general definition.} What follows may sound unnecessarily technical, but the last 15 years have indicated that this yields exactly the right definition of quantization.
Let $I\subset \R$ be the set in which $\hbar$ takes values; one usually has $I=[0,1]$, but when the phase space is compact, $\hbar$ often takes values in a countable subset of $(0,1]$.\footnote{Cf.\ Landsman (1998) and footnote \ref{FNPER}, but also see Rieffel (1989a) for the example of the noncommutative torus, where one quantizes a compact phase space for each $\hbar\in(0,1]$.} The same situation occurs in the theory of infinite systems; see Section \ref{S6}. In any case, {\it $I$ should contain zero as an accumulation point.} A continuous field of $C^*$-algebras over $I$, then, consists of a
\ca\ $\CA$, a collection of \ca s $\{\CA_{\hbar}\}_{{\hbar}\in I}$, and a surjective morphism $\phv_{\hbar}:\CA\raw\CA_{\hbar}$ for each $\hbar\in I$ , such that: \begin{enumerate}
\item The function ${\hbar}\mapsto \|\phv_{\hbar}(A)\|_{\hbar}$ is in $C_0(I)$ for all $A\in\CA$;\footnote{Here $\|\cdot\|_{\hbar}$ is the norm in the \ca\ $\CA_{\hbar}$.}
\item The norm of any $A\in\CA$ is $\| A\|=\sup_{{\hbar}\in I}\|\phv_{\hbar}(A)\|$; \item For any $f\in C_0(I)$ and $A\in\CA$ there is an element $fA\in\CA$ for which $\phv_{\hbar}(fA)=f({\hbar})\phv_{\hbar}(A)$ for all ${\hbar}\in I$. \end{enumerate}
The idea is that the family $(\CA_{\hbar})_{\hbar\in I}$ of \ca s is glued together by specifying a topology on the bundle $\coprod_{\hbar\in [0,1]}\CA_{\hbar}$ (disjoint union). However, this topology is in fact defined rather indirectly, via the specification of the space of continuous sections of the bundle.\footnote{This is reminiscent of the Gelfand--Naimark theorem for commutative \ca s, which specifies the topology on a locally compact Hausdorff space $X$ via the \ca\ $C_0(X)$. Similarly, in the theory of (locally trivial) vector bundles the Serre--Swan theorem allows one to reconstruct the topology on a vector bundle $E\stackrel{\pi}{\raw} X$ from the space $\Gm_0(E)$ of continuous sections of $E$, seen as a (finitely generated projective) $C_0(X)$-module. See, for example, Gracia-Bond\'{\i}a, V\'{a}rilly, \&\ Figueroa (2001). The third condition in our definition of a continuous field of \ca s makes $\CA$ a $C_0(I)$-module in the precise sense that there exits a nondegenerate morphism from $C_0(I)$ to the center of the multiplier of $\CA$. This property may also replace our condition 3.} Namely, a {\it continuous section} of the field is {\it by definition} an element $\{A_{\hbar}\}_{{\hbar}\in I}$ of $\prod_{{\hbar}\in I}\CA_{\hbar}$ (equivalently, a map $\hbar\mapsto A_{\hbar}$ where $A_{\hbar}\in \CA_{\hbar}$)
for which there is an $A\in \CA$ such that $A_{\hbar}=\phv_{\hbar}(A)$ for all ${\hbar}\in I$. It follows that the \ca\ $\CA$ may actually be identified with the space of continuous sections of the field: if we do so, the morphism $\phv_{\hbar}$ is just the evaluation
map at $\hbar$.\footnote{The structure of $\CA$ as a \ca\ corresponds to the operations of pointwise scalar multiplication, addition, adjointing, and operator multiplication on sections.}
Physically, $\CA_0$ is the commutative algebra of observables of the underlying classical system, and for each $\hbar>0$ the noncommutative \ca\ $\CA_{\hbar}$ is supposed to be the algebra of observables of the corresponding quantum system at value $\hbar$ of Planck's constant. The algebra $\CA_0$, then, is of the form $C_0(M)$, where $M$ is the phase space defining the classical theory. A phase space has more structure than an arbitrary topological space; it is a manifold on which a Poisson bracket $\{\, ,\,\}$ can be defined. For example, on $M=\R^{2n}$ one has the familiar expression \begin{equation} \{f,g\}=\sum_j\frac{\partial f}{\partial p_j}\frac{\partial g}{\partial q^j}-\frac{\partial f}{\partial q^j}\frac{\partial g}{\partial p_j}. \label{PBRN}\end{equation}
Technically, $M$ is taken to be a {\it Poisson manifold}. This is a manifold equipped with a Lie bracket $\{\, ,\,\}$ on $\cin(M)$ with the property that for each $f\in\cin(M)$ the map $g\mapsto \{f,g\}$ defines a derivation of the commutative algebra structure of $\cin(M)$ given by pointwise multiplication. Hence this map is given by a vector field $\xi_f$, called the {\it Hamiltonian vector field} of $f$ (i.e.\ one has $\xi_fg=\{f,g\}$).
{\it Symplectic manifolds} are special instances of Poisson manifolds, characterized by the property that the Hamiltonian vector fields exhaust the tangent bundle. A Poisson manifold is foliated by its {\it symplectic leaves}: a given symplectic leaf $L$ is characterized by the property that at each $x\in L$ the tangent space $T_xL\subset T_xM$ is spanned by the collection of all Hamiltonian vector fields at $x$. Consequently, the flow of any Hamiltonian vector field on $M$ through a given point lies in its entirety within the symplectic leaf containing that point. The simplest example of a Poisson manifold is $M=\R^{2n}$ with Poisson bracket \eqref{PBRN}; this manifold is even symplectic.\footnote{See Marsden \&\ Ratiu (1994) for a mechanics-oriented introduction to Poisson manifolds; also cf.\ Landsman (1998) or Butterfield (2005) for the basic facts. A classical mathematical paper on Poisson manifolds is Weinstein (1983). \label{PMFN}}
After this preparation, our basic definition is this:\footnote{Here $\cci(M)$ stands for the space of smooth functions on $M$ with compact support; this is a norm-dense subalgebra of $\CA_0=C_0(M)$. The question whether the maps $\CQ_{\hbar}$ can be extended from $\cci(M)$ to $C_0(M)$ has to be answered on a case by case basis. Upon such an extension, if it exists, condition \eqref{Dirac} will lose its meaning, since the Poisson bracket $\{f,g\}$ is not defined for all $f,g\in C_0(M)$.}
\begin{quote} {\it A deformation quantization of a phase space $M$ consists of a continuous field of \ca s $(\CA_{\hbar})_{\hbar\in [0,1]}$ (with $\CA_0=C_0(M)$), along with a family of self-adjoint\footnote{I.e. $\CQ_{\hbar}(\ovl{f})=\CQ_{\hbar}(f)^*$.} linear maps $\CQ_{\hbar}:\cci(M)\raw\CA_{\hbar}$, $\hbar\in(0,1]$, such that:}\end{quote} \begin{enumerate} \item For each $f\in\cci(M)$ the map defined by $0\mapsto f$ and $\hbar\mapsto\CQ_{\hbar}(f)$ ($\hbar\neq 0$) is a continuous section of the given continuous field;\footnote{Equivalently, one could extend the family $(\CQ_{\hbar})_{\hbar\in(0,1]}$ to $\hbar=0$ by $\CQ_0=\mbox{\rm id}$, and state that $\hbar\mapsto\CQ_{\hbar}(f)$ is a continuous section. Also, one could replace this family of maps by a single map $\CQ:\cci(M)\raw\CA$ and {\it define} $\CQ_{\hbar}=\phv_{\hbar}\circ \CQ:\cci(M)\raw \CA_{\hbar}$.} \item
For all $f,g\in \cci(M)$ one has \begin{equation} \lim_{\hbar\rightarrow 0}
\left\|\frac{i}{\hbar}[\CQ_{\hbar}(f),\CQ_{\hbar}(g)]-\CQ_{\hbar}(\{f,g\})\right\|_{\hbar} =0. \label{Dirac} \end{equation}\end{enumerate}
Obvious continuity properties one might like to impose, such as \begin{equation}\lim_{\hbar\rightarrow 0} \|\CQ_{\hbar}(f)\CQ_{\hbar}(g)-\CQ_{\hbar}(fg)\|=0,\end{equation} or \begin{equation}\lim_{\hbar\rightarrow 0} \|\CQ_{\hbar}(f)\|=\| f\|_{\infty},\label{normcont} \end{equation}
turn out to be an automatic consequence of the definitions.\footnote{That they are automatic should not distract from the fact that especially \eqref{normcont} is a beautiful connection between classical and \qm.
See footnote \ref{CSQP} for the meaning of $\| f\|_{\infty}$.} Condition \eqref{Dirac}, however, transcends the \ca ic setting, and is the key ingredient in proving (among other things) that the quantum dynamics converges to the classical dynamics;\footnote{This insight is often attributed to Dirac (1930), who was the first to recognize the analogy between the commutator in \qm\ and the Poisson bracket in classical mechanics.}
see Section \ref{S5}. The map $\CQ_{\hbar}$ is the quantization map at value $\hbar$ of Planck's constant; we feel it is the most precise formulation of Heisenberg's original {\it Umdeutung} of classical observables known to date. It has the same interpretation as the heuristic symbol $\CQ_{\hbar}$ used so far: the operator $\CQ_{\hbar}(f)$ is the quantum-mechanical observable whose classical counterpart is $f$.
This has turned out to be an fruitful definition of quantization, firstly because most well-understood examples of quantization fit into it (Rieffel, 1994; Landsman, 1998), and secondly because it has suggested various fascinating new ones (Rieffel, 1989a; Natsume\&\ Nest, 1999; ÊNatsume, Nest, \&\ Ingo, 2003). Restricting ourselves to the former, we note, for example, that \eqref{b2} with \eqref{pqcohst} defines a deformation quantization of the phase space $\R^{2n}$ (with standard Poisson bracket) if one takes $\CA_{\hbar}$ to be the \ca\ of compact operators on the \Hs\ $L^2(\R^n)$. This is called the {\it Berezin quantization} of $\R^{2n}$ (as a phase space);\footnote{ In the literature, Berezin quantization on $\R^{2n}$ is often called anti-Wick quantization (or ordering), whereas on compact complex manifolds it is sometimes called Toeplitz or Berezin--Toeplitz quantization. Coherent states based on other phase spaces often define deformation quantizations as well; see Landsman, 1998.}
explicitly, for $\Phi\in L^2(\R^n)$ one has
\begin{equation} \qb(f)\Phi(x)=\int_{\R^{2n}} \frac{d^np d^nqd^ny}{(2\pi\hbar)^n}\, f(p,q) \ovl{\Psi_{(p,q)}^{\hbar}(y)} \Phi(y) \Psi_{(p,q)}^{\hbar}(x) . \label{qbttsrex} \end{equation} This quantization has the distinguishing feature of positivity,\footnote{Cf.\ footnote \ref{BQP}. As a consequence, \eqref{qbttsrex} is valid not only for $f\in \cci(\R^{2n})$, but even for all $f\in L^{\infty}(\R^{2n})$, and the extension of $\qb$ from $\cci(\R^{2n})$ to $L^{\infty}(\R^{2n})$ is continuous. } a property not shared by its more famous sister called {\it Weyl quantization}.\footnote{The original reference is Weyl (1931). See, for example, Dubin, Hennings, \&\ Smith (2000) and Esposito, Marmo, \&\ Sudarshan (2004) for a modern physics-oriented yet mathematically rigorous treatment. See also Rieffel (1994) and Landsman (1998) for a discussion from the perspective of deformation quantization.} The latter is a deformation quantization of $\R^{2n}$ as well, having the same continuous field of \ca s, but differing from Berezin quantization in its quantization map \begin{equation} \qw(f)\Phi(x)=\int_{\R^{2n}}
\frac{d^npd^nq}{(2\pi\hbar)^n}\, e^{ip(x-q)/\hbar}f\left(p,\half(x+q)
\right)\Phi(q). \label{defweylq} \end{equation}
Although it lacks good positivity and hence continuity properties,\footnote{Nonetheless, Weyl quantization may be extended from $\cci(\R^{2n})$ to much larger function spaces using techniques from the theory of distributions (leaving the \Hs\ setting typical of \qm). The classical treatment is in H\"{o}rmander (1979, 1985a).}
Weyl quantization enjoys better symmetry properties than Berezin quantization.\footnote{\label{GCQQ} Weyl quantization is covariant under the affine symplectic group $\mathrm{Sp}(n,\R)\ltimes \R^{2n}$, whereas Berezin quantization is merely covariant under its subgroup $\mathrm{O}(2n)\ltimes \R^{2n}$.} Despite these differences, which illustrate the lack of uniqueness of concrete quantization procedures, Weyl and Berezin quantization both reproduce Schr\"{o}dinger's position and momentum operators \eqref{SOP}.\footnote{This requires a formal extension of the maps $\CQ_{\hbar}^W$ and $\CQ_{\hbar}^B$ to unbounded functions on $M$ like $p_j$ and $q^j$.} Furthermore, if $f\in L^1(\R^{2n})$, then $\qb(f)$ and $\qw(f)$ are trace class, with \begin{equation} \Tr \qb(f)=\Tr \qw(f)=\int_{\R^{2n}}
\frac{d^npd^nq}{(2\pi\hbar)^n}\, f(p,q).\end{equation}
Weyl and Berezin quantization are related by
\begin{equation}
\qb(f)=\qw(e^{\frac{\hbar}{4}\Delta_{2n}}f),
\end{equation}
where $\Delta_{2n}=\sum_{j=1}^n (\partial^2/\partial p_j^2 + \partial^2/\partial (q^j)^2)$, from which it may be shown that Weyl and Berezin quantization
are {\it asymptotically equal} in the sense that for any $f\in \cci(\R^{2n})$ one has
\begin{equation} \lim_{\hbar\rightarrow 0} \|\CQ_{\hbar}^B(f)-\CQ_{\hbar}^W(f)\|=0. \label{WBEQ}\end{equation}
Mackey's approach to quantization also finds its natural home in the setting of deformation quantization. Let a Lie group $G$ act on a manifold $Q$, interpreted as a {\it configuration space}, as in Subsection \ref{Mackey}. It turns out that the corresponding classical {\it phase space} is the manifold $\g^*\times Q$, equipped with the so-called {\it semidirect product Poisson structure} (Marsden, Ra\c tiu \&\ Weinstein, 1984; Krishnaprasad \&\ Marsden, 1987). Relative to a basis $(T_a)$ of the Lie algebra $\g$ of $G$ with structure constants $C_{ab}^c$ (i.e.\ $[T_a,T_b]=\sum_c C_{ab}^cT_c$), the Poisson bracket in question is given by \begin{equation} \{f,g\} = \sum_a\left(\xi^M_a f \frac{\partial g}{\partial\theta_a} - \frac{\partial f}{\partial\theta_a}\xi^M_a g\right)-\sum_{a,b,c} C_{ab}^c \theta_c \frac{\partial f}{\partial\theta_a} \frac{\partial g}{\partial\theta_b},\end{equation} where $\xi_a^M=\xi_{T_a}^M$. To illustrate the meaning of this lengthy expression, we consider a few special cases. First, take $f=X\in\g$ and $g=Y\in\g$ (seen as linear functions on the dual $\g^*$). This yields \begin{equation} \{X,Y\}=-[X,Y]. \label{PB1}\end{equation} Subsequently, assume that $g$ depends on position $q$ alone. This leads to \begin{equation} \{X,g\}=-\xi^M_X g. \label{PB2} \end{equation} Finally, assume that
$f=f_1$ and $g=f_2$ depend on $q$ only; this clearly gives \begin{equation} \{f_1,f_2\}=0. \label{PB3}\end{equation}
The two simplest physically relevant examples, already considered at the quantum level in Subsection \ref{Mackey}, are as follows. First, take $G=\R^n$ (as a Lie group) and $Q=\R^n$ (as a manifold), with $G$ acting on $Q$ by translation. Eqs.\ \eqref{PB1} - \eqref{PB3} then yield the Poisson brackets $\{p_j,p_k\}=0$, $\{p_j,q^k\}=\dl_j^k$, and $\{q^j,q^k\}=0$, showing that in this case $M=\g^*\x Q=\R^{2n}$ is the standard phase space of a particle moving in $\R^n$; cf.\ \eqref{PBRN}. Second, the case $G=E(3)$ and $Q=\R^3$ yields a phase space $M=\R^3\x\R^6$, where $\R^6$ is the phase space of a spinless particle just considered, and $\R^3$ is an additional internal space containing spin as a classical degree of freedom. Indeed, beyond the Poisson brackets on $\R^6$ just described, \eqref{PB1} - \eqref{PB3} give rise to the additional Poisson brackets $\{J_i,J_j\}=\ep_{ijk}J_k$, $\{J_i,p_j\}=\ep_{ijk}p_k$, and $\{J_i,q^j\}=\ep_{ijk}q^k$.\footnote{These are the classical counterparts of the
commutation relations for angular momentum written down in footnote \ref{AMCCR}.}
The analogy between \eqref{PB1}, \eqref{PB2}, \eqref{PB3} on the one hand, and \eqref{Gccr}, \eqref{Gccr2}, \eqref{Gccr3}, respectively, on the other, is no accident: the Poisson brackets in question {\it are} the classical counterpart of the commutation relations just referred to. This observation is made precise by the fundamental theorem relating Mackey's systems of imprimitivity to deformation quantization (Landsman, 1993, 1998): one can equip the family of \ca s \begin{eqnarray} \CA_0 &=& C_0(\g^*\times Q); \nn\\ \CA_{\hbar} &=& C^*(G,Q), \end{eqnarray}
where $C^*(G,Q)$ is the transformation grouo \ca\ defined by the given $G$-action on $Q$ (cf.\ the end of Subsection \ref{Mackey}), with the structure of a continuous field, and one can define quantization maps $\CQ_{\hbar}:\cci(\g^*\times Q)\raw C^*(G,Q)$ so as to obtain a deformation quantization of the phase space $\g^*\x Q$. It turns out that for special functions of the type $X,Y\in\g$, and $f=f(q)$ just considered, the equality
\begin{equation} \frac{i}{\hbar}[\CQ_{\hbar}(f),\CQ_{\hbar}(g)]-\CQ_{\hbar}(\{f,g\}) =0 \label{Diracexact} \end{equation}
holds exactly (and not merely asymptotically for $\hbar\raw 0$, as required in the fundamental axiom \eqref{Dirac} for deformation quantization).
This result clarifies the status of Mackey's quantization by systems of imprimitivity. The classical theory underlying the relations \eqref{impr} is not the usual phase space $T^*Q$ of a structureless particle moving on $Q$, but $M=\g^*\times Q$. For simplicity we restrict ourselves to the transitive case $Q=G/H$ (with canonical left $G$-action). Then $M$ coincides with $T^*Q$ only when $H=\{e\}$ and hence $Q=G$;\footnote{For a Lie group $G$ one has $T^*G\cong \g^*\x G$.} in general, the phase space $\g^*\times (G/H)$ is {\it locally} of the form $T^*(G/H)\x\mathfrak{h}^*$ (where $\mathfrak{h}^*$ is the dual of the Lie algebra of $H$). The internal degree of freedom described by $\mathfrak{h}^*$ is a generalization of classical spin, which, as we have seen, emerges in the case $G=E(3)$ and $H=SO(3)$. All this is merely a special case of a vast class of deformation quantizations described by Lie groupoids; see Bellisard \&\ Vittot (1990), Landsman (1998, 1999b, 2005b) and Landsman \&\ Ramazan (2001).\footnote{A similar analysis can be applied to Isham's (1984) quantization scheme mentioned in footnote \ref{Isham1}. The unitary irreducible representation s of the canonical group $G_c$ stand in bijective correspondence with the nondegenerate \rep s of the group \ca\ $C^*(G_c)$ (Pedersen, 1979), which is a deformation quantization of
the Poisson manifold $\mathfrak{g}^*_c$ (i.e.\ the dual of the Lie algebra of $G_c$). This Poisson manifold contains the coadjoint orbits of $G_c$ as ``irreducible" classical phase spaces, of which only one is the cotangent bundle $T^*(G/H)$ one initially thought one was quantizing (see Landsman (1998) for the classification of the coadjoint orbits of semidirect products). All other orbits are mere lumber that one should avoid. See also Robson (1996). If one is ready for groupoids, there is no need for the canonical group approach. \label{Isham2}} \subsection{Geometric quantization}\label{GQsection} Because of its use of abstract \ca s, deformation quantization is a fairly sophisticated and recent technique. Historically, it was preceded by a more concrete and traditional approach called {\it geometric quantization}.\footnote{\label{GQrefs} Geometric quantization was independently introduced by Kostant (1970) and Souriau (1969). Major later treatments on the basis of the original formalism are Guillemin \&\ Sternberg (1977), \'{S}niatycki (1980), Kirillov (1990), Woodhouse (1992), Puta (1993), Chernoff (1995), Kirillov (2004), and Ali \& Englis (2004). The modern era (based on the use of Dirac operators and $K$-theory) was initiated by unpublished remarks by Bott in the early 1990s; see Vergne (1994) and Guillemin, Ginzburg \& Karshon (2002).
The postmodern (i.e.\ functorial) epoch was launched in Landsman (2005a).}
Here the goal is to firstly ``quantize" a phase space $M$ by a concretely given Hilbert space ${\mathcal H}(M)$, and secondly to map the classical observables (i.e.\ the real-valued smooth functions on $M$) into self-adjoint operators on ${\mathcal H}$ (which after all play the role of observables in von Neumann's formalism of \qm).\footnote{In geometric quantization\ phase spaces are always seen as symplectic manifolds (with the sole exception of Vaisman, 1991); the reason why it is unnatural to start with the more general class of Poisson manifolds will become clear in the next subsection.} In principle, this program should align geometric quantization\ much better with the fundamental role unbounded self-adjoint operators play in \qm\ than deformation quantization, but in practice geometric quantization\ continues to be plagued by problems.\footnote{\label{GQP1} Apart from rather technical issues concerning the domains and self-adjointness properties of the operators defined by geometric quantization, the main point is that the various mathematical choices one has to make in the geometric quantization\ procedure
cannot all be justified by physical arguments, although the physical properties of the theory depend on these choices. (The notion of a polarization is the principal case in point; see also footnote \ref{GQP2} below.) Furthermore, as we shall see, one cannot quantize sufficiently many functions in standard geometric quantization. Our functorial approach to geometric quantization\ in Subsection \ref{EFQ} was partly invented to alleviate these problems.}
However, it would be wrong to see deformation quantization and geometric quantization\ as {\it competitors}; as we shall see in the next subsection, they are natural {\it allies}, forming ``complementary" parts of a conjectural quantization functor.
In fact, in our opinion geometric quantization\ is best compared and contrasted with phase space quantization in its concrete formulation of Subsection \ref{PSQ} (i.e.\ before its \ca ic abstraction and subsequent absorption into deformation quantization as indicated in Subsection \ref{DQsection}).\footnote{See also Tuynman (1987).} For geometric quantization\ equally well starts with the \Hs\ $L^2(M)$,\footnote{Defined with respect to the Liouville measure times a suitable factor $c_{\hbar}$, as in \eqref{qhnorm} etc.; in geometric quantization\ this factor is not very important, as it is unusual to study the limit $\hbar\raw 0$. For $M=\R^{2n}$ the measure on $M$ with respect to which $L^2(M)$ is defined is $d^npd^nq/(2\pi\hbar)^n$. } and subsequently attempts to construct ${\mathcal H}(M)$ from it, though typically in a different way from \eqref{pH}.
Before doing so, however, the geometric quantization\ procedure first tries to define a linear map $\CQ^{pre}_{\hbar}$ from $\cin(M)$ to the class of (generally unbounded) operators on $L^2(M)$ that formally satisfies
\begin{equation} \frac{i}{\hbar}[\CQ^{pre}_{\hbar}(f),\CQ^{pre}_{\hbar}(g)]-\CQ^{pre}_{\hbar}(\{f,g\}) =0, \label{Diracexact2}\end{equation} i.e.\ \eqref{Diracexact} with $\CQ=\CQ^{pre}_{\hbar}$, as well as the nondegeneracy property \begin{equation} \CQ^{pre}_{\hbar}(\ch_M)=1, \label{ndGQ}\end{equation} where $\ch_M$ is the function on $M$ that is identically equal to 1, and the 1 on the right-hand side is the unit operator on $L^2(M)$. Such a map is called {\it prequantization}.\footnote{The idea of prequantization predates geometric quantization; see van Hove (1951) and Segal (1960).} For $M=\R^{2n}$ (equipped with its standard Poisson bracket \eqref{PBRN}), a prequantization map is given (on $\Phi\in L^2(M)$) by \begin{equation} \CQ^{pre}_{\hbar}(f)\Phi=-i\hbar \{f,\Phi\} +\left(f-\sum_j p_j \frac{\partial f}{\partial p_j}\right)\Phi.
\label{PQRN}\end{equation} This expression is initially defined for $\Phi\in\cci(M)\subset L^2(M)$, on which domain $\CQ^{pre}_{\hbar}(f)$ is symmetric when $f$ is real-valued;\footnote{An operator $A$ defined on a dense subspace $\mathcal{D}\subset{\mathcal H}$ of a \Hs\ ${\mathcal H}$ is called {\it symmetric} when $(A\Psi,\Phi)=(\Psi,A\Phi)$ for all $\Psi,\Phi\in\mathcal{D}$.}
note that the operator in question is unbounded even when $f$ is bounded.\footnote{As mentioned, self-adjointness is a problem in geometric quantization; we will not address this issue here. Berezin quantization is much better behaved than geometric quantization\ in this respect, since it maps bounded functions into bounded operators.} This looks complicated; the simpler expression $\CQ_{\hbar}(f)\Phi=-i\hbar \{f,\Phi\}$, however, would satisfy \eqref{Diracexact} but not \eqref{ndGQ}, and the goal of the second term in \eqref{PQRN} is to satisfy the latter condition while preserving the former.\footnote{One may criticize the geometric quantization\ procedure for emphasizing \eqref{Diracexact2} against its equally natural counterpart $\CQ(fg)=\CQ(f)\CQ(g)$, which fails to be satisified by $\CQ^{pre}_{\hbar}$ (and indeed by any known quantization procedure, except the silly $\CQ(f)=f$ (as a multiplication operator on $L^2(M)$).} For example, one has \begin{eqnarray} \CQ^{pre}_{\hbar}(q^k) & =& q^k+i\hbar \frac{\partial }{\partial p_k} \nn \\ \CQ^{pre}_{\hbar}(p_j) & =& -i\hbar \frac{\partial}{\partial q^j}.\label{SOP2} \end{eqnarray}
For general phase spaces $M$ one may construct a map $\CQ^{pre}_{\hbar}$ that satisfies \eqref{Diracexact2} and \eqref{ndGQ} when $M$ is ``prequantizable"; a full explanation of this notion requires some differential geometry.\footnote{\label{GQF}A symplectic manifold $(M,\om)$ is called {\it prequantizable} at some fixed value of $\hbar$ when it admits a complex line bundle $L\raw M$ (called the {\it prequantization line bundle}) with connection $\nabla$ such that $F=-i\om/\hbar$ (where $F$ is the curvature of the connection, defined by $F(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$). This is the case iff $[\om]/2\pi\hbar\in H^2(M,\Z)$, where $[\om]$ is the de Rham cohomology class of the symplectic form. If so, prequantization is defined by the formula $\CQ^{pre}_{\hbar}(f)=-i\hbar\nabla_{\xi_f} +f$, where $\xi_f$ is the Hamiltonian vector field of $f$ (see Subsection \ref{DQsection}). This expression is defined and symmetric on the space $\cci(M,L)\subset L^2(M)$ of compactly supported smooth sections of $L$, and is easily checked to satisfy \eqref{Diracexact2} and \eqref{ndGQ}. To obtain \eqref{PQRN} as a special case, note that for $M=\R^{2n}$ with the canonical symplectic form $\om =\sum_k dp_k\wedge dq^k$ one has $[\om]=0$, so that $L$ is the trivial bundle $L= \R^{2n}\x\C$. The connection $\nabla=d+A$ with $A= -\frac{i}{\hbar}\sum_k p_kdq^k$ satisfies $F=-i\om/\hbar$, and this eventually yields \eqref{PQRN}.} Assuming this to be the case, then for one thing prequantization is a very effective tool in constructing unitary group \rep s
of the kind that are interesting for physics. Namely, suppose a Lie group $G$ acts on the phase space $M$ in ``canonical" fashion. This means that there exists a map $\mu:M\raw\g^*$, called the {\it momentum map}, such that $\xi_{\mu_X}=\xi^M_X$ for each $X\in\g$,\footnote{ Here $\mu_X\in\cin(M)$ is defined by $\mu_X(x)=\langle \mu(x),X\rangle$, and $\xi^M_X$ is the vector field on $M$ defined by the $G$-action (cf.\ footnote \ref{VFFN}). Hence this condition means that $\{\mu_X,f\}(y)=d/dt_{|t=0} [f(\exp(-tX)y)]$ for all $f\in\cin(M)$ and all $y\in M$.} and in addition $\{\mu_X,\mu_Y\}=\mu_{[X,Y]}$. See Abraham \& Marsden (1985), Marsden \&\ Ratiu (1994), Landsman (1998), Butterfield (2005), etc. On then obtains a \rep\ $\pi$ of the Lie algebra $\g$ of $G$ by skew-symmetric unbounded operators on $L^2(M)$ through \begin{equation} \pi(X)=-i\hbar \CQ^{pre}_{\hbar}(\mu_X), \label{LArep}\end{equation} which often exponentiates to a unitary \rep\ of $G$.\footnote{An operator $A$ defined on a dense subspace $\mathcal{D}\subset{\mathcal H}$ of a \Hs\ ${\mathcal H}$ is called {\it skew-symmetric} when $(A\Psi,\Phi)=-(\Psi,A\Phi)$ for all $\Psi,\Phi\in\mathcal{D}$. If one has a unitary \rep\ $U$ of a Lie group $G$ on ${\mathcal H}$, then the derived \rep\ $dU$ of the Lie algebra $\g$ (see footnote \ref{dUFN}) consists of skew-symmetric operators, making one hopeful that a given \rep\ of $\g$ by skew-symmetric operators can be integrated (or exponentiated) to a unitary \rep\ of $G$. See Barut \&\ Ra\c{c}ka (1977) or J\o rgensen \&\ Moore (1984) and references therein.}
As the name suggests, prequantization is not yet quantization. For example, the prequantization of $M=\R^{2n}$ does not reproduce Schr\"{o}dinger's wave mechanics: the operators \eqref{SOP2} are not unitarily equivalent to \eqref{SOP}. In fact, as a carrier of the \rep\ \eqref{SOP2} of the canonical commutation relations \eqref{ccr}, the \Hs\ $L^2(\R^{2n})$ contains $L^2(\R^n)$ (carrying the \rep\ \eqref{SOP}) with infinite multiplicity (Ali \&\ Emch, 1986). This situation is often expressed by the statement that ``prequantization is reducible" or that the prequantization \Hs\ $L^2(M)$ is `too large', but both claims are misleading: $L^2(M)$ is actually {\it ir}reducible under the action of $\CQ^{pre}_{\hbar}(\cin(M))$ (Tuynman, 1998), and saying that for example $L^2(\R^n)$ is ``larger" than $L^2(\R^n)$ is unmathematical in view of the unitary isomorphism of these \Hs s. What is true is that in typical examples $L^2(M)$ is generically reducible under the action of some Lie algebra where one would like it to be irreducible. This applies, for example, to \eqref{SOP}, which defines a \rep\ of the Lie algebra of the Heisenberg group. More generally, in the case where a phase space $M$ carries a transitive action of a Lie group $G$, so that one would expect the quantization of this $G$-action by unitary operators on a \Hs\ to be irreducible, $L^2(M)$ is typically highly reducible under the \rep\ \eqref{LArep} of $\g$.\footnote{This can be made precise in the context of the so-called orbit method, cf.\ the books cited in footnote \ref{GQrefs}.}
Phase space quantization encounters this problem as well. Instead of the complicated expression \eqref{PQRN}, through \eqref{berezin} it simply ``phase space prequantizes"
$f\in\cin(M)$ on $L^2(M)$ by $f$ as a multiplication operator.\footnote{For unbounded $f$ this operator is defined on the set of all $\Phi\in L^2(M)$ for which $f\Phi\in L^2(M)$.} Under this action of $\cin(M)$ the \Hs\ $L^2(M)$ is of course highly reducible.\footnote{\label{Koopman} Namely, each (measurable) subset $E\subset M$ defines a projection $\ch_E$, and $\ch_E L^2(M)$ is stable under all multiplication operators $f$. One could actually decide not to be bothered by this problem and stop here, but then one is simply doing classical mechanics in a \Hs\ setting (Koopman, 1931). This formalism even turns out to be quite useful for ergodic theory (Reed \&\ Simon, 1972).} The identification of an appropriate subspace
\begin{equation} {\mathcal H}(M)=pL^2(M) \label{pl2}\end{equation}
of $L^2(M)$ (where $p$ is a projection) as the \Hs\ carrying the ``quantization" of $M$ (or rather of $\cin(M)$) may be seen as a solution to this reducibility problem,
for if the procedure is successful, the projection $p$ is chosen such that $pL^2(M)$ is irreducible under $p\cin(M)p$. Moreover, in this way practically any function on $M$ can be quantized, albeit at the expense of \eqref{Diracexact} (which, as we have seen, gets replaced by its asymptotic version \eqref{Dirac}). See Subsection \ref{SE} for a discussion of reducibility versus irreducibility of \rep s of algebras of observables in classical and quantum theory.
We restrict our treatment of geometric quantization to situations where it adopts the same strategy as above, in assuming that the final \Hs\ has the form \eqref{pl2} as well.\footnote{\label{GQP2} Geometric quantization has traditionally been based on the notion of a polarization (cf.\ the references in footnote \ref{GQrefs}). This device produces a final \Hs\ ${\mathcal H}(M)$ which may not be a subspace of $L^2(M)$, except in the so-called (anti-) holomorphic case.} But it crucially differs from phase space quantization in that its first step is \eqref{PQRN} (or its generalization to more general phase spaces) rather than just having $f\Phi$ on the right-hand side.\footnote{It also differs from phase space quantization in the ideology that the projection $p$ ought to be constructed solely from the geometry of $M$: hence the name `geometric quantization'.} Moreover, in geometric quantization\ one merely quantizes a {\it subspace} of the set $\cin(M)$ of classical observables, consisting of those functions that satisfy \begin{equation} [\CQ^{pre}_{\hbar}(f),p]=0. \label{GQf}\end{equation} If a function $f\in\cin(M)$ satisfies this condition, then one defines the ``geometric quantization" of $f$ as \begin{equation} \CQ_{\hbar}^G(f)=\CQ^{pre}_{\hbar}(f)\rst {\mathcal H}(M).\label{QGQ}\end{equation} This is well defined, since because of \eqref{GQf} the operator $\CQ^{pre}_{\hbar}(f)$ now maps $pL^2(M)$ onto itself. Hence \eqref{Diracexact} holds
for $\CQ_{\hbar}=\CQ_{\hbar}^G$ because of \eqref{Diracexact2}; in geometric quantization\ one simply refuses to quantize functions for which \eqref{Diracexact} is {\it not} valid.
Despite some impressive initial triumphs,\footnote{Such as the orbit method for nilpotent groups and the newly understood Borel--Weil method for compact groups, cf.\ Kirillov (2004) and most other books cited in footnote \ref{GQrefs}.} there is no general method that accomplishes the goals of geometric quantization\ with guaranteed success. Therefore, geometric quantization\ has remained something like a hacker's tool, whose applicability largely depends on the creativity of the user.
In any case, our familiar example $M=\R^{2n}$ is well understood, and we illustrate the general spirit of the method in its setting, simplifying further by taking $n=1$. It is convenient to replace the canonical coordinates $(p,q)$ on $M$ by $z=p+iq$ and $\ovl{z}=p-iq$, and the mathematical toolkit of geometric quantization\ makes it very natural to look at the space of solutions within $L^2(\R^{2})$ of the equations\footnote{Using the formalism explained in footnote \ref{GQF}, we replace the 1-form $A= -\frac{i}{\hbar}\sum_k p_kdq^k$ defining the connection $\nabla=d+A$ by the gauge-equivalent form $A=\frac{i}{2\hbar}(\sum_k q^kdp_k -\sum_k p_kdq^k)=-\frac{i}{\hbar} \sum_k p_kdq^k+\frac{i}{2\hbar}d (\sum_k p_kq^k)$, which has the same curvature. In terms of this new $A$, which in complex coordinates reads $A=\sum_k(z_kd\ovl{z}_k-\ovl{z}_kdz_k)/4\hbar$, eq.\ \eqref{holz} is just $\nabla_{\partial/\partial\ovl{z}}\Phi=0$. This is an example of the so-called holomorphic polarization in the formalism of geometric quantization.} \begin{equation} \left(\frac{\partial}{\partial\ovl{z}}+\frac{z}{4\hbar} \right)\Phi(z,\ovl{z})=0.\label{holz} \end{equation}
The general solution of these equations that lies in $L^2(\R^{2})=L^2(\C)$ is
\begin{equation} \Phi(z,\ovl{z})=e^{-|z|^2/4\hbar}f(z), \end{equation} where $f$ is a holomorphic function such that \begin{equation} \int_{\C} \frac{dz d
\ovl{z}}{2\pi \hbar i}e^{-|z|^2/2\hbar}|f(z)|^2<\infty. \label{BFC}\end{equation}
The projection $p$, then, is the projection onto the closed subspace of $L^2(\C)$ consisting of these solutions.\footnote{\label{BFFN}
The collection of all holomorphic functions on $\C$ satisfying \eqref{BFC} is a \Hs\ with respect to the inner product $(f,g)=(2\pi \hbar i)^{-1} \int_{\C} dz d\ovl{z}\, \exp(-|z|^2/2\hbar) \ovl{f(z)}g(z)$,
called the {\it Bargmann--Fock space} ${\mathcal H}_{BF}$. This space may be embedded in $L^2(\C)$ by $f(z) \mapsto \exp(-|z|^2/2\hbar)f(z)$, and the image
of this embedding is of course just $pL^2(\C)$.}
The \Hs\ $pL^2(\C)$ is unitarily equivalent to $L^2(\R)$ in a natural way (i.e.\ without the choice of a basis). The condition \eqref{GQf} boils down to $\partial^2 f(z,\ovl{z})/\partial \ovl{z}_i \partial \ovl{z}_j=0$; in particular, the coordinate functions $q$ and $p$ are quantizable. Transforming to $L^2(\R)$, one finds that the operators $\CQ_{\hbar}^G(q)$ and $\CQ_{\hbar}^G(p)$ coincide with Schr\"{o}dinger's expressions \eqref{SOP}. In particular, the Heisenberg group $H_1$, which acts with infinite multiplicity on $L^2(\C)$, acts irreducibly on $pL^2(\C)$. \subsection{Epilogue: functoriality of quantization}\label{EFQ} A very important aspect of quantization is its interplay with symmetries and constraints. Indeed, the fundamental theories describing Nature (viz.\ electrodynamics, Yang--Mills theory, general relativity, and possibly also string theory) are a priori formulated as constrained systems. The classical side of constraints and reduction is well understood,\footnote{See Gotay, Nester, \&\ Hinds (1978), Binz,
\'{S}niatycki and Fischer (1988), Marsden (1992), Marsden \& Ratiu (1994),
Landsman (1998), Butterfield (2005), and Belot (2005).} a large class of important examples being codified by the procedure of symplectic reduction. A special case of this is {\it Marsden--Weinstein reduction}: if a Lie group $G$ acts on a phase space $M$ in canonical fashion with momentum map $\mu:M\raw\g^*$ (cf.\ Subsection \ref{GQsection}), one may form another phase space $M/\hspace{-1mm}/G=\mu\inv(0)/G$.\footnote{Technically, $M$ has to be a symplectic manifold, and if $G$ acts properly and freely on $\mu\inv(0)$, then $M/\hspace{-1mm}/G$ is again a symplectic manifold.} Physically, in the case where $G$ is a gauge group and $M$ is the unconstrained phase space, $\mu\inv(0)$ is the constraint hypersurface (i.e.\ the subspace of $M$ on which the constraints defined by the gauge symmetry hold), and $M/\hspace{-1mm}/G$ is the true phase space of the system that only contains physical degrees of freedom.
Unfortunately, the correct way of dealing with constrained quantum systems remains a source of speculation and controversy:\footnote{Cf.\ Dirac (1964), Sundermeyer (1982), Gotay (1986), Duval et al. (1991), Govaerts (1991), Henneaux \&\ Teitelboim (1992), and Landsman (1998) for various perspectives on the quantization of constrained systems.} practically all rigorous results on quantization (like the ones discussed in the preceding subsections) concern unconstrained systems. Accordingly, one would like to quantize a constrained system by reducing the problem to the unconstrained case. This could be done provided the following scenario applies. One first quantizes the unconstrained phase space $M$ (supposedly the easiest part of the problem), ans subsequently imposes a quantum version of symplectic reduction. Finally, one proves by abstract means that the quantum theory thus constructed is equal to the theory defined by first reducing at the classical level and then quantizing the constrained classical phase space (usually an impossible task to perform in practice).
Tragically, sufficiently powerful theorems stating that ``quantization commutes with reduction" in this sense remain elusive.\footnote{\label{GSC} The so-called Guillemin--Sternberg conjecture (Guillemin \&\ Sternberg, 1982) - now a theorem (Meinrenken, 1998, Meinrenken \&\ Sjamaar, 1999) - merely deals with the case of Marsden--Weinstein reduction where $G$ and $M$ are compact. Mathematically impressive as the ``quantization commutes with reduction" theorem already is here, it is a far call from gauge theories, where the groups and spaces are not only noncompact but even infinite-dimensional.} So far, this has blocked, for example, a rigorous quantization of Yang--Mills theory in dimension 4; this is one of the Millenium Problems of the Clay Mathematical Institute, rewarded with 1 Million dollars.\footnote{See \texttt{http://www.claymath.org/millennium/}}
On a more spiritual note, the mathematician E. Nelson famously said that `First quantization is a mystery, but second quantization is a functor.' The functoriality of `second' quantization (a construction involving Fock spaces, see Reed \&\ Simon, 1975) being an almost trivial matter, the deep mathematical and conceptual problem lies in the possible functoriality of `first' quantization, which simply means quantization in the sense we have been discussing so far. This was initially taken to mean that canonical transformations $\al$ of the phase space $M$ should be `quantized' by unitary operators $U(\al)$ on ${\mathcal H}(M)$, in such a way $U(\al)\CQ_{\hbar}(f)U(\al)\inv=\CQ(L_{\al}f)$ (cf.\ \eqref{Gcov}). This is possible only in special circumstances, e.g., when $M=\R^{2n}$ and $\al$ is a linear symplectic map, and more generally when $M=G/H$ is homogeneous and $\al\in G$ (see the end of Subsection \ref{PSQ}).\footnote{ Canonical transformations {\it can} be quantized in approximate sense that becomes precise as $\hbar\raw 0$ by means of so-called Fourier integral operators; see H\"{o}rmander (1971, 1985b) and Duistermaat (1996).} Consequently, the functoriality of quantization is widely taken to be a dead end.\footnote{See Groenewold (1946), van Hove (1951), Gotay, Grundling, \&\ Tuynman (1996), and Gotay (1999).}
However, all no-go theorems establishing this conclusion start from wrong and naive categories, both on the classical and on the quantum side.\footnote{Typically, one takes the classical category to consist of symplectic manifolds as objects and symplectomorphisms as arrows, and the quantum category to have \ca s as objects and automorphisms as arrows.} It appears very likely that one may indeed make quantization functorial by a more sophisticated choice of categories, with the additional bonus that deformation quantization and geometric quantization\ become unified: the former is the object part of the quantization functor, whereas the latter (suitably reinterpreted) is the arrow part. Amazingly, on this formulation the statement that `quantization commutes with reduction' becomes a special case of the functoriality of quantization (Landsman, 2002, 2005a).
To explain the main idea, we return to the geometric quantization\ of $M=\R^2\cong\C$ explained in the preceding subsection. The identification of $pL^2(\C)$\footnote{Or the Bargmann--Fock space ${\mathcal H}_{BF}$, see footnote \ref{BFFN}.} as the correct \Hs\ of the problem may be understood in a completely different way, which paves the way for the powerful reformulation of the geometric quantization\ program that will eventually define the quantization functor. Namely, $\C$ supports a certain linear first-order differential operator $\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}$ that is entirely defined by its geometry as a phase space, called the {\it Dirac operator}.\footnote{\label{DSFN} Specifically, this is the so-called $\mathrm{Spin}^c$ Dirac operator defined by the complex structure of $\C$, {\it coupled to the prequantization line bundle}. See Guillemin, Ginzburg, \& Karshon (2002).} This operator is given by\footnote{Relative to the Dirac matrices $\gamma^1=\left( \begin{array}{cc} 0 & i \\ i & 0\end{array} \right)$ and $\gamma^2=\left( \begin{array}{cc} 0 & -1 \\ 1 & 0\end{array} \right)$.} \begin{equation} \setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}=2\left( \begin{array}{cc} 0& -\frac{\partial}{\partial z}+\frac{\ovl{z}}{4\hbar}\\ \frac{\partial}{\partial\ovl{z}} +\frac{z}{4\hbar} & 0 \end{array} \right),\end{equation} acting on $L^2(\C)\otimes\C^2$ (as a suitably defined unbounded operator). This operator has the generic form $$\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}=\left( \begin{array}{cc} 0&\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_-\\ \setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+& 0\end{array} \right).$$ The {\it index} of such an operator is given by \begin{equation} \mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss})=[\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+)]-[\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_-)], \label{index}\end{equation} where $[\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_{\pm})]$ stand for the (unitary) isomorphism class of $\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_{\pm})$ {\it seen as a \rep\ space of a suitable algebra of operators}.\footnote{\label{indexFN} The left-hand side of \eqref{index} should really be written as $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+)$, since $\mathrm{coker}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+)=\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+^*)$ and $\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+^*=\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_-$, but since the index is naturally associated to $\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}$ as a whole, we abuse notation in writing $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss})$ for $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+)$. The usual index of a linear map $L:V\raw W$ between finite-dimensional vector spaces is defined as $\mathrm{index}(L)=\dim(\ker(L))-\dim(\mathrm{coker}(L))$, where $\mathrm{coker}(L)=W/\ran(L)$. Elementary linear algebra yields $\mathrm{index}(L)=\dim(V)-\dim(W)$. This is surprising because it is independent of $L$, whereas $\dim(\ker(L))$ and $\dim(\mathrm{coker}(L))$ quite sensitively depend on it. For, example, take $V=W$ and $L=\varepsilon\cdot 1$. If $\varepsilon\neq 0$ then $\dim(\mathrm{ker}(\varepsilon\cdot 1))=\dim(\mathrm{coker}(\varepsilon\cdot 1))=0$, whereas for $\varepsilon=0$ one has $\dim(\ker(0))=\dim(\mathrm{coker}(0))=\dim(V)$! Similarly, the usual definiton of geometric quantization\ through \eqref{holz} etc.\ is unstable against perturbations of the underlying symplectic structure, whereas the refined definition through \eqref{index} is not. To pass to the latter from the above notion of an index, we first write $\mathrm{index}(L)=[\ker(L)]-[\mathrm{coker}(L)]$, where $[X]$ is the isomorphism class of a linear space $X$ as a $\C$-module. This expression is an element of $K_0(\C)$, and we recover the earlier index through the realization that the class $[X]$ is entirely determined by $\dim(X)$, along with and the corresponding isomorphism $K_0(\C)\cong\Z$. When a more complicated finite-dimensional \ca\ $\CA$ acts on $V$ and $W$ with the property that $\ker(L)$ and $\mathrm{coker}(L)$ are stable under the $\CA$-action, one may define $[\ker(L)]-[\mathrm{coker}(L)]$ and hence $\mathrm{index}(L)$ as an element of the so-called \ca ic K-theory group $K_0(\CA)$. Under certain technical conditions, this notion of an index may be generalized to infinite-dimensional \Hs s and \ca s; see Baum, Connes \&\ Higson (1994) and Blackadar (1998). The $K$-theoretic index is best understood when $\CA=C^*(G)$ is the group \ca\ of some locally compact group $G$. In the example $M=\R^2$ one might take $G$ to be the Heisenberg group $H_1$, so that $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss})\in K_0(C^*(H_1))$.} In the case at hand, one has
$\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+)=pL^2(\C)$ (cf.\ \eqref{holz} etc.) and $\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_-)=0$, \footnote{Since $(-\frac{\partial}{\partial z}+\frac{\ovl{z}}{4\hbar})\Phi=0$ implies $\Phi(z,\ovl{z})=\exp(|z^2|/4\hbar)f(\ovl{z})$, which lies in $L^2(\C)$ iff $f=0$.} where we regard $\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_+)$ as a \rep\ space of the Heisenberg group $H_1$. Consequently, the geometric quantization\ of the phase space $\C$ is given {\it modulo unitary equivalence} by $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss})$, seen as a ``formal difference" of \rep\ spaces of $H_1$.
This procedure may be generalized to arbitrary phase spaces $M$, where $\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}$ is a certain operator naturally defined by the phase space geometry of $M$ and the demands of quantization.\footnote{ Any symplectic manifold carries an almost complex structure compatible with the symplectic form, leading to a $\mathrm{Spin}^c$ Dirac operator as described in footnote \ref{DSFN}. See, again, Guillemin, Ginzburg, \& Karshon (2002). If $M=G/H$, or, more generally, if $M$ carries a canonical action of a Lie group $G$ with compact quotient $M/G$, then $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss})$ defines an element of $K_0(C^*(G))$. See footnote \ref{indexFN}. In complete generality, $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss})$ ought to be an element of $K_0(\CA)$, where $\CA$ is the \ca\ of observables of the quantum system.} This has turned out to be the most promising formulation of geometric quantization\ - at some cost.\footnote{On the benefit side, the invariance of the index under continuous deformations of $\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}$ seems to obviate the ambiguity of traditional quantization procedures with respect to different `operator orderings' not prescribed by the classical theory.}
For the original goal of quantizing a phase space by a \Hs\ has now been replaced by a much more abstract procedure, in which the result of quantization is a formal difference of certain isomorphism classes of \rep\ spaces of the quantum algebra of observables. To illustrate the degree of abstraction involved here, suppose we ignore the action of the observables
(such as position and momentum in the example just considered). In that case the isomorphism class $[{\mathcal H}]$ of a \Hs\ ${\mathcal H}$ is entirely characterized by its dimension $\dim({\mathcal H})$, so that (in case that $\ker(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss}_-)\neq 0$) quantization (in the guise of $\mathrm{index}(\setbox0=\hbox{$D$}D\hskip-\wd0\hbox to\wd0{\hss\sl/\/\hss})$) can even be a negative number! Have we gone mad?
Not quite. The above picture of geometric quantization\ is indeed quite irrelevant to physics, {\it unless it is supplemented by deformation quantization}. It is convenient to work at some fixed value of $\hbar$ in this context, so that deformation quantization merely associates some \ca\ $\CA(P)$ to a given phase space $P$.\footnote{Here $P$ is not necessarily symplectic; it may be a Poisson manifold, and to keep Poisson and symplectic manifolds apart we denote the former by $P$ from now on, preserving the notation $M$ for the latter.} Looking for a categorical interpretation of quantization, it is therefore natural to assume that the objects of the classical category $\GC$ are phase spaces $P$,\footnote{Strictly speaking, to be an object in this category a Poisson manifold $P$ must be {\it integrable}; see Landsman (2001).} whereas the objects of the quantum category $\mathfrak{Q}$ are \ca s.\footnote{For technical reasons involving $K$-theory these have to be separable.} The object part of the hypothetical quantization functor is to be deformation quantization, symbolically written as $P\mapsto\CQ(P)$.
Everything then fits together if geometric quantization\ is reinterpreted as the arrow part of the conjectural quantization functor. To accomplish this, the arrows in the classical category $\GC$ should not be taken to be maps between phase spaces, but {\it symplectic bimodules} $P_1\law M\raw P_2$.\footnote{Here $M$ is a symplectic manifold and $P_1$ and $P_2$ are integrable Poisson manifolds; the map $M\raw P_2$ is anti-Poisson, whereas the map $P_1\law M$ is Poisson. Such bimodules (often called {\it dual pairs}) were introduced by Karasev (1989) and Weinstein (1983). In order to occur as arrows in $\GC$, symplectic bimodules have to satisfy a number of regularity conditions (Landsman, 2001).} More precisely, the arrows in $\GC$ are suitable isomorphism classes of such bimodules.\footnote{This is necessary in order to make arrow composition associative; this is given by a generalization of the symplectic reduction procedure.} Similarly, the arrows in the quantum category $\mathfrak{Q}$ are not morphisms of \ca s, as might naively be expected, but certain isomorphism classes of bimodules for \ca s, equipped with the additional structure of a generalized Dirac operator.\footnote{The category $\mathfrak{Q}$ is nothing but the category KK introduced by Kasparov, whose objects are separable \ca s, and whose arrows are the so-called Kasparov group $KK(A,B)$, composed with Kasparov's product $KK(A,B)\x KK(B,C)\raw KK(A,C)$. See Higson (1990) and Blackadar (1998).}
Having already defined the object part of the quantization map $\CQ:\GC\raw\mathfrak{Q}$ as deformation quantization, we now propose that the arrow part is geometric quantization, in the sense of a suitable generalization of \eqref{index}; see Landsman (2005a) for details. We then conjecture that $\CQ$ is a functor; in the cases where this can and has been checked, the functoriality of $\CQ$ is precisely the statement that quantization commutes with reduction.\footnote{A canonical $G$-action on a symplectic manifold $M$ with momentum map $\mu:M\raw\g^*$ gives rise to a dual pair $pt\law M\raw \g^*$, which in $\GC$ is interpreted as an arrow from the space $pt$ with one point to $\g^*$. The composition of this arrow with the arrow $\g^*\hookleftarrow 0\raw pt$ from $\g^*$ to $pt$ is $pt\law M/\hspace{-1mm}/G\raw pt$. If $G$ is connected, functoriality of quantization on these two pairs is equivalent to the Guillemin--Sternberg conjecture (cf.\ footnote \ref{GSC}); see Landsman (2005a).}
Thus Heisenberg's idea of {\it Umdeutung} finds it ultimate realization in the quantization functor. \section{The limit $\hbar\raw 0$}\label{S5}\setcounter{equation}{0} It was recognized at an early stage that the limit $\hbar\raw 0$ of Planck's constant going to zero should play a role in the explanation of the classical world from quantum theory. Strictly speaking, $\hbar$ is a dimensionful {\it constant}, but in practice one studies the semiclassical regime of a given quantum theory by forming a dimensionless combination of $\hbar$ and other parameters; this combination then re-enters the theory as if it were a dimensionless version of $\hbar$ that can indeed be varied. The oldest example is Planck's radiation formula \eqref{Planck}, with temperature $T$ as the pertinent variable. Indeed, the observation of Einstein (1905) and Planck (1906) that in the limit $\hbar\nu/kT\raw 0$ this formula converges to the classical equipartition law $E_{\nu}/N_{\nu}=kT$ may well be the first use of the $\hbar\raw 0$ limit of quantum theory.\footnote{Here Einstein (1905) put $\hbar\nu/kT\raw 0$ by letting $\nu\raw 0$ at fixed $T$ and $\hbar$, whereas Planck (1906) took $T\raw \infty$ at fixed $\nu$ and $\hbar$.}
Another example is the Schr\"{o}dinger equation \eqref{Schreq} with Hamiltonian $H=-\frac{\hbar^2}{2m}\Delta_x +V(x)$, where $m$ is the mass of the pertinent particle. Here one may pass to dimensionless parameters by introducing
an energy scale $\epsilon$ typical of $H$, like $\epsilon=\sup_x |V(x)|$, as well as a typical length scale $\lm$, such as $\lm=\epsilon/\sup_x |\nabla V(x)|$ (if these quantities are finite).
In terms of the dimensionless variable $\til{x}=x/\lm$, the rescaled Hamiltonian $\til{H}=H/\epsilon$ is then dimensionless and equal to
$\til{H}=-\til{\hbar}^2\Delta_{\til{x}} +\til{V}(\til{x})$, where $\til{\hbar}=\hbar/\lm\sqrt{2m\epsilon}$ and $\til{V}(\til{x})=V(\lm\til{x})/\epsilon$. Here $\til{\hbar}$ is dimensionless, and one might study the regime where it is small (Gustafson \&\ Sigal, 2003). Our last example will occur in the theory of large quantum systems, treated in the next Section. In what follows, whenever it is considered variable $\hbar$ will denote such a dimensionless version of Planck's constant.
Although, as we will argue, the limit $\hbar\raw 0$ cannot by itself explain the classical world, it does give rise to a number of truly pleasing mathematical results. These, in turn, render almost inescapable the conclusion that the limit in question is indeed a relevant one for the recovery of classical physics from quantum theory. Thus the present section is meant to be a catalogue of those pleasantries that might be of direct interest to researchers in the foundations of quantum theory.
There is another, more technical use of the $\hbar\raw 0$ limit, which is to perform computations in \qm\ by approximating the time-evolution of states and observables in terms of associated classical objects. This endeavour is known as {\it semiclassical analysis}. Mathematically, this use of the $\hbar\raw 0$ limit is closely related to the goal of recovering classical mechanics from \qm, but conceptually the matter is quite different. We will attempt to bring the pertinent differences out in what follows. \subsection{Coherent states revisited}\label{CSR} As Schr\"{o}dinger (1926b) foresaw, coherent states play an important role in the limit $\hbar\raw 0$. We recall from Subsection \ref{PSQ} that {\it for some fixed value $\hbar$ of Planck's constant} coherent states in a \Hs\ ${\mathcal H}$ for a phase space $M$ are defined by an injection $M\hraw {\mathcal H}$, $z\mapsto\Psi_z^{\hbar}$, such that \eqref{normcs} and \eqref{qhnorm} hold. In what follows, we shall say that $\Psi_z^{\hbar}$ is {\it centered at $z\in M$}, a terminology justified by the key example \eqref{pqcohst}.
To be relevant to the classical limit, coherent states must satisfy an additional property concerning their dependence on $\hbar$, which also largely clarifies their nature (Landsman, 1998). Namely, we require that for each $f\in C_c(M)$ and each $z\in M$ the following function from the set $I$ in which $\hbar$ takes values (i.e.\ usually $I=[0,1]$, but in any case containing zero as an accumulation point) to $\C$ is continuous: \begin{eqnarray}
\hbar &\mapsto& c_{\hbar} \int_M d\mu_L(w)\,|(\Psi_w^{\hbar},\Psi_z^{\hbar})|^2 f(w)\:\:\:(\hbar>0);\label{chca} \\ 0 &\mapsto& f(z). \label{chcb} \end{eqnarray}
In view of \eqref{b2}, the right-hand side of \eqref{chcb} is the same as $(\Psi_z^{\hbar},\CQ_{\hbar}^B(f)\Psi_z^{\hbar})$. In particular, this continuity condition implies \begin{equation} \lim_{\hbar\raw 0} (\Psi_z^{\hbar},\CQ_{\hbar}^B(f)\Psi_z^{\hbar})=f(z). \label{hnulB} \end{equation} This means that the classical limit of the quantum-mechanical expectation value of the phase space quantization \eqref{b2} of the classical observable $f$ in a coherent state centered at $z\in M$ is precisely the classical expectation value of $f$ in the state $z$. This interpretation rests on the identification of classical states with probability measures on phase space $M$, under which points of $M$ in the guise of Dirac measures (i.e.\ delta functions) are pure states.
Furthermore, it can be shown (cf.\ Landsman, 1998) that the continuity of all functions \eqref{chca} - \eqref{chcb} implies the property
\begin{equation} \lim_{\hbar\raw 0} |(\Psi_w^{\hbar},\Psi_z^{\hbar})|^2 =\dl_{wz}, \label{dlxy}\end{equation} where $\dl_{wz}$ is the ordinary Kronecker delta (i.e.\ $\dl_{wz}=0$ whenever $w\neq z$ and $\dl_{zz}=1$ for all $z\in M$). This has a natural physical interpretation as well: the classical limit of the quantum-mechanical transition probability between two coherent states centered at $w,z\in M$ is equal to the classical (and trivial) transition probability between $w$ and $z$. In other words, when $\hbar$ becomes small, coherent states at different values of $w$ and $z$ become increasingly orthogonal to each other.\footnote{See Mielnik (1968), Cantoni (1975), Beltrametti \&\ Cassinelli (1984),
Landsman (1998), and Subsection \ref{SE} below for the general meaning of the concept of a transition probability.} This has the interesting consequence that
\begin{equation} \lim_{\hbar\raw 0} (\Psi_w^{\hbar},\CQ_{\hbar}^B(f)\Psi_z^{\hbar})=0 \:\:\: (w\neq z). \label{hnulB2} \end{equation} for all $f\in C_c(M)$. In particular, the following phenomenon of the Schr\"{o}dinger cat type occurs in the classical limit: if $w\neq z$ and one has continuous functions $\hbar\mapsto c_w^{\hbar}\in\C$ and $\hbar\mapsto c_z^{\hbar}\in\C$ on $\hbar\in[0,1]$
such that \begin{equation}\Psi^{\hbar}_{w,z}=c_w^{\hbar}\Psi_w^{\hbar}+c_z^{\hbar}\Psi_z^{\hbar}\end{equation} is a unit vector for $\hbar\geq 0$ and also $|c_w^0|^2+|c_z^0|^2=1$, then \begin{equation} \lim_{\hbar\raw 0} \left(\Psi^{\hbar}_{w,z},\CQ_{\hbar}^B(f)\Psi^{\hbar}_{w,z}\right)
=|c_w^0|^2f(w)+|c_z^0|^2f(z). \label{Cat}\end{equation} Hence the family of (typically) pure states $\psi^{\hbar}_{w,z}$ (on the \ca s $\CA_{\hbar}$ in which the map $\CQ_{\hbar}^B$ takes values)\footnote{For example, for $M=\R^{2n}$ each $\CA_{\hbar}$ is equal to the \ca\ of compact operators on $L^2(\R^n)$, on which each vector state is certainly pure.} defined by the vectors $\Psi^{\hbar}_{w,z}$ in some sense converges to the mixed state on $C_0(M)$ defined by the right-hand side of \eqref{Cat}. This is made precise at the end of this subsection.
It goes without saying that Schr\"{o}dinger's coherent states \eqref{pqcohst} satisfy our axioms; one may also verify \eqref{dlxy} immediately from \eqref{tpbt}. Consequently, by \eqref{WBEQ} one has the same property \eqref{hnulB} for Weyl quantization (as long as $f\in\CS(\R^{2n})$),\footnote{Here $\CS(\R^{2n})$ is the usual Schwartz space of smooth test functions with rapid decay at infinity.} that is, \begin{equation} \lim_{\hbar\raw 0} (\Psi_z^{\hbar},\CQ_{\hbar}^W(f),\Psi_z^{\hbar})=f(z).\label{hnulW} \end{equation} Similarly, \eqref{hnulB2} holds for $\qw$ as well.
In addition, many constructions referred to as coherent states in the literature (cf.\ the references in footnote \ref{CSFNO}) satisfy \eqref{normcs}, \eqref{qhnorm}, and \eqref{dlxy}; see Landsman (1998).\footnote{For example, coherent states of the type introduced by Perelomov (1986) fit into our setting as follows (Simon, 1980). Let $G$ be a compact connected Lie group, and $\CO_{\lm}$ an integral coadjoint orbit, corresponding to a highest weight $\lm$. (One may think here of $G=SU(2)$ and $\lm=0,1/2,1,\ldots$.) Note that $\CO_{\lm}\cong G/T$, where $T$ is the maximal torus in $G$ with respect to which weights are defined. Let ${\mathcal H}^{\mbox{\tiny hw}}_{\lm}$ be the carrier space of the irreducible representation\ $U_{\lm}(G)$ with highest weight $\lm$, containing the highest weight vector $\Om_{\lm}$. (For $G=SU(2)$ one has ${\mathcal H}^{\mbox{\tiny hw}}_j=\C^{2j+1}$, the well-known \Hs\ of spin $j$, in which $\Om_j$ is the vector with spin $j$ in the $z$-direction.)
For $\hbar=1/k$, $k\in\N$, define ${\mathcal H}_{\hbar}:={\mathcal H}^{\mbox{\tiny hw}}_{\lm/\hbar}$.
Choosing a section $\sg:\CO_{\lm}\raw G$ of the projection $G\raw G/T$, one then obtains coherent states $x\mapsto U_{\lm/\hbar}(\sg(x))\Om_{\lm/\hbar}$ with respect to the Liouville measure on $\CO_{\lm}$ and $c_{\hbar}=\dim({\mathcal H}^{\mbox{\tiny hw}}_{\lm/\hbar})$. These states are obviously not defined for all values of $\hbar$ in $(0,1]$, but only for the discrete set $1/\N$.\label{FNPER}}
The general picture that emerges is that a coherent state centered at $z\in M$ is the {\it Umdeutung} of $z$ (seen as a classical pure state, as explained above) as a quantum-mechanical pure state.\footnote{This idea is also confirmed by the fact that at least Schr\"{o}dinger's coherent states are states of minimal uncertainty; cf.\ the references in footnote \ref{CSFNO}.}
Despite their wide applicability (and some would say beauty), one has to look beyond coherent states for a complete picture of the $\hbar\raw 0$ limit of \qm. The appropriate generalization is the concept of a {\it continuous field of states}.\footnote{The use of this concept in various mathematical approaches to quantization is basically folklore.
For the \ca ic setting see Emch (1984), Rieffel (1989b), Werner (1995), Blanchard (1996), Landsman (1998), and Nagy (2000).}
This is defined relative to a given deformation quantization of a phase space $M$; cf.\ Subsection \ref{DQsection}. If one now has a state $\om_{\hbar}$ on $\CA_{\hbar}$ for each $\hbar\in[0,1]$ (or, more generally, for a discrete subset of $[0,1]$ containing 0 as an accumulation point), one may call the ensuing family of states a {\it continuous field} whenever the function $\hbar\mapsto \om_{\hbar}(\CQ_{\hbar}(f))$ is continuous on $[0,1]$ for each $f\in\cci(M)$; this notion is actually intrinsically defined by the continuous field of \ca s, and is therefore independent of the quantization maps $\CQ_{\hbar}$. In particular, one has \begin{equation} \lim_{\hbar\raw 0} \om_{\hbar}(\CQ_{\hbar}(f))=\om_0(f). \label{CFS0} \end{equation} Eq. \eqref{hnulB} (or \eqref{hnulW}) shows that coherent states are indeed examples of continuous fields of states, with the additional property that each $\om_{\hbar}$ is pure. As an example where all states $ \om_{\hbar}$ are mixed, we mention the convergence of quantum-mechanical partition functions to their classical counterparts in statistical mechanics along these lines; see Lieb (1973), Simon (1980), Duffield (1990), and Nourrigat \&\ Royer (2004). Finally, one encounters the surprising phenomenon that pure quantum states may coverge to mixed classical ones. The first example of this has just been exhibited in \eqref{Cat}; other cases in point are energy eigenstates and WKB states (see Subsections \ref{PSL}, \ref{WKBS}, and \ref{QC} below). \subsection{Convergence of quantum dynamics to classical motion}\label{CEOM} Nonrelativistic \qm\ is based on the Schr\"{o}dinger equation \eqref{Schreq}, which more generally reads \begin{equation} H\Psi(t)=i\hbar \frac{\partial\Psi}{\partial t}.\label{Schreq2}\end{equation} The formal solution with initial value $\Ps(0)=\Ps$ is \begin{equation} \Psi(t)=e^{-\frac{it}{\hbar} H}\Ps. \label{SSEOM}\end{equation} Here we have assumed that $H$ is a given self-adjoint operator on the \Hs\ ${\mathcal H}$ of the system, so that this solution indeed exists and evolves unitarily by Stone's theorem; cf.\ Reed \&\ Simon (1972) and Simon (1976). Equivalently, one may transfer the time-evolution from states (Schr\"{o}dinger picture) to operators (Heisenberg picture) by putting \begin{equation} A(t)=e^{\frac{it}{\hbar} H}Ae^{-\frac{it}{\hbar} H}.\label{HSEOM}\end{equation}
We here restrict ourselves to particle motion in $\R^n$, so that ${\mathcal H}=L^2(\R^n)$.\footnote{See Hunziker \&\ Sigal (2000) for a recent survey of $N$-body
Schr\"{o}dinger operators.}
In that case, $H$ is typically given by a formal expression like \eqref{Schreq} (on some specific domain).\footnote{
One then has to prove self-adjointness (or the lack of it) on a larger domain on which the operator is closed; see the literature cited in footnote \ref{SOPrefs}.} Now, the first thing that comes to mind is {\it Ehrenfest's Theorem} (1927), which states that for any (unit) vector $\Ps\in L^2(\R^n)$ in the domain of $\CQ_{\hbar}(q^j)=x^j$ and $\partial V(x)/\partial x^j$ one has \begin{equation} m\frac{d^2}{dt^2} \langle x^j\rangle(t)=-\left\langle \frac{\partial V(x)}{\partial x^j} \right\rangle(t), \label{EhrThm} \end{equation} with the notation \begin{eqnarray} \langle x^j\rangle(t) & =& (\Ps(t),x^j\Ps(t)); \nn \\
\left\langle \frac{\partial V(x)}{\partial x^j} \right\rangle(t) & =& \left(\Ps(t), \frac{\partial V(x)}{\partial x^j}\Ps(t)\right).\label{EhrThm2} \end{eqnarray} This looks like Newton's second law for the expectation value of $x$ in the state $\ps$, with the tiny but crucial difference that Newton would have liked to see $(\partial V/\partial x^j)(\langle x\rangle(t))$ on the right-hand side of \eqref{EhrThm}. Furthermore, even apart from this point Ehrenfest's Theorem by no means suffices to have classical behaviour, since it gives no guarantee whatsoever that $\langle x\rangle(t)$ behaves like a point particle. Much of what follows can be seen as an attempt to sharpen Ehrenfest's Theorem to the effect that it {\it does} indeed yield appropriate classical equations of motion for the expectation values of suitable operators.
We assume that the quantum Hamiltonian has the more general form \begin{equation} H=h(\CQ_{\hbar}(p_j),\CQ_{\hbar}(q^j)), \label{Hh} \end{equation} where $h$ is the classical Hamiltonian (i.e.\ a function defined on classical phase space $\R^{2n}$) and $\CQ_{\hbar}(p_j)$ and $\CQ_{\hbar}(q^j)$ are the operators given in \eqref{SOP}. Whenever this expression is ambiguous (as in cases like $h(p,q)=pq$), one has to assume a specific quantization prescription such as Weyl quantization $\qw$ (cf.\ \eqref{defweylq}), so that formally one has \begin{equation} H=\CQ_{\hbar}^W(h). \label{HWeyl}\end{equation} In fact, in the literature to be cited an even larger class of quantum Hamiltonians is treated by the methods explained here. The quantum Hamiltonian $H$ carries an explicit (and rather singular) $\hbar$-dependence, and for $\hbar\raw 0$ one then expects \eqref{SSEOM} or \eqref{HSEOM} to be related in one way or another to the flow of the classical Hamiltonian $h$. This relationship was already foreseen by Schr\"{o}dinger (1926a), and was formalized almost immediately after the birth of \qm\ by the well-known WKB approximation (cf.\ Landau \&\ Lifshitz (1977) and Subsection \ref{WKBS} below). A mathematically rigorous understanding of this and analogous approximation methods only emerged much later, when a technique called {\it microlocal analysis} was adapted from its original setting of partial differential equations (H\"{o}rmander, 1965; Kohn \&\ Nirenberg, 1965; Duistermaat, 1974, 1996; Guillemin \&\ Sternberg, 1977; Howe, 1980; H\"{o}rmander, 1979, 1985a, 1985b; Grigis \&\ Sj\"ostrand, 1994) to the study of the $\hbar\raw 0$ limit of \qm. This adaptation (often called {\it semiclassical analysis}) and its results have now been explained in various reviews written by the main players, notably Robert (1987, 1998), Helffer (1988), Paul \&\ Uribe (1995), Colin de Verdi\`ere (1998), Ivrii (1998), Dimassi \&\ Sj\"ostrand (1999), and Martinez (2002) (see also the papers in Robert (1992)). More specific references will be given below.\footnote{For the heuristic theory of semiclassical asymptotics Landau \&\ Lifshitz (1977) is a goldmine.}
As mentioned before, the relationship between $H$ and $h$ provided by semiclassical analysis is double-edged. On the one hand, one obtains approximate solutions of \eqref{SSEOM} or \eqref{HSEOM}, or approximate energy eigenvalues and energy eigenfunctions (sometimes called quasi-modes) for small values of $\hbar$ in terms of classical data. This is how the results are usually presented; one computes specific properties of quantum theory in a certain regime in terms of an underlying classical theory. On the other hand, however, with some effort the very same results can often be reinterpreted as a partial explanation of the emergence of classical dynamics from \qm. It is the latter aspect of semiclassical analysis, somewhat understated in the literature, that is of interest to us. In this and the next three subsections we restrict ourselves to the simplest type of results, which nonetheless provide a good flavour of what can be achieved and understood by these means. By the same token, we just work with the usual flat phase space $M=\R^{2n}$ as before.
The simplest of all results relating classical and quantum dynamics is this:\footnote{More generally, Egorov's Theorem states that for a large class of Hamiltonians one has $\qw(f)(t) =\qw(f_t)+O(\hbar)$. See, e.g., Robert (1987), Dimassi \&\ Sj\"ostrand (1999), and Martinez (2002).} \begin{quote} {\it If the classical Hamiltonian $h(p,q)$ is at most quadratic in $p$ and $q$, and the Hamiltonian in \eqref{HSEOM} is given by \eqref{HWeyl}, then} \end{quote} \begin{equation} \qw(f)(t) =\qw(f_t). \label{QCC} \end{equation}
Here $f_t$ is the solution of the classical equation of motion $df_t/dt=\{h,f_t\}$; equivalently, one may write \begin{equation} f_t(p,q)=f(p(t),q(t)), \label{ft}\end{equation} where $t\mapsto (p(t),q(t))$ is the classical Hamiltonian flow of $h$ with initial condition $(p(0),q(0))=(p,q)$. This holds for all decent $f$, e.g., $f\in\CS(\R^{2n})$.
This result explains quantum in terms of classical, but the converse may be achieved by combining \eqref{QCC} with \eqref{CFS0}. This yields \begin{equation} \lim_{\hbar\raw 0} \om_{\hbar}(\CQ_{\hbar}(f)(t))=\om_0(f_t) \label{CFS1} \end{equation} for any continuous field of states $(\om_{\hbar})$. In particular, for Schr\"odinger's coherent states \eqref{pqcohst} one obtains \begin{equation} \lim_{\hbar\raw 0} \left(\Psi_{(p,q)}^{\hbar}, \CQ_{\hbar}(f)(t) \Psi_{(p,q)}^{\hbar}\right)=f_t(p,q). \label{hepp} \end{equation} Now, whereas \eqref{QCC} merely reflects the good symmetry properties of Weyl quantization,\footnote{Eq.\ \eqref{QCC} is equivalent to the covariance of Weyl quantization under the affine symplectic group; cf.\ footnote \ref{GCQQ}.} (and is false for $\CQ_{\hbar}^B$), eq.\ \eqref{hepp} is actually valid for a large class of realistic Hamiltonians and for any deformation quantization map $\CQ_{\hbar}$ that is asymptotically equal to $\qw$ (cf.\ \eqref{WBEQ}). A result of this type was first established by Hepp (1974); further work in this direction includes Yajima (1979), Hogreve, Potthoff \&\ Schrader (1983), Wang (1986), Robinson (1988a, 1988b), Combescure (1992), Arai (1995), Combescure \&\ Robert (1997), Robert (1998), and Landsman (1998).
Impressive results are available also in the Schr\"{o}dinger picture. The counterpart of \eqref{QCC} is that for any suitably smooth classical Hamiltonian $h$ (even a time-dependent one) that is at most quadratic in the canonical coordinates $p$ and $q$ on phase space $\R^{2n}$ one may construct generalized coherent states $\Psi^{\hbar}_{(p,q,C)}$, labeled by a set $C$ of classical parameters dictated by the form of $h$, such that \begin{equation} e^{-\frac{it}{\hbar} \qw(h)}\Ps^{\hbar}_{(p,q,C)}=e^{iS(t)/\hbar}\Ps^{\hbar}_{(p(t),q(t),C(t))}. \label{Hag} \end{equation} Here $S(t)$ is the action associated with the classical trajectory $(p(t),q(t))$ determined by $h$, and $C(t)$ is a solution of a certain system of differential equations that has a classical interpretation as well (Hagedorn, 1998). Schr\"{o}dinger's coherent states \eqref{pqcohst} are a special case for the standard harmonic oscillator Hamiltonian. For more general Hamiltonians one then has an asymptotic result (Hagedorn \&\ Joye, 1999, 2000)\footnote{See also
Paul \&\ Uribe (1995, 1996) as well as the references listed after \eqref{hepp} for analogous statements.}
\begin{equation} \lim_{\hbar\raw 0} \left\| e^{-\frac{it}{\hbar} \qw(h)}\Ps^{\hbar}_{(p,q,C)}-e^{iS(t)/\hbar}\Ps^{\hbar}_{(p(t),q(t),C(t))}\right\| =0. \label{HY}\end{equation}
Once again, at first sight such results merely contribute to the understanding of quantum dynamics in terms of classical motion. As mentioned, they may be converted into statements on the emergence of classical motion from \qm\ by taking expectation values of suitable $\hbar$-dependent obervables of the type $\qw(f)$.
For finite $\hbar$, the second term in \eqref{HY} is a good approximation to the first - the error even being as small as $\CO(\exp(-\gm/\hbar))$ for some $\gm>0$ as $\hbar\raw 0$ - whenever $t$ is smaller than the so-called {\it Ehrenfest time} \begin{equation} T_E=\lm^{\inv}\log(\hbar\inv), \label{EhrTime} \end{equation} where $\lm$ is a typlical inverse time scale of the Hamiltonian (e.g., for chaotic systems it is the largest Lyapunov exponent).\footnote{Recall that throughout this section we assume that $\hbar$ has been made dimensionless through an appropriate rescaling.} This is the typical time scale on which semiclassical approximations to wave packet solutions of the time-dependent Schr\"{o}dinger equation with a general Hamiltonian tend to be valid (Ehrenfest, 1927; Berry et al., 1979; Zaslavsky, 1981; Combescure \&\ Robert, 1997; Bambusi, Graffi, \&\ Paul, 1999; Hagedorn \&\ Joye, 2000).\footnote{ One should distinguish here between two distinct approximation methods to
the time-dependent Schr\"{o}dinger equation. Firstly, one has the semiclassical propagation of a quantum-mechanical wave packet, i.e.\ its propagation as computed from the time-dependence of the parameters on which it depends {\it according to the underlying classical equations of motion}. It is shown in the references just cited that this approximates the full quantum-mechanical propagation of the wave packet well until $t\sim T_E$. Secondly, one has the time-dependent WKB approximation (for integrable systems) and its generalization to chaotic systems (which typically involve tens of thousands of terms instead of a single one). This second approximation is valid on a much longer time scale, typically $t\sim\hbar^{-1/2}$ (O'Connor, Tomsovic, \&\ Heller, 1992; Heller \&\ Tomsovic, 1993; Tomsovic, \&\ Heller, 1993, 2002; Vanicek \&\ Heller, 2003). Adding to the confusion, Ballentine has claimed over the years that even the semiclassical propagation of a wave packet approximates its quantum-mechanical propagation for times much longer than the Ehrenfest time, typically $t\sim\hbar^{-1/2}$ (Ballentine, Yang, \&\ Zibin, 1994; Ballentine, 2002, 2003). This claim is based on the criterion that the quantum and classical (i.e.\ Liouville) probabilities are approximately equal on such time scales, but the validity of this criterion hinges on the ``statistical" or ``ensemble" interpretation of \qm. According to this interpretation, a pure state provides a description of certain statistical properties of an ensemble of similarly prepared systems, but need not provide a complete description of an individual system. See Ballentine (1970, 1986). Though once defended by von Neumann, Einstein and Popper, this interpretation has now completely fallen out of fashion.} For example, Ehrenfest (1927) himself estimated that for a mass of 1 gram a wave packet would double its width only in about $10^{13}$ years under free motion. However, Zurek and Paz (1995) have estimated the Ehrenfest time for Saturn's moon Hyperion to be of the order of 20 years! This obviously poses a serious problem for the program of deriving (the appearance of) classical behaviour from \qm, which affects all interpretations of this theory.
Finally, we have not discussed the important problem of combining the limit $t\raw\infty$ with the limit $\hbar\raw 0$; this should be done in such a way that $T_E$ is kept fixed. This double limit is of particular importance for quantum chaos; see Robert (1998) and most of the literature cited in Subsection \ref{QC}. \subsection{Wigner functions} \label{WFSEC} The $\hbar\raw 0$ limit of \qm\ is often discussed in terms of the so-called {\it Wigner function}, introduced by Wigner (1932).\footnote{The original context was quantum statistical mechanics; one may write down \eqref{WF} for mixed states as well. See Hillery et al. (1984) for a survey.} Each unit vector (i.e. wave function) $\Psi\in L^2(\R^n)$ defines such a function $W^{\hbar}_{\Psi}$ on classical phase space $M=\R^{2n}$ by demanding that for each $f\in\CS(\R^{2n})$ one has \begin{equation} \left(\Ps,\qw(f)\Ps\right)=\int_{\R^{2n}} \frac{d^np d^nq}{(2\pi)^n} \, W^{\hbar}_{\Psi}(p,q) f(p,q). \label{WF} \end{equation} The existence of such a function may be proved by writing it down explicitly as \begin{equation} W^{\hbar}_{\Ps}(p,q)=\int_{\R^n} d^nv\,e^{ipv}\ovl{\Ps(q+\half\hbar v)} \Ps(q-\half\hbar v).\label{wiganew} \end{equation} In other words, the quantum-mechanical expectation value of the Weyl quantization of the classical observable $f$ in a quantum state $\Ps$ formally equals the classical expectation value of $f$ with respect to the distribution $W_{\Ps}$.
However, the latter may not be regarded as a probability distribution because it is not necessarily positive definite.\footnote{Indeed, it may not even be in $L^1(\R^{2n})$, so that its total mass is not necessarily defined, let alone equal to 1. Conditions for the positivity of Wigner functions defined by pure states are given by Hudson (1974); see Br\"ocker \&\ Werner (1995) for the case of mixed states.} Despite this drawback, the Wigner function possesses some attractive properties. For example, one has \begin{equation} \qw(W^{\hbar}_{\Ps})=\hbar^{-n}[\Psi]. \end{equation}
This somewhat perverse result means that if the Wigner function defined by $\Ps$ is seen as a classical observable (despite its manifest $\hbar$-dependence!), then its Weyl quantization is precisely ($\hbar^{-n}$ times) the projection operator onto $\Ps$.\footnote{In other words, $W_{\Psi}$ is the Weyl symbol of the projection operator $[\Ps]$.} Furthermore, one may derive the following formula for the transition probability:\footnote{This formula is well defined since $\Ps\in\ L^2(\R^n)$ implies $W^{\hbar}_{\Psi}\in\ L^2(\R^{2n})$.} \begin{equation}
|(\Phi,\Psi)|^2=\hbar^n \int_{\R^{2n}} \frac{d^np d^nq}{(2\pi)^n} \, W^{\hbar}_{\Psi}(p,q) W^{\hbar}_{\Phi}(p,q). \label{WFTP} \end{equation} This expression has immediate intuitive appeal, since the integrand on the right-hand side is supported by the area in phase space where the two Wigner functions overlap, which is well in tune with the idea of a transition probability.
The potential lack of positivity of a Wigner function may be remedied by noting that Berezin's deformation quantization scheme (see \eqref{qbttsrex}) analogously defines functions $B^{\hbar}_{\Psi}$ on phase space by means of
\begin{equation} \left(\Ps,\qb(f)\Ps\right)=\int_{\R^{2n}} \frac{d^np d^nq}{(2\pi)^n} \, B^{\hbar}_{\Psi}(p,q) f(p,q). \label{BF} \end{equation} Formally, \eqref{qbttsrex} and \eqref{BF} immediately yield
\begin{equation} B^{\hbar}_{\Psi}(p,q)=|(\Psi_{(p,q)}^{\hbar},\Psi)|^2 \label{BFexp} \end{equation} in terms of Schr\"odinger's coherent states \eqref{pqcohst}. This expression is manifestly positive definite. The existence of $B^{\hbar}_{\Psi}$ may be proved rigorously by recalling that the Berezin quantization map $f\mapsto\qb(f)$ is {\it positive} from $C_0(\R^{2n})$ to $\CB(L^2(\R^n))$. This implies that for each (unit) vector $\Ps\in L^2(\R^n)$ the map $f\mapsto (\Ps,\qb(f)\Ps)$ is positive from $C_c(\R^{2n})$ to $\C$, so that (by the Riesz theorem of measure theory) there must be a measure $\mu_{\Psi}$ on $\R^{2n}$ such that $(\Ps,\qb(f)\Ps)=\int d\mu_{\Ps}\, f$. This measure, then, is precisely given by $d\mu_{\Ps}(p,q)=(2\pi)^{-n} d^npd^nq\, B^{\hbar}_{\Psi}(p,q)$. If $(\Ps,\Ps)=1$, then $\mu_{\Ps}$ is a probability measure. Accordingly, despite its $\hbar$-dependence, $B^{\hbar}_{\Ps}$ defines a bona fide classical probability distribution on phase space, in terms of which one might attempt to visualize \qm\ to some extent.
For finite values of $\hbar$, the Wigner and Berezin distribution functions are different, because the quantization maps $\qw$ and $\qb$ are. The connection between $B^{\hbar}_{\Psi}$ and
$W^{\hbar}_{\Psi}$ is easily computed to be
\begin{equation}
B^{\hbar}_{\Psi}=W^{\hbar}_{\Psi}*g^{\hbar},\label{CBW} \end{equation}
where $g^{\hbar}$ is the Gaussian function \begin{equation} g^{\hbar}(p,q)=(2/\hbar)^n \exp(-(p^2+q^2)/\hbar).\end{equation}
This is how physicists look at the Berezin function,\footnote{ The `Berezin' functions $B^{\hbar}_{\Psi}$ were introduced by Husimi (1940) from a different point of view, and are therefore actually called {\it Husimi functions} by physicists.} viz.\ as a Wigner function smeared with a Gaussian so as to become positive. But since $g^{\hbar}$ converges to a Dirac delta function as $\hbar\raw 0$ (with respect to the measure $(2\pi)^{-n} d^npd^nq$ in the sense of distributions), it is clear from \eqref{CBW} that as distributions one has\footnote{\label{BWD} Eq.\ \eqref{BWAE} should be interpreted as a limit of the distribution on $\mathcal{D}(\R^{2n})$ or $\mathcal{S}(\R^{2n})$ defined by $B^{\hbar}_{\Psi}-W^{\hbar}_{\Psi}$. Both functions are continuous for $\hbar>0$, but lose this property in the limit $\hbar\raw 0$, generally converging to distributions.}
\begin{equation}
\lim_{\hbar\raw 0} \left(B^{\hbar}_{\Psi}-W^{\hbar}_{\Psi}\right)=0. \label{BWAE}\end{equation} See also \eqref{WBEQ}. Hence in the study of the limit $\hbar\raw 0$ there is little advantage in the use of Wigner functions; quite to the contrary, in limiting procedures their generic lack of positivity makes them more difficult to handle than Berezin functions.\footnote{ See, however, Robinett (1993) and Arai (1995). It should be mentioned that \eqref{BWAE} expresses the asymptotic equivalence of Wigner and Berezin functions as distributions on $\hbar$-independent test functions. Even in the limit $\hbar\raw 0$ one is sometimes interested in studying $O(\hbar)$ phenomena, in which case one should make a choice.}
For example, one would like to write the asymptotic behaviour \eqref{hnulW} of coherent states in the form $\lim_{\hbar\raw 0} W^{\hbar}_{\Ps^{\hbar}_z}=\dl_z$. Although this is indeed true in the sense of distributions, the
corresponding limit \begin{equation} \lim_{\hbar\raw 0} B^{\hbar}_{\Ps^{\hbar}_z}=\dl_z, \label{lhoB} \end{equation}
exists in the sense of (probability) measures, and is therefore defined on a much larges class of test functions.\footnote{Namely those in $C_0(\R^{2n})$ rather than in $\mathcal{D}(\R^{2n})$ or $\mathcal{S}(\R^{2n})$.} Here and in what follows, we abuse notation: if $\mu^0$ is some probability measure on $\R^{2n}$
and $(\Psi^{\hbar})$ is a sequence of unit vectors in $L^2(\R^n)$ indexed by $\hbar$ (and perhaps other labels), then $B^{\hbar}_{\Ps^{\hbar}}\raw
\mu^0$ for $\hbar\raw 0$ by definition means that for any $f\in\cci(\R^{2n})$ one has\footnote{Since $\qb$ may be extended from $\cci(\R^{2n})$ to $L^{\infty}(\R^{2n})$, one may omit the stipulation that $\mu^0$ be a {\it probability} measure in this definition if one requires convergence for all $f\in L^{\infty}(\R^{2n})$, or just for all $f$ in the unitization of the \ca\ $C_0(\R^{2n})$.}
\begin{equation} \lim_{\hbar\raw 0} \left(\Psi^{\hbar},\qb(f)\Psi^{\hbar}\right)= \int_{\R^{2n}} d\mu^0\, f.\end{equation} \subsection{The classical limit of energy eigenstates} \label{PSL} Having dealt with coherent states in \eqref{lhoB}, in this subsection we discuss the much more difficult problem of computing the limit measure $\mu^0$ for eigenstates of the quantum Hamiltonian $H$. Thus we assume that $H$ has eigenvalues $E^{\hbar}_n$ labeled by $n\in\N$ (defined with or without 0 according to convenience), and also depending on $\hbar$ because of the explicit dependence of $H$ on this parameter. The associated eigenstates $\Ps^{\hbar}_{\mathsf{n}}$ then by definition satisfy \begin{equation} H\Ps^{\hbar}_{\mathsf{n}}=E^{\hbar}_n\Ps^{\hbar}_{\mathsf{n}}.\end{equation}
Here we incorporate the possibility that the eigenvalue $E^{\hbar}_n$ is degenerate, so that the label $\mathsf{n}$ extends $n$. For example, for the one-dimensional harmonic oscillator one has $E^{\hbar}_{n}=\hbar\omega (n+\half)$ ($n=0,1,2,\ldots$) without multiplicity, but for the hydrogen atom the Bohrian eigenvalues $E^{\hbar}_n=-m_e e^4/2\hbar^2 n^2$ (where $m_e$ is the mass of the electron and $e$ is its charge) are degenerate, with the well-known eigenfunctions $\Psi^{\hbar}_{(n,l,m)}$ (Landau \&\ Lifshitz, 1977). Hence in this case one has $\mathsf{n}=(n,l,m)$ with $n=1,2,3,\ldots$, subject to $l=0, 1, \ldots, n-1$, and $m=-l, \ldots, l$.
In any case, it makes sense to let ${\mathsf{n}}\raw\infty$; this certainly means $n\raw\infty$, and may in addition involve sending the other labels in $\mathsf{n}$ to infinity (subject to the appropriate restrictions on ${\mathsf{n}}\raw\infty$, as above). One then expects classical behaviour \`a la Bohr if one simultaneously lets $\hbar\raw 0$ whilst $E^{\hbar}_n\raw E^0$ converges to some `classical' value $E^0$. Depending on how one lets the possible other labels behave in this limit, this may also involve similar asymptotic conditions on the eigenvalues of operators commuting with $H$ - see below for details in the integrable case. We denote the collection of such eigenvalues (including $E^{\hbar}_n$) by $\mathsf{E}^{\hbar}_{\mathsf{n}}$. (Hence in the case where the energy levels $E^{\hbar}_n$ are nondegenerate, the label $\mathsf{E}$ is just $E$.)
In general, we denote the collective limit of the eigenvalues $\mathsf{E}^{\hbar}_{\mathsf{n}}$ as $\hbar\raw 0$ and ${\mathsf{n}}\raw\infty$ by $\mathsf{E}^0$.
For example, for the hydrogen atom one has the additional operators $J^2$ of total angular momentum as well as the operator $J_3$ of angular momentum in the $z$-direction. The eigenfunction $\Psi^{\hbar}_{(n,l,m)}$ of $H$ with eigenvalue $E^{\hbar}_n$ is in addition an eigenfunction of $J^2$ with eigenvalue $j_{\hbar}^2=\hbar^2l(l+1)$ and of $J_3$ with eigenvalue $j^{\hbar}_3=\hbar m$. Along with $n\raw\infty$ and $\hbar\raw 0$, one may then send $l\raw\infty$ and $m\raw\pm\infty$ in such a way that $j_{\hbar}^2$ and $j^{\hbar}_3$ approach specific constants.
The object of interest, then, is the measure on phase space obtained as the limit of the Berezin functions \eqref{BFexp}, i.e.\ \begin{equation} \mu_{\mathsf{E}}^0=\lim_{\hbar\raw 0,\mathsf{n}\raw\infty} B^{\hbar}_{\Ps^{\hbar}_{\mathsf{n}}}. \label{measure} \end{equation} Although the pioneers of \qm\ were undoubtedly interested in quantities like this, it was only in the 1970s that rigorous results were obtained. Two cases are well understood: in this subsection we discuss the {\it integrable} case, leaving chaotic and more generally {\it ergodic} motion to Subsection \ref{QC}.
In the physics literature, it was argued that for an integrable system the limiting measure $\mu_{\mathsf{E}}^0$ is concentrated (in the form of a $\dl$-function) on the invariant torus associated to $\mathsf{E}^0$ (Berry, 1977a).\footnote{This conclusion was, in fact, reached from the Wigner function formalism. See Ozorio de Almeida (1988) for a review of work of Berry and his collaborators on this subject.} Independently, mathematicians began to study a quantity very similar to $\mu_{\mathsf{E}}^0$, defined by limiting sequences of eigenfunctions of the Laplacian on a Riemannian manifold $M$. Here the underlying classical flow is Hamiltonian as well, the corresponding trajectories being the geodesics of the given metric (see, for example, Klingenberg (1982), Abraham \&\ Marsden (1985), Katok \&\ Hasselblatt (1995), or Landsman (1998)).\footnote{
The simplest examples of integrable geodesic motion are $n$-tori, where the geodesics are projections of lines, and the sphere, where the geodesics are great circles (Katok \&\ Hasselblatt, 1995).}
The ensuing picture largely confirms the folklore of the physicists: \begin{quote}
{\it In the integrable case the limit measure $\mu_{\mathsf{E}}^0$ is concentrated on invariant tori}. \end{quote} See Charbonnel (1986, 1988), Zelditch (1990, 1996a), Toth (1996, 1999), Nadirashvili, Toth, \&\ Yakobson (2001), and Toth \&\ Zelditch (2002, 2003a, 2003b).\footnote{These papers consider the limit $n\raw\infty$ without $\hbar\raw 0$; in fact, a physicist would say that they put $\hbar =1$. In that case $E_n \raw\infty$; in this procedure the physicists' microscopic $E\sim \CO(\hbar)$ and macroscopic $E\sim \CO(1)$ regimes correspond to $E\sim \CO(1)$ and $E\raw\infty$, respectively.} Finally, as part of the transformation of microlocal analysis to semiclassical analysis (cf.\ Subsection \ref{CEOM}), these results were adapted to \qm\ (Paul \&\ Uribe, 1995, 1996).
Let us now give some details for integrable systems (of Liouville type); these include the hydrogen atom as a special case. Integrable systems are defined by the property that on a $2p$-dimensional phase space $M$ one has $p$ independent\footnote{I.e. $df_1\wed \cdots\wed df_p\neq 0$ everywhere. At this point we write $2p$ instead of $2n$ for the dimension of phase space in order to avoid notational confusion.} classical observables $(f_1=h,f_2,\ldots, f_p)$ whose mutual Poisson brackets all vanish (Arnold, 1989). One then hopes that an appropriate quantization scheme $\CQ_{\hbar}$ exists under which the corresponding quantum observables $(\CQ_{\hbar}(f_1)=H,\CQ_{\hbar}(f_2), \ldots, \CQ_{\hbar}(f_p))$ are all self-adjoint and mutually commute (on a common core).\footnote{There is no general theory of quantum integrable systems. Olshanetsky \&\ Perelomov (1981, 1983) form a good starting point.} This is indeed the case for the hydrogen atom, where $(f_1,f_2,f_3)$ may be taken to be $(h,j^2,j_3)$ (where $j^2$ is the total angular momentum and $j_3$ is its $z$-component),\footnote{In fact, if $\mu$ is the momentum map for the standard $SO(3)$-action on $\R^3$, then $j^2=\sum_{k=1}^3 \mu_k^2$ and $j_3=\mu_3$.} $H$ is given by \eqref{HWeyl}, $J^2=\qw(j^2)$, and $J_3=\qw(j_3)$. In general, the energy eigenfunctions
$\Ps^{\hbar}_{\mathsf{n}}$ will be joint eigenfunctions of the operators
$(\CQ_{\hbar}(f_1), \ldots, \CQ_{\hbar}(f_p))$, so that $\mathsf{E}^{\hbar}_{\mathsf{n}}=(E^{\hbar}_{n_1},\ldots, E^{\hbar}_{n_p})$, with $\CQ_{\hbar}(f_k)\Ps^{\hbar}_{\mathsf{n}}=E^{\hbar}_{n_k}\Ps^{\hbar}_{\mathsf{n}}$.
We assume that the submanifolds $\cap_{k=1}^p f_k\inv(x_k)$ are compact and connected for each $x\in\R^p$, so that they are tori by the Liouville--Arnold Theorem (Abraham \&\ Marsden, 1985, Arnold, 1989).
Letting $\hbar\raw 0$ and $\mathsf{n}\raw\infty$ so that $E^{\hbar}_{n_k}\raw E_k^0$ for some point $E^0=(E^0_1,\ldots, E^0_p)\in\R^p$, it follows that the limiting measure $\mu_{\mathsf{E}}^0$ as defined in \eqref{measure} is concentrated on the invariant torus $\cap_{k=1}^p f_k\inv(E^0_k)$. This torus is generically $p$-dimensional, but for singular points
$E^0$ it may be of lower dimension. In particular, in the exceptional circumstance where the invariant torus is one-dimensional,
$\mu_{\mathsf{E}}^0$ is concentrated on a classical orbit. Of course, for $p=1$
(where any Hamiltonian system is integrable) this singular case is generic. Just think of the foliation of $\R^2$ by the ellipses that form the closed orbits
of the harmonic oscillator motion.\footnote{\label{Zelditch} It may be enlightening to consider
geodesic motion on the sphere; this example may be seen as the hydrogen atom without the radial degree of freedom (so that the degeneracy in question occurs in the hydrogen atom as well). If one sends $l\raw\infty$ and
$m\raw\infty$ in the spherical harmonics $Y^m_l$ (which are eigenfunctions of the Laplacian on the sphere) in such a way that $\lim m/l=\cos\phv$, then
the invariant tori are generically two-dimensional, and occur when $\cos\phv\neq\pm 1$; an invariant torus labeled by such a value of $\phv\neq 0,\pi$ comprises all great circles (regarded as part of phase space by adding to each point of the geodesic a velocity of unit length and direction tangent to the geodesic) whose angle with the $z$-axis is $\phv$ (more precisely, the angle in question is the one between the normal of the plane through the given great circle and the $z$-axis). For $\cos\phv=\pm 1$ (i.e.\ $m=\pm l$), however, there is only one great circle with
$\phv=0$ namely the equator (the case $\phv=\pi$ corresponds to the same equator traversed in the opposite direction). Hence in this case the invariant torus is one-dimensional. The reader may be surprised that the invariant tori explicitly depend on the choice of variables, but this feature is typical of so-called degenerate systems; see Arnold (1989), \S 51.}
What remains, then, of Bohr's picture of the hydrogen atom in this light?\footnote{We ignore coupling to the electromagnetic field here; see footnote \ref{SigalF}.} Quite a lot, in fact, confirming his remarkable physical intuition. The energy levels Bohr calculated are those given by the Schr\"{o}dinger equation, and hence remain correct in mature \qm. His orbits make literal sense only in the ``correspondence principle" limit $\hbar\raw 0$, $n\raw\infty$, where, however, the situation is even better than one might expect for integrable systems: because of the high degree of symmetry of the Kepler problem (Guillemin \&\ Sternberg, 1990), one may construct energy eigenfunctions whose limit measure $\mu^0$ concentrates on any desired classical orbit (Nauenberg, 1989).\footnote{Continuing footnote \ref{Zelditch}, for a given principal quantum number $n$ one forms the eigenfunction $\Ps^{\hbar}_{(n,n-1,n-1)}$ by multiplying the spherical harmonic $Y^{n-1}_{n-1}$ with the appropriate radial wave function. The limiting measure \eqref{measure} as $n\raw\infty$ and $\hbar\raw 0$ is then concentrated on an orbit (rather than on an invariant torus). Now, beyond what it possible for general integrable systems, one may use the $SO(4)$ symmetry of the Kepler problem and the construction in footnote \ref{FNPER} for the group-theoretic coherent states of Perelomov (1986) to find
the desired eigenfunctions. See also De Bi\`evre (1992) and De Bi\`evre et al. (1993).} In order to recover a travelling wave packet, one has to form wave packets from a very large number of energy eigenstates with very high quantum numbers, as explained in Subsection \ref{Ssection}. For finite $n$ and $\hbar$ Bohr's orbits seem to have no meaning, as already recognized by Heisenberg (1969) in his pathfinder days!\footnote{The later Bohr also conceded this through his idea that causal descriptions are complementary to space-time pictures; see Subsection \ref{compl}.} \subsection{The WKB approximation} \label{WKBS} One might have expected a section on the $\hbar\raw 0$ limit of \qm\ to be centered around the WKB approximation, as practically all textbooks base their discussion of the classical limit on this notion. Although the scope of this method is actually rather limited, it is indeed worth saying a few words about it. For simplicity we restrict ourselves to the time-independent case.\footnote{Cf.\ Robert (1998) and references therein for the time-dependent case.} In its original formulation, the time-independent WKB method involves an attempt to approximate solutions of the time-independent Schr\"{o}dinger equation $H\Ps=E\Ps$ by wave functions of the type \begin{equation} \Ps(x)=a_{\hbar}(x)e^{\frac{i}{\hbar} S(x)}, \label{WKB} \end{equation}
where $a_{\hbar}$ admits an expansion in $\hbar$ as a power series.
Assuming the Hamiltonian $H$ is of the form \eqref{Hh},
plugging the Ansatz \eqref{WKB} into the Schr\"{o}dinger equation, and expanding in $\hbar$, yields in lowest order the classical (time-independent) Hamilton--Jacobi equation
\begin{equation}
h\left( \frac{\partial S}{\partial x},x\right)=E, \label{HJE}
\end{equation} supplemented by the so-called (homogeneous) transport equation\footnote{ Only stated here for a classical Hamiltonian $h(p,q)=p^2/2m + V(q)$. Higher-order terms in $\hbar$ yield further, inhomogeneous transport equations for the expansion coefficients $a_j(x)$ in $a_{\hbar}=\sum_j a_j \hbar^j$. These can be solved in a recursive way, starting with \eqref{TPE}.} \begin{equation} \left(\half\Delta S + \sum_k\frac{\partial S}{\partial x^k}\frac{\partial }{\partial x^k}\right) a_0=0.\label{TPE}\end{equation}
In particular, $E$ should be a classically allowed value of the energy. Even when it applies (see below), in most cases of interest
the Ansatz \eqref{WKB} is only valid locally (in $x$), leading to problems with caustics. These problems turn out to be an artefact of the use of the coordinate representation that lies behind the choice of the \Hs\ ${\mathcal H}=L^2(\R^n)$,
and can be avoided (Maslov \&\ Fedoriuk, 1981): the WKB method really comes to its own in a geometric reformulation in terms of symplectic geometry. See Arnold (1989), Bates \&\ Weinstein (1995), and Dimassi \&\ Sj\"ostrand (1999) for (nicely complementary) introductory treatments, and Guillemin \&\ Sternberg (1977), H\"{o}rmander (1985a, 1985b), and Duistermaat (1974, 1996) for advanced accounts.
The basic observation leading to this reformulation is that in the rare cases that $S$ is defined globally as a smooth function on the configuration space $\R^{n}$, it defines a submanifold $\CL$ of the phase space $M=\R^{2n}$ by $\CL=\{(p=dS(x),q=x), x\in\R^n\}$. This submanifold is {\it Lagrangian} in having two defining properties: firstly, $\CL$ is $n$-dimensional, and secondly, the restriction of the symplectic form (i.e.\ $\sum_k dp_k\wedge dq^k$) to $\CL$ vanishes. The Hamilton--Jacobi equation \eqref{HJE} then guarantees that the Lagrangian submanifold $\CL\subset M$ is contained in the surface $\Sigma_E=h\inv(E)$ of constant energy $E$ in $M$. Consequently, any solution of the Hamiltonian equations of motion that starts in $\CL$ remains in $\CL$.
In general, then, the starting point of the WKB approximation is a Lagrangian submanifold $\CL\subset \Sigma_E\subset M$, rather than some function $S$ that defines it locally. By a certain adaptation of the geometric quantization procedure, one may, under suitable conditions, associate a unit vector $\Ps_{\CL}$ in a suitable \Hs\ to $\CL$, which for small $\hbar$ happens to be a good approximation to an eigenfunction of $H$ at eigenvalue $E$. This strategy is successful in the integrable case, where the nondegenerate tori (i.e. those of maximal dimension $n$) provide such Lagrangian submanifolds of $M$; the associated unit vector $\Ps_{\CL}$ then turns out to be well defined precisely when $\CL$ satisfies (generalized) Bohr--Sommerfeld quantization conditions. In fact, this is how the measures
$\mu^0_{\mathsf{E}}$ in \eqref{measure} are generally computed in the integrable case.
If the underlying classical system is not integrable, it may still be close enough to integrability for invariant tori to be defined. Such systems are called quasi-integrable or perturbations of integrable systems, and are described by the Kolmogorov--Arnold--Moser (KAM) theory; see Gallavotti (1983), Abraham \&\ Marsden (1985), Ozorio de Almeida (1988), Arnold (1989), Lazutkin (1993), Gallavotti, Bonetto \&\ Gentile (2004), and many other books. In such systems the WKB method continues to provide approximations to the energy eigenstates relevant to the surviving invariant tori (Colin de Verdi\`ere, 1977; Lazutkin, 1993; Popov, 2000), but already loses some of its appeal.
In general systems, notably chaotic ones, the WKB method is almost useless. Indeed, the following theorem of Werner (1995) shows that the measure $\mu^0_{\mathsf{E}}$ defined by a WKB function \eqref{WKB} is concentrated on the Lagrangian submanifold $\CL$ defined by $S$: \begin{quote} {\it Let $a_{\hbar}$ be in $L^2(\R^n)$ for each $\hbar>0$ with pointwise limit $a_0=\lim_{\hbar\raw 0}a_{\hbar}$ also in $L^2(\R^n)$,\footnote{This assumption is not made in Werner (1995), who directly assumes that $\Ps=a_0 \exp (iS/\hbar)$ in \eqref{WKB}.} and suppose that $S$ is almost everywhere differentiable. Then for each $f\in\cci(\R^{2n})$:} \end{quote} \begin{equation} \lim_{\hbar\raw 0} \left(a_{\hbar}e^{\frac{i}{\hbar}S},\qb(f)a_{\hbar}e^{\frac{i}{\hbar}S}\right)=
\int_{\R^n} d^nx\, |a_0(x)|^2 f\left( \frac{\partial S}{\partial x},x\right).\label{werner} \end{equation}
As we shall see shortly, this behaviour is impossible for ergodic systems, and this is enough to seal the fate of WKB for chaotic systems in general (except perhaps as a hacker's tool).
Note, however, that for a given energy level $E$ the discussion so far has been concerned with properties of the classical trajectories on $\Sigma_E$ (where they are constrained to remain by conservation of energy). Now, it belongs to the essence of \qm\ that other parts of phase space than $\Sigma_E$ might be relevant to the spectral properties of $H$ as well. For example, for a classical Hamiltonian of the simple form $h(p,q)=p^2/2m + V(q)$, this concerns the so-called {\it classically forbidden area} $\{q\in\R^n\mid V(q)>E\}$ (and any value of $p$). Here the classical motion can have no properties like integrability or ergodicity, because it does not exist. Nonetheless, and perhaps counterintuitively, it is precisely here that a slight adaptation of the WKB method tends to be most effective. For $q=x$ in the classically forbidden area, the Ansatz \eqref{WKB} should be replaced by
\begin{equation} \Ps(x)=a_{\hbar}(x)e^{-\frac{S(x)}{\hbar}}, \label{WKB2} \end{equation}
where this time $S$ obeys the Hamilton--Jacobi equation `for imaginary time', \footnote{This terminology comes from the Lagrangian formalism, where the classical action $S=\int dt\, L(t)$ is replaced by $iS$ through the substitution
$t=-i\ta$ with $\ta\in\R$.}
i.e.
\begin{equation}
h\left( i\frac{\partial S}{\partial x},x\right)=E, \label{HJE2}
\end{equation} and the transport equation \eqref{TPE} is unchanged. For example, it follows that in one dimension (with a Hamiltonian of the type \eqref{Schreq}) the WKB function \eqref{WKB2} assumes the form \begin{equation}
\Ps(x) \sim e^{-\frac{\sqrt{2m}}{\hbar}\int^{|x|} dy\,\sqrt{V(y)-E}}\end{equation} in the forbidden region, which explains both the tunnel effect in \qm\ (i.e. the propagation of the wave function into the forbidden region) {\it and} the fact that this effect disappears in the limit $\hbar\raw 0$. However, even here the use of WKB methods has now largely been superseded by techniques developed by Agmon (1982); see, for example, Hislop \&\ Sigal (1996) and Dimassi \&\ Sj\"ostrand (1999) for reviews.
\subsection{Epilogue: quantum chaos}\label{QC} Chaos in classical mechanics was probably known to Newton and was famously highlighted by Poincar\'e (1892--1899),\footnote{See also Diacu \&\ Holmes (1996) and Barrow-Green (1997) for historical background.} but its relevance for (and potential threat to) quantum theory was apparently first recognized by Einstein (1917) in a paper that was `completely ignored for 40 years' (Gutzwiller, 1992).\footnote{It was the study of the very same Helium atom that led Heisenberg to believe that a fundamentally new `quantum' mechanics was needed to replace the inadequate old quantum theory of Bohr and Sommerfeld. See Mehra \&\ and Rechenberg (1982b) and Cassidy (1992). Another microscopic example of a chaotic system is the hydrogen atom in an external magnetic field.} Currently, the study of quantum chaos is one of the most thriving businesses in all of physics, as exemplified by innumerable conference proceedings and monographs on the subject, ranging from the classic by Gutzwiller (1990) to the online {\it opus magnum} by Cvitanovic et al.\ (2005).\footnote{Other respectable books include, for example, Guhr, M\"uller-Groeling \&\ Weidenm\"uller (1998), Haake (2001) and Reichl (2004).} Nonetheless, the subject is still not completely understood, and provides a fascinating testing ground for the interplay between classical and \qm.
One should distinguish between various different goals in the field of quantum chaos. The majority of papers and books on quantum chaos is concerned with the semiclassical analysis of some concretely given quantum system having a chaotic system as its classical limit. This means that one tries to approximate (for small $\hbar$) a suitable quantum-mechanical expression in terms of data associated with the underlying classical motion. Michael Berry even described this goal as the ``Holy Grail" of quantum chaos. The methods described in Subsection \ref{CEOM} contribute to this goal, but are largely
independent of the nature of the dynamics. In this subsection we therefore concentrate on techniques and results specific to chaotic motion.
Historically, the first new tool in semiclassical approximation theory that specifically applied to chaotic systems was the so-called {\it Gutzwiller trace formula}.\footnote{This attribution is based on Gutzwiller (1971). A similar result was independently derived by Balian \&\ Bloch (1972, 1974). See also Gutzwiller (1990) and Brack \&\ Bhaduri (2003) for mathematically heuristic but otherwise excellent accounts of semiclassical physics based on the trace formula. Mathematically rigorous discussions and proofs may be found in
Colin de Verdi\`{e}re (1973), Duistermaat \&\ Guillemin (1975), Guillemin \&\ Uribe (1989), Paul \&\ Uribe (1995), and Combescure, Ralston, \&\ Robert (1999). } Roughly speaking, this formula approximates the eigenvalues of the quantum Hamiltonian in terms of the periodic (i.e.\ closed) orbits of the underlying classical Hamiltonian.\footnote{Such orbits are dense but of Liouville measure zero in chaotic classical systems. Their crucial role was first recognized by Poincar\'e (1892--1899).} The Gutzwiller trace formula does not start from the wave function (as the WKB approximation does), but from the {\it propagator} $K(x,y,t)$. Physicists write this as
$K(x,y,t)=\langle x|\exp(-itH/\hbar)|y\rangle$, whereas mathematicians see it as the Green's function in the formula \begin{equation} e^{-\frac{it}{\hbar} H}\Ps(x)=\int d^n y\, K(x,y,t)\Ps(y), \end{equation} where $\Ps\in L^2(\R^n)$. Its (distributional) Laplace transform \begin{equation} G(x,y,E)=\frac{1}{i\hbar}\int_0^{\infty} dt\, K(x,y,t)e^{\frac{itE}{\hbar} } \end{equation} contains information about both the spectrum and the eigenfunctions; for if the former is discrete, one has \begin{equation} G(x,y,E)=\sum_j \frac{\Ps_j(x)\ovl{\Ps_j(y)}}{E-E_j}.\end{equation} It is possible to approximate $K$ or $G$ itself by an expression of the type \begin{equation}
K(x,y,t)\sim (2\pi i\hbar)^{-n/2}\sum_P \sqrt{|\det V_P|}e^{\frac{i}{\hbar}S_P(x,y,t)-\half i\pi \mu_P}, \end{equation} where the sum is over {\it all} classical paths $P$ from $y$ to $x$ in time $t$ (i.e.\ paths that solve the classical equations of motion). Such a path has an associated action $S_P$, Maslov index $\mu_P$, and Van Vleck (1928) determinant $\det V_P$ (Arnold, 1989). For chaotic systems one typically has to include tens of thousands of paths in the sum, but if one does so the ensuing approximation turns out to be remarkably successful (Heller \&\ Tomsovic, 1993; Tomsovic \&\ Heller, 1993). The Gutzwiller trace formula is a semiclassical approximation to \begin{equation} g(E)=\int d^n x\, G(x,x,E)=\sum_j \frac{1}{E-E_j}, \end{equation} for a quantum Hamiltonian with discrete spectrum and underlying classical Hamiltonian having chaotic motion. It has the form \begin{equation} g(E)\sim g_0(E)+ \frac{1}{i\hbar} \sum_P \sum_{k=1}^{\infty} \frac{T_P}{2\sinh(k\ch_P/2)} e^{\frac{ik}{\hbar}S_P(E)-\half i\pi \mu_P},\label{GTF}\end{equation} where $g_0$ is a smooth function giving the mean density of states.
This time, the sum is over all (prime) {\it periodic} paths $P$ of the classical Hamiltonian at energy $E$ with associated action $S_P(E)=\oint pdq$ (where the momentum $p$ is determined by $P$, given $E$), period $T_P$, and stability exponent $\ch_P$ (this is a measure of how rapidly neighbouring trajectories drift away from $P$). Since the frustration expressed by Einstein (1917), this was the first indication that semiclassical approximations had some bearing on chaotic systems.
Another important development concerning energy levels was the formulation of two key conjectures:\footnote{Strictly speaking, both conjectures are wrong; for example, the harmonic oscillator yields a counterexamples to the first one. See Zelditch (1996a) for further information. Nonetheless, the conjectures are believed to be true in a deeper sense.} \begin{itemize} \item If the classical dynamics defined by the classical Hamiltonian $h$ is integrable, then the spectrum of $H$ is ``uncorrelated" or ``random" (Berry \&\ Tabor, 1977). \item If the classical dynamics defined by $h$ is chaotic, then the spectrum of $H$ is ``correlated" or ``regular" (Bohigas, Giannoni, \&\ Schmit, 1984). \end{itemize} The notions of correlation and randomness used here can be made precise using notions like the distribution of level spacings and the pair correlation function of eigenvalues; see Zelditch (1996a) and De Bi\`evre (2001)
for introductory treatments, and most of the literature cited in this subsection for further details.\footnote{This aspect of quantum chaos has applications to number theory and might even lead to a proof of the Riemann hypothesis; see, for example, Sarnak (1999), Berry \&\ Keating (1999), and many other recent papers. Another relevant connection, related to the one just mentioned, is between energy levels and random matrices; see especially Guhr, M\"uller-Groeling \&\ Weidenm\"uller (1998). For the plain relevance of all this to practical physics see Mirlin (2000).}
We now consider energy eigenfunctions instead of eigenvalues, and return to the limit measure \eqref{measure}. In the non (quasi-) integrable case, the key result is that \begin{quote} {\it for ergodic classical motion,\footnote{Ergodicity is the weakest property that any chaotic dynamical system possesses. See Katok \&\ Hasselblatt (1995), Emch \&\ Liu (2002), Gallavotti, Bonetto \&\ Gentile (2004), and countless other books.} the limit measure
$\mu^0_{\mathsf{E}}$ coincides with the (normalized) Liouville measure induced on the constant energy surface $\Sigma_E\equiv h\inv(E)$.}\footnote{The unnormalized Liouville measure $\mu^u_E$ on $\Sigma_E$ is defined by $\mu^u_E(B)=\int_B dS_E(x)\, (\|dh(x)\|)\inv$, where $dS_E$ is the surface element on $\Sigma_E$ and $B\subset \Sigma_E$ is a Borel set. If $\Sigma_E$ is compact, the normalized Liouville measure $\mu_E$ on $\Sigma_E$ is given by $\mu_E(B)=\mu^u_E(B)/\mu^u_E(\Sigma_E)$. It is a probability measure on $\Sigma_E$, reflecting the fact that the eigenvectors $\Ps^{\hbar}_{\mathsf{n}}$ are normalized to unit length so as to define quantum-mechanical states.} \end{quote}
This result was first suggested in the mathematical literature for ergodic geodetic motion on compact hyperbolic Riemannian manifolds (Snirelman, 1974), where it was subsequently proved with increasing generality (Colin de Verdi\`ere, 1985; Zelditch, 1987).\footnote{In the Riemannian case with $\hbar=1$ the cosphere bundle $S^*Q$ (i.e.\ the subbundle of the cotangent bundle $T^*Q$ consisting of one-forms of unit length) plays the role of $\Sigma_E$. Low-dimensional examples of ergodic geodesic motion are provided by compact hyperbolic spaces. Also cf.\ Zelditch (1992a) for the physically important case of a particle moving in an external gauge field. See also the appendix to Lazutkin (1993) by A.I. Shnirelman, and Nadirashvili, Toth, \&\ Yakobson (2001) for reviews.} For certain other ergodic systems this property was proved by Zelditch (1991), G\'erard \&\ Leichtnam (1993), Zelditch \&\ Zworski (1996), and others; to the best of our knowledge a completely general proof remains to be given.
An analogous version for Schr\"odinger operators on $\R^n$ was independently stated in the physics literature (Berry, 1977b, Voros, 1979), and was eventually proved under certain assumptions on the potential by Helffer, Martinez \&\ Robert (1987), Charbonnel (1992), and Paul \&\ Uribe (1995). Under suitable assumptions one therefore has \begin{equation} \lim_{\hbar\raw 0,\mathsf{n}\raw\infty} \left(\Ps^{\hbar}_{\mathsf{n}},\qb(f)\Ps^{\hbar}_{\mathsf{n}}\right)= \int_{\Sigma_E} d\mu_E\, f \label{HMR}\end{equation}
for any $f\in\cci(\R^{2n})$, where again $\mu_E$ is the (normalized) Liouville measure on $\Sigma_E\subset \R^{2n}$ (assuming this space to be compact). In particular, in the ergodic case $\mu_{\mathsf{E}}^0$ only depends on $E^0$ and is the same for (almost) every sequence of energy eigenfunctions $(\Ps^{\hbar}_{\mathsf{n}})$ as long as $E^{\hbar}_n\raw E^0$.\footnote{\label{scarss} The result is not necessarily valid for all sequences $(\Ps^{\hbar}_{\mathsf{n}})$ with the given limiting behaviour, but only for `almost all' such sequences (technically, for a class of sequences of density 1). See, for example, De Bi\`evre (2001) for a simple explanation of this.} Thus the support of the limiting measure is uniformly spread out over the largest part of phase space that is dynamically possible.
The result that for ergodic classical motion $\mu_{\mathsf{E}}^0$ is the Liouville measure on $\Sigma_E$ under the stated condition leaves room for the phenomenon of `scars', according to which in chaotic systems the limiting measure is sometimes concentrated on periodic classical orbits. This terminology is used in two somewhat different ways in the literature. `Strong' scars survive in the limit $\hbar\raw 0$ and concentrate on stable closed orbits;\footnote{An orbit $\gm\subset M$ is called {\it stable} when for each neighbourhood $U$ of $\gm$ there is neighbourhood $V\subset U$ of $\gm$ such that $z(t)\in U$ for all $z\in V$ and all $t$.}
they may come from `exceptional' sequences of eigenfunctions.\footnote{Cf.\ footnote \ref{scarss}.} These are mainly considered in the mathematical literature; cf.\ Nadirashvili, Toth, \&\ Yakobson (2001) and references therein.
In the physics literature, on the other hand, the notion of a scar usually refers to an anomalous concentration of the functions $B^{\hbar}_{\Ps^{\hbar}_{\mathsf{n}}}$ (cf.\ \eqref{BFexp}) near {\it un}stable closed orbits for {\it finite} values of $\hbar$; see Heller \&\ Tomsovic (1993), ÊTomsovic \&\ Heller (1993), Kaplan \&\ Heller (1998a,b), and Kaplan (1999) for surveys. Such scars turn out to be crucial in attempts to explain the energy spectrum of the associated quantum system. The reason why such scars do not survive the (double) limit in \eqref{measure} is that this limit is defined with respect to $\hbar$-independent smooth test functions. Physically, this means that one averages over more and more De Broglie wavelengths as $\hbar\raw 0$, eventually losing information about the single wavelength scale (Kaplan, 1999). Hence to pick them up in a mathematically sound way, one should redefine \eqref{measure} as a pointwise limit (Duclos \&\ Hogreve, 1993, Paul \&\ Uribe, 1996, 1998). In any case, there is no contradiction between the mathematical results cited and what physicists have found.
Another goal of quantum chaos is the identification of chaotic phenomena within a given quantum-mechanical model. Here the slight complication arises that one cannot simply copy the classical definition of chaos in terms of diverging trajectories in phase space, since (by unitarity of time-evolution) in \qm\ $\|\Ps(t)-\Phi(t)\|$ is constant in time $t$ for solutions of the Schr\"odinger equation. However, this just indicates that should intrinsic quantum chaos exist, it has to be defined differently from classical chaos.\footnote{As pointed out by Belot \&\ Earman (1997), the Koopman formulation of classical mechanics (cf.\ footnote \ref{Koopman}) excludes classical chaos if this is formulated in terms of trajectories in \Hs. The transition from classical to quantum notions of chaos can be smoothened by first reformulating the classical definition of chaos (normally put in terms of properties of trajectories in phase space).} This has now been largely accomplished in the algebraic formulation of quantum theory (Benatti, 1993; Emch et al., 1994;, Zelditch, 1996b,c; Belot \&\ Earman, 1997; Alicki \&\ Fannes, 2001; Narnhofer, 2001). The most significant recent development in this direction in the ``heuristic" literature has been the study of the quantity
\begin{equation} M(t)=|(e^{-\frac{it}{\hbar} (H+\Sigma)}\Ps,e^{-\frac{it}{\hbar} H}\Ps)|^2, \end{equation} where $\Ps$ is a coherent state (or Gaussian wave packet), and $\Sg$ is some perturbation of the Hamiltonian $H$ (Peres, 1984). In what is generally regarded as a breakthrough in the field, Jalabert \&\ Pastawski (2001) discovered that in a certain regime $M(t)$ is independent of the detailed form of $\Sg$ and decays as $\sim \exp(-\lm t)$, where $\lm$ is the (largest) Lyapunov exponent of the underlying classical system. See Cucchietti (2004) for a detailed account and further development.
In any case, the possibility that classical chaos appears in the $\hbar\raw 0$ limit of \qm\ is by no means predicated on the existence of intrinsic quantum chaos in the above sense.\footnote{Arguments by Ford (1988) and others to the effect that \qm\ is wrong because it cannot give rise to chaos in its classical limit have to be discarded for the reasons given here. See also Belot \&\ Earman (1997).
In fact, using the same argument, such authors could simultaneously have `proved' the {\it opposite} statement that any classical dynamics that arises as the classical limit of a quantum theory with non-degenerate spectrum must be ergodic. For the naive definition of quantum ergodic flow clearly is that quantum time-evolution sweeps out all states at some energy $E$; but for non-degenerate spectra this is a tautology by definition of an eigenfunction!} For even in the unlikely case that quantum dynamics would turn out to be intrinsically non-chaotic, its classical limit is sufficiently singular to admit kinds of classical motion without a qualitative counterpart in quantum theory. This possibility is not only confirmed by most of the literature on quantum chaos (little of which makes any use of notions of intrinsic quantum chaotic motion), but even more so by the possibility of {\it incomplete} motion. This is a type of dynamics in which the flow of the Hamiltonian vector field is only defined until a certain time $t_f<\infty$ (or from an initial time $t_i>-\infty$), which means that the equations of motion have no solution for $t>t_f$ (or $t<t_i$).\footnote{\label{crunch} The simplest examples are incomplete Riemannian manifolds $Q$ with geodesic flow; within this class, the case $Q=(0,1)$ with flat metric is hard to match in simplicity. Clearly, the particle reaches one of the two boundary points in finite time, and does not know what to do (or even whether its exists) afterwards. Other examples come from potentials $V$ on $Q=\R^n$ with the property that the classical dynamics is incomplete; see Reed \&\ Simon (1975) and Gallavotti (1983). On a somewhat different note, the Universe itself has incomplete dynamics because of the Big Bang and possible Big Crunch.} The point, then, is that unitary quantum dynamics, though intrinsically complete, may very well have incomplete motion as its classical limit.\footnote{\label{crunch2} The quantization of the Universe is unknown at present, but geodesic motion on Riemannian manifolds, complete or not, is quantized by $H=-\frac{\hbar^2}{2m}\Delta$ (perhaps with an additonal term proportional to the Ricci scalar $R$, see Landsman (1998)), where $\Delta$ is the Laplacian, and quantization on $Q=\R^n$ is given by the Schr\"{o}dinger equation \eqref{Schreq}, whether or not the classical dynamics is complete. In these two cases, and probably more generally, the incompleteness of the classical motion is often (but not always) reflected by the lack of essential self-adjointness of the quantum Hamiltonian on its natural initial domain $\cci(Q)$. For example, if $Q$ is complete as a Riemannian manifold, then $\Delta$ is essentially self-adjoint on $\cci(Q)$ (Chernoff, 1973, Strichartz, 1983), and if $Q$ is incomplete then the Laplacian usually fails to be essentially self-adjoint on this domain (but see Horowitz \&\ Marolf (1995) for counterexamples). One may refer to the latter property as quantum-mechanical incompleteness (Reed \&\ Simon, 1975), although a Hamiltonian that fails to be essentially self-adjoint on $\cci(Q)$ can often be extended (necessarily in a non-unique way) to a self-adjoint operator by a choice of boundary conditions (possibly at infinity). By Stone's theorem, the quantum dynamics defined by each self-adjoint extension is unitary (and therefore defined for all times). Similarly, although no general statement can be made relating (in)complete classical motion in a potential to (lack of) essential selfadjointness of the corresponding Schr\"odinger operator, it is usually the case that completeness implies essential selfadjointness, and vice versa. See Reed \&\ Simon (1975), Appendix to \S X.1, where the reader may also find examples of classically incomplete but quantum-mechanically complete motion, and vice versa. Now, here is the central point for the present discussion: as probably first noted by Hepp (1974), {\it different self-adjoint extensions have the same classical limit} (in the sense of \eqref{hepp} or similar criteria), namely the given {\it incomplete} classical dynamics. This proves that complete quantum dynamics can have incomplete motion as its classical limit. However, much remains to be understood in this area. See also Earman (2005, 2006).}
\section{The limit $N\raw\infty$}\label{S6}\setcounter{equation}{0} In this section we show to what extent classical physics may approximately emerge from quantum theory when the size of a system becomes large. Strictly classical behaviour would be an idealization reserved for the limit where this size is infinite, which we symbolically denote by ``$\lim N\raw\infty$". As we shall see, mathematically speaking this limit is a special case of the limit $\hbar\raw 0$ discussed in the previous chapter. What is more, we shall show that formally the limit $N\raw\infty$ even falls under the heading of continuous fields of \ca s and deformation quantization (see Subsection \ref{DQsection}.) Thus the `philosophical' nature of the idealization involved in assuming that a system is infinite is much the same as that of assuming $\hbar\raw 0$ in a quantum system of given (finite) size; in particular, the introductory comments in Section \ref{S1} apply here as well.
An analogous discussion pertains to the derivation of thermodynamics from statistical mechanics (Emch \&\ Liu, 2002; Batterman, 2005). For example, {\it in theory} phase transitions only occur in infinite systems, but {\it in practice} one sees them every day. Thus it appears to be valid to approximate a pot of $10^{23}$ boiling water molecules by an infinite number of such molecules. The basic point is that the distinction between microscopic and macroscopic regimes is unsharp unless one admits infinite systems as an idealization, so that one can simply say that microscopic systems are finite, whereas macroscopic systems are infinite. This procedure is eventually justified by the results it produces.
Similarly, in the context of quantum theory classical behaviour is simply not found in finite systems (when $\hbar>0$ is fixed), whereas, as we shall see, it {\it is} found in infinite ones. Given the observed classical nature of the macroscopic world,\footnote{With the well-known mesoscopic exceptions (Leggett, 2002; Brezger et al., 2002; Chiorescu et al., 2003; Marshall et al., 2003; Devoret et al., 2004). } at the end of the day one concludes that the idealization in question is apparently a valid one. One should not be confused by the fact that the error in the number of particles this approximation involves (viz.\ $\infty-10^{23}=\infty$) is considerably larger than the number of particles in the actual system. If all of the $10^{23}$ particles in question were {\it individually} tracked down, the approximation is indeed a worthless ones, but the point is rather that the limit $N\raw\infty$ is valid whenever {\it averaging} over $N=10^{23}$ particles is well approximated by averaging over an arbitrarily larger number $N$ (which, then, one might as well let go to infinity). Below we shall give a precise version of this argument.
Despite our opening comments above, the quantum theory of infinite systems has features of its own that deserve a separate section. Our treatment is complementary to texts such as Thirring (1983), Strocchi (1985), Bratteli \&\ Robinson (1987), Haag (1992), Araki (1999), and Sewell (1986, 2002), which should be consulted for further information on infinite quantum systems. The theory in Subsections \ref{MO} and \ref{PSD} is a reformulation in terms of continuous field of \ca s and deformation quantization of the more elementary parts of a remarkable series of papers on so-called quantum mean-field systems by Raggio \&\ Werner (1989, 1991), Duffield \&\ Werner (1992a,b,c), and Duffield, Roos, \&\ Werner (1992). These models have their origin in the treatment of the BCS theory of superconductivity due to Bogoliubov (1958) and Haag (1962), with important further contributions by Thirring \&\ Wehrl (1967), Thirring (1968), Hepp (1972), Hepp \&\ Lieb (1973), Rieckers (1984), Morchio \&\ Strocchi (1987), Duffner \&\ Rieckers (1988), Bona (1988, 1989, 2000), Unnerstall (1990a, 1990b), Bagarello \&\ Morchio (1992), Sewell (2002), and others.
\subsection{Macroscopic observables}\label{MO} The large quantum systems we are going to study consist of $N$ copies of a single quantum system with unital algebra of observables $\CA_1$. Almost all features already emerge in the simplest example $\CA_1=M_2(\C)$ (i.e.\ the complex $2\x 2$ matrices), so there is nothing wrong with having this case in mind as abstraction increases.\footnote{In the opposite direction of greater generality, it is worth noting that the setting below actually incorporates quantum systems defined on general lattices in $\R^n$ (such as $\mathbb{Z}^n$). For one could relabel things so as to make $\CA_{1/N}$ below the algebra of observables of all lattice points $\Lm$ contained in, say, a sphere of radius $N$. The limit $N\raw\infty$ then corresponds to the limit $\Lm\raw\mathbb{Z}^n$.} The aim of what follows is to describe in what precise sense macroscopic observables (i.e.\ those obtained by averaging over an infinite number of sites) are ``classical".
From the single \ca\ $\CA_1$, we construct a continuous field of \ca s $\CA^{\mathrm (c)}$ over \begin{equation} I=0\cup 1/\N=\{0,\ldots, 1/N,\ldots, \third,\half,1\}\subset [0,1], \label{interval}\end{equation} as follows.
We put \begin{eqnarray} \CA_0^{\mathrm (c)}&=& C(\CS(\CA_1));\nn \\ \CA_{1/N}^{\mathrm (c)}&=& \CA_1^N, \label{fibers} \end{eqnarray} where $\CS(\CA_1)$ is the state space of $\CA_1$ (equipped with the weak$\mbox{}^*$-topology)\footnote{In this topology one has $\om_{\lm}\raw\om$ when $\om_{\lm}(A)\raw\om(A)$ for each $A\in\CA_1$.}
and $\CA_1^N=\hat{\otimes}^N \CA_1$ is
the (spatial) tensor product of $N$ copies of $\CA_1$.\footnote{When $\CA_1$ is finite-dimensional the tensor product is unique. In general, one needs the {\it projective} tensor product at this point. See footnote \ref{tensorproducts}. The point is the same here: any tensor product state $\om_1\otimes\cdots\otimes\om_N$ on $\otimes^N\CA_1$ - defined on elementary tensors by $\om_1\otimes\cdots \otimes\om_N(A_1\otimes\cdots\otimes A_N)=\om_1(A_1)\cdots \om_N(A_N)$ - extends to a state on $\hat{\otimes}^N \CA_1$ by continuity. } This explains the suffix $c$ in $\CA^{\mathrm (c)}$: it refers to the fact that the limit algebra $\CA_0^{\mathrm (c)}$ is {\it c}lassical or
{\it c}ommutative.
For example, take $\CA_1=M_2(\C)$. Each state is given by a density matrix, which is of the form \begin{equation} \rh(x,y,z)=\half \left(\begin{array}{cc} 1+z & x-iy \\ x+iy & 1-z \end{array}\right), \label{gens2} \end{equation} for some $(x,y,z)\in\R^3$ satisfying $x^2+y^2+z^2\leq 1$. Hence $\CS(M_2(\C))$ is isomorphic (as a compact convex set) to the three-ball $B^3$ in $\R^3$. The pure states are precisely the points on the boundary,\footnote{\label{EBfn} The {\it extreme boundary} $\partial_e K$ of a convex set $K$ consists of all $\om\in K$ for which $\om=p\rh+(1-p)\sg$ for some $p\in (0,1)$ and $\rh,\sg\in K$ implies $\rh=\sg=\om$. If $K=\CS(\CA)$ is the state space of a \ca\ $\CA$, the extreme boundary consists of the pure states on $\CA$ (the remainder of $\CS(\CA)$ consisting of mixed states). If $K$ is embedded in a vector space, the extreme boundary $\partial_e K$ may or may not coincide with the geometric boundary $\partial K$ of $K$. In the case $K=B^3\subset \R^3$ it does, but for an equilateral triangle in $\R^2$ it does not, since $\partial_e K$ merely consists of the corners of the triangle whereas the geometric boundary includes the sides as well.}
i.e.\ the density matrices for which $x^2+y^2+z^2=1$ (for these and these alone define one-dimensional projections).\footnote{\label{SSlemma}Eq.\ \eqref{gens2} has the form $\rh(x,y,z)=\half(x\sg_x+y\sg_y+z\sg_z)$, where the $\sg_i$ are the Pauli matrices. This yields an isomorphism between $\R^3$ and the Lie algebra of $SO(3)$ in its spin-$\half$ \rep\ $\mathcal{D}_{1/2}$ on $\C^2$. This isomorphism intertwines the defining action of $SO(3)$ on $\R^3$ with its adjoint action on $M_2(\C)$. I.e., for any rotation $R$ one has $\rh(R\mathbf{x})=\mathcal{D}_{1/2}(R)\rh(\mathbf{x})\mathcal{D}_{1/2}(R)\inv$. This will be used later on (see Subsection \ref{PSD}). }
In order to define the continuous sections of the field, we introduce the {\it symmetrization maps} $j_{NM}: \CA^M_1\raw \CA^N_1$, defined by \begin{equation} j_{NM}(A_M)=S_N(A_M\otimes 1\otimes\cdots \otimes 1), \label{symmaps} \end{equation} where one has $N-M$ copies of the unit $1\in\CA_1$ so as to obtain an element of $\CA_1^N$. The symmetrization operator $S_N: \CA^N_1\raw \CA^N_1$ is given by (linear and continuous) extension of \begin{equation} S_N(B_1\otimes\cdots \otimes B_N)=\frac{1}{N!}\sum_{\sg\in \GS_N} B_{\sg(1)}\otimes\cdots\otimes B_{\sg(N)}, \label{landc} \end{equation} where $\GS_N$ is the permutation group (i.e.\ symmetric group) on $N$ elements and $B_i\in\CA_1$ for all $i=1,\ldots,N$. For example, $j_{N1}:\CA_1\raw\CA_1^N$ is given by \begin{equation} j_{N1}(B)= \ovl{B}^{(N)}=\frac{1}{N}\sum_{k=1}^N 1\otimes\cdots\otimes B_{(k)}\otimes 1\cdots \ot1,\end{equation} where $B_{(k)}$ is $B$ seen as an element of the $k$'th copy of $\CA_1$ in $\CA_1^N$. As our notation $\ovl{B}^{(N)}$ indicates, this is just the `average' of $B$ over all copies of $\CA_1$. More generally, in forming $j_{NM}(A_M)$ an operator $A_M\in\CA_1^M$ that involves $M$ sites is averaged over $N\geq M$ sites. When $N\raw\infty$ this means that one forms a {\it macroscopic} average of an $M$-particle operator.
We say that a sequence $A=(A_1,A_2,\cdots)$ with $A_N\in\CA_1^N$ is {\it symmetric} when \begin{equation} A_N=j_{NM}(A_M) \label{ass} \end{equation} for some fixed $M$ and all $N\geq M$. In other words, the tail of a symmetric sequence entirely consists of `averaged' or `intensive' observables, which become macroscopic in the limit $N\raw\infty$. Such sequences have the important property that they commute in this limit; more precisely, if $A$ and $A'$ are symmetric sequences, then \begin{equation}
\lim_{N\raw\infty} \| A_NA_N'-A_N'A_N\|=0. \label{aprc} \end{equation} As an enlightening special case we take $A_N=j_{N1}(B)$ and $A_N'=j_{N1}(C)$ with $B,C\in\CA_1$. One immediately obtains from the relation $[B_{(k)},C_{(l)}]=0$ for $k\neq l$
that \begin{equation} \left[\ovl{B}^{(N)},\ovl{C}^{(N)}\right]=\frac{1}{N}\ovl{\left[B,C\right]}^{(N)}. \label{av0} \end{equation} For example, if $\CA_1=M_2(\C)$ and if for $B$ and $C$ one takes the spin-$\half$ operators $S_j=\frac{\hbar}{2}\sg_j$ for $j=1,2,3$ (where $\sg_j$ are the Pauli matrices), then \begin{equation} \left[\ovl{S}_j^{(N)},\ovl{S}_k^{(N)}\right]=i\frac{\hbar}{N}\epsilon_{jkl} \ovl{S}_l^{(N)}. \end{equation} This shows that averaging one-particle operators leads to commutation relations formally like those of the one-particle operators in question, but with Planck's constant $\hbar$ replaced by a variable $\hbar/N$. For constant $\hbar=1$ this leads to the interval \eqref{interval} over which our continuous field of \ca s is defined; for any other constant value of $\hbar$ the field would be defined over $I=0\cup \hbar/\N$, which of course merely changes the labeling of the \ca s in question.
We return to the general case, and denote a section of the field with fibers \eqref{fibers} by a sequence $A=(A_0,A_1,A_2,\cdots)$, with $A_0\in\CA_0^{\mathrm (c)}$ and $A_N\in\CA_1^N$ as before (i.e.\ the corresponding section is $0\mapsto A_0$ and $1/N\mapsto A_N$).
We then complete the definition of our continuous field by declaring that a sequence $A$ defines a {\it continuous} section iff: \begin{itemize} \item $(A_1,A_2,\cdots)$ is {\it approximately symmetric}, in the sense that for any $\varep>0$ there is an $N_{\varep}$ and a symmetric sequence $A'$ such that
$\|A_N-A_N'\|< \varep$ for all $N\geq N_{\varep}$;\footnote{A symmetric sequence is evidently approximately symmetric.} \item $A_0(\om)=\lim_{N\raw\infty}\om^N(A_N)$, where $\om\in\CS(\CA_1)$ and $\om^N\in \CS(\CA_1^N)$ is the tensor product of $N$ copies of $\om$, defined by (linear and continuous) extension of \begin{equation} \om^N(B_1\otimes\cdots\otimes B_N)=\om(B_1)\cdots\om(B_N).\label{omN}\end{equation}
This limit exists by definition of an approximately symmetric sequence.\footnote{If $(A_1,A_2,\cdots)$ is symmetric with \eqref{ass}, one has $\om^N(A_N)=\om^M(A_M)$ for $N>M$, so that the tail of the sequence $(\om^N(A_N))$ is even independent of $N$. In the approximately symmetric case one easily proves that $(\om^N(A_N))$ is a Cauchy sequence.} \end{itemize}
It is not difficult to prove that this choice of continuous sections indeed defines a continuous field of \ca s over $I=0\cup 1/\N$ with fibers \eqref{fibers}. The main point is that \begin{equation}
\lim_{N\raw\infty} \| A_N\|=\|A_0\| \label{normeq} \end{equation} whenever $(A_0,A_1,A_2,\cdots)$ satisfies the two conditions above.\footnote{\label{Landsmanfootnote}Given \eqref{normeq}, the claim follows from Prop.\ II.1.2.3 in Landsman (1998) and the fact that the set of functions $A_0$ on $\CS(\CA_1)$ arising in the said way are dense in $C(\CS(\CA_1))$ (equipped with the supremum-norm). This follows from the Stone--Weierstrass theorem, from which one infers that the functions in question even exhaust $\CS(\CA_1)$. }
This is easy to show for symmetric sequences,\footnote{
Assume \eqref{ass}, so that $\|A_N\|=\|j_{NN}(A_N)\|$ for $N\geq M$. By the $C^*$-axiom $\|A^*A\|=\|A^2\|$ it suffices to prove \eqref{normeq} for $A_0^*=A_0$, which implies $A_M^*=A_M$ and hence $A_N^*=A_N$ for all $N\geq M$. One then has $\| A_N\|=\sup\{|\rh(A_N)|, \rh\in\CS(\CA_1^N)\}$. Because of the special form of $A_N$ one may replace the supremum over the set $\CS(\CA_1^N)$ of all states on $\CA_1^N$ by the supremum over the set $\CS^p(\CA_1^N)$ of all permutation invariant states, which in turn may be replaced by the supremum over the extreme boundary $\partial \CS^p(\CA_1^N)$ of $\CS^p(\CA_1^N)$. It is well known (St\o rmer, 1969; see also Subsection \ref{QLO}) that the latter consists of all states of the form $\rh=\om^N$, so that
$\| A_N\|=\sup\{|\om^N(A_N)|, \om\in\CS(\CA_1)\}$. This is actually equal to $\| A_M\|=\sup\{|\om^M(A_M)|\}$. Now the norm in $\CA_0^{\mathrm (c)}$ is $\|A_0\|=\sup\{|A_0(\om)|, \om\in\CS(\CA_1)\}$, and by definition of $A_0$ one has $A_0(\om)=\om^M(A_M)$. Hence
\eqref{normeq} follows.} and follows from this for approximately symmetric ones.
Consistent with \eqref{aprc}, we conclude that in the limit $N\raw\infty$ the macroscopic observables organize themselves in a commutative \ca\ isomorphic to $C(\CS(\CA_1))$. \subsection{Quasilocal observables}\label{QLO} In the \ca ic approach to quantum theory, infinite systems are usually described by means of inductive limit \ca s and the associated quasilocal observables (Thirring, 1983; Strocchi, 1985; Bratteli \&\ Robinson, 1981, 1987; Haag, 1992; Araki, 1999; Sewell, 1986, 2002). To arrive at these notions in
the case at hand, we proceed as follows (Duffield \&\ Werner, 1992c).
A sequence $A=(A_1,A_2,\cdots)$ (where $A_N\in\CA_1^N$, as before) is called {\it local} when for some fixed $M$ and all $N\geq M$ one has $A_N=A_M\otimes 1\otimes\cdots\otimes 1$
(where one has $N-M$ copies of the unit $1\in\CA_1$); cf.\ \eqref{symmaps}. A sequence is said to be {\it quasilocal} when for any $\varep>0$ there is an $N_{\varep}$ and a local sequence $A'$ such that
$\|A_N-A_N'\|< \varep$ for all $N\geq N_{\varep}$. On this basis, we define
the {\it inductive limit} \ca\ \begin{equation} \ovl{\cup_{N\in\N} \CA_1^N}\label{ILCA} \end{equation} of the family of \ca s $(\CA_1^N)$ with respect to the inclusion maps $\CA_1^N\hookrightarrow\CA_1^{N+1}$ given by $A_N\mapsto A_N\otimes 1$. As a set, \eqref{ILCA} consists of all equivalence classes $[A]\equiv A_0$ of quasilocal sequences $A$ under the equivalence relation $A\sim B$ when $\lim_{N\raw\infty}\|A_N-B_N\|=0$. The norm on $\ovl{\cup_{N\in\N} \CA_1^N}$ is
\begin{equation} \| A_0\|=\lim_{N\raw\infty} \|A_N\|, \label{normput}\end{equation} and the rest of the \ca ic structure is inherited from the quasilocal sequences in the obvious way (e.g., $A_0^*=[A^*]$ with $A^*=(A_1^*,A_2^*,\cdots)$, etc.). As the notation suggests, each $\CA_1^N$ is contained in $\ovl{\cup_{N\in\N} \CA_1^N}$ as a $C^*$-subalgebra by identifying $A_N\in \CA_1^N$ with the local (and hence quasilocal) sequence $A=(0,\cdots,0, A_N\otimes 1,A_N\otimes 1\ot1, \cdots)$, and forming its equivalence class $A_0$ in $\ovl{\cup_{N\in\N} \CA_1^N}$ as just explained.\footnote{Of course, the entries $A_1,\cdots A_{N-1}$, which have been put to zero, are arbitrary.} The assumption underlying the common idea that \eqref{ILCA} is ``the" algebra of observables of the infinite system under study is that by locality or some other human limitation the infinite tail of the system is not accessible, so that the observables must be arbitrarily close (i.e.\ in norm) to operators of the form $A_N\otimes 1\ot1, \cdots$ for some {\it finite} $N$.
This leads us to a second continuous field of \ca s $\CA^{\mathrm (q)}$ over $0\cup 1/\N$, with fibers
\begin{eqnarray} \CA^{\mathrm (q)}_0&=& \ovl{\cup_{N\in\N} \CA_1^N};\nn \\ \CA^{\mathrm (q)}_{1/N}&=& \CA_1^N. \label{fibers2} \end{eqnarray} Thus the suffix $q$ reminds one of that fact that the limit algebra $\CA^{\mathrm (q)}_0$ consists of {\it q}uasilocal or {\it q}uantum-mechanical observables. We equip the collection of \ca s \eqref{fibers2} with the structure of a
continuous field of \ca s $\CA^{\mathrm (q)}$ over $0\cup 1/\N$ by declaring that the continuous sections are of the form $(A_0,A_1,A_2,\cdots)$
where $(A_1,A_2,\cdots)$ is quasilocal and $A_0$ is defined by this quasilocal sequence as just explained.\footnote{The fact that this defines a continuous field follows from \eqref{normput} and Prop.\ II.1.2.3 in Landsman (1998); cf.\ footnote \ref{Landsmanfootnote}.} For $N<\infty$ this field has the same fibers \begin{equation} \CA^{\mathrm (q)}_{1/N}=\CA_{1/N}^{\mathrm (c)}= \CA_1^N\end{equation}
as the continuous field $\CA$ of the previous subsection, but the fiber $\CA^{\mathrm (q)}_0$ is completely different from $\CA_0^{\mathrm (c)}$. In particular, if $\CA_1$ is noncommutative then so is $\CA^{\mathrm (q)}_0$, for it contains all $\CA_1^N$.
The relationship between the continuous fields of \ca s $\CA^{\mathrm (q)}$ and $\CA^{\mathrm (c)}$ may be studied in two different (but related) ways. First, we may construct concrete \rep s of all \ca s $\CA_1^N$, $N<\infty$, as well as of $\CA_0^{\mathrm (c)}$ and $\CA_0^{\mathrm (q)}$ on a single \Hs; this approach leads to superselections rules in the traditional sense. This method will be taken up in the next subsection. Second, we may look at those families of states $(\om_1,\om_{1/2},\cdots,\om_{1/N},\cdots)$ (where $\om_{1/N}$ is a state on $\CA_1^N$) that admit limit states $\om_0^{\mathrm (c)}$ {\it and} $\om_0^{\mathrm (q)}$ on $\CA_0^{\mathrm (c)}$ and $\CA_0^{\mathrm (q)}$, respectively, such that the ensuing families of states $(\om_0^{\mathrm (c)},\om_1,\om_{1/2},\cdots)$ and $(\om_0^{\mathrm (q)},\om_1,\om_{1/2},\cdots)$ are {\it continuous} fields of states on $\CA^{\mathrm (c)}$ and on $\CA^{\mathrm (q)}$, respectively (cf.\ the end of Subsection \ref{CSR}).
Now, any state $\om_0^{\mathrm (q)}$ on $\CA^{\mathrm (q)}_0$ defines a state
$\om_{0|1/N}^{\mathrm (q)}$ on $\CA_1^N$ by restriction, and the ensuing field of states on $\CA^{\mathrm (q)}$ is clearly continuous. Conversely, any continuous field $(\om_0^{\mathrm (q)},\om_1,\om_{1/2},\ldots, \om_{1/N},\ldots)$ of states on $\CA^{\mathrm (q)}$ becomes arbitrarily close to a field of the above type for $N$ large.\footnote{ For any fixed quasilocal sequence $(A_1,A_2,\cdots)$
and $\varepsilon>0$, there is an $N_{\varepsilon}$ such that $|\om_{1/N}(A_N)-
\om_{0|1/N}^{\mathrm (q)}(A_N)|<\varepsilon$ for all $N>N_{\varepsilon}$.}
However, the restrictions $\om_{0|1/N}^{\mathrm (q)}$ of a given state $\om_0^{\mathrm (q)}$ on $\CA^{\mathrm (q)}_0$ to $\CA_1^N$ may not converge to a state $\om_0^{\mathrm (c)}$ on $\CA_0^{\mathrm (c)}$ for $N\raw\infty$.\footnote{See footnote \ref{259} below for an example}. States $\om_{0}^{\mathrm (q)}$ on
$\ovl{\cup_{N\in\N} \CA_1^N}$ that do have this property will here be called {\it classical}. In other words, $\om_{0|1/N}^{\mathrm (q)}$ is classical when there exists a probability measure $\mu_0$ on $\CS(\CA_1)$ such that \begin{equation}
\lim_{N\raw\infty} \int_{\CS(\CA_1)} d\mu_0(\rh)\, (\rh^N(A_N)-\om_{0|1/N}^{\mathrm (q)}(A_N))=0 \label{PMOQ}\end{equation} for each (approximately) symmetric sequence $(A_1, A_2,\ldots)$.
To analyze this notion we need a brief intermezzo on general \ca s and their \rep s. \begin{itemize} \item A {\it folium} in the state space $\CS(\CB)$ of a \ca\ $\CB$
is a convex, norm-closed subspace $\CF$ of $\CS(\CB)$ with the property that if $\om\in\CF$ and $B\in\CB$ such that $\om(B^*B)>0$, then the ``reduced" state $\om_B:A\mapsto \om(B^*AB)/\om(B^*B)$ must be in $\CF$ (Haag, Kadison, \&\ Kastler, 1970).\footnote{See also Haag (1992). The name `folium' is very badly chosen, since $\CS(\CB)$ is by no means foliated by its folia; for example, a folium may contain subfolia. } For example, if $\pi$ is a \rep\ of $\CB$ on a \Hs\ ${\mathcal H}$, then the set of all density matrices on ${\mathcal H}$ (i.e.\ the $\pi$-normal states on $\CB$)\footnote{A state $\om$ on $\CB$ is called $\pi$-normal when it is of the form $\om(B)=\Tr \rh\pi(B)$ for some density matrix $\rh$. Hence the $\pi$-normal states are the normal states on the von Neumann algebra $\pi(\CB)''$.} comprises a folium $\CF_{\pi}$. In particular, each state $\om$ on $\CB$ defines a folium $\CF_{\om}\equiv \CF_{\pi_{\om}}$ through its GNS-\rep\ $\pi_{\om}$. \item Two \rep s $\pi$ and $\pi'$ are called {\it disjoint}, written $\pi\bot \pi'$, if no sub\rep\ of $\pi$ is (unitarily) equivalent to a sub\rep\ of $\pi'$ and vice versa. They are said to be {\it quasi-equivalent}, written $\pi\sim \pi'$, when $\pi$ has no sub\rep\ disjoint from $\pi'$, and vice versa.\footnote{Equivalently, two \rep s $\pi$ and $\pi'$ are disjoint iff no $\pi$-normal state is $\pi'$-normal and vice versa, and quasi-equivalent iff each $\pi$-normal state is $\pi'$-normal and vice versa.} Quasi-equivalence is an equivalence relation $\sim$ on the set of \rep s. See Kadison \&\ Ringrose (1986), Ch.\ 10. \item Similarly, two states $\rh,\sg$ are called either quasi-equivalent ($\rh\sim\sg$) or disjoint ($\rh\bot\sg$) when the corresponding GNS-\rep s have these properties. \item A state $\om$ is called {\it primary} when the corresponding von Neumann algebra $\pi_{\om}(\CB)''$ is a factor.\footnote{A von Neumann algebra ${\mathcal M}$ acting on a \Hs\ is called a {\it factor} when its center ${\mathcal M}\cap{\mathcal M}'$ is trivial, i.e.\ consists of multiples of the identity.} Equivalently, $\om$ is primary iff each sub\rep\ of $\pi_{\om}(\CB)$ is quasi-equivalent to $\pi_{\om}(\CB)$, which is the case iff $\pi_{\om}(\CB)$ admits no (nontrivial) decomposition as the direct sum of two disjoint sub\rep s. \end{itemize}
Now, there is a bijective correspondence between folia in $\CS(\CB)$ and quasi-equivalence classes of \rep s of $\CB$, in that $\CF_{\pi}=\CF_{\pi'}$ iff $\pi\sim\pi'$. Furthermore (as one sees from the GNS-construction), any folium $\CF\subset \CS(\CB)$ is of the form $\CF=\CF_{\pi}$ for some \rep\ $\pi(\CB)$. Note that if $\pi$ is injective (i.e.\ faithful), then the corresponding folium is dense in $\CS(\CB)$ in the weak$\mbox{}^*$-topology by Fell's Theorem. So in case that $\CB$ is simple,\footnote{In the sense that it has no {\it closed} two-sided ideals. For example, the matrix algebra $M_n(\C)$ is simple for any $n$, as is its infinite-dimensional analogue, the \ca\ of all compact operators on a \Hs. The \ca\ of quasilocal observables of an infinite quantum systems is typically simple as well.}
any folium is weak$\mbox{}^*$-dense in the state space.
Two states need not be either disjoint or quasi-equivalent. This dichotomy does apply, however, within the class of primary states.
Hence {\it two primary states are either disjoint or quasi-equivalent}. If $\om$ is primary, then each state in the folium of $\pi_{\om}$ is primary as well, and is quasi-equivalent to $\om$. If, on the other hand, $\rh$ and $\sg$ are primary and disjoint, then $\CF_{\rh}\cap\CF_{\sg}=\emptyset$. Pure states are, of course, primary.\footnote{Since the corresponding GNS-\rep\ $\pi_{\om}$ is irreducible, $\pi_{\om}(\CB)''=\CB({\mathcal H}_{\om})$ is a factor.} Furthermore, in thermodynamics pure phases are described by primary KMS states (Emch \&\ Knops, 1970;
Bratteli \&\ Robinson, 1981; Haag, 1992; Sewell, 2002). This apparent relationship between
primary states and ``purity" of some sort is confirmed by
our description of macroscopic observables:\footnote{These claims easily follow from Sewell (2002), \S 2.6.5, which in turn relies on Hepp (1972).} {\it \begin{itemize} \item If $\om_0^{\mathrm (q)}$ is a classical primary state on $\CA^{\mathrm (q)}_0=\ovl{\cup_{N\in\N} \CA_1^N}$, then the corresponding limit state $\om_0^{\mathrm (c)}$ on $\CA_0^{\mathrm (c)}=C(\CS(\CA_1))$ is pure (and hence given by a point in $\CS(\CA_1)$). \item If $\rh_0^{\mathrm (q)}$ and $\sg_0^{\mathrm (q)}$ are classical primary states on $\CA^{\mathrm (q)}_0$, then \begin{eqnarray} \rh_0^{\mathrm (c)}=\sg_0^{\mathrm (c)} &\Leftrightarrow& \rh_0^{\mathrm (q)}\sim \sg_0^{\mathrm (q)}; \label{rssim} \\ \rh_0^{\mathrm (c)}\neq\sg_0^{\mathrm (c)} &\Leftrightarrow& \rh_0^{\mathrm (q)} \bot\, \sg_0^{\mathrm (q)}. \label{rsbot} \end{eqnarray} \end{itemize}}
As in \eqref{PMOQ}, a general classical state $\om_0^{\mathrm (q)}$ with limit state $\om_0^{\mathrm (c)}$ on $C(\CS(\CA_1))$ defines a probability measure $\mu_0$ on $\CS(\CA_1)$ by \begin{equation}
\om_0^{\mathrm (c)}(f)=\int_{\CS(\CA_1)} d\mu_0\, f, \label{probm} \end{equation} which describes the probability distribution of the macroscopic observables in that state. As we have seen, this distribution is a delta function for primary states. In any case, it is insensitive to the microscopic details of $\om_0^{\mathrm (q)}$ in the sense that local modifications of $\om_0^{\mathrm (q)}$ do not affect the limit state $\om_0^{\mathrm (c)}$ (Sewell, 2002). Namely, it easily follows from \eqref{aprc} and the fact that the GNS-\rep\ is cyclic that one can strengthen the second claim above: \begin{quote} {\it Each state in the folium $\CF_{\om_0^{\mathrm (q)}}$ of a classical state $\om_0^{\mathrm (q)}$ is automatically classical and has the same limit state on $\CA_0^{\mathrm (c)}$ as $\om_0^{\mathrm (q)}$.} \end{quote}
To make this discussion a bit more concrete, we now identify an important class of classical states on $\ovl{\cup_{N\in\N} \CA_1^N}$. We say that a state $\om$ on this \ca\ is {\it permutation-invariant} when each of its restrictions to $\CA_1^N$ is invariant under the natural action of the symmetric group $\GS_N$ on $\CA_1^N$ (i.e.\ $\sg\in\GS_N$ maps an elementary tensor $A_N=B_1\otimes\cdots\otimes B_N\in\CA_1^N$ to $B_{\sg(1)}\otimes\cdots\otimes B_{\sg(N)}$, cf.\ \eqref{landc}). The structure of the set $\CS^{\GS}$ of all permutation-invariant states in $\CS(\CA^{\mathrm (q)}_0)$ has been analyzed by St\o rmer (1969). Like any compact convex set, it is the (weak$\mbox{}^*$-closed) convex hull of its extreme boundary $\partial_e \CS^{\GS}$. The latter consists of all infinite product states $\om= \rh^{\infty}$, where $\rh\in\CS(\CA_1)$. I.e.\ if $A_0\in \CA^{\mathrm (q)}_0$ is an equivalence class $[A_1,A_2,\cdots]$, then \begin{equation} \rh^{\infty}(A_0)=\lim_{N\raw\infty} \rh^N(A_N);\end{equation} cf.\ \eqref{omN}.
Equivalently, the restriction of $\om$ to any $\CA_1^N\subset \CA^{\mathrm (q)}_0$ is given by $\otimes^N \rh$. Hence $\partial_e \CS^{\GS}$ is isomorphic (as a compact convex set) to $\CS(\CA_1)$ in the obvious way, and the primary states in $\CS^{\GS}$ are precisely the elements of $\partial_e \CS^{\GS}$.
A general state $\om^{\mathrm (q)}_0$ in $\CS^{\GS}$ has a unique decomposition\footnote{This follows because $\CS^{\GS}$ is a so-called Bauer simplex (Alfsen, 1970). This is a compact convex set $K$ whose extreme boundary $\partial_e K$ is closed and for which every $\om\in K$ has a {\it unique} decomposition as a probability measure supported by $\partial_e K$, in the sense that $a(\om)=\int_{\partial_e K} d\mu(\rh)\, a(\rh)$ for any continuous affine function $a$ on $K$. For a unital \ca\ $\CA$ the continuous affine functions on the state space $K=\CS(\CA)$ are precisely the elements $A$ of $\CA$, reinterpreted as functions $\hat{A}$ on $\CS(\CA)$ by $\hat{A}(\om)=\om(A)$. For example, the state space $\CS(\CA)$ of a commutative unital \ca\ $\CA$ is a
Bauer simplex, which consists of all (regular Borel) probability measures
on the pre state space $\CP(\CA)$.} \begin{equation} \om^{\mathrm (q)}_0(A_0)=\int_{\CS(\CA_1)} d\mu(\rh)\, \rh^{\infty}(A_0), \label{Unn} \end{equation} where $\mu$ is a probability measure on $\CS(\CA_1)$ and $A_0\in \CA^{\mathrm (q)}_0$.\footnote{ This is a quantum analogue of De Finetti's \rep\ theorem in classical probability theory (Heath \&\ Sudderth, 1976; van Fraassen, 1991); see also Hudson \&\ Moody (1975/76) and Caves et al. (2002).} The following beautiful illustration of the abstract theory (Unnerstall, 1990a,b) is then clear from \eqref{PMOQ} and \eqref{Unn}: \begin{quote}{\it
If $\om^{\mathrm (q)}_0$ is permutation-invariant, then it is classical. The associated limit state $\om_0^{\mathrm (c)}$ on $\CA_0^{\mathrm (c)}$ is characterized by the fact that
the measure $\mu_0$ in \eqref{probm} coincides with the measure $\mu$ in \eqref{Unn}.}\footnote{In fact, each state in the folium $\CF^{\GS}$ in $\CS(\CA^{\mathrm (q)}_0)$ corresponding to the (quasi-equivalence class of) the \rep\ $\oplus_{[\om\in \CS^{\GS}]}\pi_{\om}$ is classical.} \end{quote} \subsection{Superselection rules} \label{SE} Infinite quantum systems are often associated with the notion of a superselection rule (or sector), which was originally introduced by Wick, Wightman, \&\ Wigner (1952) in the setting of standard \qm\ on a \Hs\ ${\mathcal H}$. The basic idea may be illustrated in the example of the boson/fermion (or ``univalence") superselection rule.\footnote{See also Giulini (2003) for a modern mathematical treatment.} Here one has a {\it projective} unitary \rep\ $\mathcal{D}$
of the rotation group $SO(3)$ on ${\mathcal H}$, for which $\mathcal{D}(R_{2\pi})=\pm 1$ for any rotation $R_{2\pi}$ of $2\pi$ around some axis. Specifically, on bosonic states $\Ps_B$ one has $\mathcal{D}(R_{2\pi})\Ps_B=\Ps_B$, whereas on fermionic states $\Ps_F$ the rule is $\mathcal{D}(R_{2\pi})\Ps_F=-\Ps_F$. Now the argument is that a rotation of $2\pi$ accomplishes nothing, so that it cannot change the physical state of the system. This requirement evidently holds on the subspace ${\mathcal H}_B\subset {\mathcal H}$ of bosonic states in ${\mathcal H}$, but it is equally well satisfied on the subspace ${\mathcal H}_F\subset {\mathcal H}$ of fermionic states, since $\Ps$ and $z\Ps$ with $|z|=1$ describe the same physical state. However, if $\Ps=c_B\Ps_B+c_F\Ps_F$ (with
$|c_B|^2+|c_F|^2=1$), then $\mathcal{D}(R_{2\pi})\Ps=c_B\Ps_B-c_F\Ps_F$, which is not proportional to $\Ps$ and apparently describes a genuinely different physical state from $\Ps$.
The way out is to deny this conclusion by declaring that $\mathcal{D}(R_{2\pi})\Ps$ and $\Ps$ {\it do} describe the same physical state, and this is achieved by postulating that no physical {\it observables} $A$ (in their usual mathematical guise as operators on ${\mathcal H}$) exist for which $(\Ps_B,A\Ps_F)\neq 0$. For in that case one has
\begin{equation} (c_B\Ps_B\pm c_F\Ps_F,A(c_B\Ps_B\pm c_F\Ps_F))=|c_B|^2(\Ps_B,A\Ps_B)+|c_F|^2(\Ps_F,A\Ps_F) \label{SSr}\end{equation} for any {\it observable} $A$, so that $(\mathcal{D}(R_{2\pi})\Ps, A\mathcal{D}(R_{2\pi})\Ps)=(\Ps,A\Ps)$ for any $\Ps\in {\mathcal H}$. Since any quantum-mechanical prediction ultimately rests on expectation values $(\Ps,A\Ps)$ for physical observables $A$, the conclusion is that a rotation of $2\pi$ indeed does nothing to the system. This is codified by saying that superpositions of the type $c_B\Ps_B+c_F\Ps_F$ are {\it incoherent} (whereas superpositions $c_1\Ps_1+c_2\Ps_2$ with $\Ps_1,\Ps_2$ both in either ${\mathcal H}_B$ or in ${\mathcal H}_F$ are {\it coherent}). Each of the subspaces ${\mathcal H}_B$ and ${\mathcal H}_F$ of ${\mathcal H}$ is said to be a {\it superselection sector}, and the statement that $(\Ps_B,A\Ps_F)=0$ for any observbale $A$ and $\Ps_B\in{\mathcal H}_B$ and $\Ps_F\in{\mathcal H}_F$ is called a {\it superselection rule}.\footnote{In an ordinary selection rule between $\Ps$ and $\Ph$ one merely has $(\Ps,H\Ph)=0$ for the Hamiltonian $H$.}
The price one pays for this solution is that states of the form $c_B\Ps_B+ c_F\Ps_F$ with $c_B\neq 0$ and $c_F\neq 0$ are mixed, as one sees from \eqref{SSr}. More generally, if ${\mathcal H}=\oplus_{\lm\in\Lm} {\mathcal H}_{\lm}$ with $(\Ps,A\Ph)=0$ whenever $A$ is an observable, $\Ps\in {\mathcal H}_{\lm}$, $\Ph\in {\mathcal H}_{\lm'}$, and $\lm\neq\lm'$, and if in addition for each $\lm$ and each pair $\Ps,\Ph\in{\mathcal H}_{\lm}$ there exists an observable $A$ for which $(\Ps,A\Ph)\neq 0$, then the subspaces ${\mathcal H}_{\lm}$ are called superselection sectors in ${\mathcal H}$. Again a key consequence of the occurrence of superselection sectors is that unit vectors of the type $\Ps=\sum_{\lm} c_{\lm}\Ps_{\lm}$ with $\Ps\in{\mathcal H}_{\lm}$ (and $c_{\lm}\neq 0$ for at least two $\lm$'s)
define mixed states $$\ps(A)=(\Ps,A\Ps)=\sum_{\lm} |c_{\lm}|^2(\Ps_{\lm},A\Ps_{\lm})
=\sum_{\lm} |c_{\lm}|^2 \ps_{\lm}(A).$$
This procedure is rather ad hoc. A much deeper approach to superselection theory was developed by Haag and collaborators; see Roberts \&\ Roepstorff (1969) for an introduction. Here the starting point is the abstract \ca\ of observables $\CA$ of a given quantum system, and superselection sectors are reinterpreted as equivalence classes (under unitary isomorphism) of irreducible representation s of $\CA$ (satisfying a certain selection criterion - see below).
The connection between the concrete \Hs\ approach to superselection sectors discussed above and the abstract \ca ic approach is given by the following lemma (Hepp, 1972):\footnote{\label{fnhepp}Hepp proved a more general version of this lemma, in which `Two pure states $\rh,\sg$ on a \ca\ $\CB$ define different sectors iff\ldots' is replaced by `Two states $\rh,\sg$ on a \ca\ $\CB$ are disjoint iff\ldots'} \begin{quote} {\it Two pure states $\rh,\sg$ on a \ca\ $\CA$ define different sectors iff for each \rep\ $\pi(\CA)$ on a \Hs\ ${\mathcal H}$ containing unit vectors $\Ps_{\rh},\Ps_{\sg}$ such that $\rh(A)=(\Ps_{\rh},\pi(A)\Ps_{\rh})$ and $\sg(A)=(\Ps_{\sg},\pi(A)\Ps_{\sg})$ for all $A\in\CA$, one has $(\Ps_{\rh},\pi(A)\Ps_{\sg})=0$ for all $A\in\CA$.} \end{quote}
In practice, however, most irreducible representation s of a typical \ca\ $\CA$ used in physics are physically irrelevant mathematical artefacts. Such \rep s may be excluded from consideration by some {\it selection criterion}.
What this means depends on the context. For example, in quantum field theory this notion is made precise in the so-called DHR theory (reviewed by Roberts (1990), Haag (1992), Araki (1999), and Halvorson (2005)). In the class of theories discussed in the preceding two subsections, we take the algebra of observables $\CA$ to be $\CA^{\mathrm (q)}_0$ - essentially for reasons of human limitation - and for pedagogical reasons define (equivalence classes of) irreducible representation s of $\CA^{\mathrm (q)}_0$ as superselection sectors, henceforth often just called {\it sectors}, only when they are equivalent to the GNS-\rep\ given by a permutation-invariant pure state on $\CA^{\mathrm (q)}_0$. In particular, such a state is classical. On this selection criterion, the results in the preceding subsection trivially imply that there is a bijective correspondence between pure states on $\CA_1$ and sectors of $\CA_0^{\mathrm (q)}$. The sectors of the commutative \ca\ $\CA_0^{\mathrm (c)}$ are just the points of $\CS(\CA_1)$; note that a {\it mixed} state on $\CA_1$ defines a {\it pure} state on $\CA_0^{\mathrm (c)}$! The role of the sectors of $\CA_1$ in connection with those of $\CA_0^{\mathrm (c)}$ will be clarified in Subsection \ref{PSD}.
Whatever the model or the selection criterion, it is enlightening (and to some extent even in accordance with experimental practice) to consider superselection sectors entirely from the perspective of the pure states on the algebra of observables $\CA$, removing $\CA$ itself and its \rep s from the scene. To do so, we equip the space $\CP(\CA)$ of pure states on $\CA$ with the structure of a transition probability space (von Neumann, 1981; Mielnik, 1968).\footnote{See also Beltrametti \&\ Cassinelli (1984) or Landsman (1998) for concise reviews.}
A {\it transition probability} on a set $\CP$ is a function \begin{equation} p:\CP\times\CP\raw[0,1] \label{tp1} \end{equation} that satisfies \begin{equation} p(\rh,\sg)=1 \, \Longleftrightarrow \,\rh=\sg \label{tp2} \end{equation} and \begin{equation} p(\rh,\sg)=0 \, \Longleftrightarrow \, p(\sg,\rh)=0. \label{tp2half} \end{equation} A set with such a transition probability is called a {\it transition probability space}.
Now, the pure state space $\CP(\CA)$ of a \ca\ $\CA$ carries precisely this structure if we define\footnote{This definition applies to the case that $\CA$ is unital; see Landsman (1998) for the general case. An analogous formula defines a transition probability on the extreme boundary of any compact convex set.} \begin{equation} p(\rh,\sg):=\inf\{\rh(A)\mid A\in \CA, 0\leq A\leq 1, \sg(A)=1\}.\label{mtp} \end{equation} To give a more palatable formula, note that since pure states are primary, two pure states $\rh,\sg$ are either disjoint ($\rh\bot\sg$) or else (quasi, hence unitarily) equivalent ($\rh\sim\sg$). In the first case, \eqref{mtp} yields \begin{equation} p(\rh,\sg)=0\:\:\: (\rh\bot\sg).\end{equation} Ine the second case it follows from Kadison's transitivity theorem (cf.\ Thm.\ 10.2.6 in Kadison \&\ Ringrose (1986)) that the \Hs\ ${\mathcal H}_{\rh}$ from the GNS-\rep\ $\pi_{\rh}(\CA)$ defined by $\rh$ contains a unit vector $\Om_{\sg}$ (unique up to a phase) such that \begin{equation}\sg(A)=(\Om_{\sg},\pi_{\rh}(A)\Om_{\sg}).\end{equation} Eq.\ \eqref{mtp} then leads to the well-known expression
\begin{equation} p(\rh,\sg)=|(\Om_{\rh},\Om_{\sg})|^2 \:\:\: (\rh\sim\sg). \label{tpsforca} \end{equation} In particular, if $\CA$ is commutative, then \begin{equation} p(\rh,\sg)=\dl_{\rh\sg}. \label{cltp}\end{equation} For $\CA=M_2(\C)$ one obtains \begin{equation} p(\rh,\sg)=\half(1+\cos \theta_{\rh\sg}), \label{thetarhsg}\end{equation} where $\theta_{\rh\sg}$ is the angular distance between $\rh$ and $\sg$ (seen as points on the two-sphere $S^2=\partial_e B^3$, cf.\ \eqref{gens2} etc.), measured along a great circle.
Superselection sectors may now be defined for any transition probability spaces $\CP$. A family of subsets of $\CP$ is called {\it orthogonal} if $p(\rh,\sg)=0$ whenever $\rh$ and $\sg$ do not lie in the same subset. The space $\CP$ is called {\it reducible} if it is the union of two (nonempty) orthogonal subsets; if not, it is said to be {\it irreducible}. A {\it component} of $\CP$ is a subset $\CC\subset \CP$ such that $\CC$ and $\CP\backslash \CC$ are orthogonal. An irreducible component of $\CP$ is called a {\it (superselection) sector}. Thus $\CP$ is the disjoint union of its sectors. For $\CP=\CP(\CA)$ this reproduces the algebraic definition of a superselection sector (modulo the selection criterion) via the correspondence between states and \rep s given by the GNS-constructions.
For example, in the commutative case $\CA\cong C(X)$ each point in $X\cong \CP(\CA)$ is its own little sector. \subsection{A simple example: the infinite spin chain}\label{SC} Let us illustrate the occurrence of superselection sectors in a simple example, where the algebra of observables is $\CA^{\mathrm (q)}_0$ with $\CA_1=M_2(\C)$. Let ${\mathcal H}_1=\C^2$, so that ${\mathcal H}_1^N=\otimes^N\C^2$ is the tensor product of $N$ copies of $\C^2$. It is clear that $\CA_1^N$ acts on ${\mathcal H}^N_1$ in a natural way (i.e.\ componentwise). This defines an irreducible representation\ $\pi_N$ of $\CA_1^N$, which is indeed its unique irreducible representation\ (up to unitary equivalence). In particular, for $N<\infty$ the quantum system whose algebra of observables is $\CA_1^N$ (such as a chain with $N$ two-level systems) has no superselection rules. We define
the $N\raw\infty$ limit ``$(M_2(\C))^{\infty}$'' of the \ca s $(M_2(\C))^N$ as the inductive limit $\CA^{\mathrm (q)}_0$ for $\CA_1=M_2(\C)$, as introduced in Subsection \ref{QLO}; see \eqref{ILCA}. The definition of ``$\otimes^{\infty} \C^2$'' is slightly more involved, as follows (von Neumann, 1938).
For any \Hs\ ${\mathcal H}_1$, let $\Ps$ be a sequence $(\Ps_1,\Ps_2,\ldots)$ with $\Ps_n\in{\mathcal H}_1$. The space $\mathsf{H}_1$ of such sequences is a vector space in the obvious way. Now let $\Ps$ and
$\Ph$ be two such sequences, and write $(\Ps_n,\Ph_n)=\exp(i\al_n)|(\Ps_n,\Ph_n)|$. If $\sum_n |\al_n|=\infty$, we define the (pre-) inner product $(\Ps,\Ph)$ to be zero. If $\sum_n |\al_n|<\infty$, we put $(\Ps,\Ph)=\prod_n (\Ps_n,\Ph_n)$ (which, of course, may still be zero!). The (vector space) quotient of $\mathsf{H}_1$ by the space of sequences $\Ps$ for which $(\Ps,\Ps)=0$ can be completed to a \Hs\ ${\mathcal H}_1^{\infty}$ in the induced inner product, called the {\it complete} infinite tensor product of the \Hs\ ${\mathcal H}_1$ (over the index set $\N$).\footnote{Each fixed $\Ps\in{\mathcal H}_1$ defines an {\it incomplete} tensor product ${\mathcal H}_{\Ps}^{\infty}$, defined as the closed subspace of ${\mathcal H}_1^{\infty}$
consisting of all $\Ph$ for which $\sum_n|(\Ps_n,\Ph_n)-1|<\infty$. If ${\mathcal H}_1$ is separable, then so is ${\mathcal H}_{\Ps}^{\infty}$ (in contrast to ${\mathcal H}_1^{\infty}$, which is an uncountable direct sum of the ${\mathcal H}_{\Ps}^{\infty}$).} We apply this construction with ${\mathcal H}_1=\C^2$. If $(e_i)$ is some basis of $\C^2$, an orthonormal basis of ${\mathcal H}_1^{\infty}$ then consists of all different infinite strings $e_{i_1}\otimes \cdots e_{i_n}\otimes \cdots$, where $e_{i_n}$ is $e_i$ regarded as a vector in $\C^2$.\footnote{The cardinality of the set of all such strings equals that of $\R$, so that ${\mathcal H}^{\infty}_1$ is non-separable, as claimed.} We denote the multi-index $(i_1,\ldots, i_n,\ldots)$ simply by $I$, and the corresponding basis vector by $e_{I}$.
This \Hs\ ${\mathcal H}^{\infty}_1$ carries a natural faithful \rep\ $\pi$ of $\CA^{\mathrm (q)}_0$: if $A_0\in \CA^{\mathrm (q)}_0$ is an equivalence class $[A_1,A_2,\cdots]$, then $\pi(A_0)e_I=\lim_{N\raw\infty} A_Ne_i$, where $A_N$ acts on the first $N$ components of $e_I$ and leaves the remainder unchanged.\footnote{Indeed, this yields an alternative way of defining $\ovl{\cup_{N\in\N} \CA_1^N}$ as the norm closure of the union of all $\CA_1^N$ acting on ${\mathcal H}^{\infty}_1$ in the stated way.} Now the point is that although each $\CA_1^N$ acts irreducibly on ${\mathcal H}^N_1$, the \rep\ $\pi(\CA^{\mathrm (q)}_0)$ on ${\mathcal H}^{\infty}_1$ thus constructed is highly reducible. The reason for this is that by definition (quasi-) local elements of $\CA^{\mathrm (q)}_0$ leave the infinite tail of a vector in ${\mathcal H}^{\infty}_1$ (almost) unaffected, so that vectors with different tails lie in different superselection sectors. Without the quasi-locality condition on the elements of $\CA^{\mathrm (q)}_0$, no superselection rules would arise.
For example, in terms of the usual basis \begin{equation} \left\{\uparrow=\left( \begin{array}{c} 1 \\ 0 \end{array} \right), \downarrow=\left( \begin{array}{c} 0 \\ 1 \end{array} \right)\right\} \label{usualbasis}\end{equation}
of $\C^2$, the vectors $\Ps_{\uparrow}=\uparrow\otimes\uparrow\cdots \uparrow\cdots$ (i.e.\ an infinite product of `up' vectors) and $\Ps_{\downarrow}=\downarrow\otimes\downarrow\cdots \downarrow\cdots$ (i.e.\ an infinite product of `down' vectors) lie in different sectors. The reason why the inner product $(\Ps_{\uparrow}, \pi(A)\Ps_{\downarrow})$ vanishes for any $A\in\CA^{\mathrm (q)}_0$ is that for local observables $A$ one has $\pi(A)=A_M\otimes 1\otimes\cdots 1\cdots$ for some $A_M\in\CB({\mathcal H}_M)$; the inner product in question therefore involves infinitely many factors $(\uparrow, 1\downarrow)=(\uparrow,\downarrow)=0$. For quasilocal $A$ the operator $\pi(A)$ might have a small nontrivial tail, but the inner product vanishes nonetheless by an approximation argument.
More generally, elementary analysis shows that $(\Ps_u, \pi(A)\Ps_v)=0$ whenever $\Ps_u=\otimes^{\infty}u$ and $\Ps_v=\otimes^{\infty}v$ for unit vectors $u,v\in\C^2$ with $u\neq v$. The corresponding vector states $\ps_u$ and $\ps_v$ on $\CA^{\mathrm (q)}_0$ (i.e.\ $\ps_u(A)=(\Ps_u, \pi(A)\Ps_u)$ etc.) are obviously permutation-invariant and hence classical. Identifying $\CS(M_2(\C))$ with $B^3$, as in \eqref{gens2}, the corresponding limit state $(\ps_u)_0$ on $\CA_0^{\mathrm (c)}$ defined by $\ps_u$ is given by (evaluation at) the point $\til{u}=(x,y,z)$ of $\partial_e B^3=S^2$ (i.e.\ the two-sphere) for which the corresponding density matrix $\rh(\til{u})$ is the projection operator onto $u$. It follows that $\ps_u$ and $\ps_v$ are disjoint; cf.\ \eqref{rsbot}. We conclude that each unit vector $u\in\C^2$ determines a superselection sector $\pi_u$, namely the GNS-\rep\ of the corresponding state $\ps_u$, and that each such sector is realized as a subspace ${\mathcal H}_u$ of ${\mathcal H}^{\infty}_1$ (viz.\ ${\mathcal H}_u=\ovl{\pi(\CA^{\mathrm (q)}_0)\Ps_u}$). Moreover, since a permutation-invariant state on $\CA^{\mathrm (q)}_0$ is pure iff it is of the form $\ps_u$, we have found all superselection sectors of our system. Thus in what follows we may concentrate our attention on the subspace (of ${\mathcal H}^{\infty}_1$) and sub\rep\ (of $\pi$) \begin{eqnarray} {\mathcal H}_{\GS}&=&\oplus_{\til{u}\in S^2} {\mathcal H}_u; \nn\\ \pi_{\GS}(\CA^{\mathrm (q)}_0)&=& \oplus_{\til{u}\in S^2} \pi_u(\CA^{\mathrm (q)}_0),\label{repsubrep} \end{eqnarray} where $\pi_u$ is simply the restriction of $\pi$ to ${\mathcal H}_u\subset {\mathcal H}^{\infty}_1$.
In the presence of superselection sectors one may construct operators that distinguish different sectors whilst being a multiple of the unit in each sector. In quantum field theory these are typically global charges, and in our example the macroscopic observables play this role. To see this, we return to Subsection \ref{MO}.
It is not difficult to show that for any approximately symmetric sequence $(A_1,A_2,\cdots)$ the limit \begin{equation} \ovl{A}=\lim_{N\raw\infty}\pi_{\GS}(A_N) \label{slim}\end{equation} exists in the strong operator topology on $\CB({\mathcal H}_{\GS})$ (Bona, 1988). Moreover, if $A_0\in \CA_0^{\mathrm (c)}=C(\CS(\CA_1))$ is the function defined by the given sequence,\footnote{Recall that $A_0(\om)=\lim_{N\raw\infty} \om^N(A_N)$.} then the map $A_0\mapsto \ovl{A}$ defines a faithful \rep\ of $\CA_0^{\mathrm (c)}$ on ${\mathcal H}_{\GS}$, which we call $\pi_{\GS}$ as well (by abuse of notation). An easy calculation in fact shows that $\pi_{\GS}(A_0)\Ps= A_0(\til{u})\Ps$ for $\Ps\in {\mathcal H}_u$, or, in other words, \begin{equation} \pi_{\GS}(A_0)=\oplus_{\til{u}\in S^2} A_0(\til{u})1_{{\mathcal H}_u}. \end{equation} Thus the $\pi_{\GS}(A_0)$ indeed serve as the operators in question.
To illustrate how delicate all this is, it may be interesting to note that even for symmetric sequences the limit $\lim_{N\raw\infty}\pi(A_N)$ does not exist on ${\mathcal H}^{\infty}_1$, not even in the strong topology.\footnote{\label{259} For example, let us take the sequence $A_N=j_{N1}(\mathrm{diag}(1,-1))$ and the vector $\Ps=\uparrow\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\cdots,$ where a sequence of $2^N$ factors of $\uparrow$ is followed by $2^{N+1}$ factors of $\downarrow$, etc. Then the sequence $\{\pi(A_N)\Ps\}_{N\in\N}$ in ${\mathcal H}^{\infty}_1$ diverges: the subsequence where $N$ runs over all numbers $2^n$ with $n$ odd converges to $\third\Ps$, whereas the subsequence where $N$ runs over all $2^n$ with $n$ even converges to $-\third\Ps$.} On the positive side, it can be shown that
$\lim_{N\raw\infty}\pi(A_N)\Ps$ exists as an element of the von Neumann algebra $\pi(\CA^{\mathrm (q)}_0)''$ whenever the vector state $\ps$ defined by $\Ps$ lies in the folium $\CF^{\GS}$ generated by all permutation-invariant states (Bona, 1988; Unnerstall, 1990a).
This observation is part of a general theory of macroscopic observables in the setting of von Neumann algebras (Primas, 1983; Rieckers, 1984; Amann, 1986, 1987; Morchio \&\ Strocchi, 1987; Bona, 1988, 1989; Unnerstall, 1990a, 1990b; Breuer, 1994; Atmanspacher, Amann, \&\ M\"{u}ller-Herold, 1999), which complements the purely \ca ic approach of Raggio \&\ Werner (1989, 1991), Duffield \&\ Werner (1992a,b,c), and Duffield, Roos, \&\ Werner (1992) explained so far.\footnote{Realistic models have been studied in the context of both the $C^*$-algebraic and the von Neumann algebraic approach by Rieckers and his associates. See, for example, Honegger \&\ Rieckers (1994), Gerisch, M\"{u}nzner, \&\ Rieckers (1999), Gerisch, Honegger, \&\ Rieckers (2003), and many other papers. For altogether different approaches to macroscopic observables see van Kampen (1954, 1988, 1993), Wan \&\ Fountain (1998), Harrison \&\ Wan (1997), Wan et al. (1998), Fr\"{o}hlich, Tsai, \&\ Yau (2002),
and Poulin (2004).} In our opinion, the latter has the advantage that conceptually the passage to the limit $N\raw\infty$ (and thereby the idealization of a large system as an infinite one) is very satisfactory, especially in our reformulation in terms of continuous fields of \ca s. Here the commutative \ca\ $\CA_0^{\mathrm (c)}$ of macroscopic observables of the infinite system is glued to the noncommutative algebras $\CA_1^N$ of the corresponding finite systems in a continuous way, and the continuous sections of the ensuing continuous field of \ca s $\CA^{\mathrm (c)}$ exactly describe how {\it macroscopic} quantum observables of the finite systems converge to classical ones. {\it Microscopic} quantum observables of the pertinent finite systems, on the other hand, converge to quantum observables of the infinite quantum system, and this convergence is described by the continuous sections of the continuous field of \ca s $\CA^{\mathrm (q)}$. This entirely avoids the language of superselection rules, which rather displays a shocking {\it dis}continuity between finite and infinite systems: for superselection rules do not exist in finite systems!\footnote{We here refer to superselection rules in the traditional sense of inequivalent irreducible representation s of {\it simple} \ca s. For topological reasons certain finite-dimensional systems are described by (non-simple) \ca s that do admit inequivalent irreducible representation s (Landsman, 1990a,b).} \subsection{Poisson structure and dynamics}\label{PSD} We now pass to the discussion of time-evolution in infinite systems of the type considered so far. We start with the observation that the state space
$\CS(\CB)$ of a finite-dimensional \ca\ $\CB$ (for simplicity assumed unital in what follows) is a Poisson manifold (cf.\ Subsection \ref{DQsection}) in a natural way. A similar statement holds in the infinite-dimensional case, and we carry the reader through the necessary adaptations of the main argument by means of footnotes.\footnote{Of which this is the first. When $\CB$ is infinite-dimensional, the state space $\CS(\CB)$ is no longer a manifold, let alone a Poisson manifold, but a {\it Poisson space} (Landsman, 1997, 1998).
This is a generalization of a Poisson manifold, which turns a crucial {\it property} of the latter into a {\it definition}. This property is the foliation of a Poisson manifold by its symplectic leaves (Weinstein, 1983), and the corresponding definition is as follows: {\it A Poisson space $P$ is a Hausdorff space of the form $P=\cup_{\al}S_{\al}$ (disjoint union), where each $S_{\al}$ is a symplectic manifold (possibly infinite-dimensional) and each injection $\iota_{\al}: S_{\al}\hookrightarrow P$ is continuous. Furthermore, one has a linear subspace $F\subset C(P,\R)$ that separates points and has the property that the restriction of each $f\in F$ to each $S_{\al}$ is smooth. Finally, if $f,g\in F$ then $\{f,g\}\in F$, where the Poisson bracket is defined by $ \{f,g\}(\iota_{\al}(\sg))=\{\iota_{\al}^*f,\iota_{\al}^*g\}_{\al}(\sg)$.} Clearly, a Poisson manifold $M$ defines a Poisson space if one takes $P=M$, $F=\cin(M)$, and the $S_{\al}$ to be the symplectic leaves defined by the given Poisson bracket. Thus we refer to the manifolds $S_{\al}$ in the above definition as the {\it symplectic leaves} of $P$ as well.} We write $K=\CS(\CB)$.
Firstly, an element $A\in\CB$ defines a linear function $\hat{A}$ on $\CB^*$ and hence on $K$ (namely by restriction) through $\hat{A}(\om)=\om(A)$.
For such functions we define the Poisson bracket by \begin{equation} \{\hat{A},\hat{B}\}=i\widehat{[A,B]}.\label{PBSC}\end{equation} Here the factor $i$ has been inserted in order to make the Poisson bracket of two real-valed functions real-valued again; for $\hat{A}$ is real-valued on $K$ precisely when $A$ is self-adjoint, and if $A^*=A$ and $B^*=B$, then $i[A,B]$ is self-adjoint (whereras $[A,B]$ is skew-adjoint). In general, for $f,g\in\cin(K)$ we put \begin{equation} \{f,g\}(\om)=i\om([df_{\om},dg_{\om}]), \label{PBSS}\end{equation} interpreted as follows.\footnote{In the infinite-dimensional case $\cin(K)$ is defined as the intersection of the smooth functions on $K$ with respect
to its Banach manifold structure and the space $C(K)$ of weak$\mbox{}^*$-continuous functions on $K$. The differential forms $df$ and $dg$ in \eqref{PBSS} also require an appropriate definition; see Duffield \&\ Werner (1992a), Bona (2000), and Odzijewicz \&\ Ratiu (2003) for the technicalities.}
Let $\CB_{\R}$ be the self-adjoint part of $\CB$, and interpret $K$ as a subspace of $\CB_{\R}^*$; since a state $\om$ satisfies $\om(A^*)=\ovl{\om(A)}$ for all $A\in\CB$, it is determined by its values on self-adjoint elements. Subsequently, we identify the tangent space at $\om$ with \begin{equation} T_{\om}K = \{\rh\in \CB_{\R}^*\mid\rh(1)=0\}\subset\CB_{\R}^* \label{TSpace}\end{equation}
and the cotangent space at $\om$ with the quotient (of real Banach spaces)
\begin{equation} T^*_{\om}K= \CB_{\R}^{**}/\R 1, \label{Cotspace} \end{equation}
where the unit $1\in\CB$ is regarded as an element of $\CB^{**}$ through the canonical embedding $\CB\subset\CB^{**}$. Consequently, the differential forms $df$ and $dg$ at $\om\in K$ define elements of $\CB_{\R}^{**}/\R 1$.
The commutator in \eqref{PBSS} is then defined as follows: one lifts $df_{\om}\in \CB_{\R}^{**}/\R 1$ to $\CB_{\R}^{**}$, and uses the natural isomorphism
$\CB^{**}\cong \CB$ typical of finite-dimensional vector spaces.\footnote{In the infinite-dimensional case one uses the canonical identification between $\CB^{**}$ and the enveloping von Neumann algebra of $\CB$ to define the commutator.} The arbitrariness in this lift is a multiple of 1, which drops out of the commutator. Hence $i[df_{\om},dg_{\om}]$ is an element of $\CB_{\R}^{**}\cong \CB_{\R}$,
on which the value of the functional $\om$ is defined.\footnote{If $\CB$ is infinite-dimensional, one here regards $\CB^*$ as the predual of the von Neumann algebra $\CB^{**}$.} This completes the definition of the Poisson bracket; one easily recovers \eqref{PBSC} as a special case of \eqref{PBSS}.
The symplectic leaves of the given Poisson structure on $K$ have been determined by Duffield \&\ Werner (1992a).\footnote{See also Bona (2000) for the infinite-dimensional special case where $\CB$ is the \ca\ of compact operators.} Namely: \begin{quote} {\it
Two states $\rh$ and $\sg$ lie in the same symplectic leaf
of $\CS(\CB)$ iff $\rh(A)=\sg(UAU^*)$ for some unitary $U\in\CB$.} \end{quote}
When $\rh$ and $\sg$ are pure, this is the case iff the corresponding GNS-representations $\pi_{\rh}(\CB)$ and $\pi_{\sg}(\CB)$ are unitarily equivalent,\footnote{Cf.\ Thm.\ 10.2.6 in Kadison \&\ Ringrose (1986).} but in general the implication holds only in one direction: if $\rh$ and $\sg$ lie in the same leaf, then they have unitarily equivalent GNS-\rep s.\footnote{An important step of the proof is the observation that the Hamiltonian vector field $\xi_f(\om)\in T_{\om}K\subset \CA^*_{\R}$ of $f\in\cin(K)$ is given by $\langle \xi_f(\om),B\rangle=i[df_{\om},B]$, where $B\in\CB_{\R}\subset \CB_{\R}^{**}$ and $df_{\om}\in \CB_{\R}^{**}/\R 1$. (For example, this gives $\xi_{\hat{A}}\hat{B}=i\widehat{[A,B]}=\{\hat{A},\hat{B}\}$ by \eqref{PBSC}, as it should be.) If $\phv^h_t$ denotes the Hamiltonian flow of $h$ at time $t$, it follows (cf.\ Duffield, Roos, \&\ Werner (1992), Prop.\ 6.1 or Duffield \&\ Werner (1992a), Prop.\ 3.1) that $\langle\phv_h^t(\om),B\rangle=\langle\om, U_t^h B (U_t^h)^*\rangle$ for some unitary $U_t^h\in\CB$. For example, if $h=\hat{A}$ then $U_t^h=\exp(itA)$.}
It follows from this characterization of the symplectic leaves of $K=\CS(\CB)$ that the pure state space $\partial_e K=\CP(\CB)$ inherits the Poisson bracket from $K$, and thereby becomes a Poisson manifold in its own right.\footnote{More generally, a Poisson space. The structure of $\CP(\CB)$ as a Poisson space was introduced by Landsman (1997, 1998) without recourse to the full state space or the work of Duffield \&\ Werner (1992a).} This leads to an important connection between the superselection sectors of $\CB$ and the Poisson structure on $\CP(\CB)$ (Landsman, 1997, 1998): \begin{quote} {\it The sectors of the pure state space $\CP(\CB)$ of a \ca\ $\CB$ as a transition probability space coincide with its symplectic leaves as a Poisson manifold.} \end{quote}
For example, when $\CB\cong C(X)$ is commutative, the space $\CS(C(X))$ of all (regular Borel) probability measures on $X$ acquires a Poisson bracket that is identically zero, as does its extreme boundary $X$. It follows from \eqref{cltp} that the sectors in $X$ are its points, and so are its symplectic leaves (in view of their definition and the vanishing Poisson bracket). The simplest noncommutative case is $\CB=M_2(\C)$, for which the symplectic leaves of the state space $K=\CS(M_2(\C))\cong B^3$ (cf.\ \eqref{gens2}) are the spheres with constant radius.\footnote{ Equipped with a multiple of the so-called Fubini--Study symplectic structure; see Landsman (1998) or any decent book on differential geometry for this notion.
This claim is immediate from footnote \ref{SSlemma}. More generally, the pure
state space of $M_n(\C)$ is the projective space $\mathbb{P}\C^n$, which again becomes equipped with the Fubini--Study symplectic structure.
This is even true for $n=\infty$ if one defines $M_{\infty}(\C)$ as the \ca\ of compact operators on a separable \Hs\ ${\mathcal H}$:
in that case one has $\CP(M_{\infty}(\C))\cong \mathbb{P}{\mathcal H}$.
Cf.\ Cantoni (1977), Cirelli, Lanzavecchia, \&\ Mani\'{a} (1983), Cirelli, Mani\'{a}, \&\ Pizzocchero (1990), Landsman (1998), Ashtekar \&\ Schilling (1999), Marmo et al. (2005),
etc.} The sphere with radius 1 consists of points in $B^3$ that correspond to pure states on $M_2(\C)$, all interior symplectic leaves of $K$ coming from mixed states on $M_2(\C)$.
The coincidence of sectors and symplectic leaves of $\CP(\CB)$ is a compatibility condition between the transition probability structure and the Poisson structure. It is typical of the specific choices \eqref{mtp} and \eqref{PBSS}, respectively, and hence of quantum theory. In classical mechanics one has the freedom of equipping a manifold $M$ with an arbitrary Poisson structure, and yet use $C_0(M)$ as the commutative \ca\ of observables. The transition probability \eqref{cltp} (which follows from \eqref{mtp} in the commutative case) are clearly the correct ones in classical physics, but since the symplectic leaves of $M$ can be almost anything, the coincidence in question does not hold.
However, there exists a compatibility condition between the transition probability structure and the Poisson structure, which is shared by classical and quantum theory. This is the property of {\it unitarity} of a Hamiltonian flow, which in the present setting we formulate as follows.\footnote{All this can be boosted into an axiomatic structure into which both classical and quantum theory fit; see Landsman (1997, 1998).} First, in quantum theory with algebra of observables $\CB$ we define time-evolution (in the sense of an automorphic action of the abelian group $\R$ on $\CB$, i.e.\ a one-parameter group $\al$ of automorphisms on $\CB$) to be {\it Hamiltonian} when $A(t)=\al_t(A)$ satisfies the Heisenberg equation $i\hbar dA/dt=[A,H]$ for some self-adjoint element $H\in\CB$. The corresponding flow on $\CP(\CB)$ - i.e.\ $\om_t(A)=\om(A(t))$ - is equally well said to be Hamiltonian in that case. In classical mechanics with Poisson manifold $M$ we similarly say that a flow on $M$ is Hamiltonian when it is the flow of a Hamiltonian vector field $\xi_h$ for some $h\in\cin(M)$. (Equivalently, the time-evolution of the observables $f\in\cin(M)$ is given by $df/dt=\{h,f\}$; cf.\ \eqref{ft} etc.) The point is that in either case the flow is unitary in the sense that
\begin{equation} p(\rh(t),\sg(t))=p(\rh,\sg)\label{unitarityeq} \end{equation} for all $t$ and all $\rh,\sg\in P$ with $P=\CP(\CB)$ (equipped with the transition probabilities \eqref{mtp} and the Poisson bracket \eqref{PBSS}) or $P=M$ (equipped with the transition probabilities \eqref{cltp} and any Poisson bracket).\footnote{In quantum theory the flow is defined for any $t$. In classical dynamics, \eqref{unitarityeq} holds for all $t$ for which $\rh(t)$ and $\sg(t)$ are defined, cf.\ footnote \ref{crunch}.}
In both cases $P=\CP(\CB)$ and $P=M$, a Hamiltonian flow has the property (which is immediate from the definition of a symplectic leaf) that for all (finite) times $t$ a point $\om(t)$ lies in the same symplectic leaf of $P$ as $\om=\om(0)$. In particular, in quantum theory $\om(t)$ and $\om$ must lie in the same sector. In the quantum theory of infinite systems an automorphic time-evolution is rarely Hamiltonian, but one reaches a similar conclusion under a weaker assumption. Namely, if a given one-parameter group of automorphisms $\al$ on $\CB$ is {\it implemented} in the GNS-\rep\ $\pi_{\om}(\CB)$ for some $\om\in \CP(\CB)$,\footnote{This assumption means that there exists a unitary \rep\ $t\mapsto U_t$ of $\R$ on ${\mathcal H}_{\om}$ such that $\pi_{\om}(\al_t(A))=U_t \pi_{\om}(A)U_t^*$ for all $A\in\CB$ and all $t\in\R$.} then $\om(t)$ and $\om$ lie in the same sector and hence in the same symplectic leaf of $\CP(\CB)$.
To illustrate these concepts, let us return to our continuous field of \ca s $\CA^{\mathrm (c)}$; cf.\ \eqref{fibers}. It may not come as a great surprise that
the canonical \ca ic transition probabilities \eqref{mtp} on the pure state space of each fiber algebra $\CA^{\mathrm (c)}_{1/N}$ for $N<\infty$ converge to the classical transition probabilities \eqref{cltp} on the commutative limit algebra
$\CA^{\mathrm (c)}_0$. Similarly, the \ca ic Poisson structure \eqref{PBSS} on each $\CP(\CA^{\mathrm (c)}_{1/N})$ converges to zero. However, we know from the limit $\hbar\raw 0$ of \qm\ that in generating classical behaviour on the limit algebra of a continuous field of \ca s one should rescale the commutators; see Subsection \ref{DQsection} and Section \ref{S5}. Thus we replace the Poisson bracket \eqref{PBSS} for $\CA^{\mathrm (c)}_{1/N}$ by
\begin{equation} \{f,g\}(\om)=iN\om([df_{\om},dg_{\om}]). \label{PBSSN}\end{equation}
Thus rescaled, the Poisson brackets on the spaces $\CP(\CA^{\mathrm (c)}_{1/N})$ turn out to converge to the canonical Poisson bracket \eqref{PBSS}
on $\CP(\CA^{\mathrm (c)}_0)=\CS(\CA_1)$, instead of the zero bracket expected from the commutative nature of the limit algebra $\CA^{\mathrm (c)}_0$. Consequently, the symplectic leaves of the {\it full} state space $\CS(\CA_1)$ of the fiber algebra $\CA^{\mathrm (c)}_1$ become the symplectic leaves of the {\it pure} state space $\CS(\CA_1)$ of the
fiber algebra $\CA^{\mathrm (c)}_0$. This is undoubtedly indicative of the origin of classical phase spaces and their Poisson structures in quantum theory.
More precisely, we have the following result (Duffield \&\ Werner, 1992a): \begin{quote} {\it If $A=(A_0,A_1,A_2,\cdots)$ and $A'=(A_0',A_1',A_2',\cdots)$ are continuous sections of $\CA^{\mathrm (c)}$ defined by symmetric sequences,\footnote{\label{generalization}The result does not hold for all continuous sections (i.e.\ for all approximately symmetric sequences), since, for example, the limiting functions $A_0$ and $A_0'$ may not be differentiable, so that their Poisson bracket does not exist. This problem occurs in all examples of deformation quantization. However, the class of sequences for which the claim is valid is larger than the symmetric ones alone. A sufficient condition on $A$ and $B$ for \eqref{choice} to make sense is that $A_N=\sum_{M\leq N} j_{NM}(A_M^{(N)})$ (with $A_M^{(N)}\in\CA_1^M$), such that $\lim_{N\raw\infty} A_M^{(N)}$ exists (in norm)
and $\sum_{M=1}^{\infty} M \sup_{N\geq M}\{ \| A_M^{(N)}\|\}<\infty$. See Duffield \&\ Werner (1992a).} then the sequence \begin{equation} \left( \{A_0,A_0'\}, i[A_1,A_1'],\dots, iN [A_N,A_N'],\cdots\right) \label{choice}\end{equation} defines a continuous section of $\CA^{\mathrm (c)}$.} \end{quote} This follows from an easy computation. In other words, although the sequence of commutators $[A_N,A_N']$ converges to zero, the rescaled commutators $iN [A_N,A_N']\in\CA_N$ converge to the macroscopic observable $\{A_0,A_0'\}\in \CA_0^{\mathrm (c)}=C(\CS(\CA_1))$. Although it might seem perverse to reinterpret this result on the classical limit of a large quantum system in terms of quantization (which is the {\it opposite} of taking the classical limit), it is formally possible to do so (cf.\ Section \ref{DQsection}) if we put \begin{equation} \hbar=\frac{1}{N}. \label{h1N}\end{equation} Using the axiom of choice if necessary, we devise a procedure that assigns a continuous section $A=(A_0,A_1,A_2,\cdots)$ of our field to a given function $A_0\in\CA_0^{\mathrm (c)}$. We write this as $A_N=\CQ_{\frac{1}{N}}(A_0)$, and similarly $A_N'=\CQ_{\frac{1}{N}}(A_0')$. This choice need not be such that the sequence \eqref{choice} is assigned to $\{A_0,A_0'\}$, but since the latter is the unique limit of \eqref{choice}, it must be that
\begin{equation} \lim_{N\raw\infty} \left\| iN \left[\CQ_{\frac{1}{N}}(A_0),\CQ_{\frac{1}{N}}(A_0')\right]-\CQ_{\frac{1}{N}}(
\{A_0,A_0'\})\right\| =0.\end{equation} Also note that \eqref{normcont} is just \eqref{normeq}. Consequently (cf.\ \eqref{Dirac} and surrounding text): \begin{quote} {\it The continuous field of \ca s $\CA^{\mathrm (c)}$ defined by \eqref{fibers} and approximately symmetric sequences (and their limits) as continuous sections yields a deformation quantization of the phase space $\CS(\CA_1)$ (equipped with the Poisson bracket \eqref{PBSS}) for any quantization map $\CQ$.} \end{quote} For the dynamics this implies: \begin{quote} {\it Let $H=(H_0, H_1,H_2,\cdots)$ be a continuous section of $\CA^{\mathrm (c)}$ defined by a symmetric sequence,\footnote{Once again, the result in fact holds for a larger class of Hamiltonians, namely the ones satisfying the conditions specified in footnote \ref{generalization} (Duffield \&\ Werner, 1992a). The assumption that each Hamiltonian $H_N$ lies in $\CA_1^N$ and hence is bounded is natural in lattice models, but is undesirable in general.} and let $A=(A_0,A_1,A_2,\cdots)$ be an arbitrary continuous section of $\CA^{\mathrm (c)}$ (i.e.\ an approximately symmetric sequence). Then the sequence \begin{equation} \left(A_0(t), e^{iH_1 t}A_1e^{-iH_1 t},\cdots e^{iNH_N t}A_Ne^{-iNH_N t},\cdots\right), \label{DWEOM}\end{equation} where $A_0(t)$ is the solution of the equations of motion with classical Hamiltonian $H_0$,\footnote{See \eqref{ft} and surrounding text.} defines a continuous section of $\CA^{\mathrm (c)}$.} \end{quote} In other words, for bounded symmetric sequences of Hamiltonians $H_N$ the quantum dynamics restricted to macroscopic observables converges to the classical dynamics with Hamiltonian $H_0$. Compare the positions of $\hbar$ and $N$ in \eqref{HSEOM} and \eqref{DWEOM}, respectively, and rejoice in the reconfirmation of \eqref{h1N}.
In contrast, the quasilocal observables are {\it not} well behaved as far as the $N\raw\infty$ limit of the dynamics defined by such Hamiltonians is concerned. Namely, if $(A_0,A_1,\cdots)$ is a section of the continuous field $\CA^{\mathrm (q)}$, and $(H_1,H_2,\cdots)$ is any bounded symmetric sequence of Hamiltonians, then the sequence $$\left(e^{iH_1 t}A_1e^{-iH_1 t},\cdots e^{iNH_N t}A_Ne^{-iNH_N t},\cdots\right)$$ has no limit for $N\raw\infty$, in that it cannot be extended by some $A_0(t)$ to a continuous section of $\CA^{\mathrm (q)}$. Indeed, this was the very reason why macroscopic observables were originally introduced in this context (Rieckers, 1984; Morchio \&\ Strocchi, 1987; Bona, 1988; Unnerstall, 1990a; Raggio \&\ Werner, 1989; Duffield \&\ Werner, 1992a). Instead, the natural finite-$N$ Hamiltonians for which the limit $N\raw\infty$ of the time-evolution on $\CA_1^N$ exists as a one-parameter automorphism group on $\CA^{\mathrm (q)}$ satisfy an appropriate locality condition, which excludes the global averages defining symmetric sequences. \subsection{Epilogue: Macroscopic observables and the measurement problem\label{hepps}} In a renowned paper, Hepp (1972) suggested that macroscopic observables and superselection rules should play a role in the solution of the measurement problem of \qm. He assumed that a macroscopic apparatus may be idealized as an infinite quantum system, whose algebra of observables $\CA_A$ has disjoint pure states. Referring to our discussion in Subsection \ref{vNs} for context and notation, Hepp's basic idea (for which he claimed no originality) was that as a consequence of the measurement process the initial state vector $\Om_I=\sum_n c_n\Ps_n\otimes I$ of system plus apparatus evolves into a final state vector $\Om_F=\sum_n c_n\Ps_n\otimes \Ph_n$, in which each $\Ph_n$ lies in a different superselection sector of the \Hs\ of the apparatus (in other words, the corresponding states $\phv_n$ on $\CA_A$ are mutually disjoint). Consequently, although the initial state $\om_I$ is pure, the final state $\om_F$ is mixed. Moreover,
because of the disjointness of the $\om_n$ the final state $\om_F$ has a unique decomposition $\om_F=\sum_n |c_n|^2 \ps_n\otimes\phv_n$ into pure states, and therefore admits a bona fide ignorance interpretation. Hepp therefore claimed with some justification that the measurement ``reduces the wave packet", as desired in quantum measurement theory.
Even apart from the usual conceptual problem of passing from the collective of all terms in the final mixture to one actual measurement outcome, Hepp himself indicated a serious mathematical problem with this program. Namely, if the initial state is pure it must lie in a certain superselection sector (or equivalence class of states); but then the final state must lie in the very same sector if the time-evolution is Hamiltonian, or, more generally, automorphic (as we have seen in the preceding subsection). Alternatively, it follows from a more general lemma Hepp (1972) himself proved: \begin{quote} {\it If two states $\rh,\sg$ on a \ca\ $\CB$ are disjoint and $\al:\CB\raw\CB$ is an automorphism of $\CB$, then $\rh\circ\al$ and $\sg\circ\al$ are disjoint, too.} \end{quote} To reach the negative conclusion above, one takes $\CB$ to be the algebra of observables of system and apparatus jointly, and computes back in time by choosing $\al=\al_{t_F-t_I}\inv$, where $\al_t$ is the one-parameter automorphism group on $\CB$ describing the joint time-evolution of system and apparatus (and $t_I$ and $t_F$ are the initial and final times of the measurement, respectively). However, Hepp pointed out that this conclusion may be circumvented if one admits the possibility that a measurement takes infinitely long to complete. For the limit $A\mapsto \lim_{t\raw\infty} \al_t(A)$ (provided it exists in a suitable sense, e.g., weakly) does not necessarily yield an automorphism of $\CB$. Hence a state - evolving in the Schr\"{o}dinger picture by $\om_t(A)\equiv \om(\al_t(A))$ - may leave its sector in infinite time, a possibility Hepp actually demonstrated in a range of models; see also Frigerio (1974), Whitten-Wolfe \&\ Emch (1976), Araki (1980), Bona (1980), Hannabuss (1984), Bub (1988), Landsman (1991), Frasca (2003, 2004), and many other papers.
Despite the criticism that has been raised against the conclusion that a quantum-mechanical measurement requires an infinite apparatus and must take infinite time (Bell, 1975; Robinson, 1994; Landsman, 1995), and despite the fact that this procedure is quite against the spirit of von Neumann (1932), in whose widely accepted description measurements are practically instantaneous, this conclusion resonates well with the modern idea that quantum theory is universally valid and the classical world has no absolute existence; cf.\ the Introduction. Furthermore, a quantum-mechanical measurement is nothing but a specific interaction, comparable with a scattering process; and it is quite uncontroversial that such a process takes infinite time to complete. Indeed, what would it mean for scattering to be over after some finite time? Which time? As we shall see in the next section, the theory of decoherence requires the limit $t\raw\infty$ as well, and largely for the same mathematical reasons. There as well as in Hepp's approach, the limiting behaviour
actually tends to be approached very quickly (on the pertinent time scale), and one needs to let $t\raw\infty$ merely to make terms $\sim\exp-\gm t$ (with $\gm>0$) zero rather than just very small. See also Primas (1997) for a less pragmatic point of view on the significance of this limit.
A more serious problem with Hepp's approach lies in his assumption that the time-evolution on the quasilocal algebra of observables of the infinite measurement apparatus (which in our class of examples would be $\CA_0^{\mathrm (q)}$) is automorphic. This, however, is by no means always the case; cf.\ the references listed near the end of Subsection \ref{PSD}. As we have seen, for certain natural Hamiltonian (and hence automorphic) time-evolutions at finite $N$ the dynamics {\it has no limit $N\raw\infty$ on the algebra of quasilocal observables} - let alone an automorphic one.
Nonetheless, Hepp's conclusion remains valid if we use the algebra $\CA_0^{\mathrm (c)}$ of macroscopic observables, on which (under suitable assumptions - see Subsection \ref{PSD}) Hamiltonian time-evolution on $\CA_1^N$ {\it does} have a limit as $N\raw\infty$. For, as pointed out in Subsection \ref{SE}, each superselection sector of $\CA_0^{\mathrm (q)}$ defines and is defined by a pure state on $\CA_1$, which in turn defines a sector of $\CA_0^{\mathrm (c)}$. Now the latter sector is simply a point in the pure state space $\CS(\CA_1)$ of the commutative \ca\ $\CA_0^{\mathrm (c)}$, so that Hepp's lemma quoted above boils down to the claim that if $\rh\neq \sg$, then $\rh\circ\al\neq\sg\circ\al$ for any automorphism $\al$. This, of course, is a trivial property of any Hamiltonian time-evolution, and it follows once again that a transition from a pure pre-measurement state to a mixed post-measurement state on $\CA_0^{\mathrm (c)}$ is impossible in finite time. To avoid this conclusion, one should simply avoid the limt $N\raw\infty$, which is the root of the $t\raw\infty$ limit; see Janssens (2004).
What, then, does all this formalism mean for Schr\"{o}dinger's cat? In our opinion, it confirms the impression that the appearance of a
paradox rests upon an equivocation. Indeed, the problem arises because
one oscillates between two mutually exclusive interpretations.\footnote{Does {\it complementarity} re-enter through the back door?}
{\it Either} one is a bohemian theorist who, in vacant or in pensive mood, puts off his or her glasses and merely contemplates whether the cat is dead or alive. Such a person studies the cat exclusively from the point of view of its macroscopic observables, so that he or she has to use a post-measurement state $\om_F^{\mathrm (c)}$ on the algebra $\CA_0^{\mathrm (c)}$. If $\om_F^{\mathrm (c)}$ is pure, it lies in $\CP(\CA_1)$ (unless the pre-measurement state was mixed). Such a state corresponds to a single superselection sector $[\om_F^{\mathrm (q)}]$ of $\CA_0^{\mathrm (q)}$, so that the cat is dead or alive. If, on the other hand, $\om_F^{\mathrm (c)}$ is mixed (which is what occurs if Schr\"{o}dinger has his way), there is no problem in the first place: at the level of macroscopic observables one merely has a statistical description of the cat.
{\it Or} one is a hard-working experimental physicist of formidable power, who investigates the detailed microscopic constitution of the cat. For him or her the cat is always in a pure state on $\CA_1^N$ for some large $N$. This time the issue of life and death is not a matter of lazy observation and conclusion, but one of sheer endless experimentation and computation. From the point of view of such an observer, nothing is wrong with the cat being in a coherent superposition of two states that are actually quite close to each other microscopically - at least for the time being.
Either way, {\it the riddle does not exist} (Wittgenstein, TLP, \S 6.5). \section{Why classical states and observables?} \label{S7}\setcounter{equation}{0} \begin{quote} `We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own.' (Eddington, 1920, pp.\ 200--201) \end{quote}
The conclusion of Sections \ref{S5} and \ref{S6} is that quantum theory may give rise to classical behaviour {\it in certain states} and {\it with respect to certain observables}. For example, we have seen that in the limit $\hbar\raw 0$ coherent states and operators of the form $\CQ_{\hbar}(f)$, respectively, are appropriate, whereas in the limit $N\raw\infty$ one should use classical states ({\it nomen est omen}!) as defined in Subsection \ref{QLO} and macroscopic observables. If, instead, one uses superpositions of such states, or observables with the wrong limiting behaviour, no classical physics emerges. Thus the question remains why the world at large should happen to be in such states, and why we turn out to study this world with respect to the observables in question. This question found its original incarnation in the measurement problem (cf.\ Subsection \ref{vNs}), but this problem is really a figure-head for a much wider difficulty.
Over the last 25 years,\footnote{Though some say the basic idea of decoherence goes back to Heisenberg and Ludwig.} two profound and original answers to this question have been proposed.
\subsection{Decoherence}\label{DSS} The first goes under the name of {\it decoherence}. Pioneering papers include van Kampen (1954), Zeh (1970), Zurek (1981, 1982),\footnote{See also Zurek (1991) and the subsequent debate in {\it Physics Today} (Zurek, 1993), which drew wide attention to decoherence.} and Joos \&\ Zeh (1985), and some recent reviews are Bub (1999), Auletta (2001), Joos et al. (2003), Zurek (2003), Blanchard \&\ Olkiewicz (2003), Bacciagaluppi (2004) and Schlosshauer (2004).\footnote{The website \texttt{http://almaak.usc.edu/$\sim$tbrun/Data/decoherence$\mbox{}_{-}$list.html} contains an extensive list of references on decoherence.} More references will be given in due course. The existence (and excellence) of these reviews obviates the need for a detailed treatment of decoherence in this article, all the more so since at the time of writing this approach appears to be in a transitional stage, conceptually as well as mathematically (as will be evident from what follows). Thus we depart from the layout of our earlier chapters and restrict ourselves to a few personal comments. \begin{enumerate} \item \label{DP1} Mathematically, decoherence boils down to the idea of adding one more link to the von Neumann chain (see Subsection \ref{vNs}) beyond $S+A$ (i.e.\ the system and the apparatus). Conceptually, however, there is a major difference between decoherence and older approaches that took such a step: whereas previously (e.g., in the hands of von Neumann, London \&\ Bauer, Wigner, etc.)\footnote{See Wheeler \&\ Zurek (1983).} the chain {\it converged towards the observer}, in decoherence it {\it diverges away from the observer}. Namely, the third and final link is now taken to be the {\it environment} (taken in a fairly literal sense in agreement with the intuitive meaning of the word). In particular, in realistic models the environment is treated as an infinite system (necessitating the limit $N\raw\infty$), which has the consequence that (in simple models where the pointer has discrete spectrum) the post-measurement state $\sum_n c_n \Psi_n \otimes\Phi_n\otimes \ch_n$ (in which the $\ch_n$ are mutually orthogonal) is only reached in the limit $t\raw\infty$. However, as already mentioned in Subsection \ref{hepps}, infinite time is only needed mathematically in order to make terms of the type $\sim\exp-\gm t$ (with $\gm>0$) zero rather than just very small: in many models the inner products $(\ch_n,\ch_m)$ are actually negligible for $n\neq m$ within surprisingly short time scales.\footnote{Cf. Tables 3.1 and 3.2 on pp.\ 66--67 of Joos et al.\ (2003).}
If only in view of the need for limits of the type $N\raw\infty$ (for the environment) and $t\raw\infty$, in our opinion decoherence is best linked to stance 1 of the Introduction: its goal is to explain the approximate appearance of the classical world from quantum mechanics seen as a universally valid theory. However, decoherence has been claimed to support almost any opinion on the foundations of \qm; cf.\ Bacciagaluppi (2004) and Schlosshauer (2004) for a critical overview and also see Point \ref{point6} below. \item Originally, decoherence entered the scene as a proposed solution to the measurement problem (in the precise form stated at the end of Subsection \ref{vNs}). For the restriction of the state $\sum_n c_n \Psi_n \otimes\Phi_n\otimes \ch_n$ to $S+A$ (i.e.\ its trace over the degrees of freedom of the environment) is mixed in the limit $t\raw\infty$, which means that the quantum-mechanical interference between the states $ \Psi_n \otimes\Phi_n$ for different values of $n$ has become `delocalized' to the environment, and accordingly is irrelevant if the latter is not observed (i.e.\ omitted from the description). Unfortunately, the application of the ignorance interpretation of the mixed post-measurement state of $S+A$ is illegal even from the point of view of stance 1 of the Introduction. The ignorance interpretation is only valid if the environment is kept within the description {\it and} is classical (in having a commutative \ca\ of observables). The latter assumption (Primas, 1983), however, makes the decoherence solution to the measurement problem circular.\footnote{On the other hand, treating the environment {\it as if} it were classical might be an improvement on the Copenhagen ideology of treating the measurement apparatus {\it as if} it were classical (cf.\ Section \ref{S3}).}
In fact, as quite rightly pointed out by Bacciagaluppi (2004), decoherence actually {\it aggravates} the measurement problem. Where previously this problem was believed to be man-made and relevant only to rather unusual laboratory situations (important as these might be for the foundations of physics), it has now become clear that ``measurement" of a quantum system {\it by the environment} (instead of by an experimental physicist) happens everywhere and all the time: hence it remains even more miraculous than before that there is a single outcome after each such measurement. Thus decoherence as such does not provide a solution to the measurement problem (Leggett, 2002;\footnote{In fact, Leggett's argument only applies to strawman 3 of the Introduction and loses its force against stance 1. For his argument is that decoherence just removes the {\it evidence} for a given state (of Schr\"{o}dinger's cat type) to be a superposition, and accuses those claiming that this solves the measurement problem of committing the logical fallacy that removal of the evidence for a crime would undo the crime. But according to stance 1 {\it the crime is only defined relative to the evidence!} Leggett is quite right, however, in insisting on the `from `` and" to ``or" problem' mentioned at the end of the Introduction.}
Adler, 2003; Joos \&\ Zeh, 2003), but is in actual fact parasitic on such a solution.
\label{point5} \item \label{point6} There have been various responses to this insight. The dominant one has been to combine decoherence with some interpretation of \qm: decoherence then finds a home, while conversely the interpretation in question is usually enhanced by decoherence. In this context, the most popular of these has been the many-worlds interpretation, which, after decades of obscurity and derision, suddenly started to be greeted with a flourish of trumpets in the wake of the popularity of decoherence. See, for example, Saunders (1993, 1995), Joos et al.\ (2003) and Zurek (2003). In quantum cosmology circles, the consistent histories approach has been a popular partner to decoherence, often in combination with many worlds; see below.
The importance of decoherence in the modal interpretation has been emphasized by Dieks (1989b) and Bene \&\ Dieks (2002), and practically all authors on decoherence find the opportunity to pay some lip-service to Bohr in one way or another. See Bacciagaluppi (2004) and Schlosshauer (2004) for a critical assessment of all these combinations.
In our opinion, none of the established interpretations of \qm\ will do the job, leaving room for genuinely new ideas. One such idea is the {\it return of the environment}: instead of ``tracing it out", as in the original setting of decoherence theory, the environment should {\it not} be ignored! The essence of measurement has now been recognized to be the {\it redundancy} of the outcome (or ``record") of the measurement in the environment. It is this very redundancy of information about the underlying quantum object that ``objectifies" it, in that the information becomes accessible to a large number of observers without necessarily disturbing the object\footnote{Such objectification is claimed to yield an `operational definition of existence' (Zurek, 2003, p.\ 749.).} (Zurek, 2003; Ollivier, Poulin, \&\ Zurek, 2004; Blume-Kohout \&\ Zurek, 2004, 2005). This insight (called ``Quantum Darwinism") has given rise to the ``existential" interpretation of \qm\ due to Zurek (2003). \item Another response to the failure of decoherence (and indeed all other approaches) to solve the measurement problem (in the sense of failing to win a general consensus) has been of a somewhat more pessimistic (or, some would say, pragmatic) kind: all attempts to explain the quantum world are given up, yielding to the point of view that `the appropriate aim of physics at the fundamental level then becomes the representation and manipulation of information' (Bub, 2004). Here `measuring instruments ultimately remain black boxes at some level', and one concludes that all efforts to {\it understand} measurement (or, for that matter, {\sc epr}-correlations) are futile and pointless.\footnote{ It is indeed in describing the transformation of quantum information (or entropy) to classical information during measurement that decoherence comes to its own and exhibits some of its greatest strength. Perhaps for this reason such thinking pervades also Zurek (2003).} \item Night thoughts of a quantum physicist, then?\footnote{Kent, 2000. Pun on
the title of McCormmach (1982).} Not quite. Turning vice into virtue: rather than solving the measurement problem, the true significance of the decoherence program is that it gives conditions under which there is no measurement problem! Namely, foregoing an explanation of the transition from the state $\sum_n c_n \Psi_n \otimes\Phi_n\otimes \ch_n$ of $S+A+{{\mathcal E}}$ to a single one of the states $\Psi_n \otimes\Phi_n$ of $S+A$, at the heart of decoherence is the claim that each of the latter states is {\it robust} against coupling to the environment (provided the Hamiltonian is such that $\Psi_n \otimes\Phi_n$ tensored with some initial state $I_{{\mathcal E}}$ of the environment indeed evolves into $\Psi_n \otimes\Phi_n\otimes \ch_n$, as assumed so far). This implies that each state $\Psi_n \otimes\Phi_n$ remains pure after coupling to the environment and subsequent restriction to the original system plus apparatus, so that at the end of the day the environment has had no influence on it. In other words, the real point of decoherence is the phenomenon of {\it einselection} (for {\it environment-induced superselection}), where a state is `einselected' precisely when (given some interaction Hamiltonian) it possesses the stability property just mentioned. The claim, then, is that einselected states are often classical, or at least that classical states (in the sense mentioned at the beginning of this section) are classical precisely because they are robust against coupling to the environment. Provided this scenario indeed gives rise to the classical world (which remains to be shown in detail), it gives a dynamical explanation of it. But even short of having achieved this goal, the importance of the notion of einselection cannot be overstated; in our opinion, it is the most important and powerful idea in quantum theory since entanglement (which einselection, of course, attempts to undo!).
\item The measurement problem, and the associated distinction between system and apparatus on the one hand and environment on the other, can now be {\it omitted} from decoherence theory. Continuing the discussion in Subsection \ref{primas}, the goal of decoherence should simply be to find the robust or einselected states of a object $\CO$ coupled to an environment ${{\mathcal E}}$, as well as the induced dynamics thereof (given the time-evolution of $\CO+{\mathcal E}$). This search, however, must include the correct {\it identification} of the object $\CO$ within the total $\CS+{\mathcal E}$, namely as a subsystem that actually {\it has} such robust states. Thus the Copenhagen idea that the Heisenberg cut between object and apparatus be movable (cf.\ Subsection \ref{HC}) will not, in general, extend to the ``Primas--Zurek" cut between object and environment. In traditional physics terminology, the problem is to find the right ``dressing" of a quantum system so as to make at least some of its states robust against coupling to its environment (Amann \&\ Primas, 1997; Brun \&\ Hartle, 1999; Omn\`{e}s, 2002). In other words: {\it What is a system?} To mark this change in perspective, we now change notation from $\CO$ (for ``object") to $\CS$ (for ``system"). Various tools for the solution of this problem within the decoherence program have now been developed - with increasing refinement and also increasing reliance on concepts from information theory (Zurek, 2003) - but the right setting for it seems the formalism of consistent histories, see below. \label{CHD} \item Various dynamical regimes haven been unearthed, each of which leads to a different class of robust states (Joos et al., 2003; Zurek, 2003;
Schlosshauer, 2004). Here $H_{\CS}$ is the system Hamiltonian, $H_I$ is the interaction Hamiltonian between system and environment, and $H_{{\mathcal E}}$ is the environment Hamiltonian. As stated, no reference to measurement, object or apparatus need be made here. \begin{itemize} \item In the regime $H_{\CS}<<H_I$, for suitable Hamiltonians the robust states are the traditional pointer states of quantum measurement theory. This regime conforms to von Neumann's (1932) idea that quantum measurements be almost instantaneous. If, moreover, $H_{{\mathcal E}}<<H_I$ as well - with or without a measurement context - then the decoherence mechanism turns out to be universal in being independent of the details of ${\mathcal E}$ and $H_{{\mathcal E}}$ (Strunz, Haake, \&\ Braun, 2003). \item If $H_{\CS}\approx H_I$, then (at least in models of quantum Brownian motion) the robust states are coherent states (either of the traditional Schr\"{o}dinger type, or of a more general nature as defined in Subsection \ref{CSR}); see Zurek, Habib, \&\ Paz (1993) and Zurek (2003). This case is, of course, of supreme importance for the physical relevance of the results quoted in our Section \ref{S5} above, and - if only for this reason - decoherence theory would benefit from more interaction with mathematically rigorous results on quantum stochastic analysis.\footnote{Cf.\ Davies (1976),
Accardi, Frigerio, \&\ Lu (1990), Parthasarathy (1992), Streater (2000), K\"{u}mmerer (2002), Maassen (2003), etc. } \item Finally, if $H_{\CS}>> H_I$, then the robust states turn out to be eigenstates of the system Hamiltonian $H_{\CS}$ (Paz \&\ Zurek, 1999; Ollivier, Poulin \&\ Zurek, 2004). In view of our discussion of such states in Subsections \ref{WKBS} and \ref{QC}, this shows that robust states are not necessarily classical. It should be mentioned that in this context decoherence theory largely coincides with standard atomic physics, in which the atom is taken to be the system $\CS$ and the radiation field plays the role of the environment ${\mathcal E}$; see Gustafson \&\ Sigal (2003) for a mathematically minded introductory treatment and Bach, Fr\"{o}hlich, \&\ Sigal (1998, 1999) for a full (mathematical) meal. \end{itemize} \item Further to the above clarification of the role of energy eigenstates, decoherence also has had important things to say about quantum chaos (Zurek, 2003; Joos et al., 2003). Referring to our discussion of wave packet revival in Subsection \ref{Ssection}, we have seen that in atomic physics wave packets do not behave classically on long time scales. Perhaps surprisingly, this is even true for certain chaotic macroscopic systems: cf.\ the case of Hyperion mentioned in the Introduction and at the end of Subsection \ref{CEOM}.
Decoherence now replaces the underlying superposition by a classical probability distribution, which reflects the chaotic nature of the limiting classical dynamics. Once again, the transition from the pertinent pure state of system plus environment to {\it a single} observed system state remains clouded in mystery. But granted this transition, decoherence sheds new light on classical chaos and circumvents at least the most flagrant clashes with observation.\footnote{It should be mentioned, though, that any successful mechanism explaining the transition from quantum to classical should have this feature, so that at the end of the day decoherence might turn out to be a red herring here.} \item Robustness and einselection form the state side or Schr\"{o}dinger picture of decoherence. Of course, there should also be a corresponding observable side or Heisenberg picture of decoherence. But the transition between the two pictures is more subtle than in the \qm\ of closed systems. In the Schr\"{o}dinger picture, the whole point of einselection is that most pure states simply disappear from the scene. This may be beautifully visualized on the example of a two-level system with \Hs\ ${\mathcal H}_{\CS}=\C^2$ (Zurek, 2003). If $\uparrow$ and $\downarrow$ (cf.\ \eqref{usualbasis}) happen to be the robust vector states of the system after coupling to an appropriate environment, and if we identify the corresponding density matrices with the north-pole $(0,0,1)\in B^3$ and the south-pole $(0,0,-1)\in B^3$, respectively (cf.\ \eqref{gens2}), then following decoherence all other states move towards the axis connecting the north- and south poles (i.e.\ the intersection of the $z$-axis with $B^3$) as $t\raw\infty$. In the Heisenberg picture, this disappearance of all pure states except two corresponds to the reduction of the full algebra of observables $M_2(\C)$ of the system to its diagonal (and hence commutative) subalgebra $\C\oplus\C$ in the same limit. For it is only the latter algebra that contains enough elements to distinguish $\uparrow$ and $\downarrow$ without containing observables detecting interference terms between these pure states. \item To understand this in a more abstract and general way, we recall the mathematical relationship between pure states and observables (Landsman, 1998). The passage from a \ca\ $\CA$ of observables of a given system to its pure states is well known: as a set, the pure state space $\CP(\CA)$ is the extreme boundary of the total state space $\CS(\CA)$ (cf.\ footnote \ref{EBfn}). In order to reconstruct $\CA$ from $\CP(\CA)$, the latter needs to be equipped with the structure of a transition probability space (see Subsection \ref{SE}) through \eqref{mtp}. Each element $A\in\CA$ defines a function $\hat{A}$ on $\CP(\CA)$ by $\hat{A}(\om)=\om(A)$. Now, in the simple case that $\CA$ is finite-dimensional (and hence a direct sum of matrix algebras), one can show that each function $\hat{A}$ is a finite linear combination of the form $\hat{A}=\sum_i p_{\om_i}$, where $\om_i\in \CP(\CA)$ and the elementary functions $p_{\rh}$ on $\CP(\CA)$ are defined by $p_{\rh}(\sg)=p(\rh,\sg)$. Conversely, each such
linear combination defines a function $\hat{A}$ for some $A\in\CA$. Thus the elements of $\CA$ (seen as functions on the pure state space $\CP(\CA)$) are just the transition probabilities and linear combinations thereof. The algebraic structure of $\CA$ may then be reconstructed from the structure of $\CP(\CA)$ as a Poisson space with a transition probability (cf.\ Subsection \ref{PSD}). In this sense $\CP(\CA)$ uniquely determines the algebra of observables of which it is the pure state space. For example, the space consisting of two points with classical transition probabilities \eqref{cltp} leads to the commutative algebra $\CA=\C\oplus\C$, whereas the unit two-sphere in $\R^3$ with transition probabilities \eqref{thetarhsg} yields $\CA=M_2(\C)$.
This reconstruction procedure may be generalized to arbitrary \ca s (Landsman, 1998), and defines the precise connection between the Schr\"{o}dinger picture and the Heisenberg picture that is relevant to decoherence. These pictures are equivalent, but in practice the reconstruction procedure may be difficult to carry through. \item For this reason it is of interest to have a direct description of decoherence in the Heisenberg picture. Such a description
has been developed by Blanchard \&\ Olkiewicz (2003), partly on the basis of earlier results by Olkiewicz (1999a,b, 2000). Mathematically, their approach is more powerful than the Schr\"{o}dinger picture on which most of the literature on decoherence is based. Let $\CA_{\CS}=\CB({\mathcal H}_{\CS})$ and
$\CA_{{\mathcal E}}=\CB({\mathcal H}_{{\mathcal E}})$, and assume one has a total Hamiltonian $H$ acting on ${\mathcal H}_{\CS}\otimes {\mathcal H}_{{\mathcal E}}$ as well as a fixed state
of the environment, represented by a density matrix $\rh_{{\mathcal E}}$ (often taken to be a thermal equilibrium state).
If $\rh_{\CS}$ is a density matrix on ${\mathcal H}_{\CS}$ (so that the total state is
$\rh_{\CS}\otimes \rh_{{\mathcal E}}$), the Schr\"{o}dinger picture approach to decoherence (and more generally to the quantum theory of open systems) is based on the time-evolution
\begin{equation} \rh_{\CS}(t)=\Tr_{{\mathcal H}_{{\mathcal E}}}\left( e^{-\frac{it}{\hbar} H}\rh_{\CS}\otimes \rh_{{\mathcal E}}
e^{\frac{it}{\hbar} H}\right).\end{equation}
The Heisenberg picture, on the other hand, is based on the associated operator time-evolution for $A\in \CB({\mathcal H}_{\CS})$ given by
\begin{equation} A(t)=\Tr_{{\mathcal H}_{{\mathcal E}}}\left(\rh_{{\mathcal E}} e^{\frac{it}{\hbar} H}A \otimes 1\,
e^{-\frac{it}{\hbar} H}\right),\end{equation}
since this yields the equivalence of the Schr\"{o}dinger and Heisenberg pictures expressed by
\begin{equation} \Tr_{{\mathcal H}_{\CS}}\left( \rh_{\CS}(t)A\right)= \Tr_{{\mathcal H}_{\CS}}\left( \rh_{\CS}A(t)\right).\end{equation}
More generally, let $\CA_{\CS}$ and $\CA_{{\mathcal E}}$ be unital \ca s with spatial tensor product $\CA_{\CS}\otimes\CA_{{\mathcal E}}$, equipped with a time-evolution $\al_t$ and a fixed state $\om_{{\mathcal E}}$ on $\CA_{{\mathcal E}}$. This defines a conditional expectation $P_{{\mathcal E}}: \CA_{\CS}\otimes\CA_{{\mathcal E}}\raw \CA_{\CS}$ by linear and continuous extension of $P_{{\mathcal E}}(A\otimes B)=A\om_{{\mathcal E}}(B)$, and consequently a reduced time-evolution $A\mapsto A(t)$ on $\CA_{\CS}$ via
\begin{equation} A(t)=P_{{\mathcal E}}(\al_t(A\otimes 1)).\end{equation} See, for example, Alicki \&\ Lendi (1987); in our context, this generality is crucial for the potential emergence of continuous classical phase spaces; see below.\footnote{For technical reasons
Blanchard \&\ Olkiewicz (2003) assume $\CA_{\CS}$ to be a von Neumann algebra with trivial center.} Now the key point is that decoherence is described by a decomposition $\CA_{\CS}=\CA_{\CS}^{(1)}\oplus \CA_{\CS}^{(2)}$ {\it as a vector space} (not as a \ca), where $\CA_{\CS}^{(1)}$ is a \ca,
with the property that $\lim_{t\raw\infty} A(t)=0$ (weakly) for all $A\in \CA_{\CS}^{(2)}$, whereas $A\mapsto A(t)$ is an automorphism on $\CA_{\CS}^{(1)}$ for each {\it finite} $t$ . Consequently, $\CA_{\CS}^{(1)}$ is the effective algebra of observables after decoherence, and it is precisely the pure states on $\CA_{\CS}^{(1)}$ that are robust or einselected in the sense discussed before. \item For example, if $\CA_{\CS}=M_2(\C)$ and the states $\uparrow$ and $\downarrow$ are robust under decoherence, then $\CA_{\CS}^{(1)}=\C\oplus\C$ and $\CA_{\CS}^{(2)}$ consists of all $2\x 2$ matrices with zeros on the diagonal. In this example
$\CA_{\CS}^{(1)}$ is commutative hence classical, but this may not be the case in general. But if it is, the automorphic time-evolution on $\CA_{\CS}^{(1)}$
induces a classical flow on its structure space, which should be shown to be Hamiltonian using the techniques of Section \ref{S6}.\footnote{Since on the assumption in the preceding footnote $\CA_{\CS}^{(1)}$ is a commutative von Neumann algebra one should define the structure space in an indirect way; see Blanchard \&\ Olkiewicz (2003).} In any case, there will be some sort of classical behaviour of the decohered system whenever $\CA_{\CS}^{(1)}$ has a nontrivial center.\footnote{This is possible even when $\CA_{\CS}$ is a factor!} If this center is discrete, then the induced time-evolution on it is necessarily trivial, and one has the typical measurement situation where the center in question is generated by the projections on the eigenstates of a pointer observable with discrete spectrum. This is generic for the case where
$\CA_{\CS}$ is a type {\sc i} factor. However, type {\sc ii} and {\sc iii} factors may give rise to continuous classical systems with nontrivial time-evolution; see Lugiewicz \&\ Olkiewicz (2002, 2003). We cannot do justice here to the full technical details and complications involved here. But we would like to emphasize that further to quantum field theory and the theory of the thermodynamic limit, the present context of decoherence should provide important motivation for specialists in the foundations of quantum theory to learn the theory of operator algebras.\footnote{See the references in footnote \ref{Cstarlit}.} \end{enumerate}
\subsection{Consistent histories}\label{CHSS}
Whilst doing so, one is well advised to work even harder and simultaneously familiarize oneself with {\it consistent histories}. This approach to quantum theory was pioneered by Griffiths (1984) and was subsequently taken up by Omn\`{e}s (1992) and others. Independently, Gell-Mann and Hartle (1990, 1993) arrived at analogous ideas. Like decoherence, the consistent histories method has been the subject of lengthy reviews (Hartle, 1995) and even books (Omn\`{e}s, 1994, 1999; Griffiths, 2002) by the founders. See also the reviews by Kiefer (2003) and Halliwell (2004), the critiques by
Dowker \&\ Kent (1996), Kent (1998), Bub (1999), and Bassi \&\ Ghirardi (2000), as well as the various mathematical reformulations and reinterpretations of the consistent histories program (Isham, 1994, 1997; Isham \&\ Linden, 1994, 1995; Isham, Linden \&\ Schreckenberg (1994); ÊIsham \&\ Butterfield, 2000; Rudolph, 1996a,b, 2000; Rudolph \&\ Wright, 1999).
The relationship between consistent histories and decoherence is somewhat peculiar: on the one hand, decoherence is a natural mechanism through which appropriate sets of histories become (approximately) consistent, but on the other hand these approaches appear to have quite different points of departure. Namely, where decoherence starts from the idea that (quantum) systems are naturally coupled to their environments and therefore have to be treated as {\it open} systems, the aim of consistent histories is to deal with {\it closed} quantum systems such as the Universe, without a priori talking about measurements or observers. However, this distinction is merely historical: as we have seen in item \ref{CHD} in the previous subsection, the dividing line between a system and its environment should be seen as a dynamical entity to be drawn according to certain stability criteria, so that even in decoherence theory one should really study the system plus its environment as a whole from the outset.\footnote{This renders the distinction between ``open" and ``closed" systems a bit of a red herring, as even in decoherence theory the totality of the system plus its environment is treated as a closed system.} And this is precisely what consistent historians do.
As in the preceding subsection, and for exactly the same reasons, we format our treatment of consistent histories as a list of items open to discussion. \begin{enumerate} \item \label{HBC} The starting point of the consistent histories formulation of quantum theory is conventional: one has a \Hs\ ${\mathcal H}$, a state $\rh$, taken to be the initial state of the total system under consideration (realized as a density matrix on ${\mathcal H}$)
and a Hamiltonian $H$ (defined as a self-adjoint operator on ${\mathcal H}$). What is unconventional is that this total system may well be the entire Universe.
Each property $\al$ of the total system is mathematically represented by a projection $P_{\al}$ on ${\mathcal H}$; for example, if $\al$ is the property that the energy takes some value $\epsilon$, then the operator $P_{\al}$ is the projection onto the associated eigenspace (assuming $\epsilon$ belongs to the discrete spectrum of $H$). In the Heisenberg picture, $P_{\al}$ evolves in time as $P_{\al}(t)$ according to \eqref{HSEOM}; note that $P_{\al}(t)$ is once again a projection.
A {\it history} $\mathbb{H}_A$ is a chain of properties (or propositions) $(\al_1(t_1),\ldots,\al_n(t_n))$ indexed by $n$ different times $t_1<\ldots< t_n$; here $A$ is a multi-label incorporating both the properties $(\al_1,\ldots,\al_n)$ and the times $(t_1,\ldots,t_n)$. Such a history indicates that each property $\al_i$ holds at time $t_i$, $i=1,\ldots,n$. Such a history may be taken to be a collection
$\{\al(t)\}_{t\in\R}$ defined for all times, but for simplicity one usually assumes that $\al(t)\neq 1$ (where 1 is the trivial property that always holds) only for a finite set of times $t$; this set is precisely $\{t_1,\ldots,t_n\}$. An example suggested by Heisenberg (1927) is to take $\al_i$ to be the property that a particle moving through a Wilson cloud chamber may be found in a cell $\Delta_i\subset \R^6$ of its phase space; the history $(\al_1(t_1),\ldots,\al_n(t_n))$ then denotes the state of affairs in which the particle is in cell $\Delta_1$ at time $t_1$, subsequently is in cell $\Delta_2$ at time $t_2$, etcetera. Nothing is stated about the particle's behaviour at intermediate times. Another example of a history is provided by the double slit experiment, where $\al_1$ is the particle's launch at the source at $t_1$ (which is usually omitted from the description), $\al_2$ is the particle passing through (e.g.) the upper slit at $t_2$, and $\al_3$ is the detection of the particle at some location $L$ at the screen at $t_3$. As we all know, there is a potential problem with this history, which will be clarified below in the present framework.
The fundamental claim of the consistent historians seems to be that quantum theory should do no more (or less) than making predictions about the probabilities that histories occur. What these probabilities actually mean remains obscure (except perhaps when they are close to zero or one, or when reference is made to some measurement context; see Hartle (2005)), but let us first see when and how one can define them. The only potentially meaningful mathematical expression
(within \qm) for the probability of a history $\mathbb{H}_A$ with respect to a state $\rh$ is (Groenewold, 1952; Wigner, 1963)
\begin{equation} p(\mathbb{H}_A) =\Tr(C_A\rh C_A^*), \label{Wig} \end{equation}
where
\begin{equation}
C_{A}=P_{\al_n}(t_n)\cdots P_{\al_1}(t_1).\label{defCA}\end{equation} Note that $C_A$ is generally not a projection (and hence a property) itself (unless all $P_{\al_i}$ mutually commute).
In particular, when $\rh=[\Ps]$ is a pure state (defined by some unit vector $\Ps\in{\mathcal H}$), one simply has
\begin{equation} p(\mathbb{H}_A) =\| C_A\Ps\|^2=\|P_{\al_n}(t_n)\cdots P_{\al_1}(t_1)\Ps\|^2.\end{equation}
When $n=1$ this just yields the Born rule. Conversely, see Isham (1994) for a derivation of \eqref{Wig} from the Born rule.\footnote{See also Zurek (2004) for a novel derivation of the Born rule, as well as the ensuing discussion in Schlosshauer (2004).}
\item
Whatever one might think about the metaphysics of \qm, a probability makes no sense whatsoever when it is only attributed to a single history (except when it is exactly zero or one). The least one should have is something like a sample space (or event space) of histories, each (measurable) subset of which is assigned some probability such that the usual (Kolmogorov) rules are satisfied. This is a (well-known) problem even for a single time $t$ and a single projection $P_{\al}$ (i.e.\ $n=1$). In that case, the problem is solved by finding a self-adjoint operator $A$ of which $P_{\al}$ is a spectral projection, so that
the sample space is taken to be the spectrum $\sg(A)$ of $A$, with $\al\subset \sg(A)$. Given $P_{\al}$, the choice of $A$ is by no means unique, of course; different choices may lead to different and incompatible sample spaces. In practice, one usually starts from $A$ and derives the $P_{\al}$
as its spectral projections $P_{\al}=\int_{\al} dP(\lm)$, given that the spectral resolution of $A$ is $A=\int_{\R} dP(\lm)\, \lm$.
Subsequently, one may then either {\it coarse-grain} or {\it fine-grain} this sample space. The former is done by finding a partition $\sg(A)=\coprod_i \al_i$ (disjoint union), and only admitting elements of the $\sg$-algebra generated by the $\al_i$ as events (along with the associated
spectral projection $P_{\al_i}$), instead of all (measurable) subsets of $\sg(A)$. To perform fine-graining, one supplements $A$ by operators that commute with $A$ as well as with each other, so that the new sample space is the joint spectrum of the ensuing family of mutually commuting operators.
In any case, in what follows it turns out to be convenient to work with the projections $P_{\al}$ instead of the subsets $\al$ of the sample space; the above discussion then amounts to extending the given projection on ${\mathcal H}$ to some Boolean sublattice of the lattice $\CP({\mathcal H})$ of all projections on ${\mathcal H}$.\footnote{This sublattice is supposed to the unit of $\CP({\mathcal H})$, i.e.\ the unit operator on ${\mathcal H}$, as well as the zero projection. This comment also applies to
the Boolean sublattice of $\CP({\mathcal H}^N)$ discussed below.}
Any state $\rh$ then defines a probability measure on this sublattice in the usual way (Beltrametti \&\ Cassinelli, 1984).
\item
Generalizing this to the multi-time case is not a trivial task, somewhat facilitated by the following device (Isham, 1994). Put ${\mathcal H}^N=\otimes^N{\mathcal H}$, where $N$ is the cardinality of the set of all times $t_i$ relevant to the histories in the given collection,\footnote{See the mathematical references above for the case $N=\infty$.} and, for a given history $\mathbb{H}_A$, define
\begin{equation} \mathbb{C}_A =P_{\al_n}(t_n)\otimes \cdots \otimes P_{\al_1}(t_1).\end{equation}
Here $P_{\al_i}(t_i)$ acts on the copy of ${\mathcal H}$ in the tensor product ${\mathcal H}^N$
labeled by $t_i$, so to speak. Note that $ \mathbb{C}_A$
is a projection on ${\mathcal H}^N$ (whereas $C_A$ in \eqref{defCA} is generally {\it not} a projection on ${\mathcal H}$). Furthermore, given a density matrix $\rh$ on ${\mathcal H}$ as above, define the {\it decoherence functional} $d$ as a map from pairs of histories into $\C$ by
\begin{equation} d(\mathbb{H}_A,\mathbb{H}_B)=\Tr(C_A\rh C_B^*).\end{equation}
The main point of the consistent histories approach may now be summarized as follows: a collection $\{\mathbb{H}_A\}_{A\in\mathbb{A}}$ of histories can be regarded as a sample space on which a state $\rh$ defines a probability measure via \eqref{Wig}, which of course amounts to \begin{equation} p(\mathbb{H}_A)=d(\mathbb{H}_A,\mathbb{H}_A), \label{CHP} \end{equation} provided that:
\begin{enumerate} \item The operators $\{\mathbb{C}_A\}_{A\in\mathbb{A}}$ form a Boolean sublattice of the lattice $\CP({\mathcal H}^N)$ of all projections on ${\mathcal H}^N$;
\item The real part of $d(\mathbb{H}_A,\mathbb{H}_B)$ vanishes whenever $\mathbb{H}_A$ is disjoint from $\mathbb{H}_B$.\footnote{This means that $\mathbb{C}_A\mathbb{C}_B=0$; equivalently, $P_{\al_i}(t_i)P_{\bt_i}(t_i)=0$
for at least one time $t_i$. This condition guarantees that the probability \eqref{CHP} is additive on disjoint histories.} \end{enumerate} In that case, the set $\{\mathbb{H}_A\}_{A\in\mathbb{A}}$ is called {\it consistent}. It is important to realize that the possible consistency of a given set of histories depends (trivially) not only on this set, but in addition on the dynamics and on the initial state.
Consistent sets of histories generalize families of commuting projections at a single time. There is no great loss in replacing the second condition by the vanishing of $d(\mathbb{H}_A,\mathbb{H}_B)$ itself, in which case the histories $\mathbb{H}_A$ and $\mathbb{H}_B$ are said to {\it decohere}.\footnote{Consistent historians use this terminology in a different way from decoherence theorists. By definition, any two histories involving only a single time are consistent (or, indeed, ``decohere") iff condition (a) above holds; condition (b) is trivially satisfied in that case, and becomes relevant only when more than one time is considered. However, in decoherence theory the reduced density matrix at some given time does not trivially ``decohere" at all; the whole point of the (original) decoherence program was to provide models in which this happens (if only approximately) because of the coupling of the system with its environment. Having said this, within the context of models there are close links between consistency (or decoherence) of multi-time histories and decoherence of reduced density matrices, as the former is often (approximately) achieved by the same kind of dynamical mechanisms that lead to the latter.}
For example, in the double slit experiment the pair of histories $\{\mathbb{H}_A,\mathbb{H}_B\}$ where $\al_1=\bt_1$ is the particle's launch at the source at $t_1$, $\al_2$ ($\bt_2$) is the particle passing through the upper (lower) slit at $t_2$, and $\al_3=\bt_3$ is the detection of the particle at some location $L$ at the screen, is {\it not consistent}. It becomes consistent, however, when the particle's passage through either one of the slits is recorded (or measured) {\it without the recording device being included in the histories} (if it is, nothing would be gained). This is reminiscent of the von Neumann chain in quantum measurement theory, which indeed provides an abstract setting for decoherence (cf.\ item \ref{DP1} in the preceding subsection). Alternatively, the set can be made consistent by omitting $\al_2$ and $\bt_2$. See Griffiths (2002) for a more extensive discussion of the double slit experiment in the language of consistent histories.
More generally, coarse-graining by simply leaving out certain properties is often a promising attempt to make a given inconsistent set consistent; if the original history was already consistent, it can never become inconsistent by doing so. Fine-graining (by embedding into a larger set), on the other hand, is a dangerous act in that it may render a consistent set inconsistent.
\item What does it all mean? Each choice of a consistent set defines a ``universe of discourse" within which one can apply classical probability theory and classical logic (Omn\`{e}s, 1992). In this sense the consistent historians are quite faithful to the Copenhagen spirit (as most of them acknowledge): {\it in order to understand it, the quantum world has to be looked at through classical glasses}. In our opinion, no convincing case has ever been made for the absolute necessity of this Bohrian stance (cf.\ Subsection \ref{Pcl}), but accepting it, the consistent histories approach is superior to Copenhagen in not relying on measurement as an a priori ingredient in the interpretation of \qm.\footnote{See Hartle (2005) for an analysis of the connection between consistent histories and the Copenhagen interpretation and others.} It is also more powerful than the decoherence approach in turning the notion of a system
into a dynamical variable: different consistent sets describe different systems
(and hence different environments, defined as the rest of the Universe); cf.\ item \ref{CHD} in the previous subsection.\footnote{Technically, as the commutant of the projections occurring in a given history.} In other words, the choice of a consistent set boils down to a choice of ``relevant variables" against ``irrelevant" ones omitted from the description. As indeed stressed in the literature, the act of identification of a certain consistent set as a universe of discourse is itself nothing but a coarse-graining of the Universe as a whole.
\item \label{tag} But these conceptual successes come with a price tag. Firstly, {\it consistent sets turn out not to exist in realistic models} (at least if the histories in the set carry more than one time variable).
This has been recognized from the beginning of the program, the response being that one has to deal with approximately consistent sets for which (the real part of) $d(\mathbb{H}_A,\mathbb{H}_B)$ is merely very small. Furthermore, even the definition of a history often cannot be given in terms of projections. For example, in Heisenberg's cloud chamber example (see item \ref{HBC} above), because of his very own uncertainty principle it is impossible to write down the corresponding projections $P_{\al_i}$. A natural candidate would be $P_{\al}=\qb(\ch_{\Delta})$, cf.\ \eqref{b2} and \eqref{qbttsrex}, but in view of \eqref{tpbt} this operator fails to satisfy $P_{\al}^2=P_{\al}$, so that it is not a projection (although it does satisfy the second defining property of a projection $P_{\al}^*=P_{\al}$). This merely reflects the usual property $\CQ(f)^2\neq \CQ(f^2)$ of any quantization method, and necessitates the use of approximate projections (Omn\`{e}s, 1997). Indeed, this point calls for a reformulation of the entire consistent histories approach in terms of positive operators instead of projections (Rudolph, 1996a,b).
These are probably not serious problems; indeed, the recognition that classicality emerges from quantum theory only in an approximate sense (conceptually as well as mathematically) is a profound one (see the Introduction), and it rather should be counted among its blessings that the consistent histories program has so far confirmed it. \item What is potentially more troubling is that consistency by no means implies classicality {\it beyond the ability (within a given consistent set) to assign classical probabilities and to use classical logic}. Quite to the contrary, neither Schr\"{o}dinger cat states nor histories that look classical at each time but follow utterly unclassical trajectories in time are forbidden by the consistency conditions alone (Dowker \&\ Kent, 1996). But is this a genuine problem, except to those who still believe that the earth is at the centre of the Universe and/or that humans are privileged observers? It just seems to be the case that - at least according to the consistent historians - the ontological landscape laid out by quantum theory is far more ``inhuman" (or some would say ``obscure") than the one we inherited from Bohr, in the sense that most consistent sets bear no obvious relationship to the world that {\it we} observe. In attempting to make sense of these, no appeal to ``complementarity" will do now: for one, the complementary pictures of the quantum world called for by Bohr were classical in a much stronger sense than generic consistent sets are, and on top of that Bohr asked us to only think about two such pictures, as opposed to the innumerable consistent sets offered to us. Our conclusion is that, much as decoherence does not solve the measurement problem but rather aggravates it (see item \ref{point5} in the preceding subsection), also {\it consistent histories actually make the problem of interpreting \qm\ more difficult than it was thought to be before}. In any case, it is beyond doubt that the consistent historians have significantly deepened our understanding of quantum theory - at the very least by providing a good bookkeeping device! \item Considerable progress has been made in the task of identifying at least {\it some} (approximately) consistent sets that display (approximate) classical behaviour in the full sense of the word (Gell-Mann \&\ Hartle, 1993; Omn\`{e}s, 1992, 1997; Halliwell, 1998, 2000, 2004; Brun \&\ Hartle, 1999; Bosse \&\ Hartle, 2005). Indeed, in our opinion studies of this type form the main concrete outcome of the consistent histories program. The idea is to find a consistent set $\{\mathbb{H}_A\}_{A\in\mathbb{A}}$ with three decisive properties:\begin{enumerate} \item Its elements (i.e.\ histories) are strings of propositions with a classical interpretation; \item Any history in the set that delineates a classical trajectory (i.e.\ a solution of appropriate classical equations of motion) has probability \eqref{CHP} close to unity, and any history following a classically impossible trajectory has probability close to zero; \item The description is sufficiently coarse-grained to achieve consistency, but is sufficiently fine-grained to turn the deterministic equations of motion following from (b) into a closed system. \end{enumerate} When these goals are met, it is in this sense (no more, no less) that the consistent histories program can claim with some justification that it has indicated (or even explained) `How the quantum Universe becomes classical' (Halliwell, 2005).
Examples of propositions with a classical interpretation are quantized classical observables with a recognizable interpretation (such as the operators $\qb(\ch_{\Delta})$ mentioned in item \ref{tag}), macroscopic observables of the kind studied in Subsection \ref{MO}, and hydrodynamic variables (i.e.\ spatial integrals over conserved currents). These represent three different levels of classicality, which in principle are connected through mutual fine- or coarse-grainings.\footnote{The study of these connections is relevant to the program laid out in this paper, but really belongs to classical physics {\it per se}; think of the derivation of the Navier--Stokes equations from Newton's equations.} The first are sufficiently coarse-grained to achieve consistency only in the limit $\hbar\raw 0$ (cf.\ Section \ref{S5}), whereas the latter two are already coarse-grained by their very nature. Even so, also the initial state will have to be ``classical'' in some sense in order te achieve the three targets (a) - (c). \end{enumerate}
All this is quite impressive, but we would like to state our opinion that neither decoherence nor consistent histories can stand on their own in explaining the appearance of the classical world. Promising as these approaches are, they have to be combined at least with limiting techniques of the type described in Sections \ref{S5} and \ref{S6} - not to speak of the need for a new metaphysics! For even if it is granted that decoherence yields the disappearance of superpositions of Schr\"{o}dinger cat type, or that consistent historians give us consistent sets none of whose elements contain such superpositions among their properties, this by no means suffices to explain the emergence of classical phase spaces and flows thereon determined by classical equations of motion. Since so far the approaches cited in Sections \ref{S5} and \ref{S6} have hardly been combined with the decoherence and/or the consistent histories program, a full explanation of the classical world from quantum theory is still in its infancy. This is not merely true at the technical level, but also conceptually; what has been done so far only represents a modest beginning. On the positive side, here lies an attractive challenge for mathematically minded researchers in the foundations of physics! \section{Epilogue}\label{S8} As a sobering closing note, one should not forget that whatever one's achievements in identifying a ``classical realm" in \qm, the theory continues to incorporate another realm, the pure quantum world, that the young Heisenberg first gained access to, if not through his mathematics, then perhaps through the music of his favourite composer, Beethoven. This world beyond ken has never been better described than by Hoffmann (1810) in his essay on Beethoven's instrumental music, and we find it appropriate to end this paper by quoting at some length from it:\footnote{Translation copyright: Ingrid Schwaegermann (2001).} \begin{quote} Should one, whenever music is discussed as an independent art, not always be referred to instrumental music which, refusing the help of any other art (of poetry), expresses the unique essence of art that can only be recognized in it? It is the most romantic of all arts, one would almost want to say, the only truly romantic one, for only the infinite is its source. Orpheus' lyre opened the gates of the underworld.Ê Music opens to man an unknown realm, a world that has nothing in common with the outer sensual world that surrounds him, a realm in which he leaves behind all of his feelings of certainty, in order to abandon himself to an unspeakable longing. (\ldots)
Beethoven's instrumental music opens to us the realm of the gigantic and unfathomable.Ê Glowing rays of light shoot through the dark night of this realm, and we see gigantic shadows swaying back and forth, encircling us closer and closer, destroying us (\ldots) Beethoven's music moves the levers of fear, of shudder, of horror, of pain and thus awakens that infinite longing that is the essence of romanticism.Ê Therefore, he is a purely romantic composer, and may it not be because of it, that to him, vocal music that does not allow for the character of infinite longing - but, through words, achieves certain effects, as they are not present in the realm of the infinite - is harder?Ê(\ldots)
What instrumental work of Beethoven confirms this to a higher degree than his magnificent and profound Symphony in c-Minor.ÊÊ Irresistibly, this wonderful composition leads its listeners in an increasing climax towards the realm of the spirits and the infinite.ÊÊÊÊ(\ldots) Only that composer truly penetrates into the secrets of harmony who is able to have an effect on human emotions through them; to him, relationships of numbers, which, to the Grammarian, must remain dead and stiff mathematical examples without genius, are magic potions from which he lets a miraculous world emerge. (\ldots)
Instrumental music, wherever it wants to only work through itself and not perhaps for a certain dramatic purpose, has to avoid all unimportant punning, all dallying.ÊÊÊ It seeks out the deep mind for premonitions of joy that, more beautiful and wonderful than those of this limited world, have come to us from an unknown country, and spark an inner, wonderful flame in our chests, a higher expression than mere words - that are only of this earth - can spark.Ê \end{quote}
\begin{trivlist} \item Abraham, R. \&\
Marsden, J.E. (1985). \textit{Foundations of Mechanics}, 2nd ed. Addison Wesley, Redwood City. \item Accardi, L., Frigerio, A., \&\ Lu, Y. (1990). The weak coupling limit as a quantum
functional central limit. {\it Communications in Mathematical Physics} 131, 537--570. \item Adler, S.L. (2003). Why decoherence has not solved the measurement problem: A response to P.W. Anderson. {\it Studies in History and Philosophy of Modern Physics} 34B, 135--142.
\item Agmon, S. (1982). {\it Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations}. Princeton: Princeton University Press. \item Albeverio, S.A. \&\ H\o egh-Krohn, R.J. (1976).
{\it Mathematical Theory of Feynman Path Integrals}.
Berlin: Springer-Verlag. \item Alfsen, E.M. (1970). {\it Compact Convex Sets and Boundary Integrals}. Berlin: Springer. \item Ali, S.T., Antoine, J.-P., Gazeau, J.-P. \&\ Mueller, U.A. (1995). Coherent states and their generalizations: a mathematical overview. {\it Reviews in Mathematical Physics} 7, 1013--1104. \item Ali, S.T., Antoine, J.-P., \&\ Gazeau, J.-P. (2000). {\it Coherent States, Wavelets and their Generalizations}. New York: Springer-Verlag. \item Ali, S.T. \&\ Emch, G.G. (1986). Geometric quantization: modular reduction theory and coherent states. {\it Journal of Mathematical Physics} 27, 2936--2943. \item Ali, S.T \&\ Englis, M. (2004). Quantization methods: a guide for physicists and analysts. \\ \texttt{arXiv:math-ph/0405065}. \item Alicki, A. \&\ Fannes, M. (2001). {\it Quantum Dynamical Systems}. Oxford: Oxford University Press. \item Alicki, A. \&\ Lendi, K. (1987). {\it Quantum Dynamical Semigroups and Applications}. Berlin: Springer. \item Amann, A. (1986). Observables in $W^*$-algebraic \qm. {\it Fortschritte der Physik} 34, 167--215. \item Amann, A. (1987). Broken symmetry and the generation of classical observables in large systems. {\it Helvetica Physica Acta} 60, 384--393. \item Amann, A. \&\ Primas, H. (1997). What is the referent of a non-pure quantum state? {\it Experimental Metaphysics: Quantum Mechanical Studies in Honor of Abner Shimony}, S. Cohen, R.S., Horne, M.A., \&\ Stachel, J. (Eds.). Dordrecht: Kluwer Academic Publishers. \item Arai, T. (1995). Some extensions of the semiclassical limit $\hbar\raw 0$ for Wigner functions on phase space. {\em Journal of Mathematical\ Physics} 36, 622--630.
\item Araki, H. (1980). A remark on the Machida-Namiki theory of measurement. {\it Progress in Theoretical Physics } 64, 719--730. \item Araki, H. (1999). {\it Mathematical Theory of Quantum Fields}. New York: Oxford University Press. \item ÊArnold, V.I. (1989). {\it Mathematical Methods of Classical Mechanics.} Second edition. New York: Springer-Verlag. \item Ashtekar, A. \&\ Schilling, T.A. (1999). Geometrical formulation of quantum mechanics. {\it On Einstein's Path (New York, 1996)}, pp.\ 23--65.
New York: Springer. \item Atmanspacher, H., Amann, A., \&\ M\"{u}ller-Herold, U. (Eds.). (1999). {\it On Quanta, Mind and Matter: Hans Primas in Context.} Dordrecht: Kluwer Academic Publishers. \item Auletta, G. (2001). {\it Foundations and Interpretation of Quantum Mechanics}. Singapore: World Scientific. \item Bacciagaluppi, G. (1993). Separation theorems and Bell inequalities in algebraic quantum mechanics. {\it Proceedings of the Symposium on the Foundations of Modern Physics (Cologne, 1993)}, pp.\ 29--37. Busch, P., Lahti, P.J., \&\ Mittelstaedt, P. (Eds.). Singapore: World Scientific. \item Bacciagaluppi, G. (2004). The Role of Decoherence in Quantum Theory. {\it Stanford Encyclopedia of Philosophy}, (Winter 2004 Edition), Zalta, E.N. (Ed.). Online only at \\ {\texttt http://plato.stanford.edu/archives/win2004/entries/qm-decoherence/}. \item Bach, V., Fr\"{o}hlich, J., \&\ Sigal, I.M. (1998). Quantum electrodynamics of confined nonrelativistic particles. {\it Advances in Mathematics} 137, 299--395. \item Bach, V., Fr\"{o}hlich, J., \&\ Sigal, I.M. (1999). Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. {\it Communications in Mathematical Physics} 207, 249--290. \item Baez, J. (1987). Bell's inequality for $C\sp *$-algebras.
{\it Letters in Mathematical Physics} 13, 135--136.
\item Bagarello, F. \&\ Morchio, G. (1992). Dynamics of mean-field spin models from basic results in abstract differential equations. {\it Journal of Statistical Physics} 66, 849--866. \item Ballentine, L.E. (1970). The statistical interpretation of quantum mechanics. {\it Reviews of Modern Physics} 42, 358--381. \item Ballentine, L.E. (1986). Probability theory in quantum mechanics. {\it American Journal of Physics} 54, 883--889. \item Ballentine, L.E. (2002). Dynamics of quantum-classical differences for chaotic systems. {\it Physical Review} A65, 062110-1--6. \item Ballentine, L.E. (2003). The classical limit of \qm\ and its implications for the foundations of \qm. {\it Quantum Theory: Reconsideration of Foundations -- 2}, pp.\ 71--82. Khrennikov, A. (Ed.). V\"{a}xj\"{o}: V\"{a}xj\"{o} University Press. \item Ballentine, L.E., Yang, Y. \&\ Zibin, J.P. (1994). Inadequacy of Ehrenfest's theorem to characterize the classical regime. {\it Physical Review} A50, 2854--2859.
\item Balian, R. \&\ Bloch, C. (1972). Distribution of eigenfrequencies for the wave equation in a finite domain. III. Eigenfrequency density oscillations. {\it Annals of Physics } 69, 76--160.
\item Balian, R. \&\ Bloch, C. (1974). Solution of the Schr\"odinger equation in terms of classical paths. {\it Annals of Physics } 85, 514--545. \item Bambusi, D., Graffi, S., \&\ Paul, T. (1999).
Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time.
{\it Asymptotic Analysis} 21, 149--160. \item Barrow-Green, J. (1997). {\it Poincar\'e and the Three Body Problem}.
Providence, RI: (American Mathematical Society. \item Barut, A.O. \&\ Ra\c{c}ka, R. (1977). \textit{Theory of Group Representations and Applications}. Warszawa: PWN. \item Bassi, A. \&\ Ghirardi, G.C. (2000).
Decoherent histories and realism.
{\it Journal of Statistical Physics } 98, 457--494. Reply by Griffiths, R.B. (2000).
{\it ibid.} 99, 1409--1425. Reply to this reply by
Bassi, A. \&\ Ghirardi, G.C. (2000). {\it ibid.} 99, 1427. \item Bates, S. \&\ Weinstein, A. (1995). {\it Lectures on the Geometry of Quantization}. {\it Berkeley Mathematics Lecture Notes} 8. University of California, Berkeley. Re-issued by the American Mathematical Society. \item Batterman, R.W. (2002). {\it The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence}. Oxford: Oxford University Press. \item Batterman, R.W. (2005). Critical phenomena and breaking drops: Infinite idealizations in physics.
{\it Studies in History and Philosophy of Modern Physics} 36, 225--244. \item Baum, P., Connes, A. \&\ Higson, N. (1994).
Classifying space for proper actions and K-theory of group $C^*$-algebras. {\it Contemporary\ Mathematics} 167, 241--291.
\item Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. \&\
Sternheimer, D. (1978). Deformation theory and quantization I, II. {\it Annals of Physics \ } 110, 61--110, 111--151. \item Bell, J.S. (1975). On wave packet reduction in the Coleman--Hepp model. {\it Helvetica Physica Acta} 48, 93--98. \item Bell, J.S. (1987). {\it Speakable and Unspeakable in Quantum Mechanics}. Cambridge: Cambridge University Press. \item Bell, J.S. (2001). {\it John S. Bell on the Foundations of Quantum Mechanics}. Singapore: World Scientific. \item Beller, M. (1999). {\it Quantum Dialogue}. Chicago: University of Chicago Press. \item Bellissard, J. \&\ Vittot, M. (1990).
Heisenberg's picture and noncommutative geometry of the semiclassical limit in quantum mechanics. {\it
Annales de l' Institut Henri Poincar\'{e} - Physique Th\'{e}orique} 52, 175--235. \item Belot, G. (2005). Mechanics: geometrical. This volume. \item Belot, G. \&\ Earman, J. (1997). Chaos out of order: \qm, the correspondence principle and chaos. {\it Studies in History and Philosophy of Modern Physics} 28B, 147--182. \item Beltrametti, E.G. \&\ Cassinelli, G. (1984). \textit{The Logic of Quantum Mechanics}. Cambridge: Cambridge University Press. \item Benatti, F. (1993).
{\it Deterministic Chaos in Infinite Quantum Systems}. Berlin:
Springer-Verlag. \item Bene, G. \& Dieks, D. (2002). A Perspectival Version of the Modal Interpretation of Quantum Mechanics and the Origin of Macroscopic Behavior. {\it Foundations of Physics } 32, 645-671. \item Berezin, F.A. (1974). Quantization. {\it Mathematical\ USSR Izvestia} 8, 1109--1163. \item Berezin, F.A. (1975a). Quantization in complex symmetric spaces. {\it Mathematical\ USSR Izvestia} 9, 341--379. \item Berezin, F.A. (1975b). General concept of quantization. {\it Communications in \ Mathematical\ Physics } 40, 153--174. \item Berry, M.V. (1977a). Semi-classical mechanics in phase space: a study of Wigner's function. {\it Philosophical Transactions of the Royal Society} 287, 237--271. \item Berry, M.V. (1977b). Regular and irregular semi-classical wavefunctions. {\it Journal of Physics } A10, 2083--2091. \item Berry, M.V., Balazs, N.L., Tabor, M., \&\ Voros, A. (1979). Quantum maps. {\it Annals of Physics } 122, 26--63. \item Berry, M.V. \& Tabor, M. (1977). Level clustering in the regular spectrum. {\it Proceedings of the Royal Society} A356, 375--394. \item Berry, M.V. \&\ Keating, J.P. (1999). The Riemann zeros and eigenvalue asymptotics. {\it SIAM Review} 41, 236--266. \item Binz, E., J., \'{S}niatycki, J. \&\ Fischer, H. (1988). {\it The Geometry of Classical Fields}. Amsterdam: North--Holland. \item Birkhoff, G. \&\ von Neumann, J. (1936). The logic of quantum mechanics. {\it Annals of Mathematics} (2) 37, 823--843. \item Bitbol, M. (1996). {\it Schr\"{o}dinger's Philosophy of Quantum Mechanics}. Dordrecht: Kluwer Academic Publishers. \item Bitbol, M. \&\ Darrigol, O. (Eds.) (1992). {\it Erwin Schr\"{o}dinger: Philosophy and the Birth of Quantum Mechanics}. Dordrecht: Kluwer Academic Publishers. \item Blackadar, B. (1998). {\it $K$-Theory for Operator Algebras}. Second edition. Cambridge: Cambridge University Press. \item Blair Bolles, E. (2004). {\it Einstein Defiant: Genius versus Genius in the Quantum Revolution}. Washington: Joseph Henry Press. \item Blanchard, E. (1996). Deformations de $C^*$-algebras de Hopf. \textit{Bulletin de la Soci\'{e}t\'{e} math\'{e}matique de France} 124, 141--215. \item Blanchard, Ph. \&\ Olkiewicz, R. (2003). Decoherence induced transition from quantum to classical dynamics. {\it Reviews in Mathematical Physics} 15, 217--243. \item Bohigas, O., Giannoni, M.-J., \&\ Schmit, C. (1984). Characterization of chaotic quantum spectra and universality of level fluctuation laws. {\it Physical Review Letters} 52, 1--4. \item Blume-Kohout, R. \&\ Zurek, W.H. (2004). A simple example of "Quantum Darwinism": Redundant information storage in many-spin environments. {\it Foundations of Physics}, to appear.\\ \texttt{arXiv:quant-ph/0408147}. \item Blume-Kohout, R. \&\ Zurek, W.H. (2005). Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information. {\it Physical Review} A, to appear. \texttt{arXiv:quant-ph/0505031}. \item Bohr, N. (1927) The quantum postulate and the recent development of atomic theory. {\it Atti del Congress Internazionale dei Fisici (Como, 1927)}. Reprinted in Bohr (1934), pp.\ 52--91. \item Bohr, N. (1934). {\it Atomic Theory and the Description of Nature}. Cambridge: Cambridge University Press. \item Bohr, N. (1935). Can quantum-mechanical description of physical reality be considered complete? {\it Physical Review} 48, 696--702. \item Bohr, N. (1937). Causality and complementarity. {\it Philosophy of Science} 4, 289--298. \item Bohr, N. (1949). Discussion with Einstein on epistemological problems in atomic physics. {\it Albert Einstein: Philosopher-Scientist}, pp.\ 201--241. P.A. Schlipp (Ed.). La Salle: Open Court. \item Bohr, N. (1958). {\it Atomic Physics and Human Knowlegde}. New York: Wiley. \item Bohr, N. (1985). {\it Collected Works. Vol.\ 6: Foundations of Quantum Physics {\sc i} (1926--1932)}. Kalckar, J. (Ed.). Amsterdam: North-Holland. \item Bohr, N. (1996). {\it Collected Works. Vol.\ 7: Foundations of Quantum Physics {\sc ii} (1933--1958)}. Kalckar, J. (Ed.). Amsterdam: North-Holland. \item Bogoliubov, N.N. (1958). On a new method in the theory of superconductivity. {\it Nuovo Cimento} 7, 794--805. \item Bona, P. (1980). A solvable model of particle detection in quantum theory. {\it Acta Facultatis Rerum Naturalium Universitatis Comenianae Physica} XX, 65--94. \item Bona, P. (1988). The dynamics of a class of mean-field theories. {\it Journal of Mathematical Physics} 29, 2223--2235. \item Bona, P. (1989). Equilibrium states of a class of mean-field theories. {\it Journal of Mathematical Physics} 30, 2994--3007. \item Bona, P. (2000). Extended quantum mechanics. {\it Acta Physica Slovaca} 50, 1--198.
\item Bonechi, F. \&\ De Bi\`{e}vre, S. (2000). Exponential mixing and $\ln \hbar$ time scales in quantized hyperbolic maps on the torus. {\it Communications in Mathematical Physics} 211, 659--686. \item Bosse, A.W. \&\ Hartle, J.B. (2005). Representations of spacetime alternatives and their classical limits. \texttt{arXiv:quant-ph/0503182}. \item Brack, M. \&\ Bhaduri, R.K. {\it Semiclassical Physics}. Boulder: Westview Press. \item Bratteli, O. \&\ Robinson, D.W. (1987). {\it Operator Algebras and Quantum Statistical Mechanics. Vol.\ I: $C^*$- and $W^*$-Algebras, Symmetry Groups, Decomposition of States}. 2nd Ed. Berlin: Springer. \item Bratteli, O. \&\ Robinson, D.W. (1981). {\it Operator Algebras and Quantum Statistical Mechanics. Vol.\ II: Equilibrium States, Models in Statistical Mechanics}. Berlin: Springer. \item Brezger, B., Hackerm\"{u}ller, L., Uttenthaler, S., Petschinka, J., Arndt, M., \&\ Zeilinger, A. (2002). Matter-Wave Interferometer for Large Molecules. {\it Physical Review Letters} 88, 100404. \item Breuer T. (1994). {\it Classical Observables, Measurement, and Quantum Mechanics}. Ph.D.\ Thesis, University of Cambridge. \item Br\"ocker, T. \&\ Werner, R.F. (1995).
Mixed states with positive Wigner functions.
{\it Journal of Mathematical Physics} 36, 62--75.
\item Brun, T.A. \&\ Hartle, J.B. (1999). Classical dynamics of the quantum harmonic chain. {\it Physical Review} D60, 123503-1--20.
\item Brush, S.G. (2002). Cautious revolutionaries: Maxwell, Planck, Hubble. {\it American Journal of Physics} 70, 119-Ð127. \item Bub, J. (1988). How to Solve the Measurement Problem of Quantum Mechanics. {\it Foundations of Physics} 18,Ê 701--722. \item Bub, J. (1999). {\it Interpreting the Quantum World}. Cambridge: Cambridge University Press. \item Bub, J. (2004). Why the quantum? {\it Studies in History and Philosophy of Modern Physics } 35B, 241--266. \item Busch, P., Grabowski, M. \&\ Lahti, P.J. (1998). {\it Operational Quantum Physics}, 2nd corrected ed.
Berlin: Springer.
\item Busch, P., Lahti, P.J., \&\ Mittelstaedt, P. (1991). {\it The Quantum Theory of Measurement}. Berlin: Springer. \item Butterfield, J. (2002). Some Worlds of Quantum Theory. R.Russell, J. Polkinghorne et al (Ed.). {\it Quantum Mechanics} (Scientific Perspectives on Divine Action vol 5), pp.\ 111-140. Rome: Vatican Observatory Publications, 2. \texttt{arXiv:quant-ph/0105052}; PITT-PHIL-SCI00000204. \item Butterfield, J. (2005). On symmetry, conserved quantities and symplectic reduction in classical mechanics. {\it This volume}. \item B\"{u}ttner, L., Renn, J., \&\ Schemmel, M. (2003). Exploring the limits of classical physics: Planck, Einstein, and the structure of a scientific revolution. {\it Studies in History and Philosophy of Modern Physics} 34B, 37--60. \item Camilleri, K. (2005). {\it Heisenberg and Quantum Mechanics: The Evolution of a Philosophy of Nature}. Ph.D.\ Thesis, University of Melbourne.
\item Cantoni, V. (1975). Generalized ``transition probability''. \textit{Communications in Mathematical\ Physics }
44, 125--128.
\item Cantoni, V. (1977). The Riemannian structure on the states of quantum-like systems.
\textit{Communications in Mathematical\ Physics } 56, 189--193. \item Carson, C. (2000). Continuities and discontinuities in Planck's {\it Akt der Verzweiflung}. {\it Annalen der Physik} 9, 851--960. \item Cassidy, D.C. (1992). {\it Uncertainty: the Life and Science of Werner Heisenberg}. New York: Freeman. \item Castrigiano, D.P.L. \&\ Henrichs, R.W. (1980). Systems of covariance and subrepresentations of induced representations. {\it Letters in Mathematical
Physics} 4, 169-175. \item Cattaneo, U. (1979). On Mackey's imprimitivity theorem.
{\it Commentari Mathematici Helvetici} 54, 629-641. \item Caves, C.M., Fuchs, C.A., \&\ Schack, R. (2002). Unknown quantum states: the quantum de Finetti representation. Quantum information theory. {\it Journal of Mathematical Physics} 43, 4537--4559. \item Charbonnel, A.M. (1986). Localisation et d\'{e}veloppement asymptotique des \'{e}l\'{e}ments du spectre conjoint d' op\'{e}rateurs psuedodiff\'{e}rentiels qui commutent. {\it Integral Equations Operator Theory} 9, 502--536. \item Charbonnel, A.M. (1988). Comportement semi-classiques du spectre conjoint d' op\'{e}rateurs psuedodiff\'{e}rentiels qui commutent. {\it Asymptotic Analysis} 1, 227--261. \item Charbonnel, A.M. (1992). Comportement semi-classiques des syst\`{e}mes ergodiques. {\it Annales de l' Institut Henri Poincar\'{e} - Physique Th\'{e}orique} 56, 187--214. \item Chernoff, P.R. (1973). Essential self--adjointness of powers of generators of hyperbolic equations. \textit{Journal of Functional Analysis} 12, 401--414. \item Chernoff, P.R. (1995). Irreducible representations of infinite dimensional transformation groups and Lie algebras I. {\it Journal of Functional \ Analysis} 130, 255--282. \item Chevalley, C. (1991). Introduction: Le dessin et la couleur. {\it Niels Bohr: Physique Atomique et Connaissance Humaine}. (French translation of Bohr, 1958). Bauer, E. \&\ Omn\`{e}s, R. (Eds.), pp.\ 17--140. Paris: Gallimard. \item Chevalley, C. (1999). Why do we find Bohr obscure? \textit{Epistemological and Experimental Perspectives on Quantum Physics}, pp.\ 59--74.
Greenberger, D., Reiter, W.L., \&\ Zeilinger, A. (Eds.).
Dordrecht: Kluwer Academic Publishers. \item Chiorescu, I., Nakamura, Y., Harmans, C.J.P.M., \&\ Mooij, J.E. (2003). Coherent Quantum Dynamics of a Superconducting Flux Qubit. {\it Science} 299, Issue 5614, 1869--1871. \item Cirelli, R., Lanzavecchia, P., \&\ Mania, A. (1983). Normal pure states of the von Neumann algebra of bounded operators as a K\"{a}hler manifold. {\it Journal of Physics } A16, 3829--3835. \item Cirelli, R.,
Mani\'{a}, A., \&\ Pizzocchero, L. (1990). Quantum mechanics as an infinite-dimensional Hamiltonian system with uncertainty structure. I, II. \textit{Journal of Mathematical\ Physics } 31, 2891--2897, 2898--2903. \item Colin de Verdi\`{e}re, Y. (1973). Spectre du laplacien et longueurs des g\'{e}od\'{e}siques p\'{e}riodiques. I, II. {\it Compositio Mathematica} 27, 83--106, 159--184. \item Colin de Verdi\`ere, Y. (1977). Quasi-modes sur les vari\'et\'es Riemanniennes. {\it Inventiones Mathematicae} 43, 15--52. \item Colin de Verdi\`ere, Y. (1985). Ergodicit\'e et fonctions propres du Laplacien. {\it Communications in Mathematical Physics} 102, 497--502. \item Colin de Verdi\`ere, Y. (1998). Une introduction ˆ la mŽcanique semi-classique. {\it l'Enseignement Mathematique (2)} 44, 23--51. \item Combescure, M. (1992). The squeezed state approach of the semiclassical limit of the time-dependent Schr\"{o}dinger equation. {\it Journal of Mathematical\ Physics } 33, 3870--3880. \itemÊCombescure, M., Ralston, J., \&\ Robert, D. (1999). A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition. {\it Communications in Mathematical\ Physics } 202, 463--480. \itemÊCombescure, M. \&\ Robert, D. (1997). Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. {\it Asymptotic Analysis} 14, 377--404. \item Corwin, L. \&\ Greenleaf, F.P. (1989). \textit{Representations of Nilpotent Lie Groups and Their Applications}, Part I. Cambridge: Cambridge University Press. \item Cucchietti, F.M. (2004). {\it The Loschmidt Echo in Classically Chaotic Systems: Quantum Chaos, Irreversibility and Decoherence}. Ph.D. Thesis, Universidad Nacional de C\'{o}rdobo. \texttt{arXiv:quant-ph/0410121}. \item Cushing, J.T. (1994). {\it Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony}. Chicago: University of Chicago Press. \item Cvitanovic, P. et al. (2005). {\it Classical and Quantum Chaos}. \texttt{http://ChaosBook.org}. \item ÊCycon, H. L., Froese, R. G., Kirsch, W., \&\ Simon, B. (1987). {\it Schr\"{o}dinger Operators with Application to Quantum Mechanics and Global Geometry}. Berlin: Springer-Verlag. \item Darrigol, O. (1992). {\it From c-Numbers to q-Numbers}. Berkeley: University of California Press. \item Darrigol, O. (2001). The Historians' Disagreements over the Meaning of Planck's Quantum. {\it Centaurus} 43, 219--239. \item Davidson, D. (2001). {\it Subjective, Intersubjective, Objective}. Oxford: Clarendon Press \item
Davies, E.B. (1976). {\it Quantum Theory of Open Systems}. London: Academic Press. \item De Bi\`evre, S. (1992).
Oscillator eigenstates concentrated on classical trajectories. {\it Journal of Physics} A25, 3399-3418. \item De Bi\`evre, S. (2001). Quantum chaos: a brief first visit. {\it Contemporary Mathematics} 289,
161--218. \item De Bi\`{e}vre, S. (2003). Local states of free bose fields. Lectures given at the Summer School on Large Coulomb Systems, Nordfjordeid. \item De Bi\`evre, S., Irac-Astaud, M, \&\ Houard, J.C. (1993). Wave packets localised on closed classical trajectories. {\it Differential Equations and Applications in Mathematical Physics}, pp.\
25--33. Ames, W.F. \&\ Harrell, E.M., \&\ Herod, J.V. (Eds.). New York:
Academic Press. \item De Muynck, W.M. (2002) {\it Foundations of Quantum Mechanics: an Empiricist Approach}. Dordrecht: Kluwer Academic Publishers. \item Devoret, M.H., Wallraff, A., \&\ Martinis, J.M. (2004). Superconducting Qubits: A Short Review. \texttt{arXiv:cond-mat/0411174}. \item Diacu, F. \&\ Holmes, P. (1996). {\it Celestial Encounters. The Origins of Chaos and Stability}. Princeton: Princeton University Press. \item Dickson, M. (2005). Non-relativistic \qm. {\it This Volume}. \item Dieks, D. (1989a). Quantum mechanics without the projection postulate and its realistic interpretation. {\it Foundations of Physics} 19, 1397--1423. \item Dieks, D. (1989b). Resolution of the measurement problem through decoherence of the quantum state. {\it Physics Letters} 142A, 439--446. \item Dimassi, M. \&\ Sj\"ostrand, J. (1999). {\it Spectral Asymptotics in the Semi-Classical Limit}. Cambridge: Cambridge University Press. \item Dirac, P.A.M. (1926). The fundamental equations of \qm. {\it Proceedings of the Royal Society} A109, 642--653. \item Dirac, P.A.M. (1930). {\it The Principles of Quantum Mechanics}. Oxford: Clarendon Press. \item Dirac, P.A.M. (1964). \textit{Lectures on Quantum Mechanics}. New York: Belfer School of Science, Yeshiva University. \item Dixmier, J. (1977). {\it $C^*$-Algebras}. Amsterdam: North--Holland. \item Doebner, H.D. \&\ J. Tolar (1975). Quantum mechanics on homogeneous spaces. {\em Journal of Mathematical Physics} 16, 975--984. \item Dowker, F. \&\ Kent, A. (1996). On the Consistent Histories Approach to Quantum Mechanics. {\it Journal of Statistical Physics } 82, 1575--1646. \item Dubin, D.A., Hennings, M.A., \&\ Smith, T.B. (2000). {\it Mathematical Aspects of Weyl Quantization and Phase}. Singapore: World Scientific. \item ÊDuclos, P. \&\ Hogreve, H. (1993). On the semiclassical localization of the quantum probability. {\it Journal of Mathematical Physics} 34, 1681--1691. \item Duffield, N.G. (1990). Classical and thermodynamic limits for generalized quantum spin systems. {\em Communications in Mathematical\ Physics } 127, 27--39. \item Duffield, N.G. \&\ Werner, R.F. (1992a). Classical Hamiltonian dynamics for quantum Hamiltonian mean-field limits. {\it Stochastics and Quantum Mechanics: Swansea, Summer 1990}, pp.\ 115--129. Truman, A. \&\ Davies, I.M. (Eds.). Singapore: World Scientific. \item Duffield, N.G. \&\ Werner, R.F. (1992b). On mean-field dynamical semigroups on \ca s. {\it Reviews in Mathematical Physics} 4, 383--424. \item Duffield, N.G. \&\ Werner, R.F. (1992c). Local dynamics of mean-field quantum systems. {\it Helvetica Physica Acta} 65, 1016--1054. \item Duffield, N.G., Roos, H., \&\ Werner, R.F. (1992). Macroscopic limiting dynamics of a class of inhomogeneous mean field quantum systems. {\it Annales de l' Institut Henri Poincar\'{e} - Physique Th\'{e}orique} 56, 143--186. \item ÊDuffner, E. \&\ Rieckers, A. (1988). On the global quantum dynamics of multilattice systems with nonlinear classical effects. {\it Zeitschrift f\"{u}r Naturforschung} A43, 521--532. \item Duistermaat, J.J. (1974). Oscillatory integrals, Lagrange immersions and unfolding of singularities. {\it Communications in Pure and Applied Mathematics} 27, 207--281. \item Duistermaat, J.J. \&\ Guillemin, V. (1975). The spectrum of positive elliptic operators and periodic bicharacteristics. {\it Inventiones Mathematicae} 29, 39--79. \item Duistermaat, J.J. (1996). {\it Fourier Integral Operators}. Original Lecture Notes from 1973. Basel: Birkh\"auser. \item ÊDuval, C., Elhadad, J., Gotay, M.J., \'Sniatycki, J., \&\ Tuynman, G.M. (1991). Quantization and bosonic BRST theory. {\it Annals of Physics} 206, 1--26. \item Earman, J. (1986). {\it A Primer on Determinism}. Dordrecht: Reidel. \item Earman, J. (2005). Aspects of determinism in modern physics. {\it This volume}. \item Earman, J. (2006). Essential self-adjointness: implications for determinism and the classical-quantum correspondence. {\it Synthese}, to appear. \item Echterhoff, S., Kaliszewski, S., Quigg, J., \&\ Raeburn, I. (2002). A categorical approach to imprimitivity theorems for C*-dynamical systems. \texttt{arXiv:math.OA/0205322}. \item Eddington, A.S. (1920). {\it Space, Time, and Gravitation: An Outline of the General Relativity Theory}. Cambridge: Cambridge University Press. \item Effros, E.G. \&\ Hahn, F. (1967). Locally compact transformation groups and $C\sp{*} $- algebras. {\it Memoirs of the American Mathematical Society} 75. \item Ehrenfest, P. (1927). Bemerkung \"{u}ber die angen\"{a}herte Gultigkeit der klassischen Mechanik innerhalb der Quantenmechanik. {\it Zeitschrift f\"{u}r Physik} 45, 455--457. \item Einstein, A. (1905). \"{U}ber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtpunkt. {\it Annalen der Physik} 17, 132--178. \item Einstein, A. (1917). Zum Quantensatz von Sommerfeld und Epstein. {\it Verhandlungen der deutschen Physikalischen Geselschaft (2)} 19, 82--92. \item Einstein, A. (1949). Remarks to the essays appearing in this collective volume. (Reply to criticisms).
{\it Albert Einstein: Philosopher-Scientist}, pp.\ 663--688. Schilpp, P.A. (Ed.). La Salle: Open Court. \item ÊEmch, G.G. \&\ Knops, H.J.F. (1970). Pure thermodynamical phases as extremal KMS states. {\it Journal of Mathematical Physics} 11, 3008--3018. \item ÊEmch, G.G. (1984) {\it Mathematical and conceptual foundations of 20th-century physics}. Amsterdam: North-Holland. \item Emch, G.G. \&\ Liu, C. (2002). {\it The Logic of Thermostatistical Physics}. Berlin: Springer-Verlag. \item Emch, G.G., Narnhofer, H., Thirring, W., \&\ Sewell, G. (1994).
Anosov actions on noncommutative algebras.
{\it Journal of Mathematical Physics} 35, 5582--5599. \item d'Espagnat, B. (1995). {\it Veiled Reality: An Analysis of Present-Day Quantum Mechanical Concepts}. Reading (MA): Addison-Wesley. \item Esposito, G., Marmo, G., \&\ Sudarshan, G. (2004). {\it From Classical to Quantum Mechanics: An Introduction to the Formalism, Foundations and Applications}. Cambridge: Cambridge University Press. \item Enz, C.P. (2002). {\it No Time to be Brief: A Scientific Biography of Wolfgang Pauli}. Oxford: Oxford University Press. \item Everett, H. {\sc iii} (1957). ``Relative state" formulation of \qm. {\it Reviews in Modern Physics} 29, 454--462. \item Faye, J. (1991). {\it Niels Bohr: His Heritage and Legacy. An Anti-Realist View of Quantum Mechanics}. Dordrecht: Kluwer Academic Publishers. \item Faye, J. (2002). Copenhagen Interpretation of Quantum Mechanics. {\it The Stanford Encyclopedia of Philosophy (Summer 2002 Edition)}. Zalta, E.N.Ê(Ed.). \\ \texttt{http://plato.stanford.edu/archives/sum2002/entries/qm-copenhagen/}. \item Faye, J. \&\ Folse, H. (Eds.) (1994). {\it Niels Bohr and Contemporary Philosophy}. Dordrecht: Kluwer Academic Publishers. \item Fell, J.M.G. \&\ Doran, R.S. (1988). {\it Representations of $\mbox{}^*$-Algebras, Locally Compact Groups and Banach $\mbox{}^*$-Algebraic Bundles, Vol.\ 2}. Boston: Academic Press. \item Feyerabend, P. (1981). Niels Bohr's world view. {\it Realism, Rationalism \&\ Scientific Method: Philosophical Papers Vol.\ 1}, pp.\ 247--297. Cambridge: Cambridge University Press. \item Fleming, G. \&\ Butterfield, J. (2000). Strange positions. {\it From Physics to Philosophy}, pp.\ 108--165. Butterfield, J. \&\ Pagonis, C. (Eds.). Cambridge: Cambridge University Press. \item Folse, H.J. (1985). {\it The Philosophy of Niels Bohr}. Amsterdam: North-Holland. \item Ford, J. (1988). Quantum chaos. Is there any? {\it Directions in Chaos, Vol. 2}, pp.\ 128--147. Bai-Lin, H. (Ed.). Singapore: World Scientific. \item Frasca, M. (2003). General theorems on decoherence in the thermodynamic limit. {\it Physics Letters} A308, 135--139. \item Frasca, M. (2004). Fully polarized states and decoherence. \texttt{arXiv:cond-mat/0403678}. \item Frigerio, A. (1974). Quasi-local observables and the problem of measurement in quantum mechanics. {\it Annales de l' Institut Henri Poincar\'{e}} A3, 259--270. \item Fr\"{o}hlich, J., Tsai, T.-P., \&\ Yau, H.-T. (2002). On the point-particle (Newtonian) limit of the non-linear Hartree equation. {\it Communications in Mathematical Physics} 225, 223--274. \item Gallavotti, G. (1983). {\it The Elements of Mechanics}. Berlin: Springer-Verlag. \item Gallavotti, G., Bonetto, F., \&\ Gentile, G. (2004). {\it Aspects of Ergodic, Qualitative and Statistical Theory of Motion}. New York: Springer. \item Gell-Mann, M. \&\ Hartle, J.B. (1990). Quantum mechanics in the light of quantum cosmology. {\it Complexity, Entropy, and the Physics of Information}, pp.\ 425--458. Zurek, W.H. (Ed.). Reading, Addison-Wesley. \item Gell-Mann, M. \&\ Hartle, J.B. (1993). Classical equations for quantum systems.
{\it Physical Review} D47, 3345--3382. \item G\'erard, P. \&\ Leichtnam, E. (1993). Ergodic properties of eigenfunctions for the Dirichlet problem. {\it Duke Mathematical Journal} 71, 559--607. \item Gerisch, T., M\"{u}nzner, R., \&\ Rieckers, A. (1999).
Global $C^*$-dynamics and its KMS states of weakly inhomogeneous bipolaronic superconductors. {\it Journal of Statistical Physics } 97, 751--779. \item Gerisch, T., Honegger, R., \&\ Rieckers, A. (2003). Algebraic quantum theory of the Josephson microwave radiator. {\it Annales Henri Poincar\'{e}} 4, 1051--1082. \item Geyer, B., Herwig, H., \&\ Rechenberg, H. (Eds.) (1993). {\it Werner Heisenberg: Physiker und Philosoph}. Leipzig: Spektrum. \item Giulini, D. (2003). Superselection rules and symmetries. {\it Decoherence and the Appearance of a Classical World in Quantum Theory}, pp.\ 259--316. Joos, E. et al. (Eds.). Berlin: Springer. \item Glimm, J. \&\ Jaffe, A. (1987). {\it Quantum Physics. A Functional Integral Point of View}. New York: Springer-Verlag. \item Gotay, M.J. (1986). Constraints, reduction, and quantization. {\it Journal of Mathematical\ Physics } 27, 2051--2066. \item Gotay, M.J. (1999). On the Groenewold-Van Hove problem for $\mathbf{R}\sp {2n}$. {\it Journal of Mathematical Physics} 40, 2107--2116. \item Gotay, M.J., Grundling, H.B., \&\ Tuynman, G.M. (1996). Obstruction results in quantization theory. {\it Journal of Nonlinear Science} 6, 469--498. \item Gotay, M.J.,
Nester, J.M., \&\ Hinds, G. (1978). Presymplectic manifolds and the Dirac--Bergmann theory of constraints. {\it Journal of Mathematical\ Physics } 19, 2388--2399. \item G\"{o}tsch, J. (1992). {\it Erwin Schr\"{o}dinger's World View: The Dynamics of Knowlegde and Reality}. Dordrecht: Kluwer Academic Publishers.
\item Govaerts, J. (1991). {\it Hamiltonian Quantization and Constrained Dynamics}. Leuven: Leuven University Press.
\item Gracia-Bond\'{\i}a, J.M., V\'{a}rilly, J.C., \&\ Figueroa, H. (2001). \textit{Elements of Noncommutative Geometry}. Boston: Birkh\"{a}user. \item Griesemer, M., Lieb, E.H., \&\ Loss, M. (2001). Ground states in non-relativistic quantum electrodynamics. {\it Inventiones Mathematicae} 145, 557--595.
\item Griffiths, R.B. (1984). Consistent histories and the interpretation of \qm. {\it Journal of Statistical Physics} 36, 219--272. \item Griffiths, R.B. (2002). {\it Consistent Quantum Theory}. Cambridge: Cambridge University Press. \item Grigis, A. \&\ Sj\"ostrand, J. (1994). {\it Microlocal Analysis for Differential Operators.} Cambridge: Cambridge University Press. \item Groenewold, H.J. (1946). On the principles of elementary \qm. {\it Physica} 12, 405--460. \item Groenewold, H.J. (1952). Information in quantum measurements. {\it Proceedings Koninklijke Nederlandse Akademie van Wetenschappen} B55, 219--227. \item Guhr, T., M\"uller-Groeling, H., \&\ Weidenm\"uller, H. (1998). Random matrix theories in quantum physics: common concepts. {\it Physics Reports} 299, 189--425. \item Guillemin, V., Ginzburg, V. \& Karshon, Y. (2002). {\it Moment Maps, Cobordisms, and Hamiltonian Group Actions}. Providence (RI): American Mathematical Society. \item Guillemin, V. \&\ Sternberg, S. (1977). {\it Geometric Asymptotics}. Providence (RI): American Mathematical Society. \item Guillemin, V. \&\ Sternberg, S. (1990). {\it Variations on a Theme by Kepler}. Providence (RI): American Mathematical Society. \item Guillemin, V. \&\ Uribe, A. (1989). Circular symmetry and the trace formula. {\it Inventiones Mathematicae} 96, 385--423. \item Gustafson, S.J. \&\ Sigal, I.M. (2003). {\it Mathematical concepts of quantum mechanics}. Berlin: Springer. \item Gutzwiller, M.C. (1971). Periodic orbits and classical quantization conditions. {\it Journal of Mathematical Physics} 12, 343--358. \item Gutzwiller, M.C. (1990). {\it Chaos in Classical and Quantum Mechanics}. New York: Springer-Verlag. \item Gutzwiller, M.C. (1992). Quantum chaos. {\it Scientific American} 266, 78--84. \item Gutzwiller, M.C. (1998). Resource letter ICQM-1: The interplay between classical and quantum mechanics. {\it American Journal of Physics } 66, 304--324. \item Haag, R. (1962). The mathematical structure of the Bardeen--Cooper--Schrieffer model. {\it Nuovo Cimento} 25, 287--298. \item Haag, R., Kadison, R., \&\ Kastler, D. (1970). Nets of \ca s and classification of states. {\it Communications in Mathematical Physics} 16, 81--104. \item Haag, R. (1992). {\it Local Quantum Physics: Fields, Particles, Algebras}. Heidelberg: Springer-Verlag. \item Haake, F. (2001). {\it Quantum Signatures of Chaos}. Second Edition. New York: Springer-Verlag. \item Hagedorn, G.A. (1998). Raising and lowering operators for semiclassical wave packets. {\it Annals of Physics} 269, 77--104. \item Hagedorn, G.A. \&\ Joye, A. (1999). Semiclassical dynamics with exponentially small error estimates. {\it Communications in Mathematical Physics} 207, 439--465. \item Hagedorn, G.A. \&\ Joye, A. (2000). Exponentially accurate semiclassical dynamics: propagation, localization, Ehrenfest times, scattering, and more general states. {\it Annales Henri Poincar\'e} 1, 837--883. \item Halliwell, J.J. (1998). Decoherent histories and hydrodynamic equations. {\it Physical Review} D58, 105015-1--12. \item ÊHalliwell, J.J. (2000). The emergence of hydrodynamic equations from quantum theory: a decoherent histories analysis. {\it International Journal of Theoretical Physics } 39, 1767--1777. \item Halliwell, J.J. (2004).Some recent developments in the decoherent histories approach to quantum theory. {\it Lecture Notes in Physics} 633, 63--83.
\item ÊHalliwell, J.J. (2005). How the quantum Universe becomes classical.
\texttt{arXiv:quant-ph/0501119}. \item ÊHalvorson, H. (2004). Complementarity of representations in quantum mechanics. {\it Studies in History and Philosophy of Modern Physics } B35, 45--56. \item ÊHalvorson, H. (2005). Algebraic quantum field theory. This volume. \item Halvorson, H. \&\ Clifton, R. (1999). Maximal beable subalgebras of quantum-mechanical observables. {\it International Journal of Theoretical Physics } 38, 2441--2484. \item Halvorson, H. \&\ Clifton, R. (2002). Reconsidering Bohr's reply to {\sc epr}. {\it Non-locality and Modality}, pp.\ 3--18. Placek, T. \&\ Butterfield, J. (Eds.). Dordrecht: Kluwer Academic Publishers. \item Hannabuss, K.C. (1984). Dilations of a quantum measurement. {\it Helvetica Physica Acta} 57, 610--620. \item Harrison, F.E. \&\ Wan, K.K. (1997). Macroscopic quantum systems as measuring devices: dc SQUIDs and superselection rules. {\it Journal of Physics} A30, 4731--4755. \item Hartle, J.B. (1995). Spacetime quantum mechanics and the quantum mechanics of spacetime. {\it Gravitation et Quantifications (Les Houches, 1992)}, pp.\ 285--480. Amsterdam: North-Holland. \item Hartle, J.B. (2005).
What connects different interpretations of quantum mechanics? {\it Quo Vadis Quantum Mechanics}, pp.\ 73-82. Elitzur, A., Dolev, S., \&\ Kolenda, N. (Eds.). Heidelberg: Springer-Verlag. \texttt{arXiv:quant-ph/0305089}. \item Heath, D. \&\ Sudderth, W. (1976). De Finetti's theorem on exchangeable variables. {\it American Statistics} 30, 188--189. \item Heelan, P. (1965). {\it Quantum Mechanics and Objectivity: A Study of the Physical Philosophy of Werner Heisenberg}. Den Haag: Martinus Nijhoff. \item Heilbron, J. (2000). {\it The Dilemmas of an Upright Man: Max Planck as a Spokesman for German Science}. Second Edition. Los Angeles: University of California Press. \item Heisenberg, W. (1925). \"{U}ber die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. {\it Zeitschrift f\"{u}r Physik} 33, 879-893. \item Heisenberg, W. (1927). \"{U}ber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. {\it Zeitschrift f\"{u}r Physik} 43, 172--198. \item Heisenberg, W. (1930). {\it The Physical Principles of the Quantum Theory}. Chicago: University of Chicago Press. \item Heisenberg, W. (1942). Ordnung der Wirklichkeit. In Heisenberg (1984a), pp.\ 217--306.
Also available at \texttt{http://werner-heisenberg.unh.edu/Ordnung.htm}. \item Heisenberg, W. (1958). {\it Physics and Philosophy: The Revolution in Modern Science}. London: Allen \&\ Unwin. \item Heisenberg, W. (1969). {\it Der Teil und das Ganze: Gespr\"{a}che im Umkreis der Atomphysik}. M\"{u}nchen: Piper. English translation as Heisenberg (1971). \item Heisenberg, W. (1971). {\it Physics and Beyond}. New York: Harper and Row. Translation of Heisenberg (1969). \item Heisenberg, W. (1984a). {\it Gesammelte Werke. Series C: Philosophical and Popular Writings, Vol {\sc i}: Physik und Erkenntnis 1927--1955}. Blum, W., D\"{u}rr, H.-P., \&\ Rechenberg, H. (Eds.). M\"{u}nchen: Piper. \item Heisenberg, W. (1984b). {\it Gesammelte Werke. Series C: Philosophical and Popular Writings, Vol {\sc ii}: Physik und Erkenntnis 1956--1968}. Blum, W., D\"{u}rr, H.-P., \&\ Rechenberg, H. (Eds.). M\"{u}nchen: Piper. \item Heisenberg, W. (1985). {\it Gesammelte Werke. Series C: Philosophical and Popular Writings, Vol {\sc iii}: Physik und Erkenntnis 1969--1976}. Blum, W., D\"{u}rr, H.-P., \&\ Rechenberg, H. (Eds.). M\"{u}nchen: Piper. \item Held, C. (1994). The Meaning of Complementarity. {\it Studies in History and Philosophy of Science} 25, 871--893. \item Helffer, B. (1988) {\it Semi-classical Analysis for the Schr\"{o}dinger Operator and Applications}. {\it Lecture Notes in Mathematics} 1336. Berlin: Springer-Verlag. \item Heller, E.J. \&\ Tomsovic, S. (1993). Postmodern quantum mechanics. {\it Physics Today} July, 38--46. \item Hendry, J. (1984). {\it The Creation of Quantum Mechanics and the Bohr-Pauli Dialogue}. Dordrecht: D. Reidel. \item ÊHenneaux, M. \&\ Teitelboim, C. (1992). {\it Quantization of Gauge Systems}. Princeton: Princeton University Press. \item Hepp, K. (1972). Quantum theory of measurement and macroscopic observables. {\it Helvetica Physica Acta} 45, 237--248.
\item Hepp, K. (1974). The classical limit of quantum mechanical correlation functions. {\it Communications in Mathematical\ Physics } 35, 265--277.
\item Hepp, K. \&\ Lieb, E. (1974). Phase transitions in reservoir driven open systems with applications to lasers and superconductors. {\it Helvetica Physica Acta} 46, 573--602. \item Higson, N. (1990).
A primer on $KK$-theory. {\it Operator Theory: Operator Algebras and Applications. Proceedings Symposia in Pure Mathematical, 51, Part 1}, pp.\ 239--283. Providence, RI:
American Mathematical Society \item Hillery, M.,
O'Connel, R.F., Scully, M.O., \&\ Wigner, E.P. (1984). Distribution
functions in physics -- Fundamentals. \textit{Physics \ Reports} 106,
121--167. \item Hislop, P. D. \& Sigal, I. M. (1996). {\it Introduction to Spectral Theory. With Applications to Schr\"{o}dinger Operators}. New York: Springer-Verlag. \item Hoffmann, E.T.A. (1810). {\it Musikalische Novellen und Aufs\"{a}tze}. Leipzig: Insel-B\"{u}cherei. \item Holevo, A.S. (1982). {\it Probabilistic and Statistical Aspects of Quantum Theory}. Amsterdam: North-Holland Publishing Co. \item Hogreve, H., Potthoff, J., \&\ Schrader, R. (1983).
Classical limits for quantum particles in external Yang--Mills
potentials. {\it Communications in Mathematical\ Physics } 91, 573--598. \item Honegger, R. \&\ Rieckers, A. (1994). Quantized radiation states from the infinite Dicke model. {\it Publications of the Research Institute for Mathematical Sciences (Kyoto)} 30, 111--138. \item Honner, J. (1987). {\it The Description of Nature: Niels Bohr and the Philosophy of Quantum Physics}. Oxford: Oxford University Press. \item Hooker, C.A. (1972). The nature of quantum mechanical reality: Einstein versus Bohr. {\it Paradigms \&\ Paradoxes: The Philosophical Challenges of the Quantum Domain}, pp.\ 67--302. Colodny, J. (Ed.). Pittsburgh: University of Pittsburgh Press. \item H\"{o}rmander, L. (1965). Pseudo-differential operators. {\it Communications in Pure Applied Mathematical} 18, 501--517 \item H\"{o}rmander, L. (1979). The Weyl calculus of
pseudo-differential operators. {\it Communications in Pure Applied\ Mathematical}
32, 359--443. \item H\"{o}rmander, L. (1985a). {\it The Analysis of Linear
Partial Differential Operators, Vol.\ III}. Berlin: Springer-Verlag. \item H\"{o}rmander, L. (1985b). {\it The Analysis of Linear
Partial Differential Operators, Vol.\ IV}. Berlin: Springer-Verlag. \item Horowitz,
G.T. \&\ Marolf, D. (1995). Quantum probes of spacetime
singularities. \textit{Physical Review} D52, 5670--5675. \item H\"{o}rz, H. (1968). {\it Werner Heisenberg und die Philosophie}. Berin: VEB Deutscher Verlag der Wissenschaften. \item Howard, D. (1990). `Nicht sein kann was nicht sein darf', or the Prehistory of {\sc epr}, 1909-1935: Einstein's early worries about the \qm\ of composite systems. {\it
Sixty-Two Years of Uncertainty}, pp.\ 61--11. Miller, A.I. (Ed.). New
York: Plenum. \item Howard, D. (1994). What makes a classical concept classical? Towards a reconstruction of Niels Bohr's philosophy of physics. {\it Niels Bohr and Contemporary Philosophy}, pp.\ 201--229. Faye, J. \&\ Folse, H. (Eds.). Dordrecht: Kluwer Academic Publishers. \item Howard, D. (2004). Who Invented the Copenhagen Interpretation? {\it Philosophy of Science} 71, 669-682. \item Howe, R. (1980). Quantum mechanics and partial differential equations. {\it Journal of Functional Analysis} 38, 188--254 \item Hudson, R.L. (1974).
When is the Wigner quasi-probability density non-negative?
{\it Reports of Mathematical Physics} 6, 249--252. \item Hudson, R.L. \&\ Moody, G.R. (1975/76).
Locally normal symmetric states and an analogue of de Finetti's theorem. {\it Z. Wahrscheinlichkeitstheorie und Verw. Gebiete} 33, 343--351 \item Hunziker, W. \&\ Sigal, I. M. (2000). The quantum $N$-body problem. {\it Journal of Mathematical Physics} 41, 3448--3510. \item Husimi, K. (1940). Some formal
properties of the density matrix. \textit{Progress of the Physical and Mathematical Society of Japan} 22, 264--314.
\item ÊIsham, C.J. (1984). Topological and global aspects of quantum theory. {\it Relativity, Groups and Topology, II (Les Houches, 1983)}, 1059--1290. Amsterdam: North-Holland.
\item Isham, C.J. (1994). Quantum logic and the histories approach to quantum theory. {\it Journal of Mathematical Physics} 35, 2157--2185.
\item Isham, C.J. (1997). Topos theory and consistent histories: the internal logic of the set of all consistent sets. {\it International Journal of Theoretical Physics } 36, 785--814.
Ê\item Isham, C.J. \&\ Butterfield, J. (2000). Some possible roles for topos theory in quantum theory and quantum gravity. {\it Foundations of Physics } 30, 1707--1735.
\item Isham, C.J. \&\ Linden, N. (1994). Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory. {\it Journal of Mathematical Physics} 35, 5452--5476.
\item ÊIsham, C.J. \&\ Linden, N. (1995). Continuous histories and the history group in generalized quantum theory. {\it Journal of Mathematical Physics} 36, 5392--5408.
\item ÊIsham, C.J., Linden, N., \&\ Schreckenberg, S. (1994). The classification of decoherence functionals: an analog of Gleason's theorem. {\it Journal of Mathematical Physics} 35, 6360--6370. \item Ivrii, V. (1998). {\it Microlocal Analysis and Precise Spectral Asymptotics}. New York: Springer-Verlag. \item Jalabert, R.O. \&\ Pastawski, H.M. (2001). Environment-Independent Decoherence Rate in Classically Chaotic Systems. {\it Physical Review Letters} 86, 2490--2493. \item Jammer, M. (1966). {\it The Conceptual Development of Quantum Mechanics}. New York: McGraw-Hill. \item Jammer, M. (1974). {\it The Philosophy of Quantum Mechanics}. New York: Wiley. \item Janssens, B. (2004). {\it Quantum Measurement: A Coherent Description}. M.Sc.\ Thesis, Radboud Universiteit Nijmegen. \item Jauch, J.M. (1968). {\it Foundations of Quantum Mechanics}. Reading (MA): Addison-Wesley. \item Joos, E. \&\ Zeh, H.D. (1985). The emergence of classical properties through interaction with the environment. {\it Zeitschrift f\"{u}r Physik} B59, 223--243. \item Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, J., \&\ Stamatescu, I.-O. (2003). {\it Decoherence and the Appearance of a Classical World in Quantum Theory}. Berlin: Springer-Verlag. \item J\o rgensen, P.E.T. \&\ Moore, R.T. (1984). {\it Operator Commutation Relations}. Dordrecht: Reidel. \item ÊKaplan, L. \&\ Heller, E.J. (1998a). Linear and nonlinear theory of eigenfunction scars. {\it Annals of Physics} 264, 171--206. \item Kaplan, L. \&\ Heller, E.J. (1998b). Weak quantum ergodicity. {\it Physica D} 121, 1--18. \item Kaplan, L. (1999). Scars in quantum chaotic wavefunctions. {\it Nonlinearity} 12, R1--R40. \item Katok, A. \&\ Hasselblatt, B. (1995). {\it Introduction to the Modern Theory of Dynamical Systems}. Cambridge: Cambridge University Press. \item Kadison, R.V. \&\ Ringrose, J.R. (1983). {\it Fundamentals of the theory of operator algebras. Vol. 1: Elementary Theory}. New York: Academic Press. \item Kadison, R.V. \&\ Ringrose, J.R. (1986). {\it Fundamentals of the theory of operator algebras. Vol. 2: Advanced Theory}. New York: Academic Press. \item Karasev, M.V. (1989). The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds. I, II. {\it Selecta Mathematica Formerly Sovietica } 8, 213--234, 235--258. \item Kent, A. (1990). Against Many-Worlds Interpretations. {\it International Journal of Modern Physics } A5 (1990) 1745. \item Kent, A. (1997). Consistent sets yield contrary inferences in quantum theory. {\it Physical Review Letters} 78, 2874--2877. Reply by Griffiths, R.B. \&\ Hartle, J.B. (1998). {\it ibid.} 81, 1981. Reply to this reply by Kent, A. (1998). {\it ibid.} 81, 1982. \item Kent, A. (1998). Quantum histories. {\it Physica Scripta} T76, 78--84. \item Kent, A. (2000). Night thoughts of a quantum physicist. {\it Philosophical Transactions of the Royal Society of London} A 358, 75--88. \item Kiefer, C. (2003). Consistent histories and decoherence. {\it Decoherence and the Appearance of a Classical World in Quantum Theory}, pp.\ 227--258. Joos, E. et al. (Eds.). Berlin: Springer-Verlag. \item Kirchberg, E. \&\
Wassermann, S. (1995). Operations on continuous bundles of
$C^*$-algebras. {\it Mathematische Annalen} 303, 677--697. \item
Kirillov, A.A. (1990). Geometric Quantization. {\it Dynamical Systems IV}, pp.\ 137--172. Arnold, V.I. \&\ S.P. Novikov (Eds.). Berlin: Springer-Verlag. \item Kirillov, A.A. (2004). {\it Lectures on the Orbit Method}. Providence, RI: American Mathematical Society. \item Klauder, J.R. \&\ B.-S. Skagerstam (Eds.). (1985). {\it Coherent
States}. Singapore: World Scientific. \item
Klingenberg, W. (1982). \textit{Riemannian Geometry}. de Gruyter,
Berlin. \item Kohn, J.J. \&\ Nirenberg, L. (1965). An algebra of pseudo-differential operators. {\it Communications in Pure and Applied Mathematics} 18 269--305. \item Koopman, B.O. (1931). Hamiltonian systems and transformations in \Hs. {\it Proceedings of the National Academy of Sciences} 18, 315--318. \item Kostant, B. (1970). Quantization and unitary
representations. {\it Lecture Notes in Mathematics} 170,
87--208. \item Krishnaprasad, P. S. \&\ Marsden, J. E. (1987). Hamiltonian structures and stability for rigid bodies with flexible attachments. {\it Archive of Rational Mechanics and Analysis} 98, 71--93. \item Kuhn, T. S. (1978). {\it Black-body Theory and the Quantum Discontinuity: 1894Ð1912}. New York: Oxford University Press. \item K\"{u}mmerer, B. (2002). Quantum Markov processes. {\it Coherent Evolution in Noisy Environment (Lecture Notes in Physics Vol. 611)}, Buchleitner, A. \&\ Hornberger, K. (Eds.), pp.\ 139-198. Berlin: Springer-Verlag. \item Lahti, P. \&\ Mittelstaedt, P. (Eds.) (1987). {\it The Copenhagen Interpretation 60 Years After the Como Lecture}. Singapore: World Scientific. \item Landau, L.D. \&\ Lifshitz, E.M. (1977). {\it Quantum Mechanics: Non-relativistic Theory}. 3d Ed. Oxford: Pergamon Press. \item Landsman, N.P. (1990a).
Quantization and superselection sectors I. Transformation group
$C^*$-algebras. \textit{Reviews in Mathematical Physics } 2, 45--72. \item
Landsman, N.P. (1990b). Quantization and superselection sectors
II. Dirac Monopole and Aharonov--Bohm effect. \textit{Reviews in Mathematical
Physics } 2, 73--104. \item Landsman, N.P. (1991). Algebraic theory of superselection sectors and the measurement problem in quantum mechanics. {\it International Journal of Modern Physics} A30, 5349--5371. \item Landsman, N.P. (1992) Induced representations, gauge fields, and
quantization on homogeneous spaces. \textit{Reviews in Mathematical Physics }
4, 503--528. \item Landsman, N.P. (1993). Deformations of algebras
of observables and the classical limit of quantum mechanics. {\it
Reviews in Mathematical Physics } 5, 775--806. \item Landsman, N.P. (1995). Observation and superselection in quantum mechanics. {\it Studies in History and Philosophy of Modern Physics } 26B, 45--73. \item Landsman, N.P. (1997). Poisson spaces with a
transition probability. \textit{Reviews in Mathematical Physics } 9,
29--57. \item Landsman, N.P. (1998). {\it Mathematical Topics Between Classical and Quantum Mechanics}. New York: Springer-Verlag. \item Landsman, N.P. (1999a). Quantum Mechanics on phase Space. {\it Studies in History and Philosophy of Modern Physics } 30B, 287--305. \item Landsman, N.P. (1999b). Lie groupoid $C\sp *$-algebras and Weyl quantization. {\it Communications in Mathematical Physics} 206, 367--381. \item Landsman, N.P. (2001). Quantized reduction as a tensor product. {\it Quantization of Singular Symplectic Quotients}, pp.\ 137--180. Landsman, N.P., Pflaum, M.J., \&\ Schlichenmaier, M. (Eds.). Basel: Birkh\"{a}user. \item Landsman, N.P. (2002). Quantization as a functor. {\it Contemporary Mathematics} 315, 9--24. \item Landsman, N.P. (2005a). Functorial quantization and the Guillemin--Sternberg conjecture. {\it Twenty Years of Bialowieza: A Mathematical Anthology}, pp.\ 23--45. Ali, S.T., Emch, G.G., Odzijewicz, A., Schlichenmaier, M., \&\ Woronowicz, S.L. (Eds). Singapore: World Scientific. \texttt{arXiv:math-ph/0307059}. \item Landsman, N.P. (2005b). Lie Groupoids and Lie algebroids in physics and noncommutative geometry. {\it Journal of Geom. Physics }, to appear. \item Landsman, N.P. (2006). When champions meet: Rethinking the Bohr--Einstein debate. {\it Studies in History and Philosophy of Modern Physics}, to appear. \texttt{arXiv:quant-ph/0507220}. \itemÊLandsman, N.P. \&\ Ramazan, B. (2001). Quantization of Poisson algebras associated to Lie algebroids. {\it Contemporary Mathematics} 282, 159--192. \item Laurikainen, K.V. (1988). {\it Beyond the Atom: The Philosophical Thought of Wolfgang Pauli}. Berlin: Springer-Verlag. \item Lazutkin, V.F. (1993). {\it KAM Theory and Semiclassical Approximations to Eigenfunctions}. Berlin: Springer-Verlag. \item Leggett, A.J. (2002). Testing the limits of quantum mechanics: motivation, state of play, prospects. {\it Journal of Physics: Condensed Matter} 14, R415--R451. \item Liboff, R.L. (1984). The correspondence principle revisited. {\it Physics Today} February, 50--55. \item Lieb, E.H. (1973). The classical limit of quantum spin systems.
{\it Communications in Mathematical Physics} 31, 327--340. \item Littlejohn, R.G. (1986). The semiclassical evolution of wave packets. {\it Physics Reports} 138, 193--291. \item Ludwig, G. (1985). {\it An Axiomatic Basis for Quantum Mechanics. Volume 1:
Derivation of Hilbert Space Structure}. Berlin: Springer-Verlag. \item ÊLugiewicz, P. \&\ Olkiewicz, R. (2002). Decoherence in infinite quantum systems. {\it Journal of Physics } A35, 6695--6712. \item ÊLugiewicz, P. \&\ Olkiewicz, R. (2003). Classical properties of infinite quantum open systems. Communications in Mathematical Physics 239, 241--259. \item Maassen, H. (2003). Quantum probability applied to the damped harmonic oscillator. {\it Quantum Probability Communications, Vol. XII (Grenoble, 1998)}, pp.\ 23--58.
River Edge, NJ: World Scientific Publishing. \item Mackey, G.W. (1962). {\it The Mathematical Foundations of Quantum Mechanics}. New York: Benjamin. \itemÊMackey, G.W. (1968). {\it Induced Representations of Groups and Quantum Mechanics}. New York: W. A. Benjamin; Turin: Editore Boringhieri. \itemÊMackey, G.W. (1978). {\it Unitary Group Representations in Physics, Probability, and Number Theory}. Reading, Mass.: Benjamin/Cummings Publishing Co. \itemÊMackey, G.W. (1992). {\it The Scope and History of Commutative and Noncommutative Harmonic Analysis}. Providence, RI: American Mathematical Society. \item Majid, S. (1988). Hopf algebras for physics at the Planck scale. {\it Classical \&\ Quantum Gravity} 5, 1587--1606. \item Majid, S. (1990). Physics for algebraists: noncommutative and noncocommutative Hopf algebras by a bicrossproduct construction. {\it Journal of Algebra} 130, 17--64. \item Marmo, G., Scolarici, G., Simoni, A., \&\ Ventriglia, F. (2005). The quantum-classical transition: the fate of the complex structure. {\it International Journal of Geometric Methods in Physics} 2, 127--145. \item Marsden, J.E. (1992). {\it Lectures on Mechanics}. Cambridge: Cambridge University Press.
\item Marsden, J.E. \&\ T.S. Ratiu (1994). {\it Introduction to Mechanics and Symmetry}. New York: Springer-Verlag.
\item Marsden, J.E., Ra\c tiu, T., \&\ Weinstein, A. (1984). Semidirect products and reduction in mechanics. {\it Transactions of the American Mathematical Society} 281, 147--177. \item Marshall, W., Simon, C., Penrose, R., \&\ Bouwmeester, D. (2003). Towards quantum superpositions of a mirror. {\it Physical Review Letters} 91, 130401-1--4.
\item Martinez, A. (2002). {\it An Introduction to Semiclassical and Microlocal Analysis}. New York: Springer-Verlag.
\item Maslov, V.P. \&\ Fedoriuk, M.V. (1981). {\it Semi-Classical Approximation in Quantum Mechanics}. Dordrecht: Reidel.
\item McCormmach, R. (1982). {\it Night Thoughts of a Classical Physicist}.
Cambridge (MA): Harvard University Press. \item Mehra, J. \&\ and Rechenberg, H. (1982a). {\it The Historical Development of Quantum Theory. Vol.\ 1: The Quantum Theory of Planck, Einstein, Bohr, and Sommerfeld: Its Foundation and the Rise of Its Difficulties}. New York: Springer-Verlag. \item Mehra, J. \&\ and Rechenberg, H. (1982b). {\it The Historical Development of Quantum Theory. Vol.\ 2: The Discovery of Quantum Mechanics }. New York: Springer-Verlag. \item Mehra, J. \&\ and Rechenberg, H. (1982c). {\it The Historical Development of Quantum Theory. Vol.\ 3: The formulation of matrix mechanics and its modifications, 1925-1926.} New York: Springer-Verlag. \item Mehra, J. \&\ and Rechenberg, H. (1982d). {\it The Historical Development of Quantum Theory. Vol.\ 4: The fundamental equations of quantum mechanics 1925-1926. The reception of the new quantum mechanics.} New York: Springer-Verlag. \item Mehra, J. \&\ and Rechenberg, H. (1987). {\it The Historical Development of Quantum Theory. Vol.\ 5: Erwin Schr\"{o}dinger and the Rise of Wave Mechanics.} New York: Springer-Verlag. \item Mehra, J. \&\ and Rechenberg, H. (2000). {\it The Historical Development of Quantum Theory. Vol.\ 6: The Completion of Quantum Mechanics 1926--1941. Part 1: The probabilistic Interpretation and the Empirical and Mathematical Foundation of Quantum Mechanics, 1926-1936.} New York: Springer-Verlag. \item Mehra, J. \&\ and Rechenberg, H. (2001). {\it The Historical Development of Quantum Theory. Vol.\ 6: The Completion of Quantum Mechanics 1926--1941. Part 2: The Conceptual Completion of Quantum Mechanics.} New York: Springer-Verlag. \item Meinrenken, E. (1998). Symplectic surgery and the $\mathrm{Spin}^c$-Dirac operator. {\it Adv.\ Mathematical}
134, 240--277. \item Meinrenken, E. \&\ Sjamaar, R. (1999). Singular reduction and quantization. {\it Topology} 38, 699--762. \item Mermin, N.D. (2004). What's wrong with this quantum world? {\it Physics Today} 57 (2), 10. \item Mielnik, B. (1968). Geometry of quantum states. {\it Communications in Mathematical\ Physics } 9, 55--80. \item Miller, A.I. (1984). {\it
Imagery in Scientific Thought: Creating 20th-Century Physics}. Boston: Birkh\"{a}user. \item Mirlin, A.D. (2000). Statistics of energy levels and eigenfunctions in disordered systems. {\it Physics Reports} 326, 259--382. \item Mittelstaedt, P. (2004). {\it The Interpretation of Quantum Mechanics and the Measurement Process}. Cambridge: Cambridge University Press. \item Moore, G.E. (1939). Proof of an external world. {\it Proceedings of the British Academy} 25, 273--300. Reprinted in {\it Philosophical Papers} (George, Allen and Unwin, London, 1959) and in {\it Selected Writings} (Routledge, London, 1993). \item Moore, W. (1989). {\it Schr\"{o}dinger: Life and Thought}. Cambridge: Cambridge University Press. \item Morchio, G. \&\ Strocchi, F. (1987). Mathematical structures for long-range dynamics and symmetry breaking. {\it Journal of Mathematical Physics} 28, 622--635. \item Muller, F.A. (1997). The equivalence myth of quantum mechanics I, II. {\it Studies in History and Philosophy of Modern Physics } 28 35--61, 219--247. \item Murdoch, D. (1987). {\it Niels BohrÕs Philosophy of Physics}. Cambridge: Cambridge University Press. \item Nadirashvili, N., Toth, J., \&\ Yakobson, D. (2001).
Geometric properties of eigenfunctions. {\it Russian Mathematical Surveys} 56, 1085--1105. \item ÊNagy, G. (2000). A deformation quantization procedure for $C^*$-algebras. {\it Journal of Operator Theory} 44, 369--411. \item Narnhofer, H. (2001). Quantum K-systems and their abelian models. {\it Foundations of Probability and Physics}, pp.\ 274--302. River Edge, NJ: World Scientific. \item ÊNatsume, T. \&\ Nest, R. (1999). Topological approach to quantum surfaces. {\it Communications in Mathematical Physics} 202, 65--87. \item ÊNatsume, T., Nest, R., \&\ Ingo, P. (2003). Strict quantizations of symplectic manifolds. {\it Letters in Mathematical Physics} 66, 73--89.
\item Nauenberg, M. (1989). Quantum wave packets on Kepler elliptic orbits. {\it Physical Review} A40, 1133--1136. \item Nauenberg, M., Stroud, C., \&\ Yeazell, J. (1994). The classical limit of an atom. {\it Scientific American } June, 24--29. \item Neumann, J. von (1931). Die Eindeutigkeit der Schr\"{o}dingerschen Operatoren. {\it Mathematische Annalen} 104, 570--578. \item Neumann, J. von (1932). {\it Mathematische Grundlagen der Quantenmechanik.} Berlin: Springer--Verlag. English translation (1955): {\it Mathematical Foundations of Quantum Mechanics}. Princeton: Princeton University Press. \item Neumann, J. von (1938). On infinite direct products. {\it Compositio Mathematica} 6, 1--77. \item Neumann, J. von (1981). Continuous geometries with a transition probability. \textit{Memoirs of the American Mathematical Society} 252, 1--210. (Edited by I.S. Halperin. MS from 1937). \item Neumann, H. (1972). Transformation properties of observables. {\it Helvetica Physica Acta} 25, 811-819. \item Nourrigat, J. \&\ Royer, C. (2004). Thermodynamic limits for Hamiltonians defined as pseudodifferential operators. {\it Communications in Partial Differential Equations} 29, 383--417. \item O'Connor, P.W., Tomsovic, S., \&\ Heller, E.J. (1992). Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier. {\it Physica } D55, 340--357. \item Odzijewicz, A. (1992). Coherent states and geometric quantization. {\it Communications in Mathematical Physics} 150, 385--413. \item Odzijewicz, A. \&\ Ratiu, T.S. (2003). Banach Lie-Poisson spaces and reduction. {\it Communications in Mathematical Physics} 243, 1--54. \item ÊOlkiewicz, R. (1999a). Dynamical semigroups for interacting quantum and classical systems. {\it Journal of Mathematical Physics} 40, 1300--1316. \item Olkiewicz, R. (1999b). Environment-induced superselection rules in Markovian regime. {\it Communications in in Mathematical Physics} 208, 245--265. \item Olkiewicz, R. (2000). Structure of the algebra of effective observables in quantum mechanics. {\it Annals of Physics} 286, 10--22. \item Ollivier, H., Poulin, D., \&\ Zurek, W.H. (2004). Environment as witness: selective proliferation of information and emergence of objectivity. \texttt{arXiv: quant-ph/0408125}. \item Olshanetsky, M.A. \&\ Perelomov, A.M. (1981). Classical integrable finite-dimensional systems related to Lie algebras. {\it Physics Reports} 71, 313--400. \item Olshanetsky, M.A. \&\ Perelomov, A.M. (1983). Quantum integrable systems related to Lie algebras. {\it Physics Reports} 94, 313--404.
\item Ê Omn\`{e}s, R. (1992). Consistent interpretations of quantum mechanics. {\it Reviews of Modern Physics } 64, 339--382.
\item Omn\`{e}s, R. (1994). {\it The Interpretation of Quantum Mechanics}. Princeton: Princeton University Press. \item Omn\`{e}s, R. (1997). Quantum-classical correspondence using projection operators. {\it Journal of Mathematical Physics} 38, 697--707. \item Omn\`{e}s, R. (1999). {\it Understanding Quantum Mechanics}. Princeton: Princeton University Press. \item \O rsted, B. (1979). Induced representations and a new proof of the imprimitivity theorem. {\it Journal of Functional \ Analysis} 31, 355--359. \item Ozorio de Almeida, A.M. (1988). {\it Hamiltonian Systems: Chaos and Quantization}. Cambridge: Cambridge University Press. \item Pais, A. (1982). {\it Subtle is the Lord: The Science and Life of Albert Einstein}. Oxford: Oxford University Press. \item Pais, A. (1991). {\it Niels BohrÕs Times: In Physics, Philosophy, and Polity}. Oxford: Oxford University Press. \item Pais, A. (1997). {\it A Tale of Two Continents: A Physicist's Life in a Turbulent World}.
Princeton: Princeton University Press. \item Pais, A. (2000). {\it The Genius of Science}. Oxford: Oxford University Press. \item Parthasarathy, K.R. (1992). {\it An Introduction to Quantum Stochastic Calculus}. Basel: Birkh\"{a}user. \item Paul, T. \&\ Uribe, A. (1995). The semi-classical trace formula and propagation of wave packets. {\it Journal of Functional Analysis} 132, 192--249. \item Paul, T. \&\ Uribe, A. (1996). On the pointwise behavior of semi-classical measures. {\it Communications in Mathematical Physics} 175, 229--258. \item Paul, T. \&\ Uribe, A. (1998). A. Weighted trace formula near a hyperbolic trajectory and complex orbits. {\it Journal of Mathematical Physics} 39, 4009--4015. \item Pauli, W. (1925). \"{U}ber den Einflu\ss\ der Geschwindigkeitsabh\"{a}ngigkeit der Elektronenmasse auf den Zeemaneffekt. {\it Zeitschrift f\"{u}r Physik} 31, 373--385. \item Pauli, W. (1933). {\it Die allgemeinen Prinzipien der Wellenmechanik}. Fl\"{u}gge, S. (Ed.). {\it Handbuch der Physik}, Vol. V, Part I. Translated as Pauli, W. (1980). {\it General Principles of Quantum Mechanics}. Berlin: Springer-Verlag. \item Pauli, W. (1949). Die philosophische Bedeutung der Idee der Komplementarit\"{a}t. Reprinted in von Meyenn, K. (Ed.) (1984). {\it Wolfang Pauli: Physik und Erkenntnistheorie}, pp.\ 10--23. Braunschweig: Vieweg Verlag. English translation in Pauli (1994). \item Pauli, W. (1979). {\it Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg. Vol 1: 1919--1929}. Hermann, A., von Meyenn, K., \&\ Weisskopf, V. (Eds.). New York: Springer-Verlag. \item Pauli, W. (1985). {\it Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg. Vol 2: 1930--1939}. von Meyenn, K. (Ed.). New York: Springer-Verlag. \item Pauli, W. (1994). {\it Writings on Physics and Philosophy}. Enz, C.P. \&\ von Meyenn, K. (Eds.). Berlin: Springer-Verlag. \item Paz, J.P. \&\ Zurek, W.H. (1999). Quantum limit of decoherence: environment induced superselection of energy eigenstates. {\it Physical Review Letters} 82, 5181--5185. \item Pedersen, G.K. (1979.) {\it \ca s and their Automorphism Groups}.
London: Academic Press. \item Pedersen, G.K. (1989). {\it Analysis Now}. New York: Springer-Verlag.
\item Perelomov, A. (1986.) {\it Generalized Coherent States and their Applications}. Berlin: Springer-Verlag. \item Peres, A. (1984). Stability of quantum motion in chaotic and regular systems. {\it Physical Review} A30, 1610--1615. \item Peres, A. (1995). {\it Quantum Theory: Concepts and Methods}. Dordrecht: Kluwer Academic Publishers. \item Petersen, A. (1963). The Philosophy of Niels Bohr. {\it Bulletin of the Atomic Scientists} 19, 8--14. \item Pitowsky, I. (1989). {\it Quantum Probability - Quantum Logic}. Berlin: Springer-Verlag. \item Planck, M. (1906). {\it Vorlesungen \"{U}ber die Theorie der W\"armestrahlung} Leipzig: J.A. Barth. \item Poincar\'e, H. (1892--1899). {\it Les M\'ethodes Nouvelles de la M\'echanique C\'eleste}. Paris: Gauthier-Villars. \item Popov, G. (2000). Invariant tori, effective stability, and quasimodes with exponentially small error terms. I \&\ II. {\it Annales Henri Poincar\'e} 1, 223--248 \&\ 249--279. \item Poulin, D. (2004). Macroscopic observables. \texttt{arXiv:quant-ph/0403212}. \item Poulsen, N.S. (1970). {\it Regularity Aspects of the Theory of Infinite-Dimensional Representations of Lie Groups}. Ph.D Thesis, MIT. \item Primas, H. (1983). {\it Chemistry, Quantum Mechanics and Reductionism}. Second Edition. Berlin: Springer-Verlag. \item Primas, H. (1997). The representation of facts in physical theories. {\it Time, Temporality, Now}, pp.\ 241-263. Atmanspacher, H. \&\ Ruhnau, E. (Eds.). Berlin: Springer-Verlag. \item Prugovecki, E. (1971). {\it Quantum Mechanics in Hilbert Space}. New York: Academic Press. \item Puta, M. (1993). {\it Hamiltonian Dynamical Systems and Geometric Quantization}. Dordrecht: D. Reidel. \item Raggio, G.A. (1981). {\it States and Composite Systems in $W^*$-algebras Quantum Mechanics}. Ph.D Thesis, ETH Z\"{u}rich. \item Raggio, G.A. (1988).
A remark on Bell's inequality and decomposable normal states.
{\it Letters in Mathematical Physics} 15, 27--29.
\item Raggio, G.A. \&\ Werner, R.F. (1989). Quantum statistical mechanics
of general mean field systems. {\it Helvetica Physica Acta} 62, 980--1003.
\item Raggio, G.A. \&\ Werner, R.F. (1991). The Gibbs variational principle for inhomogeneous mean field systems. {\it Helvetica Physica Acta} 64, 633--667. \item Raimond, R.M., Brune, M., \&\ Haroche, S. (2001). Manipulating quantum entanglement with atoms and photons in a cavity. {\it Reviews of Modern Physics} 73, 565--582. \item R\'{e}dei, M. (1998). {\it Quantum logic in algebraic approach}. Dordrecht: Kluwer Academic Publishers. \item R\'{e}dei, M. \&\ St\"{o}ltzner, M. (Eds.). (2001). {\it John von Neumann and the Foundations of Modern Physics}. Dordrecht: Kluwer Academic Publishers. \item Reed, M. \&\ Simon, B. (1972). {\it Methods of Modern Mathematical Physics. Vol I. Functional Analysis.} New York: Academic Press. \item Reed, M. \&\ Simon, B. (1975). {\it Methods of Modern Mathematical Physics. Vol II. Fourier Analysis, Self-adjointness.} New York: Academic Press. \item Reed, M. \&\ Simon, B. (1979). {\it Methods of Modern Mathematical Physics. Vol III. Scattering Theory}. New York: Academic Press. \item Reed, M. \&\ Simon, B. (1978). {\it Methods of Modern Mathematical Physics. Vol IV. Analysis of Operators.} New York: Academic Press. \item Reichl, L.E. (2004). {\it The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations}. Second Edition. New York: Springer-Verlag. \item Rieckers, A. (1984). On the classical part of the mean field dynamics for quantum lattice systems in grand canonical representations. {\it Journal of Mathematical Physics} 25, 2593--2601. \item Rieffel, M.A. (1989a). Deformation quantization of Heisenberg manifolds. {\it Communications in Mathematical Physics} 122, 531--562. \item Rieffel, M.A. (1989b). Continuous fields of $C^*$-algebras coming from group cocycles and actions. {\it Mathematical Annals} 283, 631--643. \item Rieffel, M.A. (1994).
Quantization and $C\sp *$-algebras. {\it Contemporary Mathematics} 167, 66--97. \item Riesz, F. \&\ Sz.-Nagy, B. (1990). {\em Functional Analysis}. New York: Dover. \item Robert, D. (1987) {\it Autour de l'Approximation
Semi-Classique}. Basel: Birkh\"{a}user. \item Robert, D. (Ed.). (1992) {\it M\'{e}thodes Semi-Classiques}. {\it Ast\'{e}risque} 207, 1--212, {\it ibid.}\ 210, 1--384. \item Robert, D. (1998). Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results. {\it Helvetica Physica Acta} 71, 44--116. \item Roberts, J.E. (1990). Lectures on algebraic quantum field theory. {\it The Algebraic Theory of Superselection Sectors. Introduction and Recent Results}, pp.\ 1--112. Kastler, D. (Ed.). River Edge, NJ: World Scientific Publishing Co. \item Roberts, J.E. \&\ Roepstorff, G. (1969).
Some basic concepts of algebraic quantum theory. {\it Communications in Mathematical Physics } 11, 321--338. \item Robinett, R.W. (2004). Quantum wave packet revival. {\it Physics Reports} 392, 1-119. \item Robinson, D. (1994). Can Superselection Rules Solve the Measurement Problem? {\it British Journal for the Philosophy of Science} 45, 79-93. \item Robinson, S.L. (1988a). The semiclassical limit of quantum mechanics. I. Time evolution. {\it Journal of Mathematical\ Physics } 29, 412--419. \item Robinson, S.L. (1988b). The semiclassical limit of quantum mechanics. II. Scattering theory. {\it
Annales de l' Institut Henri Poincar\'{e}}
A48, 281--296. \item Robinson, S.L. (1993). Semiclassical mechanics for time-dependent Wigner functions. \textit{Journal of Mathematical\ Physics } 34, 2185--2205. \item Robson, M.A. (1996). Geometric quantization of reduced cotangent bundles. {\it Journal of Geometry and Physics} 19, 207--245. \item Rosenfeld, L. (1967). Niels Bohr in the Thirties. Consolidation and extension of the conception of complementarity. {\it Niels Bohr: His Life and Work as Seen by His Friends and Colleagues}, pp.\ 114--136. Rozental, S. (Ed.). Amsterdam: North-Holland. Ê\item Rudolph, O. (1996a). Consistent histories and operational quantum theory. {\it International Journal of Theoretical Physics } 35, 1581--1636. \item Rudolph, O. (1996b). On the consistent effect histories approach to quantum mechanics. {\it Journal of Mathematical Physics} 37, 5368--5379. \item Rudolph, O. (2000). The representation theory of decoherence functionals in history quantum theories. {\it International Journal of Theoretical Physics } 39, 871--884. \item Rudolph, O. \&\ Wright, J.D. M. (1999). Homogeneous decoherence functionals in standard and history quantum mechanics. {\it Communications in Mathematical Physics} 204, 249--267. \item Sarnak, P. (1999). Quantum chaos, symmetry and zeta functions. I. \&\ II. {\it Quantum Chaos. Current Developments in Mathematics, 1997 (Cambridge, MA)}, pp.\ 127--144 \&\ 145--159. Boston: International Press. \item Saunders, S. (1993). Decoherence, relative states, and evolutionary adaptation. {\it Foundations of Physics} 23, 1553--1585.
\item Saunders, S. (1995) Time, quantum mechanics, and decoherence. {\it Synthese} 102, 235--266.
\item Saunders, S. (2004). Complementarity and Scientific Rationality. \texttt{arXiv:quant-ph/0412195}. \item Scheibe, E. (1973). {\it The Logical Analysis of Quantum Mechanics}. Oxford: Pergamon Press. \item Scheibe, E. (1991). J. v. Neumanns und J. S. Bells Theorem. Ein Vergleich. {\it Philosophia Naturalis} 28, 35--53. English translation in Scheibe, E. (2001). {\it Between Rationalism and Empiricism: Selected Papers in the Philosophy of Physics}. New York: Springer-Verlag. \item Scheibe, E. (1999). {\it Die Reduktion Physikalischer Theorien.
Ein Beitrag zur Einheit der Physik. Teil II: Inkommensurabilit\"{a}t und Grenzfallreduktion.} Berlin: Springer-Verlag. \item Schlosshauer, M. (2004). Decoherence, the measurement problem, and interpretations of quantum mechanics. {\it Reviews of Modern Physics } 76, 1267--1306. \item Schm\"{u}dgen, K. (1990). {\it Unbounded Operator Algebras and Representation Theory}. Basel: Birkh\"{a}user Verlag. \item Schr\"{o}dinger, E. (1926a). Quantisierung als Eigenwertproblem. I.-IV. {\it Annalen der Physik } 79, 361--76, 489--527, {\it ibid.} 80, 437--90, {\it ibid.} 81, 109--39. English translation in Schr\"{o}dinger (1928). \item Schr\"{o}dinger, E. (1926b). Der stetige \"{U}bergang von der Mikro-zur Makromekanik. {\it Die Naturwissenschaften} 14, 664--668. English translation in Schr\"{o}dinger (1928). \item Schr\"{o}dinger, E. (1926c). \"{U}ber das Verhaltnis der Heisenberg--Born--Jordanschen Quantenmechanik zu der meinen. {\it Annalen der Physik} 79, 734--56. English translation in Schr\"{o}dinger (1928). \item Schr\"{o}dinger, E. (1928). {\it Collected Papers on Wave Mechanics}. London: Blackie and Son. \item Schroeck, F.E., Jr. (1996). {\it Quantum Mechanics on Phase Space}. Dordrecht: Kluwer Academic Publishers. \item Scutaru, H. (1977). Coherent states and induced representations. {\it Letters in Mathematical
Physics} 2, 101-107. \item Segal, I.E. (1960). Quantization of nonlinear systems. {\it Journal of Mathematical Physics} 1, 468--488. \item Sewell, G. L. (1986). {\it Quantum Theory of Collective Phenomena}. New York:
Oxford University Press. \item Sewell, G. L. (2002). {\it Quantum Mechanics and its Emergent Macrophysics}. Princeton: Princeton University Press. \item Simon, B. (1976). Quantum dynamics: from automorphism to Hamiltonian. {\it Studies in Mathematical Phyiscs: Essays in Honour of Valentine Bargmann}, pp.\ 327--350. Lieb, E.H., Simon, B. \&\ Wightman, A.S. (Eds.). Princeton: Princeton University Press. \item Simon, B. (1980). The classical limit of quantum partition functions.
{\it Communications in Mathematical Physics} 71, 247--276. \item ÊSimon, B. (2000) Schr\"{o}dinger operators in the twentieth century. {\it Journal of Mathematical Physics} 41, 3523--3555. \item \'{S}niatycki, J. (1980). {\it Geometric Quantization and Quantum Mechanics}. Berlin: Springer-Verlag. \item Snirelman, A.I. (1974). Ergodic properties of eigenfunctions. {\it Uspekhi Mathematical Nauk} 29, 181--182. \item Souriau, J.-M. (1969). {\it Structure des syst\`{e}mes dynamiques}. Paris: Dunod. Translated as Souriau, J.-M. (1997). \item Souriau, J.-M. (1997). {\it Structure of Dynamical Systems. A Symplectic View of Physics}. Boston: Birkh\"{a}user. \item Stapp, H.P. (1972). The Copenhagen Interpretation. {\it American Journal of Physics} 40, 1098--1116. \item Steiner, M. (1998). {\it The Applicability of Mathematics as a Philosophical Problem}. Cambridge (MA): Harvard University Press. \item Stinespring, W. (1955). Positive functions on $C^*$-algebras. {\it Proceedings of the American Mathematical Society} 6, 211--216. \item St\o rmer, E. (1969). Symmetric states of infinite tensor products of \ca s. {\it Jornal of Functional Analysis} 3, 48--68. \item Strawson, P.F. (1959). {\it Individuals: An Essay in Descriptive Metaphysics}. London: Methuen. \item Streater, R.F. (2000). Classical and quantum probability. {\it Journal of Mathematical Physics} 41, 3556--3603. \item Strichartz, R.S. (1983). Analysis of the Laplacian on a complete Riemannian manifold. \textit{Journal of Functional \ Analysis} 52, 48--79. \item ÊStrocchi, F. (1985). {\it Elements of Quantum mechanics of Infinite Systems}. Singapore: World Scientific. \item Strunz, W.T., Haake, F., \&\ Braun, D. (2003). Universality of decoherence for macroscopic quantum superpositions. {\it Physical Review} A 67, 022101--022114. \item Summers, S.J. \&\ Werner, R. (1987). BellÕs inequalities and quantum field theory, I, II. {\it Journal of Mathematical Physics} 28, 2440--2447, 2448--2456. \item Sundermeyer, K. (1982). {\it Constrained Dynamics}. Berlin: Springer-Verlag. \item Takesaki, M. (2003). {\it Theory of Operator Algebras. Vols.\ I-III}. New York: Springer-Verlag. \item Thirring, W. \&\ Wehrl, A. (1967). On the mathematical structure of the BCS model. I. {\it Communications in Mathematical Physics} 4, 303--314. \item Thirring, W. (1968). On the mathematical structure of the BCS model. II. {\it Communications in Mathematical Physics} 7, 181--199. \item Thirring, W. (1981). {\it A Course in Mathematical Physics. Vol.\ 3: Quantum Mechanics of Atoms and Molecules}. New York: Springer-Verlag. \item Thirring, W. (1983). {\it A Course in Mathematical Physics. Vol.\ 4: Quantum Mechanics of Large Systems}. New York: Springer-Verlag. \item ÊTomsovic, S. \&\ Heller, E.J. (1993). Long-time semiclassical dynamics of chaos: The stadium billiard. {\it Physical Review} E47, 282--299. \item ÊTomsovic, S. \&\ Heller, E.J. (2002). Comment on" Ehrenfest times for classically chaotic systems". {\it Physical Review} E65, 035208-1--2. \item Toth, J.A. (1996). Eigenfunction localization in the quantized rigid body.
{\it Journal of Differential Geometry} 43, 844--858. \item Toth, J.A. (1999). On the small-scale mass concentration of modes. {\it Communications in Mathematical Physics} 206, 409--428. \item Toth, J.A. \&\ Zelditch, S. (2002). Riemannian manifolds with uniformly bounded eigenfunctions. {\it Duke Mathematical Journal} 111, 97--132. \item Toth, J.A. \&\ Zelditch, S. (2003a). $L^p$ norms of eigenfunctions in the completely integrable case. {\it Annales Henri Poincar\'e} 4, 343--368. \item Toth, J.A. \&\ Zelditch, S. (2003b). Norms of modes and quasi-modes revisited. {\it Contemporary Mathematics} 320, 435--458. \item ÊTuynman, G.M. (1987). Quantization: towards a comparison between methods.
{\it Journal of Mathematical Physics} 28, 2829--2840. \item ÊTuynman, G.M. (1998). Prequantization is irreducible. {\it Indagationes Mathematicae (New Series)} 9, 607--618. \item Unnerstall, T. (1990a). Phase-spaces and dynamical descriptions of infinite mean-field quantum systems. {\it Journal of Mathematical Physics} 31, 680--688. \item Unnerstall, T. (1990b). Schr\"{o}dinger dynamics and physical folia of infinite mean-field quantum systems. {\it Communications in Mathematical\ Physics } 130, 237--255. \item Vaisman, I. (1991). On the geometric quantization of Poisson manifolds. {\it Journal of Mathematical Physics} 32, 3339--3345. \item van Fraassen, B.C. (1991). {\it Quantum Mechanics: An Empiricist View}. Oxford: Oxford University Press. \item van Hove, L. (1951). Sur certaines repr\'{e}sentations unitaires d'un
groupe infini de transformations. \textit{Memoires de l'Acad\'{e}mie Royale de
Belgique, Classe des Sciences} 26, 61--102. \item Van Vleck, J.H. (1928). The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics. {\it Proceedings of the National Academy of Sciences} 14, 178--188. \item van der Waerden, B.L. (Ed.). (1967). {\it Sources of Quantum Mechanics}. Amsterdam: North-Holland. \item van Kampen, N. (1954). Quantum statistics of irreversible processes. {\it Physica} 20, 603--622. \item van Kampen, N. (1988). Ten theorems about quantum mechanical measurements. {\it Physica} A153, 97--113. \item van Kampen, N. (1993). Macroscopic systems in \qm. {\it Physica} A194, 542--550. \item Vanicek, J. \&\ Heller, E.J. (2003). Semiclassical evaluation of quantum fidelity {\it Physical Review} E68, 056208-1--5. \item Vergne, M. (1994). Geometric quantization and equivariant cohomology. {\it First European Congress in Mathematics, Vol.\ 1}, pp.\ 249--295. Boston: Birkh\"{a}user. \item Vermaas, P. (2000). {\it A Philosopher's Understanding of Quantum Mechanics: Possibilities and Impossibilities of a Modal Interpretation}. Cambridge: Cambridge University Press. \item Vey, J. (1975). D\'{e}formation du crochet de Poisson sur une vari\'{e}t\'{e} symplectique. {\it Commentarii Mathematici Helvetici} 50, 421--454. \item Voros, A. (1979). Semi-classical ergodicity of quantum eigenstates in the Wigner representation. {\it Stochastic Behaviour in Classical and Quantum Hamiltonian Systems}. {\it Lecture Notes in Physics} 93, 326--333. \item Wallace, D. (2002). Worlds in the Everett interpretation. {\it Studies in History and Philosophy of Modern Physics } 33B, 637--661. \item Wallace, D. (2003). Everett and structure. {\it Studies in History and Philosophy of Modern Physics} 34B, 87--105. \item ÊWan, K.K. \&\ Fountain, R.H. (1998). Quantization by parts, maximal symmetric operators, and quantum circuits. {\it International Journal of Theoretical Physics } 37, 2153--2186. \item Wan, K.K., Bradshaw, J., Trueman, C., \&\ Harrison, F.E. (1998). Classical systems, standard quantum systems, and mixed quantum systems in Hilbert space. {\it Foundations of Physics } 28, 1739--1783. \item Wang, X.-P. (1986). Approximation semi-classique de l'equation de Heisenberg. {\it Communications in Mathematical\ Physics } 104, 77--86. \item Wegge-Olsen, N.E. (1993). \textit{K-theory and \ca s}. Oxford: Oxford University Press. \item Weinstein, A. (1983). The local structure of Poisson manifolds. {\it Journal of Differential Geometry} 18, 523--557. \item Werner, R.F. (1995). The classical limit of quantum theory. \texttt{arXiv:quant-ph/9504016}. \item Weyl, H. (1931). {\it The Theory of Groups and Quantum Mechanics}.
New York: Dover. \item Wheeler, J.A. \&\ and Zurek, W.H. (Eds.) (1983). {\it Quantum Theory and Measurement}. Princeton: Princeton University Press. \item Whitten-Wolfe, B. \&\ Emch, G.G. (1976). A mechanical quantum measuring process. {\it Helvetica Physica Acta} 49, 45-55. \item Wick, C.G., Wightman, A.S., \&\ Wigner, E.P. (1952). The intrinsic parity of elementary particles. {\it Physical Review} 88, 101--105. \item Wightman, A.S. (1962) On the localizability of quantum mechanical systems. {\it Reviews of Modern Physics} 34, 845-872. \item Wigner, E.P. (1932). On the quantum correction for
thermodynamic equilibrium. \textit{Physical Review} 40, 749--759. \item Wigner, E.P. (1939) Unitary representations of the inhomogeneous Lorentz group. {\it Annals of Mathematics} 40, 149-204. \item Wigner, E.P. (1963). The problem of measurement.
{\it American Journal of Physics} 31, 6--15. \item Williams, F.L. (2003). {\it Topics in Quantum Mechanics}. Boston: Birkh\"{a}user. \item ÊWoodhouse, N. M. J. (1992). {\it Geometric Quantization}. Second edition. Oxford: The Clarendon Press. \item Yajima, K. (1979). The quasi-classical limit of quantum scattering theory.
{\it Communications in Mathematical\ Physics } 69, 101--129. \item Zaslavsky, G.M. (1981). Stochasticity in quantum systems. {\it Physics Reports} 80, 157--250. \item Zeh, H.D. (1970). On the interpretation of measurement in quantum theory. {\it Foundations of Physics} 1, 69--76. \item Zelditch, S. (1987). Uniform distribution of eigenfunctions on compact hyperbolic surfaces. {\it Duke Mathematical J.} 55, 919--941. \item Zelditch, S. (1990). Quantum transition amplitudes for ergodic and for completely integrable systems. {\it Journal of Functional Analysis} 94, 415--436. \item Zelditch, S. (1991). Mean Lindel\"of hypothesis and equidistribution of cusp forms and Eisenstein series. {\it Journal of Functional Analysis} 97, 1--49. \item Zelditch, S. (1992a). On a ``quantum chaos'' theorem of R. Schrader and M. Taylor. {\it Journal of Functional Analysis} 109, 1--21. \item Zelditch, S. (1992b). Quantum ergodicity on the sphere. {\it Communications in Mathematical\ Physics } 146, 61--71. \item Zelditch, S. (1996a). Quantum dynamics from the semiclassical viewpoint. Lectures at the Centre E. Borel. Available at \texttt{http://mathnt.mat.jhu.edu/zelditch/Preprints/preprints.html}. \item Zelditch, S. (1996b). Quantum mixing. {\it Journal of Functional Analysis} 140, 68--86. \item Zelditch, S. (1996c). Quantum ergodicity of $C\sp *$ dynamical systems. {\it Communications in Mathematical Physics} 177, 507--528. \item Zelditch, S. \&\ Zworski, M. (1996). Ergodicity of eigenfunctions for ergodic billiards. {\it Communications in Mathematical Physics} 175, 673--682. \item Zurek, W.H. (1981). Pointer basis of quantum apparatus: into what mixture does the wave packet collapse? {\it Physical Review} D24, 1516--1525. \item Zurek, W.H. (1982) Environment-induced superselections rules. {\it Physical Review} D26, 1862--1880. \item Zurek, W.H. (1991). Decoherence and the transition from quantum to classical. {\it Physics Today} 44 (10), 36--44. \item Zurek, W.H. (1993). Negotiating the tricky border between quantum and classical. {\it Physics Today} 46 (4), 13--15, 81--90. \item Zurek, W.H. (2003). Decoherence, einselection, and the quantum origins of the classical. {\it Reviews of Modern Physics} 75, 715--775. \item Zurek, W.H. (2004). Probabilities from entanglement, Born's rule from envariance.\\ \texttt{arXiv:quant-ph/0405161}. \item Zurek, W.H., Habib, S., \&\ Paz, J.P. (1993). Coherent states via decoherence. {\it Physical Review Letters} 70, 1187--1190. \item Zurek, W.H. \&\ Paz, J.P. (1995). Why We Don't Need Quantum Planetary Dynamics: Decoherence and the Correspondence Principle for Chaotic Systems. {\it Proceedings of the Fourth Drexel Meeting}. Feng, D.H. et al. (Eds.). New York: Plenum. \texttt{arXiv:quant-ph/9612037}.
\end{trivlist}
\end{document} | arXiv |
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x2. The adjective which corresponds to squaring is quadratic.
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1)2 = x2 + 2x + 1.
One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers x), the square of x is the same as the square of its additive inverse −x. That is, the square function satisfies the identity x2 = (−x)2. This can also be expressed by saying that the square function is an even function.
In real numbers
The squaring operation defines a real function called the square function or the squaring function. Its domain is the whole real line, and its image is the set of nonnegative real numbers.
The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a monotonic function on the interval [0, +∞). On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on (−∞,0]. Hence, zero is the (global) minimum of the square function. The square x2 of a number x is less than x (that is x2 < x) if and only if 0 < x < 1, that is, if x belongs to the open interval (0,1). This implies that the square of an integer is never less than the original number x.
Every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square is the original number.
No square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. The lack of real square roots for the negative numbers can be used to expand the real number system to the complex numbers, by postulating the imaginary unit i, which is one of the square roots of −1.
The property "every non-negative real number is a square" has been generalized to the notion of a real closed field, which is an ordered field such that every non-negative element is a square and every polynomial of odd degree has a root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which may be expressed in first-order logic (that is expressed by a formula in which the variables that are quantified by ∀ or ∃ represent elements, not sets), is true for every real closed field, and conversely every property of the first-order logic, which is true for a specific real closed field is also true for the real numbers.
In geometry
There are several major uses of the square function in geometry.
The name of the square function shows its importance in the definition of the area: it comes from the fact that the area of a square with sides of length l is equal to l2. The area depends quadratically on the size: the area of a shape n times larger is n2 times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that is manifested physically by the inverse-square law describing how the strength of physical forces such as gravity varies according to distance.
The square function is related to distance through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a smooth function: the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance (denoted d2 or r2), which has a paraboloid as its graph, is a smooth and analytic function.
The dot product of a Euclidean vector with itself is equal to the square of its length: v⋅v = v2. This is further generalised to quadratic forms in linear spaces via the inner product. The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size (length).
There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle.
In abstract algebra and number theory
The square function is defined in any field or ring. An element in the image of this function is called a square, and the inverse images of a square are called square roots.
The notion of squaring is particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number p. A non-zero element of this field is called a quadratic residue if it is a square in Z/pZ, and otherwise, it is called a quadratic non-residue. Zero, while a square, is not considered to be a quadratic residue. Every finite field of this type has exactly (p − 1)/2 quadratic residues and exactly (p − 1)/2 quadratic non-residues. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.
More generally, in rings, the square function may have different properties that are sometimes used to classify rings.
Zero may be the square of some non-zero elements. A commutative ring such that the square of a non zero element is never zero is called a reduced ring. More generally, in a commutative ring, a radical ideal is an ideal I such that $x^{2}\in I$ implies $x\in I$. Both notions are important in algebraic geometry, because of Hilbert's Nullstellensatz.
An element of a ring that is equal to its own square is called an idempotent. In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in integral domains. However, the ring of the integers modulo n has 2k idempotents, where k is the number of distinct prime factors of n. A commutative ring in which every element is equal to its square (every element is idempotent) is called a Boolean ring; an example from computer science is the ring whose elements are binary numbers, with bitwise AND as the multiplication operation and bitwise XOR as the addition operation.
In a totally ordered ring, x2 ≥ 0 for any x. Moreover, x2 = 0 if and only if x = 0.
In a supercommutative algebra where 2 is invertible, the square of any odd element equals zero.
If A is a commutative semigroup, then one has
$\forall x,y\in A\quad (xy)^{2}=xyxy=xxyy=x^{2}y^{2}.$
In the language of quadratic forms, this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling. The doubling method was formalized by A. A. Albert who started with the real number field $\mathbb {R} $ and the square function, doubling it to obtain the complex number field with quadratic form x2 + y2, and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson construction, and has been generalized to form algebras of dimension 2n over a field F with involution.
The square function z2 is the "norm" of the composition algebra $\mathbb {C} $, where the identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras.
In complex numbers
See also: Exponentiation § Powers of complex numbers
On complex numbers, the square function $z\to z^{2}$ is a twofold cover in the sense that each non-zero complex number has exactly two square roots.
The square of the absolute value of a complex number is called its absolute square, squared modulus, squared magnitude, or squared norm.[1] It is the product of the complex number with its complex conjugate, and equals the sum of the squares of the real and imaginary parts of the complex number.
The absolute square of a complex number is always a nonnegative real number, that is zero if and only if the complex number is zero. It is easier to compute than the absolute value (no square root), and is a smooth real-valued function. Because of these two properties, the absolute square is often preferred to the absolute value for explicit computations and when methods of mathematical analysis are involved (for example optimization or integration).
Other uses
Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in physics where many units are defined using squares and inverse squares: see below.
Least squares is the standard method used with overdetermined systems.
Squaring is used in statistics and probability theory in determining the standard deviation of a set of values, or a random variable. The deviation of each value xi from the mean ${\overline {x}}$ of the set is defined as the difference $x_{i}-{\overline {x}}$. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the variance, and its square root is the standard deviation.
See also
• Exponentiation by squaring
• Polynomial SOS, the representation of a non-negative polynomial as the sum of squares of polynomials
• Hilbert's seventeenth problem, for the representation of positive polynomials as a sum of squares of rational functions
• Square-free polynomial
• Cube (algebra)
• Metric tensor
• Quadratic equation
• Polynomial ring
• Sums of squares (disambiguation page with various relevant links)
Related identities
Algebraic (need a commutative ring)
• Difference of two squares
• Brahmagupta–Fibonacci identity, related to complex numbers in the sense discussed above
• Euler's four-square identity, related to quaternions in the same way
• Degen's eight-square identity, related to octonions in the same way
• Lagrange's identity
Other
• Pythagorean trigonometric identity
• Parseval's identity
Related physical quantities
• acceleration, length per square time
• cross section (physics), an area-dimensioned quantity
• coupling constant (has square charge in the denominator, and may be expressed with square distance in the numerator)
• kinetic energy (quadratic dependence on velocity)
• specific energy, a (square velocity)-dimensioned quantity
Footnotes
1. Weisstein, Eric W. "Absolute Square". mathworld.wolfram.com.
Further reading
• Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1, ISBN 0-8218-4402-4
• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
| Wikipedia |
\begin{document}
\title[On Explicit Random-like Tournaments] {On Explicit Random-like Tournaments}
\maketitle
\begin{abstract} We give a new theorem describing a relation between the quasi-random property of regular tournaments and their spectra. This provides many solutions to a constructing problem mentioned by Erd\H{o}s and Moon (1965) and Spencer (1985). \end{abstract}
\section{Introduction} \label{intro} A {\it tournament} is an oriented complete graph. {\it Random tournaments} $\mathcal{T}_n$ with $n$ vertices are obtained by choosing a direction of each edge of a complete graph with $n$ vertices with probability $1/2$, independently. We say that random tournaments {\it asymptotically almost surely} ({\it a.a.s.}) satisfy a property $\mathcal{P}$ if the probability of the event that tournaments satisfy $\mathcal{P}$ tends to $1$ when $n$ goes to infinity. In graph theory, there have been many problems focusing on deterministic tournaments satisfying properties which random tournaments a.a.s satisfy; see e.g. \cite{AS16}, \cite{B09}, \cite{CG91}, \cite{CR17}, \cite{KS13}. \par In this paper, as such a property, we mainly focus on the {\it quasi-random property} proposed by Chung-Graham~\cite{CG91}. Our main result is to give a new theorem describing a relation between the quasi-random property and spectra of regular tournaments. This result also provides many solutions to a problem, proposed by Erd\H{o}s-Moon~\cite{EM65} and Spencer~\cite{S85} (see also \cite[Section 9.1]{AS16}), on explicit constructions of tournaments with a small number of consistent edges. It is well-known that Paley tournaments have the quasi-random property (e.g. \cite{CG91}). Moreover, by proving that Paley tournaments have a property stronger than the quasi-random property, Alon-Spencer~\cite{AS16} showed that they provide solutions to the problem by Erd\H{o}s, Moon and Spencer. We note that the proof in \cite{AS16} contains a part (Lemma 9.1.2 in \cite{AS16}) depending on the definition of Paley tournaments. Remarkably, we generalize their discussion to all regular tournaments by using a digraph-version of the {\it expander-mixing lemma} proved by Vu~\cite{V08}. \par The rest of this paper is organized as follows. In Section~\ref{sect:Rank}, we recap the quasi-random property and introduce some related known facts. In Section~\ref{sect:Main}, we introduce our main result and give its proof. In Section~\ref{sect:Const}, we provide some examples of regular tournaments satisfying the quasi-random property which are also solutions to the problem by Erd\H{o}s, Moon and Spencer. At last, in Section~\ref{sec:Shutte}, we discuss another random-like property defined as an adjacency property.
\section{The quasi-random property and related facts} \label{sect:Rank} In this section, we review the quasi-random property and some related known facts. For a digraph $D$, let $V(D)$ and $E(D)$ be the vertex and the edge set of $D$, respectively. For two distinct vertices $x$ and $y$, let the ordered pair $(x, y)$ denote the edge directed from $x$ to $y$. \par First, we give the definition of the quasi-random property of tournaments which was formulated by Chung-Graham~\cite{CG91}. \begin{definition}[The quasi-random property, \cite{CG91}] \label{def:quasi} Let $T$ be a tournament with $n$ vertices. Let $\sigma$ be a bijection from $V(T)$ to $\{1, 2, \ldots, n\}$. An edge $(x, y)$ of $T$ is called {\it consistent} with $\sigma$ if $\sigma(x)<\sigma(y)$. Let $C(T, \sigma)$ be the number of consistent edges with $\sigma$ and $C(T)=\max_{\sigma}C(T, \sigma)$. Then, $T$ has the {\it quasi-random property} if $T$ satisfies \begin{equation} \label{eq:quasi} C(T) \leq (1+o(1))\frac{n^2}{4}. \end{equation} \end{definition} Surprisingly, Chung-Graham~\cite{CG91} gave some other properties which are seemingly unrelated, but actually equivalent with (\ref{eq:quasi}). The interested reader is referred to \cite{CG91}. \par Consistent edges of tournaments was originally investigated by Erd\H{o}s-Moon~\cite{EM65}. Their work was from paired comparisons (e.g. \cite{KS40}). It is reasonable to find suitable rankings, that is, bijections with many consistent edges. First observe that for every tournament $T$ with $n$ vertices, \begin{equation}
\frac{1}{2}\binom{n}{2} \leq C(T)\leq \binom{n}{2}. \end{equation} The lower bound of $C(T)$ is obtained by the following simple fact$\colon$ \begin{equation} \label{eq:plus} C(T, \sigma)+C(T, \sigma')=\binom{n}{2}, \end{equation} where $\sigma'$ is the reversed ranking of $\sigma$ which is defined as $\sigma'(v)=n+1-\sigma(v)$ for each $v \in V(T)$. For the upper bound of $C(T)$, the equality holds if and only if $T$ is a transitive tournament. On the other hand, it is non-trivial to check the tightness of the lower bound of $C(T)$. In \cite{EM65}, it was proved that there exist tournaments $T$ such that $C(T)\leq (1+o(1))\binom{n}{2}/2$ by a probabilistic argument. Moreover Spencer~\cite{S71}, \cite{S80} and de la Vega~\cite{d83} proved that random tournament $\mathcal{T}_n$ a.a.s satisfies the following property which is stronger than the quasi-random property$\colon$ \begin{equation} \label{eq:stongquasi} C(\mathcal{T}_n) \leq \frac{1}{2}\binom{n}{2}+O(n^{\frac{3}{2}}). \end{equation} \par Erd\H{o}s-Moon~\cite{EM65} and Spencer~\cite{S85} mentioned the problem on explicit constructions of tournaments $T$ such that $C(T)$ is close to the lower bound. At present, such a construction of tournaments $T$ giving the best known ^^ ^^ constructive" upper bound of $C(T)$ is obtained by Alon-Spencer~\cite{AS16}. For a prime $p \equiv 3 \pmod{4}$, the {\it Paley tournament} $T_p$ is the tournament with vertex set $\mathbb{F}_p$, the finite field of $p$ elements, and edge set formed by all edges $(x, y)$ such that $x-y$ is a non-zero square of $\mathbb{F}_p$. In \cite[Theorem 9.1.1]{AS16}, it was proved that \begin{equation} \label{eq:paley} C(T_p) \leq \frac{1}{2}\binom{p}{2}+O(p^{\frac{3}{2}}\log p). \end{equation} In Section~\ref{sect:Const}, by applying the main theorem proved in the next section, we give some new explicit constructions of regular tournaments $T$ with $n$ vertices such that $C(T)$ is close to the lower bound.
\section{Main theorem} \label{sect:Main} In this section, we prove our main theorem. We first give the definition of regular digraphs and the adjacency matrix of a digraph. A digraph is said to be {\it $d$-regular} if in-degree and out-degree of each vertex is $d$. Especially a tournament with $n$ vertices is simply said to be {\it regular} if it is $(n-1)/2$-regular. The {\it adjacency matrix} $M_D$ of a digraph $D$ with vertices is the $\{0, 1\}$-square matrix of size $n$ whose rows and columns are indexed by the vertices of $D$ and the $(x, y)$-entry is equal to $1$ if and only if $(x, y) \in E(D)$.
The following is our main theorem. \begin{theorem} \label{thm:main}
Let $T$ be a regular tournament with $n$ vertices. Suppose that the adjacency matrix $M_T$ of $T$ has eigenvalues such that $(n-1)/2=\lambda_1, \lambda_2, \cdots, \lambda_{n}$. Let $\lambda(T)=\max_{2\leq i \leq n}|\lambda_i|$. Then, \begin{equation} C(T)\leq \frac{1}{2}\binom{n}{2}+\lambda(T) \cdot n \log_2 (2n). \end{equation} \end{theorem}
\begin{remark} Theorem~\ref{thm:main} implies that every regular tournament $T$ with $n$ vertices such that $\lambda(T)=o(n/\log n)$ has the quasi-random property. It should be remarked that Kalyanasundaram-Shapira~\cite{KS13} shows a stronger result; a proof of Lemma 2.3 and the first concluding remark in \cite{KS13} implies that a regular tournament $T$ with $n$ vertices has the quasi-random property if and only if $T$ satisfies that $\lambda(T)=o(n)$. (In \cite{KS13}, the authors considered the eigenvalues of the $\{0, \pm 1\}$-matrix $2M_T-J_n+I_n$, but these eigenvalues can be directly computed from ones of $M_T$.)
On the other hand, Theorem~\ref{thm:main} not only gives a spectral condition for the quasi-random property, but also implies that estimating eigenvalues of $M_T$ provides better upper bounds of $C(T)$ than the bound (\ref{eq:quasi}). Thus, considering (\ref{eq:stongquasi}), Theorem~\ref{thm:main} provides a spectral condition for a property, which random tournaments a.a.s. satisfy, stronger than the quasi-random property; for example, if $T$ satisfies $\lambda(T)=o(n/\log n)$, then Theorem~\ref{thm:main} implies that $C(T) \leq \binom{n}{2}/2+o(n^2)$, which immediately implies the quasi-random property. \end{remark}
In the proof of Theorem~\ref{thm:main}, we use the {\it expander-mixing lemma} for normal regular digraphs proved by Vu~\cite{V08}. A digraph $D$ is said to be {\it normal} if $M_D$ and its transpose $M_D^t$ are commutative. In other word, $D$ is normal if $|N^+(x,y)|=|N^-(x,y)|$ for any two distinct vertices $x$ and $y$ where $N^+(x, y)$ (resp. $N^-(x, y)$) is the set of vertices $z$ such that $(x, z) , (y, z) \in E(D)$ (resp. $(z, x) , (z, y) \in E(D)$).
Now we are ready to introduce the expander-mixing lemma for normal regular digraphs.
\begin{lemma}[Expander-mixing lemma, \cite{V08}] \label{lem:exp}
Let $D$ be a normal $d$-regular digraph with $n$ vertices and $\lambda(D)=\max_{2\leq i \leq n}|\lambda_i|$. For two disjoint subsets $A, B \subset V(D)$, let \[
e(A, B):=\bigl|\{(a, b)\in E(D) \mid a \in A, \; b \in B\} \bigr|. \] Then for every pair of two disjoint subsets $A, B \subset V(D)$, it holds that \begin{align}
\Bigl|e(A, B)-\frac{d}{n} \cdot |A|\cdot |B| \Bigr| \leq \lambda(D)\sqrt{|A|\cdot |B|}. \end{align} \end{lemma} From this lemma, we can easily obtain the following corollary. \begin{corollary} \label{cor:exp} Let $D$ be a normal $d$-regular digraph with $n$ vertices. Then for every pair of two disjoint subsets $A, B \subset V(D)$, \begin{align}
|e(A, B)-e(B, A)| \leq 2\lambda(D)\sqrt{|A|\cdot |B|}. \end{align} \end{corollary} \begin{proof} From the triangle inequality, we see that \begin{align*}
|e(A, B)-e(B, A)|&=\Bigl| \Bigl(e(A, B)-\frac{d}{n} \cdot |A|\cdot |B|\Bigr)-\Bigl(e(B, A)-\frac{d}{n} \cdot |B|\cdot |A|\Bigr) \Bigr|\\
&\leq \Bigl|e(A, B)-\frac{d}{n} \cdot |A|\cdot |B| \Bigr|+\Bigl|e(B, A)-\frac{d}{n} \cdot |B|\cdot |A|\Bigr|. \end{align*} Thus, by Lemma~\ref{lem:exp}, we get the corollary. \end{proof}
By Corollary~\ref{cor:exp}, we get the following lemma. \begin{lemma} \label{cor:exp2} Let $T$ be a regular tournament with $n$ vertices and let $\sigma$ be a bijection from $V(T)$ to $\{1, 2, \ldots, n\}$. Then \begin{equation} C(T, \sigma)-C(T, \sigma') \leq 2\lambda(T) \cdot n \log_2 (2n). \end{equation} \end{lemma} \begin{proof}[Proof of Lemma~\ref{cor:exp2}] The lemma follows by combining Corollary~\ref{cor:exp} and the argument in \cite[pp.150-151]{AS16} to prove the bound (\ref{eq:paley}) for Paley tournaments. It should be noted (see also \cite{BG72}) that every regular tournament $T$ with $n$ vertices is normal since it holds that $M_T^t=J_n-I_n-M_T$, where $I_n$ and $J_n$ are the identity matrix and the all-one matrix of order $n$, respectively.
\par \par Fix a bijection $\sigma$. Let $r$ be the smallest integer such that $n \leq 2^r$. Let $n=a_1+a_2$, where $a_1$ and $a_2$ are positive integers with $a_1, a_2 \leq 2^{r-1}$. Consider a partition of $V(T)$, say $A_1$ and $A_2$, such that $A_1$ is the set of ^^ ^^ highly ranked" $a_1$ vertices in $\sigma$ and $A_2$ is the remaining $a_2$ vertices. It follows from Corollary~\ref{cor:exp} that \begin{align} \label{eq2:thm2} e(A_1, A_2)-e(A_2, A_1) \leq 2\lambda(T)\sqrt{a_1 a_2} \leq 2\lambda(T) \cdot 2^{r-1}. \end{align} Next, let $a_1=a_{11}+a_{12}$, where $a_{11}$ and $a_{12}$ are positive integers with $a_{11}, a_{12} \leq 2^{r-2}$, and similarly for $a_2=a_{21}+a_{22}$. As above, divide $A_1$ into two subsets, say $A_{11}$ and $A_{12}$, where $A_{11}$ is the set of ^^ ^^ highly ranked" $a_{11}$ vertices of $A_1$ in $\sigma$ and $A_{12}$ is the remaining $a_{12}$ vertices of $A_1$. For $a_{21}$ and $a_{22}$, two subsets $A_{21}$ and $A_{22}$ of $A_2$ are defined in the same way as $A_{11}, A_{12}$. It then follows from Corollary~\ref{cor:exp} that \begin{align*} &e(A_{11}, A_{12})-e(A_{12}, A_{11}) + e(A_{21}, A_{22})-e(A_{22}, A_{21}) \\ &\leq 2\lambda(T) \sqrt{a_{11} a_{12}}+2\lambda(T) \sqrt{a_{21} a_{22}} \\ &\leq 2 \cdot 2\lambda(T) \cdot 2^{r-2}. \end{align*} Then iterate such estimation from the first to the $r$-th step. In the $i$-th step, $V(T)$ is partitioned into $2^i$ subsets, say $A_{\boldsymbol{\varepsilon}1}$ and $A_{\boldsymbol{\varepsilon}2}$ ($\boldsymbol{\varepsilon} \in \{1, 2\}^{i}$), such that each $A_{\boldsymbol{\varepsilon}j}$ ($j=1, 2$) contains at most $2^{r-i}$ vertices which are consecutive in $\sigma$. It follows from Corollary~\ref{cor:exp} that \begin{align} \label{eq3:thm2} \sum_{\boldsymbol{\varepsilon} \in \{1, 2\}^{i-1}}\{e(A_{\boldsymbol{\varepsilon} 1}, A_{\boldsymbol{\varepsilon}2})-e(A_{\boldsymbol{\varepsilon}2}, A_{\boldsymbol{\varepsilon}1})\} \leq 2^{i-1} &\cdot 2\lambda(T) \cdot 2^{r-i}=2\lambda(T) \cdot2^{r-1}. \end{align}
On the other hand, it turns out from the construction of partitions that \begin{align} \label{eq4:thm2} \sum_{1 \leq i \leq r}\sum_{\boldsymbol{\varepsilon}\in \{1, 2\}^{i-1}}\{e(A_{\boldsymbol{\varepsilon} 1}, A_{\boldsymbol{\varepsilon}2})-e(A_{\boldsymbol{\varepsilon}2}, A_{\boldsymbol{\varepsilon}1})\} =C(T, \sigma)-C(T, \sigma'). \end{align} Thus by combining (\ref{eq3:thm2}) and (\ref{eq4:thm2}), it follows that \[ C(T, \sigma)-C(T, \sigma') \leq r \cdot 2\lambda(T) \cdot 2^{r-1} \leq 2\lambda(T) \cdot n \log_2 (2n). \] \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}] The theorem is a direct consequence of the equality (\ref{eq:plus}) and Lemma~\ref{cor:exp2}. \end{proof}
\begin{remark} It should be noted that for every regular tournament $T$ with $n$ vertices, $\lambda(T) \cdot n \log_2 (2n)$ cannot be less than $\sqrt{n^{3}+n} \log_2 (2n)/2$. In fact, for every such tournament $T$, it holds that \begin{equation} \label{eq:lambda}
\lambda(T) \geq \frac{\sqrt{n+1}}{2}. \end{equation} Indeed, for every strongly-connected normal $d$-regular digraph $D$ with $n$ vertices, it holds that \[
nd = E(D)= Tr(M_DM_D^t) =\sum_{i=1}^n|\lambda_i|^2 \leq d^2+(n-1)\lambda(D)^2, \] which follows from the hand shaking lemma and the Perron-Frobenius theorem (see e.g. \cite{LW01}). The idea of the above inequality can be found in \cite[p.217]{KS06}. Also note that every regular tournament $T$ is strongly connected, which follows from the Perron-Frobenius theorem and facts that $T$ is normal and every eigenvalue of $M_T$ corresponding to eigenvectors distinct to the all-one vector has the real part equal to $-1/2$ (see also \cite{BG68}). \end{remark}
\section{Examples of quasi-random regular tournaments} \label{sect:Const} In this section, we give some examples of regular tournaments $T$ with $n$ vertices and $\lambda(T)=o(n/\log n)$. As will be shown below, we can construct such tournaments for almost all positive integers $n$.
\par First we consider the following tournaments constructed from finite fields which are variants of cyclotomic tournaments (see e.g. \cite{S16} and reference therein). Let $m$ be a positive even integer and $p\equiv m+1 \pmod {2m}$ be a prime. Note that there exist infinitely many such primes by the Dirichlet's theorem on arithmetic progressions and the fact that $m+1$ and $2m$ are coprime when $m$ is even. Recall that $\mathbb{F}_p$ is the finite field of order $p$. Let $g$ be a primitive element of $\mathbb{F}_p$. For even $m$, the multiplicative group of $\mathbb{F}_p$, which is denoted by $\mathbb{F}_p^*$, is divided into $m$ cosets $S_0, S_1, \ldots, S_{m-1}$ where $S_i:=\{g^t \mid t \equiv i \pmod{m} \}$ for each $0 \leq i \leq m-1$. Note that $S_j = -S_i$ if $j \equiv -i \pmod{m}$.
\begin{definition} Let $\boldsymbol{i}=(i_1, i_2, \ldots, i_{m/2}) \in \{0, 1, \ldots, m-1\}^{m/2}$ such that $S_{\boldsymbol{i}}=S_{i_1} \cup \cdots \cup S_{i_{m/2}}$ and $\mathbb{F}_p^* \setminus S=-S$. Then the tournament $T_p^m(S_{\boldsymbol{i}})$ is defined as follows: \begin{equation} \begin{split} &V(T_p^m(S_{\boldsymbol{i}}))=\mathbb{F}_p, \\ &E(T_p^m(S_{\boldsymbol{i}}))=\{(x, y)\in \mathbb{F}_p^2 \mid x-y \in S_{\boldsymbol{i}} \}. \end{split} \end{equation} \end{definition}
This is a direct generalization of Paley tournament since $T_p^m(S_{\boldsymbol{i}})$ is exactly $T_p$ in the case of $m=2$. Moreover from the definition, it is not so hard to see that $T_p^m(S_{\boldsymbol{i}})$ is a regular tournament with $p$ vertices. \par Now we obtain the following corollary.
\begin{corollary} \label{cor:Tpm} \begin{equation} C\bigl(T_p^m(S_{\boldsymbol{i}})\bigr) \leq \frac{1}{2}\binom{p}{2}+O(p^{\frac{3}{2}} \log p). \end{equation} \end{corollary}
Corollary~\ref{cor:Tpm} is proved by combining Lemma~\ref{cor:exp2} and the following evaluation of $\lambda(T_p^m(S_{\boldsymbol{i}}))$.
\begin{lemma} \label{lem:eigen} \begin{align} \lambda \bigl(T_p^m(S_{\boldsymbol{i}})\bigr) \leq \frac{m\sqrt{p}}{2}. \end{align} \end{lemma} \begin{proof} First, by a simple calculation, it can be shown that the set of eigenvalue of $M_{T_p^m(S_{\boldsymbol{i}})}$ is \[ \Bigl\{ \sum_{s\in S_{\boldsymbol{i}}}\psi(s) \mid \text{$\psi$ is an additive character of $\mathbb{F}_p$} \Bigr\}. \] Since $S_{i}=g^{i}S_0$ for each $1 \leq i \leq m-1$, we see that \begin{align} \label{eq:eigen1} \sum_{s\in S_i}\psi(s)=\sum_{s\in g^iS_0}\psi(s)=\sum_{s\in S_0}\psi(g^i s). \end{align} Since $S_0$ is the set of non-zero $m$-th power elements and each non-zero $m$-th power residue appears exactly $m$ times in the sequence $(x^m)_{x \in \mathbb{F}_{p}^*}$, \begin{align} \label{eq:eigen2} \sum_{s\in S_0}\psi(g^i s)=\frac{1}{m}\sum_{x \in \mathbb{F}_p^*}\psi(g^i x^m). \end{align} At last, we use the following known estimation (see e.g. \cite[p.44]{S76}); \begin{align} \label{eq:eigen3}
\Bigl|\sum_{x \in \mathbb{F}_p}\psi(a x^m) \Bigr| \leq (m-1)\sqrt{p}, \end{align} for any non-trivial additive character $\psi$ and $a \neq 0$. By combining (\ref{eq:eigen1}), (\ref{eq:eigen2}) and (\ref{eq:eigen3}), \begin{align*} \lambda \bigl(T_p^m(S_{\boldsymbol{i}})\bigr) \leq \frac{m}{2}\cdot \frac{1}{m} \cdot \{(m-1)\sqrt{p}+1\} =\frac{(m-1)\sqrt{p}+1}{2} \leq \frac{m\sqrt{p}}{2}. \end{align*} \end{proof}
The second example is doubly regular tournament which has been extensively studied in algebraic combinatorics and related areas (e.g. \cite{RB72}). \begin{definition} A tournament $T$ with $n$ vertices is called a {\it doubly regular tournament} if $T$ is a regular tournament such that for any distinct two vertices $x$ and $y$, $N^+(x, y)=N^-(x, y)=(n-3)/4$. \end{definition} Let $DRT_n$ denote a doubly regular tournament with $n$ vertices.
\begin{corollary} \label{cor:dr} \begin{equation} C(DRT_n)\leq \frac{1}{2}\binom{n}{2}+O(n^{\frac{3}{2}} \log n). \end{equation} \end{corollary} Corollary~\ref{cor:dr} is proved by the following well-known evaluation of $\lambda(DRT_n)$ which also shows that the inequality (\ref{eq:lambda}) is tight. \begin{lemma}[e.g. \cite{dGKPM92}] \label{lem:eigen2} \begin{align} \lambda(DRT_n)=\frac{\sqrt{n+1}}{2}. \end{align} \end{lemma}
\begin{proof} We give a proof for the reader's convenience. Let $M=M_{DRT_n}$. Then by the definition, it holds that \begin{equation} MM^t=\frac{n+1}{4}I_n+\frac{n-3}{4}J_n. \end{equation} Since $M+M^t=J_n-I_n$, we obtain the following equality. \begin{equation} M^2+M+\frac{n+1}{4}I_n-\frac{n+1}{4}J_n=O. \end{equation} Since $DRT_n$ is regular, we see that $(n-1)/2$ is an eigenvalue of $M$ and a corresponding eigenvector is the all-one eigenvector $\boldsymbol{1}$. Since $DRT_n$ is normal, each eigenvalue $\theta$ except for $(n-1)/2$ has an eigenvector $\boldsymbol{v}$ which is orthogonal to $\boldsymbol{1}$. Thus, \begin{equation} \Bigl(\theta^2+\theta+\frac{n+1}{4} \Bigr)\boldsymbol{v}=\boldsymbol{0}. \end{equation} Since $\boldsymbol{v}\neq \boldsymbol{0}$, we get \begin{equation} \Bigl(\theta^2+\theta+\frac{n+1}{4} \Bigr)=0, \end{equation} completing the proof. \end{proof}
\begin{remark} We remark that Corollary~\ref{cor:dr} is a generalization of the bound (\ref{eq:paley}) because Paley tournaments are also doubly-regular tournaments. For other non-isomorphic examples of doubly regular tournaments, see e.g. \cite{IO94} and \cite{S69}. As shown in, for example, \cite{HW14} and \cite{RB72}, there are some known constructions of doubly regular tournaments such that the number of vertices is non-prime (and non-prime power). Especially, constructions of complex codebooks in \cite{HW14} provide $DRT_n$ for every integer $n$ such that each prime factor $f$ of $n$ is the form of $f \equiv 3 \pmod{4}$. \end{remark}
\begin{remark} By the definition of $DRT_n$, $n$ must be a positive integer of the form $n \equiv 3 \pmod{4}$. On the other hand, as an analogue of $DRT_n$ for integers $n$ of the form $n\equiv 1 \pmod{4}$, Savchenko~\cite{S16} introduced the notion of a nearly-doubly-regular tournament $CNDR_n$ with $n$ vertices which is a certain regular tournament with exactly four eigenvalues distinct to $(n-1)/2$ with multiplicity $(n-1)/4$. According to \cite{S16}, it holds that $\lambda(CNDR_n)=(\sqrt{n}+1)/2$. Thus if there exists $CNDR_n$ for infinitely many $n \equiv 1\pmod{4}$, then it holds that \[ C(CNDR_n)\leq \frac{1}{2}\binom{n}{2}+O(n^{\frac{3}{2}}\log n). \] It is conjectured in \cite{S16} (see also \cite{S17}) that there exists a $CNDR_n$ for every $n \equiv 1 \pmod{4}$. Interestingly, Savchenko~\cite{S16} also found examples of $CNDR_p$ for primes $p=5, 13, 29, 53, 173, 229, 293$ and $733$ from the class of $T_p^4(S_{(0, 1)})$ in the first example, and thus Lemma~\ref{lem:eigen} can be improved for these examples. (It is shown in \cite{S16} that for every prime $p \equiv 5 \pmod{8}$, $T_p^4(S_{(0, 1)})$ has exactly four eigenvalues distinct to $(p-1)/2$ with multiplicity $(p-1)/4$.) It would be interesting to prove or disprove the existence of infinitely many primes $p \equiv 5 \pmod{8}$ such that the tournament $T_p^4(S_{(0, 1)})$ is in the class of $CNDR_p$. \end{remark}
The third example is based on a construction of pseudo-random graphs due to Shparlinski~\cite{S08}. For related facts on eliptic curves, see \cite[Section 2.1]{S08}. For a prime $p$, let $n \in [p+1-2\sqrt{p}, p+1+2\sqrt{p}]$ be an odd integer. It is known (e.g. \cite{BS04}, \cite{D41}) that there exists an eliptic curve $E$ over $\mathbb{F}_p$ such that the number of $\mathbb{F}_p$-rational points of $E$ is $n$. It is also known (e.g. \cite{S95}) that all $\mathbb{F}_p$-rational points of $E$ form an abelian group $G$ of order $n$ under an operation $\oplus$. Let $0_G$ be the identity of $G$. For an element $s \in G$ and a subset $S \subset G$, the inverse of $s$ is denoted by $\ominus s$ and let $\ominus S=\{\ominus s \mid s \in S\}$.
\begin{definition}
Let $S \subset G$ be a subset such that $S \cup \ominus S \cup \{0_G \}=G$ and $|S|=(n-1)/2$. Then the tournament $T_{p,n}(S)$ is defined as follows. \begin{equation} \begin{split} &V(T_{p,n}(S))=G, \\ &E(T_{p,n}(S))=\{(x, y)\in G^2 \mid x \ominus y \in S \}. \end{split} \end{equation} \end{definition} By the definition, $T_{p,n}(S)$ is a regular tournament with $n$ vertices.
\begin{corollary} \label{cor:eliptic} There exists a subset $S \subset G$ such that \begin{equation} C(T_{p,n}(S)) \leq \frac{1}{2}\binom{n}{2}+O(n^{\frac{3}{2}} \log^2 n). \end{equation} \end{corollary}
Corollary~\ref{cor:eliptic} is obtained by Lemma~\ref{cor:exp2} and the following evaluation of $\lambda(T_{p,n}(S))$ which follows from \cite[Theorem 1]{S08}. \begin{lemma}[\cite{S08}] There exists a subset $S \subset G$ such that \label{lem:eigen3} \begin{align} \lambda(T_{p,n}(S))=O(\sqrt{n}\log n). \end{align} \end{lemma} For the details of a construction of such a subset $S$, see \cite{S08}.
\begin{remark} It is worth noting that as shown in \cite{S08}, almost all positive integers are in the interval $[p+1-2\sqrt{p}, p+1+2\sqrt{p}]$ for some prime $p$. Indeed, it holds (\cite{S08}) that \[
\lim_{N \to \infty}\frac{|\{n \leq N \mid \text{$\exists$ prime $p$ s.t. $n$ is odd and $n \in [p+1-2\sqrt{p}, p+1+2\sqrt{p}]$}\}|}{\lceil\frac{N}{2} \rceil}=1. \] Thus the third example provides regular tournaments $T$ with $n$ vertices and small $\lambda(T)$ for almost all positive integers $n$. \end{remark}
\section{Sh\"{u}tte's problem for tournaments} \label{sec:Shutte} At last, in this section, we focus on another random-like property. \begin{definition} \label{def:ec} Let $k$ be a positive integer. A tournament $T$ has the property $S_k$ if for every $A \subset V(T)$ of size $k$, there exists a vertex $z \notin A$ directing to all members of $A$. \end{definition} The {\it Sh\"{u}tte's problem} asks the existence of tournaments satisfying this property (see \cite{E63} and \cite{M68}). As shown by Erd\H{o}s~\cite{E63}, random tournaments a.a.s. satisfy $S_k$ for any $k\geq 1$. On the other hand, the problem of explicit constructions has been considered in graph theory. For example, Graham-Spencer~\cite{GS71} showed that the Paley tournament $T_p$ satisfies $S_k$ if $p>k^22^{2k-2}$ for each $k \geq 1$. From the digraphs constructed in \cite{AHLNVW15}, we can also construct tournaments satisfying $S_k$ for every $k$ by adding some edges. At present, there seems to be almost no explicit constructions of tournaments satisfying both of the quasi-random property and $S_k$ except for Paley tournaments. The following proposition and Corollary~\ref{cor:Tpm} show that the tournament $T_p^m(S_{\boldsymbol{i}})$ has the quasi-random property and $S_k$.
\begin{proposition} \label{prop:shutte1} Let $m$ be an even positive integer. Then for every $k \geq 1$, there exists a prime $p_m(k)$ such that for every prime $p>p_m(k)$, the tournament $T_p^m(S_{\boldsymbol{i}})$ has the property $S_k$. \end{proposition} Proposition~\ref{prop:shutte1} is proved by a direct generalization of the discussion in \cite{GS71} and \cite{AC06}, so we omit the proof here. Moreover, it is not so hard to prove that $T_p^m(S_{\boldsymbol{i}})$ has the existentially closed property (see e.g. \cite{B09}). \par We also note that doubly regular tournaments constructed in \cite{S69} satisfy both of the quasi-random property and $S_2$, which follows from Corollary~\ref{cor:dr} and the corollary in \cite[p.277]{S69}.
\end{document} | arXiv |
\begin{document}
\title{Improving sum uncertainty relations with the quantum Fisher information} \author{Shao-Hen Chiew} \affiliation{Laboratoire Kastler Brossel, ENS-Universit\'{e} PSL, CNRS, Sorbonne Universit\'{e}, Coll\`{e}ge de France, 24 Rue Lhomond, 75005, Paris, France} \affiliation{Department of Physics, National University of Singapore, Singapore 117551, Singapore} \affiliation{Centre for Quantum Technologies, National University of Singapore, Singapore 117551, Singapore} \affiliation{Mines-ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75006 Paris, France} \author{Manuel Gessner} \email{[email protected]} \affiliation{Laboratoire Kastler Brossel, ENS-Universit\'{e} PSL, CNRS, Sorbonne Universit\'{e}, Coll\`{e}ge de France, 24 Rue Lhomond, 75005, Paris, France} \affiliation{ICFO-Institut de Ci\`{e}ncies Fot\`{o}niques, The Barcelona Institute of Science and Technology, Avinguda Carl Friedrich Gauss 3, 08860, Castelldefels (Barcelona), Spain} \date{\today}
\begin{abstract} We show how preparation uncertainty relations that are formulated as sums of variances may be tightened by using the quantum Fisher information to quantify quantum fluctuations. We apply this to derive stronger angular momentum uncertainty relations, which in the case of spin-$1/2$ turn into equalities involving the purity. Using an analogy between pure-state decompositions in the Bloch sphere and the moment of inertia of rigid bodies, we identify optimal decompositions that achieve the convex- and concave-roof decomposition of the variance. Finally, we illustrate how these results may be used to identify the classical and quantum limits on phase estimation precision with an unknown rotation axis. \end{abstract}
\maketitle
\section{Introduction} The uncertainty principle expresses that independent measurements of non-commuting observables cannot be arbitrarily sharp when the system is prepared in the same quantum state. The aim of preparation uncertainty relations is to make precise, quantitative statements about the fluctuations of incompatible observables for a given quantum state. The most well known uncertainty relation is the Heisenberg-Robertson inequality \cite{Heisenberg,PhysRev.34.163}: For arbitrary quantum states $\rho$ and observables $A$ and $B$, we have \begin{equation} \label{eq:robertson}
(\Delta A)_{\rho} ^2 (\Delta B)_{\rho} ^2 \geq \frac{1}{4} \left\langle i[A,B] \right\rangle_\rho^2, \end{equation} where $[A,B] = AB - BA$ is the commutator and $(\Delta A)^2_{\rho}=\langle A^2\rangle_{\rho}-\langle A\rangle_{\rho}^2$ is the variance, with $\langle A\rangle_{\rho}=\mathrm{Tr}\{A\rho\}$. The Heisenberg-Robertson inequality, however, does not make general statements about the set of tuples $((\Delta A)_{\rho} ^2,(\Delta B)_{\rho} ^2)$ that are compatible with the laws of quantum mechanics since the lower bound depends on the quantum state $\rho$ and becomes trivial for states whose expectation value for the commutator is zero. Nevertheless, constraints on these measurement fluctuations exist even when the right-hand side in~(\ref{eq:robertson}) vanishes~\cite{Hofmann,RivasLuisPRA2008,HePRA2011,HuangPRA2012,Busch_inequality,MacconePRL2014,Dammeier_2015,BagchiPRA2016,tightqubit,MondalPRA2017,GiordaPRA2019,KarolJPA2020}.
Sum uncertainty relations may avoid these problems by providing state-independent lower bounds on sums of variances~\cite{Hofmann,HePRA2011,Dammeier_2015,tightqubit}. For instance, any quantum state $\rho$ of a spin-$s$ system satisfies for the three components $L_{\vec{n}_1},L_{\vec{n}_2},L_{\vec{n}_3}$ of the SU(2) angular momentum algebra \cite{Hofmann,Dammeier_2015} \begin{equation} \label{eq:varsumreln3}
(\Delta L_{\vec{n}_1})_{\rho} ^2 + (\Delta L_{\vec{n}_2})_{\rho} ^2 + (\Delta L_{\vec{n}_3})_{\rho} ^2 \geq s \end{equation} and~\cite{HePRA2011,Dammeier_2015} \begin{equation} \label{eq:varsumreln2}
(\Delta L_{\vec{n}_1})_{\rho} ^2 + (\Delta L_{\vec{n}_2})_{\rho} ^2 \geq c(s), \end{equation} where $c(s)$ are constants. For small $s$ these constants are known analytically whereas for larger $s$ they can be determined numerically and show an asymptotic scaling of $c(s)\sim s^{2/3}$ \cite{HePRA2011,Dammeier_2015}. For the case of qubits $(s=1/2)$, the constant reads $c(1/2) = 1/4$, and a state-independent bound can be stated for two observables with arbitrary orientations $\vec{a}$ and $\vec{b}$ as \cite{Busch_inequality} \begin{equation} \label{eq:Busch_ineq}
(\Delta L_{\vec{a}})_{\rho} ^2 + (\Delta L_{\vec{b}})_{\rho} ^2 \geq \frac{1}{4}(1-|\vec{a}\cdot\vec{b}|). \end{equation}
While standard formulations of uncertainty relations are based on variances, they can also be stated and improved in terms of more general quantifiers of fluctuation such as entropies \cite{PhysRevLett.50.631,MaassenUffink,RMP2017} and the quantum Fisher information (QFI)~\cite{Braunstein&Caves,BraunsteinCavesMilburn,T_th_2014,Pezze_QTOPE,fisherrobertson,YadinNatCommun2021}. By quantifying the sensitivity of a quantum state $\rho$ to small perturbations generated by $A$~\cite{Braunstein&Caves}, the QFI $F_Q[\rho,A]$ plays a fundamental role in identifying the precision limits of measurements in quantum metrology~\cite{HelstromBOOK,ParisIntJQI2009,Pezze_QTOPE,Advances,T_th_2014}. For example, a tighter formulation of the Heisenberg-Robertson inequality is implied by a lower bound on the QFI~\cite{Kholevo,HottaOzawa,PS09,fisherrobertson,Pezze_QTOPE,GessnerPRL2019}: \begin{equation} \label{eq:robertson2} F_Q[\rho,A] (\Delta B)_{\rho} ^2
\geq \left\langle i[A,B] \right\rangle_\rho^2. \end{equation} Since $4(\Delta A)_{\rho} ^2\geq F_Q[\rho,A]$ holds for arbitrary states $\rho$ (and is saturated by all pure states)~\cite{T_th_2014,Pezze_QTOPE}, Eq.~(\ref{eq:robertson2}) is, indeed, a tighter condition than Eq.~(\ref{eq:robertson}) but inherits the drawback of being state dependent.
In this article, we show how sum uncertainty relations of the kind~(\ref{eq:varsumreln3}),~(\ref{eq:varsumreln2}), and ~(\ref{eq:Busch_ineq}) can be tightened with the QFI. Using the fact that the QFI is the convex roof of the variance~\cite{roof,decomposition}, we demonstrate how any variance-based sum uncertainty relation can be improved by replacing one of the variances with the QFI. For the qubit case we prove a set of tight relations for the QFI and variances as a function of the state's purity. By establishing an analogy between pure-state decompositions and the classical moment of inertia of rigid bodies, we identify optimal state decompositions that achieve the corresponding convex and concave roof constructions. Finally, we use these results to identify classical and quantum sensitivity limits for the estimation of angular parameters with an unknown rotation axis.
\section{Variance and quantum Fisher information} We begin by reviewing some of the most important properties of the variance and the QFI---the two quantities of central interest in this article for the quantification of fluctuations. Interestingly, the two quantities represent opposite extrema of the average variance of pure-state decompositions. A generic quantum state $\rho$ can be represented in terms of inequivalent decompositions $\{p_k, \ket{\Psi_k}\}$ consisting of a probability distribution $p_k$ and a set of pure states $\Psi_k=\ket{\Psi_k}\bra{\Psi_k}$ that are not necessarily pairwise orthogonal but yield $\rho = \sum_{k} p_k \ket{\Psi_k}\bra{\Psi_k}$. The variance is the concave roof of itself \cite{roof}: \begin{equation} \label{eq:varroof}
(\Delta A)_{\rho}^2 = \max_{\{p_k, \ket{\Psi_k}\}} \sum_{k} p_k (\Delta A)_{\Psi_k} ^2; \end{equation} that is, it corresponds to the decomposition that maximizes the average pure-state variance. The QFI satisfies the opposite property; that is, it is the convex roof~\cite{roof,decomposition}, \begin{equation} \label{eq:mindecomp}
\frac{1}{4}F_Q[\rho,A] = \min_{\{p_k, \ket{\Psi_k}\}} \sum_{k} p_k (\Delta A)_{\Psi_k} ^2. \end{equation} Optimal decompositions that achieve either the maximum~(\ref{eq:varroof}) or the minimum~(\ref{eq:mindecomp}) can always be found~\cite{decomposition}. These results further imply for any mixture $\rho=\sum_kp_k\rho_k$ the following sequence of bounds: \begin{equation} \label{eq:QFI<4Var}
\frac{1}{4}F_Q[\rho,A] \leq \frac{1}{4}\sum_{k} p_kF_Q[\rho_k,A]\leq \sum_{k} p_k (\Delta A)_{\rho_k} ^2\leq (\Delta A)_{\rho}^2, \end{equation} and all terms coincide if $\rho$ is a pure state. In particular, we observe the convexity of the QFI and concavity of the variance.
The QFI is a quantity of central interest in the theory of quantum metrology~\cite{Pezze_QTOPE,Advances,T_th_2014}. Quantum phase estimation, for instance, describes the estimation of a phase parameter $\theta$ that is imprinted by a unitary evolution into a quantum state via $\rho(\theta)=e^{-iA\theta}\rho e^{iA\theta}$. The quantum Cram\'{e}r-Rao bound states that the variance of arbitrary unbiased estimators $\theta_{\rm{est}}$ for $\theta$ cannot be lower than the inverse of the QFI \cite{HelstromBOOK,Braunstein&Caves,Pezze_QTOPE,Advances,T_th_2014}: \begin{equation} \label{eq:CRB} (\Delta\theta_{\rm{est}})^2\geq F_Q[\rho,A]^{-1}. \end{equation}
Besides convexity~(\ref{eq:QFI<4Var}), the QFI satisfies additivity~\cite{T_th_2014,Pezze_QTOPE}:
\begin{equation} \label{eq:addQFI}
F_Q[\rho^{(1)} \otimes \rho^{(2)}, A^{(1)} \otimes \mathbb{1} + \mathbb{1} \otimes A^{(2)}] = F_Q[\rho^{(1)},A^{(1)}]+F_Q[\rho^{(2)},A^{(2)}],
\end{equation} where $\mathbb{1}$ denotes the local identity operator. Moreover, we note that the QFI vanishes if and only if the state $\rho$ commutes with the generator $A$: \begin{equation} \label{eq:iff0}
[\rho,A] = 0
\iff
F_Q[\rho,A] = 0. \end{equation} \begin{proof} For the forward direction, note that the QFI can be computed with a closed expression \cite{Pezze_QTOPE,T_th_2014}. Expanding $\rho$ in its eigenbasis so that $\rho = \sum_{k} \lambda_k \ket{\Phi_k}\bra{\Phi_k}$, the QFI reads \begin{equation} \label{eq:explicitfisher}
F_Q[\rho,A] = 2 \sum_{k,l} \frac{(\lambda_k-\lambda_l)^2}{\lambda_k+\lambda_l} |\bra{\Phi_k}A\ket{\Phi_l}|^2. \end{equation} The $k=l$ terms vanish. The $k \neq l$ terms also vanish, since $\rho$ and $A$ commute and are thus simultaneously diagonalizable, so the off diagonal terms $\bra{\Phi_k}A\ket{\Phi_l}$ are zero.
For the backward direction, we start with the decomposition $\rho = \sum_{k} p_k \ket{\Psi_k}\bra{\Psi_k}$ that reaches the minimum of Eq.~(\ref{eq:mindecomp}), so we have \begin{equation}
\frac{1}{4}F_Q[\rho,A] = \sum_{k} p_k (\Delta A)_{\Psi_k} ^2 = 0. \end{equation} Since the $p_k$ are positive, each $(\Delta A)_{\Psi_k} ^2$ in the sum must vanish, which implies that the $\ket{\Psi_k}$ are eigenvectors of $A$. Thus, $\rho$ is diagonal in the eigenbasis of $A$ and consequently commutes with $A$. \end{proof}
\section{Improving sum uncertainty relations} \subsection{General result} We first introduce a general result before applying it to sum uncertainty relations in the next subsection. \begin{proposition}[Tightening variance inequalities with the QFI]\label{prop:1} Let $f$ be a convex function of the set of quantum states, i.e., $f(\rho) \leq \sum_{k} p_k f(\rho_k)$ for $\rho=\sum_kp_k\rho_k$ and let $A$ be an arbitrary observable. If for any pure state $\Psi=\ket{\Psi}\bra{\Psi}$,
\begin{equation} \label{eq:one}
(\Delta A)_{\Psi} ^2 \geq f(\Psi)
\end{equation} holds, then for any mixed state $\rho$ we obtain the inequality
\begin{equation}\label{eq:resultprop1}
\frac{1}{4}F_Q [\rho, A] \geq f(\rho),
\end{equation} where $F_Q[\rho,A]$ is the QFI. \end{proposition} \begin{proof} Let $\{p_k, \ket{\Psi_k}\}$ be the decomposition of $\rho$ that achieves the minimum in Eq.~(\ref{eq:mindecomp}). Using~(\ref{eq:one}) and the convexity of $f$, we obtain \begin{align}
\frac{1}{4}F_Q [\rho, A] = \sum_{k} p_k (\Delta A)_{{\Psi_k}} ^2 \geq
\sum_{k} p_k f(\Psi_k) \geq
f(\rho).\notag \end{align} \end{proof}
Combining the result~(\ref{eq:resultprop1}) with~(\ref{eq:QFI<4Var}), we obtain the weaker bound $(\Delta A)_{\rho} ^2 \geq f(\rho)$. This bound follows directly from the concavity of the variance~(\ref{eq:QFI<4Var}) and the convexity of $f$: \begin{equation}
(\Delta A)_{\rho} ^2 \geq
\sum_{k} p_k (\Delta A)_{\Psi_k} ^2 \geq
\sum_{k} p_k f(\Psi_k) \geq
f(\rho). \end{equation} However, the tighter bound~(\ref{eq:resultprop1}) involving the QFI requires the existence of a decomposition that achieves the minimum in~(\ref{eq:mindecomp}). Moreover, an inequality involving only pure states is sufficient to establish inequalities for mixed states.
We also note that a dual result involving any concave function $g$, i.e., $g(\rho) \geq \sum_{k} p_k g(\rho_k)$, can be obtained with a similar proof, exploiting the fact that the variance is the concave roof of itself, Eq.~(\ref{eq:varroof}). That is, suppose the following inequality holds for any pure state $\Psi$: \begin{equation}
(\Delta A)_{\Psi} ^2 \leq g(\Psi). \end{equation} Then for all mixed states $\rho$ we also have: \begin{equation}
(\Delta A)_{\rho} ^2 \leq g(\rho). \end{equation} In combination with~(\ref{eq:QFI<4Var}) we obtain the weaker bound $\frac{1}{4}F_Q[\rho,A] \leq g(\rho)$, which follows immediately from the convexity of the QFI~(\ref{eq:QFI<4Var}) and the concavity of $g$.
\subsection{Tighter sum uncertainty relations} Let us now apply Proposition~\ref{prop:1} to derive tighter sum uncertainty relations. We consider the quantum mechanical angular momentum algebra on a $(2s+1)$-dimensional Hilbert space of a spin $s$ system with $s=1/2,1,3/2,\dots$. We denote the angular momentum operator along a direction $\vec{n}\in\mathbb{R}^3$ by $L_{\vec{n}}$. Throughout this article we assume that $\{\vec{n}_1, \vec{n}_2, \vec{n}_3\}$ is an arbitrary orthonormal basis of $\mathbb{R}^3$.
\begin{proposition}[Application to sum uncertainty relations] Arbitrary quantum states $\rho$ of a spin $s$ system satisfy \begin{equation} \label{eq:qfisum3} \frac{1}{4}F_Q [\rho, L_{\vec{n}_1}] + (\Delta L_{\vec{n}_2})_\rho ^2 + (\Delta L_{\vec{n}_3})_\rho ^2\geq s, \end{equation} and \begin{equation} \label{eq:qfisum2} \frac{1}{4}F_Q [\rho, L_{\vec{n}_1}] + (\Delta L_{\vec{n}_2})_\rho ^2 \geq c(s), \end{equation} where the $c(s)$ are the same constants that appear in~(\ref{eq:varsumreln2}). For $s=1/2$, we also have: \begin{equation}
\frac{1}{4}F_Q [\rho, L_{\vec{a}}] + (\Delta L_{\vec{b}})_{\rho} ^2 \geq \frac{1}{4}(1-|\vec{a}\cdot\vec{b}|). \end{equation}
\end{proposition}
\begin{proof} Equations~(\ref{eq:varsumreln3}), (\ref{eq:varsumreln2}), and (\ref{eq:Busch_ineq}) imply that inequality~(\ref{eq:one}) holds for $A=L_{\vec{n}_1}$ or $A=L_{\vec{a}}$ with the lower bounds \begin{equation}
f_1(\Psi) = s - (\Delta L_{\vec{n}_2})_{\Psi} ^2 - (\Delta L_{\vec{n}_3})_{\Psi} ^2, \end{equation} \begin{equation}
f_2(\Psi) = c(s) - (\Delta L_{\vec{n}_2})_{\Psi} ^2, \end{equation} and \begin{equation}
f_3(\Psi) = \frac{1}{4}(1-|\vec{a}\cdot\vec{b}|) - (\Delta L_{\vec{b}})_{\Psi} ^2 \end{equation} respectively. Convexity of $f_1$, $f_2$, and $f_3$ follows from the concavity of the variance, Eq.~(\ref{eq:QFI<4Var}). Applying Proposition \ref{prop:1} then leads to the results. \end{proof}
Plotting values of variance and the QFI of randomly generated mixed states allows us to illustrate these results and to reveal additional features of interest. We provide the uncertainty plots of $s=1/2$ and $s=1$ for $L_x$, $L_y$, and $L_z$ in Figs.~\ref{fig:s_half} and \ref{fig:s_one}, respectively.
\begin{figure}
\caption{Uncertainty diagrams for $s=1/2$, where projections on three planes are included for visualisation. Each point in the colored regions (except the projections) represents the fluctuations of a possible quantum state. The left plot shows the variance-based uncertainty relation. Since the tuples $((\Delta L_x)_\rho ^2,(\Delta L_y)_\rho ^2),(\Delta L_z)_\rho ^2)$ obey relation~(\ref{eq:var1/2}), the fluctuations for pure states all lie in a plane (purple), whereas mixed states occupy a region of finite volume (red tetrahedron). The excluded area near the origin reflects the uncertainty relation~(\ref{eq:varsumreln3}). In the right plot, one of the variances has been replaced by the quantum Fisher information, which is generally bounded from above by the variance. The tuples $((\Delta L_x)_\rho ^2,(\Delta L_y)_\rho ^2,\frac{1}{4}F_Q[\rho, L_z])$ of pure and mixed states (green) lie in a single plane, as predicted by Eq.~(\ref{eq : qfivareq}).}
\label{fig:s_half}
\end{figure}
\begin{figure}
\caption{Uncertainty diagrams for $s=1$. Each point in the colored region represents the fluctuations of a possible quantum state. (a) and (b) display $((\Delta L_x)_\rho ^2,(\Delta L_y)_\rho ^2, (\Delta L_z)_\rho ^2)$ tuples in two and three dimensions respectively, while (c) and (d) display $((\Delta L_x)_\rho ^2,(\Delta L_y)_\rho ^2,\frac{1}{4}F_Q[\rho, L_z]$ tuples in 2D and 3D respectively. The occupied regions in (b) and (d) are bounded below by the planes defined by Eqs.~(\ref{eq:varsumreln3}) and (\ref{eq:qfisum3}), respectively, which results in empty regions near the origin. Also, the green $((\Delta L_x)_\rho ^2,(\Delta L_y)_\rho ^2,\frac{1}{4}F_Q[\rho, L_z])$ tuples occupy a region previously unoccupied by the red $((\Delta L_x)_\rho ^2,(\Delta L_y)_\rho ^2,(\Delta L_z)_\rho ^2)$ tuples. This region can be occupied by only mixed states [in particular, mixed states such as~(\ref{eq:diagonal})].}
\label{fig:s_one}
\end{figure}
For $s = 1$, a feature that arises when a variance is replaced with its corresponding QFI is the occupation of a region that is not occupied if we consider only tuples of variances. This region can be observed in Fig.~\ref{fig:s_one} by comparing the green and red plots. For mixed states, the difference between $(\Delta A)_\rho ^2$ and $\frac{1}{4}F_Q [\rho, A]$ is bounded from above \cite{puritybound}: \begin{equation}\label{eq:varQFIdiff}
(\Delta A)_\rho ^2 - \frac{1}{4}F_Q [\rho, A] \leq \frac{1-\mathrm{Tr}(\rho^2)}{2}(\lambda_{\max}(A)-\lambda_{\min}(A))^2, \end{equation} where $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$ are the largest and smallest eigenvalues of $A$, respectively. Thus, for mixed states, it is possible for a coordinate of a tuple to decrease when a corresponding variance in this direction is replaced by its QFI. To understand this phenomenon intuitively, it is instructive to consider an extreme example. Take, for instance, incoherent mixtures of $L_z$ eigenstates, i.e., density matrices of the form \begin{align} \label{eq:diagonal} \rho = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0\\ 0 & 0 & 1-a-b\\ \end{pmatrix}, \end{align} with real, non-negative parameters $a$ and $b$ such that $a+b\leq1$. It is straightforward to verify that $ (\Delta L_z)_\rho ^2=1 - b - (2 a + b - 1)^2 \geq 0$ and equality is reached only when $\rho$ is pure. However, we can interpret the state as being a classical mixture of pure states that have zero fluctuations in $L_z$. Hence, the nonzero variance is entirely due to the classical ignorance or the mixing process. Indeed, the concave-roof property of the variance ensures that the effect of classical mixing is large. In contrast, the QFI, by construction, minimizes these effects and consequently yields $F_Q[\rho,L_z]=0$ [recall also Eq.~(\ref{eq:iff0})]. This reflects the interpretation that the state is, in principle, prepared in a zero-variance state, we are just (classically) unsure which one.
Moreover, the variances of the states~(\ref{eq:diagonal}) along the other directions yield $(\Delta L_x)_\rho ^2 = (\Delta L_y)_\rho ^2 = \frac{1+b}{2}$. Thus, states that are diagonal in $L_z$ are exactly the points that lie in the straight line between $(0.5,0.5,0)$ and $(1,1,0)$ of the uncertainty diagram in Fig.~\ref{fig:s_one}(d).
\section{The qubit case} For the special case of $s=1/2$, i.e., qubits, we prove stronger conditions by working with the Bloch representation. By expressing the QFI of a mixed state in terms of Bloch vectors, it can be interpreted as an analog of the moment of inertia of classical mechanics. This motivates us to prove analogs of the parallel and perpendicular axis theorems (\ref{eq:parallel_axis}) and (\ref{eq:perpendicular_axis}), as well as stronger sum uncertainty relations (\ref{eq:var1/2})-(\ref{eq:qfisum}). Furthermore, the Bloch representation, together with these results, provides a geometrical picture for decompositions $\{p_k, \ket{\Psi_k}\}$ of a mixed state $\rho$, which allows us to explicitly construct optimal decompositions that achieve the extrema (\ref{eq:varroof}) and (\ref{eq:mindecomp}).
\subsection{Uncertainty equalities} For qubits, the angular momentum operator along the direction $\vec{n}\in\mathbb{R}^3$ can be written as $L_{\vec{n}}=\frac{1}{2}\vec{n}\cdot\vec{\sigma}$, where $\vec{\sigma} = (\sigma_x,\sigma_y,\sigma_z)^\intercal$ is a vector of Pauli matrices. Arbitrary quantum states are fully characterized by their Bloch vector $\vec{r}\in\mathbb{R}^3$ via $\rho(\vec{r}) = \frac{1}{2}(\mathbb{1}+\vec{r}\cdot\vec{\sigma})$. The QFI and variance of a spin-$1/2$ state $\rho$ can be determined analytically as a function of the Bloch vector $\vec{r}$. For a state $\rho(\theta)$ with Bloch vector $\vec{r}$ that depends in an arbitrary way on a parameter $\theta$, the QFI is given by \cite{bloch} \begin{equation} \label{eq:qfibloch}
F_Q[\rho(\theta)] =
\begin{cases}
\lvert \partial_\theta \vec{r}\rvert^2 + \frac{(\partial_\theta\vec{r} \cdot \vec{r})^2}{{1-\lvert \vec{r}\rvert}^2} & \text{if } \lvert\vec{r}\rvert < 1, \\
\lvert \partial_\theta \vec{r}\rvert^2 & \text{if } \lvert\vec{r}\rvert = 1.
\end{cases} \end{equation}
Assuming that the parameter $\theta$ is imprinted by a unitary evolution, generated by the operator $L_{\vec{n}}$, i.e., $i \partial_{\theta} \rho = [L_{\vec{n}},\rho]$, we obtain $\partial_{\theta} \vec{r} = \vec{n} \times \vec{r}$. Inserting this into Eq.~(\ref{eq:qfibloch}) yields \begin{equation}\label{eq:QFI_cross}
F_Q[\rho,L_{\vec{n}}] = \lvert \vec{n} \times \vec{r} \rvert^2 = |\vec{r}|^2 - (\vec{n} \cdot \vec{r})^2, \end{equation} and the orthonormality of $\{\vec{n}_1,\vec{n}_2,\vec{n}_3\}$ allows us to further write \begin{align}\label{eq:qfin} F_Q[\rho,L_{\vec{n}_1}]=(\vec{n}_2\cdot\vec{r})^2+(\vec{n}_3\cdot\vec{r})^2. \end{align} On the other hand, the variance \begin{align} \label{eq:varn} (\Delta L_{\vec{n}})_{\rho}^2 & = \frac{1 - (\vec{n} \cdot \vec{r})^2}{4} \end{align}
follows from $\langle L_{\vec{n}}\rangle_{\rho} = \frac{1}{2}\vec{n}\cdot\vec{r}$ and $\langle L_{\vec{n}}^2\rangle = \frac{1}{4}$, where we used $|\vec{n}|^2=1$. Finally, we recall that the purity of a qubit state is given by \begin{equation} \label{eq:purity}
\mathrm{Tr}\rho^2 = \frac{1}{2}(1+|\vec{r}|^2) = \frac{1}{2}\left[ 1+(\vec{n}_1\cdot\vec{r})^2+(\vec{n}_2\cdot\vec{r})^2+(\vec{n}_3\cdot\vec{r})^2 \right]. \end{equation} Combining Eqs.~(\ref{eq:qfin}), (\ref{eq:varn}), and (\ref{eq:purity}), we can immediately prove the following result: \begin{proposition}[Equalities for spin-$1/2$ systems] Any quantum state $\rho$ of a spin-$1/2$ system satisfies \begin{align} \label{eq:var1/2} (\Delta L_{\vec{n}_1})_\rho ^2 + (\Delta L_{\vec{n}_2})_\rho ^2 + (\Delta L_{\vec{n}_3})_\rho ^2 &= 1-\frac{1}{2} \mathrm{Tr}\rho^2 . \end{align} \end{proposition}
We note that the difference between the variance and the QFI for the qubit operator $L_{\vec{n}}$~\cite{puritybound} \begin{equation}\label{eq:diffqubit} (\Delta L_{\vec{n}})_{\rho}^2 - \frac{1}{4}F_Q[\rho,L_{\vec{n}}] = \frac{1}{2}(1-\mathrm{Tr}\rho^2) \end{equation} can also be easily obtained with the above expressions. Equation~(\ref{eq:diffqubit}) then allows us to express the condition~(\ref{eq:var1/2}) as equalities involving the QFI, i.e., \begin{align} \frac{1}{4}F_Q [\rho, L_{\vec{n}_1}] + (\Delta L_{\vec{n}_2})_\rho ^2 + (\Delta L_{\vec{n}_3})_\rho ^2 &= \frac{1}{2} \label{eq : qfivareq}, \\
\frac{1}{4}F_Q[\rho, L_{\vec{n}_1}] + \frac{1}{4}F_Q[\rho, L_{\vec{n}_2}] + (\Delta L_{\vec{n}_3})_\rho ^2 &= \frac{1}{2}\mathrm{Tr}\rho^2, \\
\frac{1}{4} F_Q [\rho, L_{\vec{n}_1}] + \frac{1}{4} F_Q[\rho, L_{\vec{n}_2}] + \frac{1}{4} F_Q[\rho, L_{\vec{n}_3}] &= \mathrm{Tr}\rho^2 - \frac{1}{2} \label{eq:qfisum}. \end{align}
\begin{figure}\label{purityplot}
\end{figure}
We observe that relation~(\ref{eq:qfisum3}) is saturated by all qubit states, as is revealed by Eq.~(\ref{eq : qfivareq}). We illustrate Eq.~(\ref{eq:qfisum}) in Fig.~\ref{purityplot}. We note also that the bounds \begin{eqnarray}
1-|\vec{r}|^2 \leq\: & 4(\Delta L_{\vec{n}})_{\rho}^2\:&\leq 1,\\
0 \leq\: & F_Q[\rho,L_{\vec{n}}] \:&\leq |\vec{r}|^2 \end{eqnarray} hold for any $\rho$ and can be saturated by a suitable $\vec{n}$. Using them together with~(\ref{eq:var1/2}) and~(\ref{eq:diffqubit}) then yields upper and lower bounds on the fluctuations of qubit states: \begin{align}
\frac{1}{2}-\frac{|\vec{r}|^2}{4} \leq && (\Delta L_{\vec{n}_1})_\rho ^2+ (\Delta L_{\vec{n}_2})_\rho ^2 \leq && \frac{1}{2} \label{eq:var_twosum} ,\\
\frac{1}{4} \leq && \frac{1}{4}F_Q[\rho, L_{\vec{n}_1}] + (\Delta L_{\vec{n}_2})_\rho ^2 \leq && \frac{1+|\vec{r}|^2}{4} ,\\
\frac{|\vec{r}|^2}{4} \leq && \frac{1}{4}F_Q[\rho, L_{\vec{n}_1}] + \frac{1}{4}F_Q[\rho, L_{\vec{n}_2}] \leq && \frac{|\vec{r}|^2}{2},\label{eq:QFI_twosum} \end{align}
and we recall that $0\leq |\vec{r}|^2\leq 1$ can be expressed as a function of the purity using~(\ref{eq:purity}). Note that Eq.~(\ref{eq:var_twosum}) can be interpreted as a generalization of the state-independent bound for $s=1/2$, Eq.~(\ref{eq:varsumreln2}), that takes additional information about purity into account. The lower bound takes its smallest value for a pure state, when Eq.~(\ref{eq:varsumreln2}) is recovered.
\subsection{Moment of inertia analogy} In the following, we establish an analogy between the QFI and the moment of inertia of a fictitious rigid body. Besides linking to the intuitive picture from classical mechanics, this analogy turns out to be useful for identifying optimal state decompositions. First, note that any decomposition $\{p_k, \ket{\Psi_k}\}$ of a mixed state $\rho$ into $n$ pure states can be geometrically represented in the Bloch sphere as the $n$ vertices of a polygon/polyhedron. Its vertices are given by the Bloch vectors $\vec{r}_k$ of the pure states $\ket{\Psi_k}$ on the surface of the sphere. For example, when there are only two elements, the decompositions \begin{equation} \label{eq:chord_decomp}
\rho(\vec{r}) = p\rho(\vec{r}_1) + (1-p)\rho(\vec{r}_2) \end{equation} represent chords with end points $\vec{r}_1$ and $\vec{r}_2$ that intersect $\vec{r} = p\vec{r}_1 + (1-p)\vec{r}_2$ (see Fig. \ref{bloch_1}). \begin{figure}
\caption{ Bloch sphere showing the qubit $\rho(\vec{r})$ (black dot). The red chord represents a possible decomposition, $\rho(\vec{r}) = p\rho(\vec{r}_1) + (1-p)\rho(\vec{r}_2)$.}
\label{bloch_1}
\end{figure}
Any such distribution $\{p_k, \ket{\Psi_k}\}$ of pure states can thus be interpreted as a rigid body $\{p_k, \vec{r}_k\}$ with point masses $p_k$, located at positions $\vec{r}_k$ that sum up to unity, with their center of mass at $\vec{r}$. When rotated around the axis $\vec{n}$, this body has a moment of inertia of \begin{align}\label{eq:Ipknk} I(\{p_k, \vec{r}_k\},\vec{n})=\sum_kp_k r_{k,\perp}^2, \end{align} where $r_{k,\perp}^2 = 1-(\vec{r}_k\cdot\vec{n})^2$ is the squared perpendicular distance between the point mass at $\vec{r}_k$ and the rotation axis $\vec{n}$. According to Eq.~(\ref{eq:QFI_cross}), $F_Q[\rho,L_{\vec{n}}]$ is the squared perpendicular distance between the axis $\vec{n}$ and the vector $\vec{r}$ in the Bloch sphere. One can thus interpret this as the moment of inertia of a unit mass located at $\vec{r}$. Moreover, this allows us to express Eq.~(\ref{eq:Ipknk}) as \begin{align}\label{eq:I_rigidbody}
I(\{p_k, \vec{r}_k\},\vec{n})=\sum_kp_k F_Q[|\Psi_k\rangle\langle\Psi_k|,L_{\vec{n}}]. \end{align}
We are now in a position to state an analog of the parallel axis theorem. Recall that in classical mechanics, this theorem allows us to determine the moment of inertia for rotations about an axis that is shifted by the distance $l$ from the center of mass. For a rigid body of unit mass, it reads \begin{align}\label{eq:pat} I(\{p_k, \vec{r}_k\},\vec{n}) - l^2= I_{\mathrm{cm}}(\{p_k, \vec{r}_k\},\vec{n}), \end{align} where $I_{\mathrm{cm}}(\{p_k, \vec{r}_k\},\vec{n})$ is the moment of inertia for a rotation about an axis parallel to $\vec{n}$ that passes through the center of mass $\vec{r}$ of the rigid body $\{p_k, \vec{r}_k\}$ and $l^2$ is the squared perpendicular distance between $\vec{n}$ and $\vec{r}$. Invoking the above analogy with the QFI, we obtain $l^2=F_Q[\rho,L_{\vec{n}}]$. Moreover, the squared perpendicular distance between $\vec{r}_k$ and the axis of rotation parallel to $\vec{n}$ passing through the center of mass is given by $\lvert \vec{n} \times \vec{r}'_k \rvert^2$, where $\vec{r}'_k \equiv \vec{r}_k - \vec{r}$, as can be verified from elementary geometric considerations, leading to $I_{\mathrm{cm}}(\{p_k, \vec{r}_k\},\vec{n})=\sum_k p_k \lvert \vec{n} \times \vec{r}'_k \rvert^2$. We thus obtain the following result for pure-state decompositions for the QFI, in direct correspondence with~(\ref{eq:pat}): \begin{proposition} [Parallel axis theorem]
For a mixed quantum state $\rho$ of a spin-$1/2$ system, its QFI, $F_Q [\rho, L_{\vec{n}}]$ is related to the average QFI of a decomposition $\{p_k, \ket{\Psi_k}\}$, $\sum_kp_k F_Q[|\Psi_k\rangle\langle\Psi_k|,L_{\vec{n}}]$, by \begin{equation} \label{eq:parallel_axis}
\sum_kp_k F_Q[|\Psi_k\rangle\langle\Psi_k|,L_{\vec{n}}] - F_Q[\rho,L_{\vec{n}}] = \sum_k p_k \lvert \vec{n} \times \vec{r}'_k \rvert^2, \end{equation} where we have introduced $\vec{r}'_k \equiv \vec{r}_k - \vec{r}$ as the separation vector connecting $\vec{r}$ to the Bloch vector $\vec{r}_k$ of $\ket{\Psi_k}$. \end{proposition} \begin{proof} The result is proven straightforwardly with Eq.~(\ref{eq:QFI_cross}), using the fact that $\sum_k p_k \vec{r}'_k = 0$ by construction. \end{proof}
We can also rewrite Eq.~(\ref{eq:parallel_axis}) in terms of variances with Eq.~(\ref{eq:diffqubit}) to yield \begin{align} \label{eq:parallel_axis_var}
(\Delta L_{\vec{n}})_{\rho}^2 - \sum_{k} p_k (\Delta L_{\vec{n}})_{\Psi_k} ^2 &= \frac{1}{4}[(1-|\vec{r}|^2)-\sum_k p_k \lvert \vec{n} \times \vec{r}'_k \rvert^2] \notag\\ & = \frac{1}{4}\sum_k p_k(\vec{n} \cdot \vec{r}'_k)^2. \end{align}
For any pure-state decomposition $\{p_k,|\Psi_k\rangle\}$ of $\rho$, the central inequality in~(\ref{eq:QFI<4Var}) is saturated. From Eqs.~(\ref{eq:parallel_axis}) and (\ref{eq:parallel_axis_var}) we can then quantify the deviation between the average pure-state variance and the lower and upper bounds in~(\ref{eq:QFI<4Var}), i.e., the QFI and variance of the mixed state, respectively.
Similarly, we can formulate a result analogous to the classical perpendicular axis theorem. If all $\vec{r}_k$ lie in a plane, spanned, e.g., by $\vec{n}_2$-$\vec{n}_3$, the perpendicular axis theorem states that: \begin{equation} \label{eq:perpendicular_axis_classical} I(\{p_k, \vec{r}_k\},\vec{n}_1) = I(\{p_k, \vec{r}_k\},\vec{n}_2) + I(\{p_k, \vec{r}_k\},\vec{n}_3), \end{equation} where we assumed that $\vec{n}_1$ is an axis perpendicular to the plane. In terms of the QFI, we have the following: \begin{proposition} [Perpendicular axis theorem] For a mixed quantum state $\rho$ of a spin-$1/2$ system, its QFI in orthogonal directions $\{\vec{n}_1, \vec{n}_2, \vec{n}_3\}$ are related by \begin{equation} \label{eq:perpendicular_axis} F_Q [\rho, L_{\vec{n}_1}] = F_Q [\rho, L_{\vec{n}_2}] + F_Q [\rho, L_{\vec{n}_3}] - 8\langle L_{\vec{n}_1} \rangle_{\rho}^2. \end{equation}
Furthermore, if the Bloch vectors of all $|\Psi_k\rangle$ of a decomposition $\{p_k,|\Psi_k\rangle\}$ of $\rho$ lie in a plane perpendicular to $\vec{n}_1$, then Eq.~(\ref{eq:perpendicular_axis_classical}) holds for the average Fisher information defined in Eq.~(\ref{eq:I_rigidbody}). \end{proposition}
\begin{proof} To show Eq.~(\ref{eq:perpendicular_axis}), note that from Eq.~(\ref{eq:qfin}), we have \begin{align}
F_Q[\rho,L_{\vec{n}_1}] &= (\vec{n}_2\cdot\vec{r})^2+(\vec{n}_3\cdot\vec{r})^2
\notag \\ & = \lvert \vec{n}_2 \times \vec{r} \rvert^2 + \lvert \vec{n}_3 \times \vec{r} \rvert^2 - 2(\vec{n}_1\cdot\vec{r})^2
\notag\\ & = F_Q [\rho, L_{\vec{n}_2}] + F_Q [\rho, L_{\vec{n}_3}] - 8\langle L_{\vec{n}_1} \rangle_{\rho}^2. \end{align}
If the Bloch vectors of all $|\Psi_k\rangle$ lie in the plane perpendicular to $\vec{n}_1$, we have $\langle L_{\vec{n}_1} \rangle_{\Psi_k} = 0$ and \begin{equation}
F_Q [|\Psi_k\rangle\langle\Psi_k|, L_{\vec{n}_1}] = F_Q [|\Psi_k\rangle\langle\Psi_k|, L_{\vec{n}_2}] + F_Q [|\Psi_k\rangle\langle\Psi_k|, L_{\vec{n}_3}]. \end{equation} With Eq.~(\ref{eq:I_rigidbody}) this leads to Eq.~(\ref{eq:perpendicular_axis_classical}). \end{proof}
\subsection{Construction of optimal decompositions and the properties of eigendecompositions}
We now discuss optimal decompositions, i.e., decompositions $\{p_k,|\Psi_k\rangle\}$ that achieve the minimum~(\ref{eq:mindecomp}) or maximum~(\ref{eq:varroof}) average variance. The parallel axis theorem~(\ref{eq:parallel_axis}) allows us to see that there is always a unique minimal decomposition and a set of maximal decompositions that are orthogonal to one another (the precise sense will be discussed below).
\begin{figure*}\label{subfigB2}
\label{subfigB3}
\end{figure*}
To construct the minimal decomposition, note that from Eq.~(\ref{eq:parallel_axis}), the average variance $\sum_kp_k F_Q[|\Psi_k\rangle\langle\Psi_k|,L_{\vec{n}}]$ reaches its minimum and hence achieves $F_Q[\rho,L_{\vec{n}}]$ when the term $\sum_k p_k \lvert \vec{n} \times \vec{r}'_k \rvert^2$ is zero. This can be uniquely achieved by choosing the two-element decomposition such that the separation vectors $\vec{r}'_k \equiv \vec{r}_k - \vec{r}$ are both parallel to $\vec{n}$, representing the chord parallel to $\vec{n}$. Explicitly, all two-element decompositions (\ref{eq:chord_decomp}) satisfy \begin{align} \vec{r}'_1 &= -\vec{r}'_2, \label{two-element1}\\
p &= \frac{|\vec{r}'_2|}{|\vec{r}'_1|+|\vec{r}'_2|},\\
|\vec{r}'_1| &= \sqrt{(\vec{r}'_1 \cdot \vec{r})^2 - |\vec{r}|^2 + 1} - (\vec{r}'_1 \cdot \vec{r}),\\
|\vec{r}'_2| &= \sqrt{(\vec{r}'_2 \cdot \vec{r})^2 - |\vec{r}|^2 + 1} + (\vec{r}'_2 \cdot \vec{r}) \label{two-element2}. \end{align} With the choice $\vec{r}'_1 = -\vec{r}'_2=\vec{n}$, the sum $\sum_k p_k \lvert \vec{n} \times \vec{r}'_k \rvert^2$ therefore vanishes. Any decomposition with more than two elements is nonoptimal since it will necessarily lead to nonzero terms in the sum.
To construct the maximal decompositions, from Eq.~(\ref{eq:parallel_axis_var}) we instead minimize $\sum_k p_k(\vec{n} \cdot \vec{r}'_k)^2$, which can be achieved by choosing the $\vec{r}'_k$ perpendicular to $\vec{n}$. This choice is no longer unique.
As examples, consider the operator $L_z$, with states $\rho_1(\vec{r}_1 = 0) = \frac{1}{2} \mathbb{1}$ and $\rho_2(\vec{r}_2 = \frac{1}{2}\vec{x})$ (Fig. \ref{subfigB2}, \ref{subfigB3}). The unique chord that reaches $F_Q[\rho,L_z]$ is parallel to $\vec{z}$, so $\rho_1 = \frac{1}{2}(\rho(\vec{z}) + \rho(-\vec{z}))$ and $\rho_2 = \frac{1}{2}\rho(\frac{1}{2}\vec{x} + \sqrt{\frac{3}{4}}\vec{z}) + \frac{1}{2}\rho(\frac{1}{2}\vec{x} - \sqrt{\frac{3}{4}}\vec{z})$ are the minimal decompositions. On the other hand, among two-element decompositions, the chords that reach $(\Delta L_{z})_{\rho}^2$ are not unique. Any chord that intersects both $\vec{r}$ and two points of the circle with an axis parallel to $\vec{n}$ reaches the variance.
\begin{figure}
\caption{The positive real number line showing the relative positions of the QFI, average variance of the eigendecomposition, and variance of a mixed qubit state $\rho$. The difference between the variance and QFI of $\rho$ is given by Eq.~(\ref{eq:diffqubit}) as $2(1-\mathrm{Tr}\rho^2)=1-|\vec{r}|^2$, and the average QFI of any decomposition of $\rho$ is located within this range. In particular, the average variance of the eigendecomposition of $\rho$ separates this range with the ratio $\lvert \vec{n} \times \vec{r} \rvert^2/(\vec{n} \cdot \vec{r})^2$.}
\label{numberline}
\end{figure}
The eigendecomposition of any qubit $\rho$ besides the maximally mixed state is a unique two-element decomposition. The following result shows how its average QFI is related to its optimal decompositions.
\begin{corollary}
Let $\rho$ be a nondegenerate mixed qubit state and $\{\lambda_k, \ket{\Phi_k}\}$ be its unique eigendecomposition. The average QFI of the eigendecomposition of $\rho$, $\sum_k \lambda_k F_Q[|\Phi_k\rangle\langle\Phi_k|,L_{\vec{n}}]$, is related to its QFI and variance by \begin{align}
4(\Delta L_{\vec{n}})_{\rho}^2 - 4\sum_{k} \lambda_k (\Delta L_{\vec{n}})_{\Phi_k} ^2 &= (1-|\vec{r}|^2)(\vec{n} \cdot \vec{r})^2\\
\sum_k \lambda_k F_Q[|\Phi_k\rangle\langle\Phi_k|,L_{\vec{n}}] - F_Q[\rho,L_{\vec{n}}] &= (1-|\vec{r}|^2)\lvert \vec{n} \times \vec{r} \rvert^2 \end{align} \end{corollary} \begin{proof} The eigendecomposition of $\rho$ is the two-element decomposition where both $\vec{n}_i$ and $\vec{n}'_i$ are parallel to $\vec{r}$. Applying Eqs.~(\ref{two-element1})--(\ref{two-element2}) to explicitly determine the differences~(\ref{eq:parallel_axis}) and~(\ref{eq:parallel_axis_var}) yields the desired results. \end{proof}
The relation between the eigendecomposition and the optimal decompositions is summarized in Fig.~\ref{numberline}. Additionally, one can also choose $\vec{r}$ or $\vec{n}$ such that the eigendecomposition of $\rho$ corresponds to its minimal or maximal decomposition, namely, by requiring that $\lvert \vec{n} \times \vec{r} \rvert = 0$ or $(\vec{n} \cdot \vec{r}) = 0$, respectively.
Finally, we can relate the minimal decomposition in one direction with the maximal decomposition in directions that are perpendicular to it due to the following result. \begin{corollary} Let $\rho$ be a mixed qubit state. A decomposition $\{p_k, \ket{\Psi_k}\}$ of $\rho$ minimizes the convex sum of variances in one direction, $\frac{1}{4}F_Q[\rho,\Delta L_{\vec{n}_1}]=\sum_{k} p_k (\Delta L_{\vec{n}_1})_{\Psi_k} ^2$, if and only if it maximizes the same sum in both the other two orthogonal directions, $(\Delta L_{\vec{n}_2})_{\rho}^2=\sum_{k} p_k (\Delta L_{\vec{n}_2})_{\Psi_k} ^2$ and $(\Delta L_{\vec{n}_3})_{\rho}^2=\sum_{k} p_k (\Delta L_{\vec{n}_3})_{\Psi_k} ^2$.
\end{corollary} \begin{proof} Let $\{p_k, \ket{\Psi_k}\}$ be the decomposition of $\rho$ that achieves the minimum in Eq.~(\ref{eq:mindecomp}) for the operator $L_{\vec{n}_1}$. For each $\ket{\Psi_k}$, we obtain from Eq.~(\ref{eq:var1/2}) \begin{equation} \label{eq:puresum}
(\Delta L_{\vec{n}_1})_{\ket{\Psi_k}} ^2 + (\Delta L_{\vec{n}_2})_{\ket{\Psi_k}} ^2 + (\Delta L_{\vec{n}_3})_{\ket{\Psi_k}} ^2 = \frac{1}{2}. \end{equation} Multiplying Eq.~(\ref{eq:puresum}) by $p_k$ and summing over $k$, we obtain \begin{equation} \label{eq:puresum2} \begin{split}
\frac{1}{4}F_Q[\rho, L_{\vec{n}_1}] + \sum_{k} p_k (\Delta L_{\vec{n}_2})_{\ket{\Psi_k}} ^2 + \sum_{k} p_k (\Delta L_{\vec{n}_3})_{\ket{\Psi_k}} ^2 = \frac{1}{2}.
\end{split} \end{equation} Taking the difference between Eqs.~(\ref{eq : qfivareq}) and (\ref{eq:puresum2}) yields \begin{equation}
[(\Delta L_{\vec{n}_2})_\rho ^2 - \sum_{k} p_k (\Delta L_{\vec{n}_2})_{\ket{\Psi_k}} ^2]+ [(\Delta L_{\vec{n}_3})_\rho ^2 - \sum_{k} p_k (\Delta L_{\vec{n}_3})_{\ket{\Psi_k}} ^2]= 0. \end{equation} Since the two terms in brackets are positive due to the concavity of the variance, they vanish separately, proving the result.
For the reverse direction, we suppose there exists a decomposition that achieves the maximum of Eq.~(\ref{eq:varroof}) for both $L_{\vec{n}_1}$ and $L_{\vec{n}_2}$. Following steps analogous to those used before, we can write \begin{equation} \begin{split}
\sum_{k} p_k (\Delta L_{\vec{n}_1})_{\ket{\Psi_k}} ^2 + (\Delta L_{\vec{n}_2})_{\rho} ^2 + (\Delta L_{\vec{n}_3})_{\rho} ^2 = \frac{1}{2},
\end{split} \end{equation} and taking the difference with Eq.~(\ref{eq : qfivareq}) now yields \begin{equation}
\sum_{k} p_k (\Delta L_{\vec{n}_1})_{\ket{\Psi_k}} ^2 = \frac{1}{4}F_Q[\rho, L_{\vec{n}_1}], \end{equation} proving the statement. \end{proof}
\section{Quantum and classical sensitivity limits for phase estimation with an unknown axis} In the context of quantum metrology, the quantum Cram\'{e}r-Rao bound (\ref{eq:CRB}) allows us to interpret the QFI as the quantum limit on the precision that can be achieved for an estimation of the parameter $\theta$, generated by $A$, using the state $\rho$~\cite{Pezze_QTOPE}. This limit can be attained by an optimal choice of the measurement observable and the estimator~\cite{Braunstein&Caves}. A suitable preparation of the probe state $\rho$ can additionally improve the sensitivity up to the ultimate quantum limit~\cite{GiovannettiPRL2006}, where increasing sensitivity typically demands larger amounts of multipartite entanglement~\cite{PS09,HyllusPRA2012,TothPRA2012,RenPRL2021,T_th_2014,Pezze_QTOPE}.
The choice of the optimal quantum state $\rho$ that maximizes $F_Q[\rho,A]$, however, depends on the precise knowledge of the phase-imprinting generator $A$. Suppose that $\theta$ describes a phase shift generated by $L_{\vec{n}}$, but the direction $\vec{n}$ of the rotation is unknown at the moment where the state $\rho$ is prepared. One approach in such a situation, would be to maximize the average sensitivity~\cite{BartlettRMP2007,TothPRA2012,LiPRA2013,platonic,GoldbergPRA2018,YadinNatCommun2021,GoldbergJPPhot2021,GoldbergPRL2021}. With the convexity~(\ref{eq:QFI<4Var}) and additivity~(\ref{eq:addQFI}) properties of the QFI, sums containing only the variance or the QFI such as Eqs.~(\ref{eq:var1/2}), (\ref{eq:qfisum}), (\ref{eq:var_twosum}), and (\ref{eq:QFI_twosum}) directly imply bounds on the average sensitivity of $N$-qubit separable states that were first derived in Ref.~\cite{TothPRA2012}; see also Ref.~\cite{Hofmann} for similar methods based on the average variance. Here, we focus on an alternative figure of merit, given by the ``worst-case", i.e., the minimal sensitivity that can be achieved in all possible directions; see also Refs.~\cite{GirolamiPRL2014,WolfNatCommun2019} for similar approaches. In the following, we focus on $N$-qubit systems with collective spin $N/2$ and use our results from the previous sections to identify the ultimate classical and quantum limits on the minimal sensitivity as well as the respective optimal quantum states that achieve them.
Formally, we define these limits as \begin{equation} B(\mathcal{R},\Omega) := \max_{\rho\in\mathcal{R}}\min_{\vec{n} \in \Omega} F_Q[\rho,L_{\vec{n}}], \end{equation} where $\mathcal{R}$ is a set of quantum states and $\Omega$ describes the set of possible rotation axes. We are interested in the situations in which the rotation axis $\vec{n}$ is limited to a plane, i.e., $\Omega=\mathbb{R}^2$, or it can be chosen arbitrarily in three dimensions, $\Omega=\mathbb{R}^3$. The quantum limit corresponds to the unconstrained maximization over $\mathcal{R}=\mathcal{S}$, where $\mathcal{S}$ is the set of all quantum states. The classical limit is obtained by maximizing only over the class of separable states $\mathcal{R}=\mathcal{S}_{\mathrm{sep}}$.
Identifying the classical bounds on measurable properties of quantum states, such as their sensitivity or fluctuations, naturally leads to entanglement witnesses, see Refs.~\cite{PS09,HyllusPRA2012,TothPRA2012,RenPRL2021,T_th_2014,Pezze_QTOPE,GuehneToth,squeezing_ent,Hofmann,LiPRA2013} for other examples following this approach. By construction, any state whose properties violate this bound must necessarily be entangled.
We first focus on the quantum limit for $\Omega=\mathbb{R}^2$. We make use of basic properties of the quantum Fisher matrix (see, e.g., ~\cite{PhysRevA.82.012337,GessnerPRL2018}), i.e., $F_Q[\rho,L_{\vec{n}}] = \vec{n}^\intercal \mathbf{F}[\rho,\vec{L}] \vec{n}$ and $\mathbf{F}[\rho,\vec{L}] \leq 4\mathbf{\Gamma}[\rho,\vec{L}]$, where $\vec{L}=\{L_x,L_y,L_z\}^\intercal$ is a vector of angular momentum operators and we introduced the covariance matrix with elements $\mathbf{\Gamma}[\rho,\vec{L}]_{ij} = \frac{1}{2} \left\langle L_i L_j + L_j L_i \right\rangle_\rho - \langle L_i \rangle_\rho \langle L_j \rangle_\rho$, where $i,j \in \{x,y,z\}$.
Without loss of generality, we consider $\Omega$ to be the $xy$ plane of $\mathbb{R}^3$. The minimal sensitivity corresponds to the smallest eigenvalue of the corresponding $2\times 2$ block of $\mathbf{F}[\rho,\vec{L}]$. We obtain \begin{align}\label{eq:optim} \min_{\vec{n} \in \mathbb{R}^2} F_Q[\rho,L_{\vec{n}}] & = \frac{1}{2} \bigg( F_Q[\rho, L_x] + F_Q[\rho, L_y] \notag\\&\qquad- \sqrt{(F_Q[\rho, L_x]-F_Q[\rho, L_y])^2 + 4\mathbf{F}[\rho,\vec{L}]_{xy}^2 } \bigg) \notag \\ & \stackrel{\mathrm{(i)}}{\leq} 2\bigg((\Delta L_x)_{\rho}^2 + (\Delta L_y)_{\rho}^2 \notag\\&\qquad- \sqrt{(\Delta L_x)_{\rho}^2-(\Delta L_y)_{\rho}^2+4\mathbf{\Gamma}[\rho,\vec{L}]_{xy}^2} \bigg) \notag\\
& \stackrel{\mathrm{(ii)}}{\leq} 2[(\Delta L_x)_{\rho} ^2 + (\Delta L_y)_{\rho} ^2]\notag \\
& \stackrel{\mathrm{(iii)}}{\leq} 2 \left( \langle L_x^2 \rangle_{\rho} + \langle L_y^2 \rangle_{\rho} \right)\notag \\
& = 2 \left(\frac{N(N+2)}{4} - \langle L_z^2 \rangle_{\rho} \right)\notag \\
& \stackrel{\mathrm{(iv)}}{\leq} \frac{N(N+2)}{2}. \end{align} An optimal state that achieves this limit must saturate all inequalities in this derivation. This can be achieved by a state that (i) is pure, (ii) has zero covariances and equal variances for $L_x$ and $L_y$, (iii) has zero expectation values for $L_x$ and $L_y$, and (iv) has zero expectation values for $\langle L_z^2 \rangle_{\rho}$. When $N$ is even, a state that unites all of these conditions is the so-called twin-Fock state, i.e., an eigenstate of $L_z$ with zero eigenvalue or, equivalently, a Dicke state containing the same number of spin-up and spin-down particles.
If the state is assumed to be separable, i.e., $\rho_{\mathrm{sep}}=\sum_{\gamma}p_{\gamma}\rho_{\gamma}^{(1)}\otimes\cdots\otimes\rho_{\gamma}^{(N)}$, where $p_{\gamma}$ is a probability distribution and $\rho_{\gamma}^{(i)}$ are local quantum states for the $i$th particle, we can make use of the convexity~(\ref{eq:QFI<4Var}) and additivity~(\ref{eq:addQFI}) properties of the QFI. Moreover, the angular momentum observables may be decomposed as $L_{\vec{n}}=\sum_{i=1}^NL_{\vec{n}}^{(i)}$, where $L_{\vec{n}}^{(i)}=\frac{1}{2}\vec{n}\cdot\vec{\sigma}^{(i)}$ acts on the $i$th qubit. We obtain the limit: \begin{align} \label{eq:optim2} \min_{\vec{n} \in \mathbb{R}^2} F_Q[\rho_{\mathrm{sep}},L_{\vec{n}}]
& \stackrel{\mathrm{(i)}}{\leq} \frac{1}{2} (F_Q[\rho_{\mathrm{sep}}, L_x] + F_Q[\rho_{\mathrm{sep}}, L_y])\notag \\
& \stackrel{\mathrm{(ii)}}{\leq} \frac{1}{2} \sum_{\gamma} p_{\gamma} (F_Q[\rho_{\gamma}^{(1)}\otimes\cdots\otimes\rho_{\gamma}^{(N)}, L_x]\notag\\
&\hspace{1.5cm}+ F_Q[\rho_{\gamma}^{(1)}\otimes\cdots\otimes\rho_{\gamma}^{(N)}, L_y]) \notag\\
& = \frac{1}{2} \sum_{\gamma} p_{\gamma} \sum\limits_{i=1}^{N} (F_Q[\rho_{\gamma}^{(i)}, L_x^{(i)}] + F_Q[\rho_{\gamma}^{(i)}, L_y^{(i)}]) \notag\\
& \stackrel{\mathrm{(iii)}}{\leq} \sum_{\gamma} p_{\gamma} \sum\limits_{i=1}^{N} \left[2\mathrm{Tr}\{(\rho_{\gamma}^{(i)})^2\}-1\right]\notag\\
& \stackrel{\mathrm{(iv)}}{\leq} \sum_{\gamma} p_{\gamma} \sum\limits_{i=1}^{N} 1\notag\\
& = N, \end{align} where in (iii) we used Eq.~(\ref{eq:QFI_twosum}). All the above inequalities are saturated by a pure product state [(ii) and (iv)], with a diagonal QFI matrix of equal $x$ and $y$ diagonal elements (i), and maximum variance in both the $x$ and $y$ directions (iii). Such states are given by eigenstates of $L_z$ with the extremal eigenvalue $\pm N/2$, and they correspond to products of $N$ identical qubit states, each one polarized along the $\pm z$ direction. We note that in the proof, Eq.~(\ref{eq:varsumreln2}) for variances can be used in place of Eq.~(\ref{eq:QFI_twosum}) in inequality (iii) to obtain the same result.
Next, we extend $\Omega$ to the entire $\mathbb{R}^3$. Since the minimal eigenvalue is bounded from above by the average eigenvalue, we obtain \begin{align}\label{eq:minFR3} \min_{\vec{n} \in \mathbb{R}^3} F_Q[\rho,L_{\vec{n}}] & \leq \frac{1}{3} \mathrm{Tr}\mathbf{F}[\rho,\vec{L}] \notag\\ &\leq\frac{4}{3} \mathrm{Tr}\mathbf{\Gamma}[\rho,\vec{L}] \notag\\ &\leq \frac{4}{3} (\langle L_x^2\rangle_{\rho}+\langle L_y^2\rangle_{\rho} +\langle L_z^2\rangle_{\rho} ) \notag\\
& =\frac{N(N+2)}{3}. \end{align} This bound is achieved by pure states that have a diagonal covariance matrix with zero first moments and equal second moments for all three angular momentum observables. These conditions are satisfied by so-called anti-coherent states~\cite{Zimba}, which are known to optimize the average sensitivity in all three directions~\cite{platonic}, and the average sensitivity of Euler angles~\cite{GoldbergPRA2018}. Note that due to the saturation of the first inequality by these optimal states, Eq.~(\ref{eq:minFR3}) is equivalent to the average sensitivity that was considered in Refs.~\cite{platonic,TothPRA2012}. This bound for the average QFI is also saturated by Greenberger-Horne-Zeilinger states and Dicke states~\cite{TothPRA2012}, but these states do not satisfy the symmetry requirements under the exchange of axes to saturate also the first inequality in~(\ref{eq:minFR3}).
For separable states, we obtain, following steps analogous to the ones used before, the classical limit \begin{align} &\quad\min_{\vec{n} \in \mathbb{R}^3} F_Q[\rho_{\mathrm{sep}},L_{\vec{n}}] \notag\\
&\leq\frac{1}{3} \sum_{\gamma} p_{\gamma} \sum\limits_{i=1}^{N} (F_Q[\rho_{\gamma}^{(i)}, L_x^{(i)}] + F_Q[\rho_{\gamma}^{(i)}, L_y^{(i)}]+F_Q[\rho_{\gamma}^{(i)}, L_z^{(i)}]) \notag\\
& = \frac{1}{3} \sum_{k} p_{\gamma} \sum\limits_{i=1}^{N} \left[ 4\mathrm{Tr}\{(\rho_{\gamma}^{(i)})^2\} - 2\right]\notag\\
&\leq \frac{2}{3}N. \end{align} where we have used Eq.~(\ref{eq:qfisum}) in the second step. The bound is saturated by pure product states with a diagonal covariance matrix and equal variances in all three directions. Again, the same bound can be obtained from the variance bound~(\ref{eq:varsumreln3}). This bound coincides with the classical limit on the average sensitivity that was derived in Ref.~\cite{TothPRA2012}.
To find the states that fulfill these constraints, it is instructive to first express the elements of the covariance matrix in terms of the Bloch vectors of the individual qubits $\vec{r_k}=(r^{(k)}_x, r^{(k)}_y, r^{(k)}_z)^\intercal$, which yields $\mathbf{\Gamma}[\rho,\vec{L}]_{ii} = \frac{1}{4} (1 - \sum\limits_{k=1}^{N} (r^{(k)}_i)^2)$ on the diagonal and $\mathbf{\Gamma}[\rho,\vec{L}]_{ij} = - \frac{1}{4} \sum\limits_{k=1}^{N} r^{(k)}_i r^{(k)}_j$ for $i\neq j$. The constraints can then be written in terms of the individual Bloch vectors as \begin{align} (r^{(k)}_x)^2 + (r^{(k)}_y)^2 + (r^{(k)}_z)^2 &= 1 \qquad\text{$\forall k$},\\ \sum\limits_{k=1}^{N} (r^{(k)}_x)^2 = \sum\limits_{k=1}^{N} (r^{(k)}_y)^2 &= \sum\limits_{k=1}^{N} (r^{(k)}_z)^2, \\ \sum\limits_{k=1}^{N} r^{(k)}_x r^{(k)}_y = \sum\limits_{k=1}^{N} r^{(k)}_x r^{(k)}_z = \sum\limits_{k=1}^{N} r^{(k)}_y r^{(k)}_z&=0. \end{align} These constraints can be satisfied only for $N>2$. For example, if $N$ is a multiple of 3, a state that saturates this bound is the product of the states with Bloch vectors $(1,0,0)^\intercal$, $(0,1,0)^\intercal$, and $(0,0,1)^\intercal$, repeated $N/3$ times. If $N$ is a multiple of 4, we take the product of the states with Bloch vectors $(-1,1,1)^\intercal/\sqrt{3}$, $(1,-1,1)^\intercal/\sqrt{3}$, $(1,1,-1)^\intercal/\sqrt{3}$, and $(1,1,1)^\intercal/\sqrt{3}$, repeated $N/4$ times.
In summary, we observe that \begin{align} B(\mathcal{S},\mathbb{R}^2)&=\frac{N}{2}(N+2),\\ B(\mathcal{S}_{\mathrm{sep}},\mathbb{R}^2)&=N, \label{b2_sep}\\ B(\mathcal{S},\mathbb{R}^3)&=\frac{N}{3}(N+2), \label{b3_all}\\ B(\mathcal{S}_{\mathrm{sep}},\mathbb{R}^3)&=\frac{2}{3}N. \label{b3_sep} \end{align} Previous studies have focused on the average sensitivity~\cite{platonic,TothPRA2012,LiPRA2013,GoldbergPRA2018}, which always implies bounds on the minimum. In particular, the upper bounds for the minimum sensitivity~(\ref{b3_all}) and (\ref{b3_sep}) follow directly from known bounds on the average sensitivity~\cite{TothPRA2012}. However, saturability of the minimum bounds is less clear, and the above derivations explicitly outline the conditions for states that saturate them. Notably, states that saturate the bound for the average sensitivity need not saturate the bound for the minimum, as they can be asymmetric, with unequal variances in different directions.
Finally, the bound~(\ref{b2_sep}) can be directly obtained by applying the separable limit $F_Q[\rho_{\mathrm{sep}}, L_{\vec{n}}] \leq N$ \cite{PS09} to both directions in the plane individually, i.e., $\min_{\vec{n} \in \mathbb{R}^2} F_Q[\rho_{\mathrm{sep}},L_{\vec{n}}] \leq \frac{1}{2} (F_Q[\rho_{\mathrm{sep}}, L_x] + F_Q[\rho_{\mathrm{sep}}, L_y]) \leq N$. Our derivation confirms that this bound is indeed tight. Note that the same procedure for rotations in all three directions yields the weaker bound of $\min_{\vec{n} \in \mathbb{R}^3} F_Q[\rho_{\mathrm{sep}},L_{\vec{n}}] \leq N$ compared to the tight bound~(\ref{b3_sep}).
\section{Conclusions} The convex-roof property~(\ref{eq:mindecomp}) allows us to interpret the quantum Fisher information as the minimal average variance of all pure-state decompositions. This gives rise to an alternative measure of quantum fluctuations for mixed states, constructed in a way that the contribution of the classical uncertainty about the state due to the mixing of pure states, is minimal. In contrast, the standard variance of a mixed state maximizes this contribution, as is shown by its concave-roof property~(\ref{eq:varroof}).
Employing the quantum Fisher information as a measure of quantum fluctuations leads to state-independent preparation uncertainty relations that are sharper than relations involving only variances. For the case of a spin-$1/2$ particle, we provided exact equalities that express the deviation from the minimal uncertainty in terms of the purity. Using the geometry of the Bloch sphere, we further demonstrated how extremal pure-state decompositions can be constructed and interpreted, revealing an analogy with the classical moment of inertia of rigid bodies. Finally, we used these results to derive the classical and quantum limits on the estimation of a phase parameter generated by an unknown rotation axis.
\section*{Note added} Independently of our work, the convex-roof property of the QFI was used to derive uncertainty relations by T\'oth and Fr\"owis~\cite{TothFrowis}.
\begin{acknowledgments}
We thank A. Smerzi and R. F. Werner for stimulating discussions. This work was funded by the LabEx ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*. This work received funding from Ministerio de Ciencia e Innovaci\'{o}n (MCIN) /
Agencia Estatal de Investigaci\'on (AEI) for Project No. PID2020-115761RJ-I00 and support of a fellowship from ``la Caixa” Foundation (ID 100010434) and from the European Union’s Horizon 2020 research and innovation program under Marie Sk\l{}odowska-Curie Grant Agreement No. 847648, fellowship code LCF/BQ/PI21/11830025. \end{acknowledgments}
\end{document} | arXiv |
\begin{definition}[Definition:Integral Operator]
Let $X, Y$ be sets of mappings, real or complex numbers.
Let $A : X \to Y$ be a mapping such that $A$ is expressible as an integral of elements of $X$ (possibly involving weight functions and kernels).
Then $A$ is known as an '''integral operator'''.
Category:Definitions/Operator Theory
\end{definition} | ProofWiki |
A simple structural estimator of disclosure costs
E. Cheynel ORCID: orcid.org/0000-0001-5763-32531 &
M. Liu-Watts2
Review of Accounting Studies (2020)Cite this article
This study recovers a simple firm-level measure of disclosure costs implied by the voluntary disclosure theory of Verrecchia (Journal of Accounting and Economics 12(4), 365–380, 1990). The measure does not require knowledge by the researcher of the distribution of private information and can be implemented with three simple observable inputs: the minimum, average, and frequency of disclosure. We document a positive association of disclosure costs with proxies for existing and potential competition, information asymmetry, and insider trading. Higher values of disclosure costs are associated with lower contemporaneous and future disclosures as well as lower propensity to disclose in holdout samples. Overall, we provide future researchers with an easy-to-implement procedure to structurally estimate unobserved firm-level disclosure costs.
Following Verrecchia (1990), we have left aside considerations of risk-aversion modeled in the original Verrecchia (1983) model; that is, we assume that the set of potential investors is large, and the disclosure is about a diversifiable component such that there is risk-neutral pricing of the disclosed value (Cheynel 2013). For a more comprehensive empirical analysis of settings with a small investor base, see Armstrong et al. (2011). Note that investor risk-aversion increases the willingness to disclose for a given cost, since markets discount more risky non-disclosure. Therefore risk-aversion would directionally predict higher disclosure costs than those under risk-neutrality.
In other words, if the market price is \(P(\mathcal {I})= \alpha \mathbb {E}(e|\mathcal {I})\), estimated variables can be rescaled by α, which can be estimated empirically. We omit α since it plays no further role in the estimation.
Within product market theories, information may be proprietary, and thus c is positive because disclosure will benefit existing competitors (e.g., Verrecchia 1983; Dye 1986). At the same time, depending on the effect of disclosure on the aggressiveness of potential competitors, disclosure can entail some benefits (e.g., Darrough and Stoughton 1990). Disclosure may mitigate agency costs stemming from diverging interests between owners and managers and affect investment efficiency (e.g., Stocken and Verrecchia 2004; Liang and Wen 2007). Or disclosure may reduce the risk of a shareholder lawsuit (e.g., Skinner 1994; Skinner 1997) or reduce information asymmetry (e.g., Diamond and Verrecchia 1991). Other disclosure costs can take the form of processing, communicating and producing the information to outsiders as well as psychological costs and reputational concerns. The cost c is positive in aggregate.
The model is laid out as if the manager paid the cost directly, mainly to avoid having to redefine the price equation to include the costs and burdening the exposition.
Our cost estimation approach still works even if there are multiple equilibria as long as the same equilibrium is implemented during the sample period. More assumptions are typically needed to guarantee the existence of a unique equilibrium. For example, if x is logconcave with sub-exponential lower tail, there exists a unique interior equilibrium. The proof of uniqueness follows immediately from the definition of logconcavity. Without sub-exponential lower-tail, there may be equilibria featuring unravelling even if the cost is non-zero (e.g., Laplace distributions); we refer to Bertomeu and Cheynel (2018) for a proof of a unique interior equilibrium as long as the lower tail becomes small at a rate greater than exponential rate. As a special case, this is always true if the distribution is bounded from below. Note that, strictly speaking, we can lift the requirement of logconcavity and sub-exponential tail as long as we assume that the entire sample is generated from players coordinating on one interior equilibrium.
The validity of your approach is contingent upon the equilibrium being characterized by a threshold and some relations between the cost and x could undermine that equilibrium characterization.
We can approximate \(\mathbb {E}(x|x\leq \tau )\) by computing the average earnings surprise for firm quarters for which there is no disclosure in the data, but this estimation procedure entails many caveats. First, it assumes that the private information is formulated in terms of posterior expectations, such that \(\mathbb {E}(e|ND)=\mathbb {E}(x|ND)\) by the law of iterated expectations. Second, empirically, because we do not know when the manager receives his information, we cannot measure accurately the consensus and need to make choices when to pick the consensus, which is likely to add noise in the estimation. The estimation procedure that we adopt is not sensitive to these choices. We express \(\mathbb {E}(x|ND)=-\frac {p}{1-p}\mathbb {E}(x|x>\tau )\) such that both p and \(\mathbb {E}(x|x>\tau )\) can be measured directly in the data.
In Appendix A, we derive the asymptotic variance of the BBT and NP estimators.
While our empirical analyses will focus on the NP estimator, we show in Appendix B that the model can serve as a starting point for richer settings; we also discuss how to modify (or sometimes reinterpret) the analysis in three plausible alternative settings. Specifically, our NP estimator derived under pure disclosure costs continues to hold after including other disclosure frictions. For example, it is robust to a probability that the manager does not receive private information or to a probability that the manager receives some information that she would like to disclose, but she cannot convey it credibly, as in Dye (1985). It can also incorporate an exogenous probability that forces the firm to disclose or a probability that the manager does not care about maximizing the price and never discloses.
Everything else held equal, a greater frequency \(\hat {p}\) implies a higher proprietary cost. This last property is seemingly counterintuitive, that is, one might expect a firm that discloses more often to have low proprietary costs. To see why this occurs, consider the case when c is small; then \(\hat {p}/(1-\hat {p})\) becomes large but, at the same time, the average forecast surprise \(\mathbb {E}(x|x>\tau )\rightarrow \mathbb {E}(x)=0\). The two effects offset each other; a high forecast frequency, controlling for the forecast surprise, tends to indicate a higher cost. The frequency effect on the NP-cost estimator stands in contrast to the effect on the BBT estimator. This observation illustrates that a negative correlation between the NP-cost estimator and disclosure is not by construction but will hold under the assumptions of the theoretical model.
Einhorn (2007) develops a more general voluntary disclosure model, where the manager might either maximize or minimize the price at a cost if he decides to disclose. In equilibrium, low and high outcomes are disclosed, and intermediate earnings are never disclosed. However, this intermediate non-disclosure region does not appear to be consistent with the observed management forecasts.
The cost here is a personal cost, in line with the presentation used in the main analysis. The manager solely minimizes the perception of the price similarly to Einhorn (2007). However, we could also assume that the manager minimizes the total cash flows, i.e., subtracting the cost from price via a decrease in future earnings, and this alternative formulation would result in minor changes.
Note that this effect is not driven by small-sample deviations from asymptotic theory, as the asymptotic standard-error of the estimator increases when trying to estimate small costs as shown in Appendix A.
We select this period for two reasons. First, the implementation of Regulation Fair Disclosure (Reg. FD) in the United States closed private channels of communication to analysts and thus greatly increased the number of forecasting firms for reasons unrelated to disclosure costs. Second, in previous years, management forecasts were not systematically collected by the First Call Company Issued Guidance (CIG) database, but after 2003, the requirement by Sarbanes-Oxley to record transcripts of conference calls greatly improved forecast archives. Prior to 2003, many forecasts were made during unrecorded conference calls, thus leading to systematic omitted forecast data for smaller firms that are less likely to trigger follow-up press releases.
Because management forecasts of EPS are typically adjusted, we use the ratio of unadjusted to adjusted EPS to convert these forecasts to raw forecasts. Adjusted forecasts are problematic because, for firms that had stock splits, the variance of adjusted forecasts will decline over time. In cases of zero adjusted EPS (which can occur because earnings are zero or because of two-digit rounding given very large splits), we use the nearest available adjustment factor, thereby dropping observations that have no adjustment factor.
Since CIG reports adjusted EPS forecasts, we recover unadjusted forecasts using the adjustment factor, i.e., the ratio of unadjusted to adjusted EPS, to convert these forecasts to raw forecasts.
We can make no theoretical predictions about the correct variable to measure and scale surprises. For example, in the context of management earnings forecasts, one might use any variable capturing market change in expectations, such as short-window market response (Kasznik and Lev 1995), earnings per share (Cheong and Thomas 2011), or earnings surprise scaled by lagged assets or prices.
Li (2010) classifies MKTS into two categories as a measure of existing competition as well as potential rivals. Firms generating high sales may operate in environments where the number of existing rivals is larger, and they may avoid disclosing information that competitors might use to their advantage.
We define high litigation industries as those with SIC code 2833-2836, 8731-8734 (biotech), 3570-3577 (computer hardware), 3600-3674 (electronics), 7371-7379 (computer software), 5200-5961 (retail), 4812-4813, 4833, 4841, 4899 (communications), or 4911, 4922-4924, 4931, 4941 (utilities), as defined by Ajinkya et al. (2005).
While we conduct our analysis using both logit and OLS regressions, we tabulate the OLS regressions, because these are less sensitive to the inclusion of fixed effects compared to logit and with marginal effects that are simpler to interpret (Angrist and Pischke 2008).
Our results are qualitatively similar if the holdout period consists only of the firm-quarters in 2016.
The four-digit Standard Industrial Classification (SIC) codes used by government agencies to classify industry areas remain quite popular, but this is being supplemented by the six-digit North American Industry Classification System (NAICS) codes. The Global Industry Classifications Standard (GICS) system that has been jointly developed by Standard & Poor's and Morgan Stanley Capital International (MSCI) is popular among financial practitioners, whereas the 48 Fama and French classification tends to be more popular in academic research.
Not all of the 48 industries are tabulated because some of them did not contain the minimum requirement of five firms.
In untabulated results, we examine the correlations between \(\hat {c}_{NPALT}\) and the competition variables. This measure \(\hat {c}_{NPALT}\) correlates with proxies capturing competition from potential rivals, with the exception of R&D. The evidence is more mixed with measures of existing competition. The alternative measure correlates positively with the HHI variable, whereas the correlation with NUM is insignificant. Lastly, the correlation is negative with CAPX, consistent with the existence of barriers to entry.
Federal laws govern insider trading in the United States, and several pieces of legislation concerning insider trading include the following: the Securities Act of 1933, the Securities and Exchange Act of 1934, and the Sarbanes-Oxley Act of 2002. On August 10, 2000, the Securities and Exchange Commission (SEC) adopted Rules 10b5-1 and 10b5-2 that clarify certain principles of insider trading while simultaneously announcing the adoption of Regulation FD (Fair Disclosure), which prohibits public companies from selectively disclosing information.
In untabulated tests, we also include lagged insider trading to control for unobservable factors that affect insider trading. The results are qualitatively similar.
In fact, this equation is exactly the same as in Jung and Kwon (1988), subtracting the cost from the disclosing firm price, i.e., changing τ into τ − c.
Ajinkya, B., Bhojraj, S., Sengupta, P. (2005). The association between outside directors, institutional investors and the properties of management earnings forecasts. Journal of Accounting Research, 43(3), 343–376.
Angrist, J.D., & Pischke, J.S. (2008). Mostly harmless econometrics: an empiricist's companion. Princeton: Princeton University Press.
Anilowski, C., Feng, M., Skinner, D.J. (2007). Does earnings guidance affect market returns? the nature and information content of aggregate earnings guidance. Journal of Accounting and Economics, 44(1), 36–63.
Armstrong, C.S., Core, J.E., Taylor, D.J., Verrecchia, R.E. (2011). When does information asymmetry affect the cost of capital? Journal of Accounting Research, 49(1), 1–40.
Berger, P.G. (2011). Challenges and opportunities in disclosure research - A discussion of the financial reporting environment: Review of the recent literature. Journal of Accounting and Economics, 51(1), 204–218.
Bertomeu, J., & Cheynel, E. (2018). PhD Notes, available at https://sites.google.com/site/jeremybertomeu/ph-d-course.
Bertomeu, J., Beyer, A., Taylor, D.J. (2016). From casual to causal inference in accounting research: the need for theoretical foundations. Foundations and Trends in Accounting, 10(2-4), 262–313.
Bertomeu, J., Ma, P., Marinovic, I. (2015a). How often do managers withhold information?. Stanford GSB Working Paper.
Bertomeu, J., Marinovic, I., Terry, S.J., Varas, F. (2015b). The dynamics of concealment: CEO Myopia and information withholding, Stanford GSB Working Paper.
Beyer, A., Cohen, D.A., Lys, T.Z., Walther, B.R. (2010). The financial reporting environment: Review of the recent literature. Journal of Accounting and Economics, 50(2-3), 296–343.
Botosan, C.A. (1997). Disclosure level and the cost of equity capital. The Accounting Review, 72(3), 323–349.
Bresnahan, T.F. (1989). Empirical studies of industries with market power. Handbook of Industrial Organization, 2, 1011–1057.
Caskey, J., Hughes, J., Liu, J. (2015). Strategic informed trades, diversification, and cost of capital. Accounting Review, 90(5), 1811–37.
Cheong, F.S., & Thomas, J. (2011). Why do EPS forecast error and dispersion not vary with scale? Implications for analyst and managerial behavior. Journal of Accounting Research, 49(2), 359–401.
Cheynel, E. (2013). A theory of voluntary disclosure and cost of capital. Review of Accounting Studies, 18(4), 987–1020.
Chung, K.H., McInish, T.H., Wood, R.A., Wyhowski, D.J. (1995). Production of information, information asymmetry, and the bid-ask spread: Empirical evidence from analysts' forecasts. Journal of Banking & Finance, 19(6), 1025–1046.
Cotter, J., Tuna, I., Wysocki, P.D. (2006). Expectations management and beatable targets: How do analysts react to explicit earnings guidance?. Contemporary Accounting Research, 23(3), 593–624.
Darrough, M. (1993). Disclosure policy and competition: Cournot vs. Bertrand. The Accounting Review, 68(3), 534–561.
Darrough, M., & Stoughton, N.M. (1990). Financial disclosure policy in an entry game. Journal of Accounting and Economics, 12(1-3), 219–243.
Demsetz, H. (1968). The cost of transacting. The Quarterly Journal of Economics, 82(1), 33–53.
Diamond, D.W., & Verrecchia, R.E. (1991). Disclosure, liquidity, and the cost of capital. The Journal of Finance, 46(4), 1325–1359.
Dye, R.A. (1985). Disclosure of nonproprietary information. Journal of Accounting Research, 23(1), 123–145.
Dye, R.A. (1986). Proprietary and nonproprietary disclosures. Journal of Business, 59(2), 331–366.
Dye, R.A. (2001). An evaluation of 'essays on disclosure' and the disclosure literature in accounting. Journal of Accounting and Economics, 32(1–3), 181–235.
Einhorn, E. (2007). Voluntary disclosure under uncertainty about the reporting objective. Journal of Accounting and Economics, 43(2-3), 245–274.
Einhorn, E., & Ziv, A. (2008). Intertemporal dynamics of corporate voluntary disclosures. Journal of Accounting Research, 46(3), 567–589.
Gaver, J.J., & Gaver, K.M. (1993). Additional evidence on the association between the investment opportunity set and corporate financing, dividend, and compensation policies. Journal of Accounting and Economics, 16(1), 125–160.
Hughes, J., Liu, J., Liu, J. (2009). On the relation between expected returns and implied cost of capital. Review of Accounting Studies, 14(2-3), 246–259.
Jensen, M.C. (1986). Agency costs of free cash flow, corporate finance, and takeovers. The American Economic Review, 76(2), 323–329.
Jensen, M.C., & Meckling, W.H. (1976). Theory of the firm: Managerial behavior, agency costs and ownership structure. Journal of Financial Economics, 3(4), 305–360.
Jung, W.-O., & Kwon, Y.K. (1988). Disclosure when the market is unsure of information endowment of managers. Journal of Accounting Research, 26(1), 146–153.
Kasznik, R., & Lev, B. (1995). To warn or not to warn: Management disclosures in the face of an earnings surprise. The Accounting Review, 70(1), 113–134.
Lang, M., & Lundholm, R. (1993). Cross-sectional determinants of analyst ratings of corporate disclosures. Journal of Accounting Research, 31(2), 246–271.
Lang, M., & Sul, E. (2014). Linking industry concentration to proprietary costs and disclosure: Challenges and opportunities. Journal of Accounting and Economics, 58(2), 265–274.
Lang, M.H., & Lundholm, R.J. (1996). Corporate disclosure policy and analyst behavior. The Accounting Review, 71(4), 467–492.
Li, F., Lundholm, R., Minnis, M. (2013). A measure of competition based on 10-K filings. Journal of Accounting Research, 51(2), 399–436.
Li, X. (2010). The impacts of product market competition on the quantity and quality of voluntary disclosures. Review of Accounting Studies, 15(3), 663–711.
Liang, P.J., & Wen, X. (2007). Accounting measurement basis, market mispricing, and firm investment efficiency. Journal of Accounting Research, 45(1), 155–197.
Milgrom, P.R. (1981). Good news and bad news: Representation theorems and applications. Bell Journal of Economics, 12(2), 380–391.
Rogers, J., & Van Buskirk, A. (2009). Bundled forecasts and selective disclosure of good news. Unpublished paper available at http://papers.ssrn.com/sol3/papers.cfm.
Shaked, A., & Sutton, J. (1987). Product differentiation and industrial structure. The Journal of Industrial Economics, 36(2), 131–146.
Skinner, D.J. (1994). Why firms voluntarily disclose bad news. Journal of Accounting Research, 32(1), 38–60.
Skinner, D.J. (1997). Earnings disclosures and stockholder lawsuits. Journal of Accounting and Economics, 23(3), 249–282.
Soffer, L.C., Thiagarajan, S.R., Walther, B.R. (2000). Earnings preannouncement strategies. Review of Accounting Studies, 5(1), 5–26.
Stocken, P.C., & Verrecchia, R.E. (2004). Financial reporting system choice and disclosure management. The Accounting Review, 79(4), 1181–1203.
Sutton, J. (1991). Sunk costs and market structure: price competition, advertising, and the evolution of concentration. Cambridge: MIT Press.
Verrecchia, R.E. (1983). Discretionary disclosure. Journal of Accounting and Economics, 5(1), 179–194.
Verrecchia, R.E. (1990). Information quality and discretionary disclosure. Journal of Accounting and Economics, 12(4), 365–380.
Verrecchia, R.E. (2001). Essays on disclosure. Journal of Accounting and Economics, 32(1-3), 97–180.
Yohn, T.L. (1998). Information asymmetry around earnings announcements. Review of Quantitative Finance and Accounting, 11(2), 165–182.
Zhou, F. (2016). Disclosure dynamics and investor learning. Working paper.
We warmly thank Jeremy Bertomeu, Moritz Hieman, Hui Chen, Ivan Marinovic, Korok Ray, Paul Fischer, Michael Kirschenheiter, and participants at the Stanford Summer Camp (2015), the Burton Workshop at Columbia Business School (2015), the Annual American Accounting Association (2016), the Rady School of Management Workshop (2016), and the tenth Accounting Research Workshop at Basel (2017) for helpful feedback.
Rady School of Management, University of California San Diego, Wells Fargo Hall Hall, 9500 Gilman Drive, La Jolla, CA, 92093, USA
E. Cheynel
Department of Economics and Accounting, Hunter College CUNY, 695 Park Avenue, Room 1535HW, New York, NY, 10065, USA
M. Liu-Watts
Search for E. Cheynel in:
Search for M. Liu-Watts in:
Correspondence to E. Cheynel.
Appendix: A
Asymptotic properties of the estimators
We derive the asymptotic properties of the BBT-cost and NP-cost estimators.
If c > 0, the estimator cBBT is consistent (\(plim \hat {c}_{BBT}=c\)) with asymptotic variance given by
$$ \sqrt{N}(\hat{c}_{BBT}-c)\rightarrow_{d} N(0,\sigma_{BBT}^{2}), $$
where \(\sigma _{BBT}^{2}=(H^{\prime }(p))^{2}p (1-p)\) and \(H^{\prime }(p)=-\frac {1}{\phi ({\Phi }^{-1}(1-p))}-\frac {\phi ^{\prime }({\Phi }^{-1}(1-p))}{(1-p)(\phi ({\Phi }^{-1}(1-p)))}+\frac {\phi ({\Phi }^{-1}(1-p))}{(1-p)^{2}}\).
Likewise, cNP is consistent (satisfies \(plim \hat {c}_{NP}=c\)) with asymptotic variance given by
$$ \sqrt{N}(\hat{c}_{NP}-c)\rightarrow_{d} N(0,\sigma_{NP}^{2}), $$
where \(\sigma _{NP}^{2}= \frac {p (p m + m (1 - p))^{2} + (1 - p) p v_{x}} {(1 - p)^{3}}\) and \(v_{x}=Var(\tilde {x}|\tilde {x}\geq \tau )\).
The asymptotic variances of the estimators \(\hat {c}_{BBT}\) and \(\hat {c}_{NP}\) can be easily estimated using sample moments, that is, replacing all elements of the asymptotic variance by their sample estimates, i.e.,
$$ \begin{array}{@{}rcl@{}} \hat{\sigma}_{BBT}^{2}&=&(H^{\prime}(\hat{p}))^{2}\hat{p} (1-\hat{p}),\\ {and} \hat{\sigma}_{NP}^{2}&=& \frac {\hat{p} (\hat{p} \hat{m} + \hat{m} (1 - \hat{p}))^{2} + (1 - \hat{p}) \hat{p} \hat{v}_{x}} {(1 - \hat{p})^{3}}, \end{array} $$
where \(\hat {v}_{x}\) is the sample variance of forecasts, then \(\hat {\sigma }^{2}_{BBT}\) and \(\hat {\sigma }^{2}_{NP}\) are respectively consistent estimators of \(\sigma ^{2}_{BBT}\) and \(\sigma _{NP}^{2}\).
Proof of Proposition 1:
The proof of consistency is immediate. By continuity, the estimator \(\hat {c}_{BBT}\) is consistent. Likewise, by continuity, the estimator \(\hat {c}_{NP}\) is consistent, satisfying
$$ plim \hat{c}_{NP}=plim\hat{\tau}+\frac{plim\hat{p}}{1-plim\hat{p}}plim\hat{m}=\tau+\frac{p_{}}{1-p}\mathbb{E}(x|x\geq \tau)=c^{}. $$
Let us derive the asymptotic variances next.
$$ \begin{array}{@{}rcl@{}} \sqrt{N}(\hat{p}-p)\rightarrow_{d} N(0,p(1-p)) \end{array} $$
Applying the Delta method,
$$ \sqrt{N}(\hat{c}_{BBT}-c)\rightarrow_{d} N(0,(H^{\prime}(p))^{2}p (1-p) ), $$
where \(H(p)={\Phi }^{-1}(1-p)+\frac {\phi ({\Phi }^{-1}(1-p))}{1-p}\). Taking the derivative of H(p) completes the proof.
We denote xi as the manager's information and di as the disclosure for each observation i, where di = 1 if a forecast is issued or di = 0 otherwise.
$$ \hat{c}_{NP}=\hat{\tau}+\frac{\hat{p}}{1-\hat{p}}\hat{m}=\hat{\tau}+\frac{\sum d_{i}/N}{1-\hat{p}}\frac{\sum d_{i} x_{i}}{\sum d_{i}}=\hat{\tau}+\frac{1}{1-\hat{p}}\underbrace{\frac{\sum d_{i} x_{i}}{N}}_{\hat{w}}. $$
In what follows, let us denote \(\tilde {x}\) as the random variable, corresponding to the manager's private information, and \(\tilde {d}=1\) if a forecast is issued or d = 0 otherwise. The associated moments to these random variables are denoted \(m= \mathbb {E}(\tilde {x}|\tilde {x}\geq \tau )\), \(v_{x}= Var(\tilde {x}|\tilde {x}\geq \tau ),\) and \(\mathbb {E}(\tilde {d}\tilde {x})=p \mathbb {E}(\tilde {x})=pm\). Note that \(\hat {p}\) and \(\hat {w}\) are sample means. Therefore, by the central limit theorem,
$$ \sqrt{N}\left( \left( \begin{array}{l} \hat{p} \\ \hat{w} \end{array}\right)-\left( \begin{array}{l} p \\pm \end{array}\right)\right) \rightarrow_{d} N(\mathbf{0}_{2},\underbrace{\left( \begin{array}{ll} Var(\tilde{d}) & cov(\tilde{d},\tilde{d}\tilde{x}) \\ cov(\tilde{d},\tilde{d}\tilde{x}) & Var(\tilde{d}\tilde{x}) \end{array}\right)}_{{\mathbf{V}}_{\mathbf{0}}}. $$
Simplifying this variance-covariance matrix and denoting \(m= \mathbb {E}(\tilde {x}|\tilde {x}\geq \tau )\) and \(v_{x}= Var(\tilde {x}|\tilde {x}\geq \tau )\),
$$ {\mathbf{V}_{\mathbf{0}}}=\left( \begin{array}{ll} p(1-p) & (1-p)p m \\ (1-p)p m & p(v_{x }+(1-p)m^{2}) \end{array}\right) $$
$$ \begin{array}{@{}rcl@{}} \text{because} Var(\tilde{d})&=& p(1-p);\\ cov(\tilde{d},\tilde{d}\tilde{x})&=& \mathbb{E}(\tilde{d}^{2}\tilde{x})- \mathbb{E}(\tilde{d})\mathbb{E}(\tilde{d}\tilde{x})\\ &=& \mathbb{E}(\tilde{d}\tilde{x})- \mathbb{E}(\tilde{d})\mathbb{E}(\tilde{d}\tilde{x})=(1-p)\mathbb{E}(\tilde{d}\tilde{x})\\ &=& (1-p)p \mathbb{E}(\tilde{x}|\tilde{x}\geq \tau)=(1-p)p m; \end{array} $$
$$ \begin{array}{@{}rcl@{}} \text{and} Var(\tilde{d}\tilde{x})&=& \mathbb{E}(\tilde{d}^{2}(\tilde{x})^{2})- \mathbb{E}(\tilde{d}\tilde{x})^{2}\\ &=& \mathbb{E}(\tilde{d}(\tilde{x})^{2})-(p \mathbb{E}(\tilde{x}|\tilde{x}\geq \tau))^{2}\\ &=& p \mathbb{E}((\tilde{x})^{2}|\tilde{x}\geq \tau)-(p \mathbb{E}(\tilde{x}|\tilde{x}\geq \tau))^{2} \\ &=& p(Var(\tilde{x}|\tilde{x}\geq \tau)+ \mathbb{E}(\tilde{x}|\tilde{x}\geq \tau)^{2})-(p \mathbb{E}(\tilde{x}|\tilde{x}\geq \tau))^{2}\\ &=& p(Var(\tilde{x}|\tilde{x}\geq \tau)+ (\mathbb{E}(\tilde{x}|\tilde{x}\geq \tau))^{2} (1-p))=p(v_{x}+(1-p)m^{2}). \end{array} $$
Next, note that \(\hat {c}_{NP}=G(\hat {p},\hat {w})\) where \(G(X)=\hat {\tau }+\frac {z}{1-y}\) and X = (y,z). Hence, applying the delta method,
$$ \sqrt{N}(\hat{c}_{NP}^{}-c^{})\rightarrow_{d} N(0,\underbrace{A V_{0} A^{\prime}}_{\sigma_{NP}^{2}}) $$
such that \(A=\frac {\partial G}{\partial X^{\prime }}|_{X=(p,pm)}=(\frac {pm}{(1-p)^{2}}, \frac {1}{1-p})\). Therefore,
$$ \begin{array}{@{}rcl@{}} \sigma_{NP}^{2}&=&(\frac{pm}{(1-p)^{2}}, \frac{1}{1-p})\left( \begin{array}{ll} p(1-p) & (1-p)p m \\ (1-p)p m & p(v_{x}+(1-p)m^{2}) \end{array}\right) \left( \begin{array}{l} \frac{pm}{(1-p)^{2}} \\ \frac{1}{1-p} \end{array}\right)\\ &=& \frac {p (p m + m (1 - p))^{2} + (1 - p) p v_{x}} {(1 - p)^{3}}. \end{array} $$
To complete the proof, we further show that \(\hat {\tau }\) converges at a rate greater than \(\sqrt {N}\). We define J(.), the distribution of x|x > τ and let t > 0, b ∈ (0,p) and α ∈ (1/2, 1). Letting m be the number of disclosures, we can decompose the probability \(Prob(N^{\alpha }(\hat {\tau }-\tau )\leq t)\) as follows:
$$ \begin{array}{@{}rcl@{}} \forall t>0,Prob(N^{\alpha}(\hat{\tau}-\tau)\leq t)=Prob(m< bN)Prob(N^{\alpha}(\hat{\tau}-\tau)\leq t|m<bN)\\ +Prob(m> b N)Prob(N^{\alpha}(\hat{\tau}-\tau)\leq t|m>bN+1),\\ \geq Prob(N^{\alpha}(\hat{\tau}-\tau)\leq t|m>[bN])+o(1/N), \end{array} $$
where the inequality follows from the fact that Prob(m < bN) converges to zero as N becomes large and \(Prob(N^{\alpha }(\hat {\tau }-\tau )\leq t|m)\) is decreasing in m. The probability in the right-hand side is, up to a negligible term, equal to
$$ \begin{array}{@{}rcl@{}} 1-\left( 1-J\left( \frac{t}{N^{\alpha}}+\tau\right)\right)^{bN} &=&1-\exp \left( bN log\left( 1-J\left( \frac{t}{N^{\alpha}}+\tau\right)\right)\right). \end{array} $$
Taking the limit in \(N\rightarrow +\infty ,\)
$$ \begin{array}{@{}rcl@{}} lim_{N\rightarrow+\infty}\frac{ log(1-J(\frac{t}{N^{\alpha}}+\tau))}{\frac{t}{N^{\alpha}}}=-\frac{J^{\prime}(\tau)}{1-J(\tau)}=-J^{\prime}(\tau)<0. \end{array} $$
Hence \(lim_{N\rightarrow +\infty }1-\exp (bN log(1-J(\frac {t}{N^{\alpha }}+\tau )))= {lim_{N}\rightarrow +\infty }1-\exp (bNt \frac { log(1-J(\frac {t}{N^{\alpha }}+\tau ))}{\frac {t}{N^{\alpha }}})=1\). We conclude: \(lim_{N\rightarrow +\infty }Prob(N^{\alpha }(\hat {\tau }-\tau )\leq t)=1\). □
Appendix: B
We prove next that the NP estimator derived under pure disclosure costs continues to hold, even if there is an uncertain information endowment coupled with an inability to credibly communicate a lack of information, as in Dye (1985), or a probability that managers do not care about market perceptions, which would induce them to never engage in costly disclosure.
In addition to a disclosure cost c, the manager may not be informed with some probability q > 0 and then cannot disclose. Although this theoretical assumption is not parsimonious, the reader may note that we are trying to take frictions that may simultaneously exist in the data (Einhorn and Ziv 2008) and hence speak to the fair concern that what we identify as a disclosure cost may, in fact, be uncertain information endowment. In this joint model, the equation for the disclosure threshold remains
$$ \tau-c=\mathbb{E}(x|ND), $$
where the right-hand side is the expected value conditional on not disclosing and is now a function of p as in Jung and Kwon (1988).Footnote 27
Even though the non-disclosure expectation will be different in this model, we can denote p = (1 − q)(1 − F(τ)), i.e., the probability to disclose, where F(.) is the cumulative probability density of x, and apply the law of total expectations as in the baseline:
$$ \begin{array}{@{}rcl@{}} 0=\mathbb{E}(x)=p\mathbb{E}(x|x\geq\tau)+(1-p)\mathbb{E}(x|ND). \end{array} $$
Solving for E(x|ND) and plugging this equation into (B) implies the same estimator as NP,
$$ c=\tau+\frac{p}{1-p}\mathbb{E}(x|x\geq \tau)). $$
Hence the NP estimator derived under pure disclosure costs continues to hold even if there is uncertainty about information endowment although (as we have shown) this other friction affects the disclosure threshold. The key to this finding is that the probability of disclosure p is endogenous and changes as q increases; we only need to observe this endogenous variable to recover the cost c.
Appendix C: Variable definitions
Cheynel, E., Liu-Watts, M. A simple structural estimator of disclosure costs. Rev Account Stud (2020) doi:10.1007/s11142-019-09511-1
DOI: https://doi.org/10.1007/s11142-019-09511-1
Voluntary disclosures
Disclosure costs
Proprietary costs
Structural estimation
Management forecasts | CommonCrawl |
Measure of spread of a multivariate normal distribution
What is a good measure of spread for a multivariate normal distribution?
I was thinking about using an average of the component standard deviations; perhaps the trace of the covariance matrix divided by the number of dimensions, or a version of that. Is that any good?
normal-distribution multivariate-analysis
Kristian D'AmatoKristian D'Amato
$\begingroup$ as such, the spread of multivariate gaussian doesn't make sense. However, depending on your needs, there might exists approaches to answer your question. Trace of the matrix is one of the many ways, but you would be ignoring correlations, which may make a huge difference. Eigen values, PCA, etc. might be much better. Therefore, could you please elaborate on your needs? $\endgroup$
– suncoolsu
$\begingroup$ As such, I want an analog of the standard deviation to a multi-dimensional space. Yes, the trace would ignore the correlations, which is what I fear. Having said that, this does not need to be mathematically exact. Basically, a good indication of spread would be the hypervolume size of the hyperellipse defined by 1 std. deviation from the mean. But a nice, handy formula without deriving the exact volume would be much appreciated. $\endgroup$
– Kristian D'Amato
$\begingroup$ Seems like PCA could answer your question. $\endgroup$
What about the determinant of the sample variance-covariance matrix: a measure of the squared volume enclosed by the matrix within the space of dimension of the measurement vector. Also, an often used scale invariant version of that measure is the determinant of the sample correlation matrix: the volume of the space occupied within the dimensions of the measurement vector.
whuber♦
schenectadyschenectady
$\begingroup$ +1 Yes, the determinants are directly related to the "hypervolume...of the ellipse defined by 1 sd from the mean." $\endgroup$
– whuber ♦
$\begingroup$ So that's the determinant of the covariance matrix, right? $\endgroup$
$\begingroup$ @Kristian The square root of the determinant of the covariance matrix tells you the hypervolume, incorporating both shape (correlation) and size (standard deviation) information. It is the product of the standard deviations of the principal components. The determinant of the correlation matrix is basically a shape factor only, ranging from 0 for degenerate distributions up to 1 when all components are uncorrelated. $\endgroup$
$\begingroup$ @whuber, what if I'd like to have a separate measurement of shape and size? (I'm actually interested in the size only, I think.) $\endgroup$
– Atcold
$\begingroup$ @Atcold You would need to establish a quantitative definition of "size". This would be equivalent to establishing what a unit-size distribution is for each given shape. (By definition, "shape" is whatever properties a distribution may have that are unchanged by translation or rescaling.) There are innumerable ways to do that, so ultimately the issue comes down to choosing a suitable definition for your particular analysis. This is one reason there cannot be a universal definition of size (or "spread") for any distribution family that comprises multiple shapes. $\endgroup$
I would go with either trace or determinant with a preference towards trace depending on the application. They're both good in that they're invariant to representation and have clear geometric meanings.
I think there is a good argument to be made for Trace over Determinant.
The determinant effectively measures the volume of the uncertainty ellipsoid. If there is any redundancy in your system however then the covariance will be near-singular (the ellipsoid is very thin in one direction) and then the determinant/volume will be near-zero even if there is a lot of uncertainty/spread in the other directions. In a moderate to high-dimensional setting this occurs very frequently
The trace is geometrically the sum of the lengths of the axes and is more robust to this sort of situation. It will have a non-zero value even if some of the directions are certain.
Additionally, the trace is generally much easier to compute.
MRocklinMRocklin
$\begingroup$ +1 Good points. This gets me thinking: any symmetric function of the $n$ eigenvalues would qualify as "good." All such polynomial functions are polynomials in the $n$ elementary symmetric functions, which include the determinant and the trace. $\endgroup$
$\begingroup$ Yes, the sum (trace) isn't necessarily the best way to go. You're right that you could imagine lots of mixtures here depending on the application. I wonder if there is some standard family of functions that would be good here.... $\endgroup$
– MRocklin
$\begingroup$ @MR I'm not aware of anybody attempting to use a single statistic to compute the spread of a multivariate normal distribution (except, of course, when independence of all components is assumed). This leads me to believe there may be no such standard family. $\endgroup$
Another (closely related) quantity is the entropy of the distribution: for a multivariate Gaussian this is the log of the determinant of the covariance matrix, or
$\frac{1}{2} \log |(2\pi e)\Lambda|$
where $\Lambda$ is the covariance matrix. The advantage of this choice is that it can be compared to the "spread" of points under other (e.g., non-Gaussian) distributions.
(If we want to get technical, this is the differential entropy of a Gaussian).
jpillowjpillow
Not the answer you're looking for? Browse other questions tagged normal-distribution multivariate-analysis or ask your own question.
Estimating multivariate normal distribution by observing variance in different directions
2D multivariate normal coverage probability
Multivariate logistic distribution
Metropolis random walk using multivariate normal
Multivariate normal with singular covariance
Single index to measure the randomness of multivariate distributions (or data)
Computing Highest Density Region given multivariate normal distribution with dimension $d$ > 3
How can a covariance matrix for a normal distribution not be quadratic? | CommonCrawl |
Lignocellulose integration to 1G-ethanol process using filamentous fungi: fermentation prospects of edible strain of Neurospora intermedia
Ramkumar B. Nair1,2,
Osagie A. Osadolor1,
Vamsi K. Ravula1,
Patrik R. Lennartsson1 &
Mohammad J. Taherzadeh1
Integration of first- and second-generation ethanol processes is one among the alternate approaches that efficiently address the current socio-economic issues of the bioethanol sector. Edible filamentous fungus capable of utilizing pentoses from lignocelluloses and also possessing biomass application as potential animal feed component was used as the fermentation strain for the integration model. This study presents various fermentation aspects of using edible filamentous fungi in the integrated first and second generation ethanol process model.
Fermentation of edible strain of N. intermedia on the integrated first and second-generation ethanol substrate (the mixture of dilute acid pretreated and enzymatically hydrolyzed wheat straw and thin stillage from the first-generation ethanol process), showed an ethanol yield maximum of 0.23 ± 0.05 g/g dry substrate. The growth of fungal pellets in presence of fermentation inhibitors (such as acetic acid, HMF and furfural) resulted in about 11 to 45% increase in ethanol production as compared to filamentous forms, at similar growth conditions in the liquid straw hydrolysate. Fungal cultivations in the airlift reactor showed strong correlation with media viscosity, reaching a maximum of 209.8 ± 3.7 cP and resulting in 18.2 ± 1.3 g/L biomass during the growth phase of fungal pellets.
N. intermedia fermentation showed high sensitivity to the dilute acid lignocellulose pretreatment process, with improved fermentation performance at milder acidic concentrations. The rheological examinations showed media viscosity to be the most critical factor influencing the oxygen transfer rate during the N. intermedia fermentation process. Mycelial pellet morphology showed better fermentation efficiency and high tolerance towards fermentation inhibitors.
The second generation (2G) lignocellulose-to-ethanol process that intends to reduce the dependence of first generation (1G) ethanol on the food grains, has gained much attention by researchers and industry, over the past few decades [1]. However, the existing 1G-ethanol processes continue to be favored over the 2G processes due to the surplus production of grains (wheat or corn), especially in Europe and the USA, together with the associated technical and economic challenges of 2G processes. A smart alternative approach to address this issue is to develop a model 'integrating both first- and second-generation ethanol processes'. The model first described by Lennartsson et al. [2] especially for the grain-based first- and second-generation ethanol processes, proposed the use of grain-derived lignocellulose waste such as straw, stover, bran, or stillage (waste stream from the 1G-ethanol process), as the 2G-ethanol substrate. The integration model hence opens-up a new avenue for converting the existing 1G-ethanol plants to a lignocellulose based 'biorefinery', utilizing the existing infrastructure facilities such as distillation column, reactors, evaporators, etc. [3, 4]. This could also possibly cut down the investment cost and the associated risks that are currently faced by the stand-alone 2G lignocellulose-to-ethanol plants. The ease of lignocellulose availability at the vicinity of the 1G plant also creates an advantage towards the collection and transportation (logistics) system, which otherwise is a challenging issue faced by the 2G-ethanol process [4,5,6].
It is estimated that in 2017 nearly 202, 383 and 71 first generation (sugar/ starch based) ethanol refineries with a total production capacity (of the facilities) of about 59.3, 39.6 and 8.5 billion liters per year, currently exist in the USA, Brazil, and Europe, respectively [7]. Hence, introducing the lignocellulose integration model at these 1G-ethanol refineries could successfully render a technically and economically sustainable 2G-ethanol process [8,9,10]. However, the use of 1G-ethanol fermenting microorganisms, such as Saccharomyces cerevisiae (yeast), hinders the fermentation of pentose sugar from the lignocellulose biomass. It should be considered that the use of genetically modified pentose consumers can create socio-regulatory issues as well as it affects the quality of the animal feed product, DDGS (distiller's dried grains with solubles) that contributes to a major share of the economics at the 1G-ethanol facility [11]. Therefore, finding the right microorganism for ethanol fermentation, capable of consuming pentose sugars and simultaneously maintaining the quality of DDGS narrows the options down to using the edible strains of filamentous fungi as the fermentation microbe [2].
The choice of filamentous fungi as a key player in 'ethanol biorefinery' has been initiated almost 15 years ago, when Rhizopus oryzae was used to produce ethanol from paper pulp sulfite liquor (a waste stream from pulp and paper industry) [12]. Since then several research and pilot scale studies have been carried out to explore the ethanol fermenting potential of various strains of edible filamentous fungi [13,14,15]. Many filamentous fungal strains are considered as GRAS (generally regarded as safe) microorganisms under Sections 201(s) and 409 of the Federal Food, Drug, and Cosmetic Act of the Food and Drug Administration (US-FDA) and also meet the requirements of being in the list of Microbial food cultures (MFC) of the EU regulation described under the Qualified Presumption of Safety (QPS) protocol introduced by the European Food Safety Authority (EFSA) [16]. This could be an important aspect should the fungal biomass obtained from the integration model be used either as an animal feed component or enrich the DDGS quality. However, the use of edible strains of filamentous fungi as the fermenting microbe in an integrated first and second-generation ethanol process model has not been studied previously. Hence, a deeper understanding of the process conditions such as the media fluid-rheology, fungal growth pattern and the effect of fermentation inhibitors, needs to be investigated and optimized before the desired outcome of the integration process model can be achieved.
In this study, Neurospora intermedia an edible strain of filamentous fungi was hence used as a model organism for the filamentous fungi based lignocellulose (wheat straw) integration to the existing first-generation wheat grain-to-ethanol process. The effect of lignocellulose pretreatment conditions, together with the inhibitor's effect, on the fungal fermentation was determined. The morphological and rheological aspects were also investigated for an optimized ethanol and biomass production in the integrated first and second-generation ethanol process model.
Fungal strain
Neurospora intermedia CBS 131.92 (Centraalbureau voor Schimmelcultures, The Netherlands), an edible filamentous ascomycete, was used as the model-fermentation microorganism in the present study. The fungal strain was maintained and the inoculum preparation was followed as previously described [17]. Inoculum in the form of (a) fungal spores, 3–5 mL spore suspension (per L medium) with a spore concentration of 5.7 ± 1.8 × 105 spores/ml; (b) mycelial filamentous biomass (0.2 to 0.3 g/L wet weight content) and (c) fungal pellets (0.1 to 0.8 g/L wet weight content), obtained following the methods specified by Nair et al. [17], were used throughout the cultivations.
Wheat straw (92.4% dry content) used in the demonstration scale pretreatment experiments and thin stillage (a residual product from the wheat based first generation ethanol facility) used for the integration experiments were supplied by Lantmännen Agroetanol (Norrköping, Sweden). Straw, with the composition (g/g, dry basis) arabinan 0.048 ± 0.013; galactan 0.0053 ± 0.0015; glucan 0.315 ± 0.061; mannan 0.0047 ± 0.0011; and xylan 0.24 ± 0.08, was milled (0.2–0.25 mm size) using a rotor beater mill before use. Thin stillage with a natural pH of 3.5 was characterized with the composition of total solids (% w/v) 9.2 ± 0.4 and suspended solids (% w/v) 2.2 ± 0.6 and (g/L) total nitrogen 4.8 ± 0.5; xylose 0.8 ± 0.1; arabinose 1.5 ± 0.1; glycerol 7.0 ± 0.; lactic acid 1.8 ± 0.1; acetic acid 0.21 ± 0.01 and ethanol 1.2 ± 0.2.
Pretreatment and hydrolysis: Preparing the fermentation substrate
The dilute-phosphoric acid pretreatment of wheat straw was carried out in a 30-L one-step vertical plug-flow continuous reactor at a Biorefinery Demo Plant (RISE, Örnsköldsvik, Sweden). Based on the preliminary laboratory study [18], three pretreaments were carried out at the demonstration plant at different conditions (Table 1). Based on the pretreatment temperature, the straw pre-hydrolysates obtained from the demo-plant, were designed as P201, P195, and P190. The chemical characteristics of the pretreated slurry are depicted in Table 1. Pretreatment was carried out as explained in a previous study [18]. The pretreated straw slurry P201, P195, and P190 (with slurry pH, 3.5 2.9 and 3.2 respectively) was subjected to enzymatic hydrolysis without any pre-washing at solid loading 3.5, and 7.0%, at pH 5.0 ± 0.3 (adjusted with 2 M NaOH) at 50.0 ± 0.2 °C water bath at an enzyme loading of 10 FPU/g substrate dry weight. Cellulase enzyme Cellic CTec2 (Novozymes, Denmark) with 134 FPU /mL activity was used for the hydrolysis. The hydrolysate obtained after the enzyme hydrolysis of pretreated slurry P201, P195, and P190 were designated as hydrolysate H201, H195, and H190, respectively.
Table 1 Characteristics of the dilute-phosphoric acid pretreated wheat straw pre-hydrolysate from the demonstration facility
Neurospora intermedia fermentation for the integration process
The general schematic for the integrated first and second-generation ethanol process is shown in Fig. 1. The hydrolyzed wheat straw H190, H195, and H201 at different solid loading concentrations (w/v) of 3.5, and 7.0%, were used for the fungal fermentation. The hydrolysates, either in the form of slurry (solid and liquid) or as liquid supernatant (obtained after centrifugation of the hydrolysate- slurry at 15,000 g), were mixed with thin stillage (total solids 8%) at ratio 1:1 to form the fermentation media. Cultivations were made in 250 ml Erlenmeyer flasks (100 ml liquid volume) at 35 °C and 150 rpm in an orbital shaking water bath (Grant OLS-Aqua pro, UK), for 120 h. Control fermentation experiments were carried out separately in straw hydrolysates (both liquid and slurry) and thin stillage.
Integration model for the first and second generation bioethanol process at the existing wheat based ethanol facilities using edible filamentous fungus, N. intermedia. Fermentation media for the integration model was developed using a) whole lignocellulosic slurry or b) liquid part of the slurry hydrolysate (modified from [2])
N. intermedia pellets fermentation in airlift reactor
The enzymatic hydrolysis of the pretreated slurry (P201) was carried out with 7.0% w/v total solids at pH 5.5 ± 0.1 for 48 h. Hydrolysate slurry was further subjected to centrifugation at 15,000 g, and the liquid supernatant was mixed with thin stillage (at 1:1 ratio) to form the fermentation media. The cultivation was carried out in a 4.5 L airlift reactor (Belach Bioteknik, Stockholm, Sweden), with the liquid volume of 3.5 L, for 120 h at 35 °C with an aeration of 1.4 vvm (volumeair /volumemedia /min) following the protocol described in a previous study [19]. The media pH was maintained at 3.5 ± 0.3 throughout the cultivation using 2 M HCl, attributing to the optimum pellet growth condition. N. intermedia mycelial pellets, 0.39 ± 0.04 g wet weight (obtained from a pre-culture), was used as the inoculum.
Inhibitor effect on fungal growth
Cultivations of N. intermedia were carried out aerobically in semi synthetic PDB (potato dextrose broth) media containing 20 g/L glucose and 4 g/L potato extract, with varying concentrations (based on the substrate slurry composition) of inhibitors such as., acetic acid (0.5–3.0 g/L), furfural (0.5–2.0 g/L), and hydroxymethylfurfural (HMF, 0.2–5.0 g/L). Batch fermentation in 250 ml Erlenmeyer flasks was carried out for 120 h in a shaking water bath at 35 °C and 150 rpm with samples taken every 24 h. Initial culture pH was adjusted to 3.5 ± 0.1 or 5.5 ± 0.2 with 2 M HCl or 2 M NaOH respectively.
The content of total solids (TS), suspended solids (SS), ash, starch, lignin, and sugars present in the lignocellulosic materials were quantified according to NREL (National Renewable Energy Laboratory) protocols [20,21,22,23,24,25]. The pH was measured with a digital pH-meter (Philips, PW-9420). Spore concentration was measured using a Bürker counting chamber (with a depth of 0.1 mm) under the light microscope (Carl Zeiss Axiostar plus, Germany). The spore solution was diluted ten times before the measurement, and the spores were counted in a volume of 1/250 μl each. HPLC (Waters 2695, Waters Corporation, USA.) was used to analyze all liquid fractions. A hydrogen-based ion-exchange column (Aminex HPX-87H, Bio-Rad Hercules, CA, U.S.A.) at 60 °C with a Micro-Guard cation-H guard column (Bio-Rad) and 0.6 mL/min 5 mM H2SO4, used as eluent, was used for the analyses of glucose, ethanol, glycerol and acetic acid, furfural, and 5-hydroxymethyl-furfural. For the separation of glucose, mannose, galactose, cellobiose, xylose, and arabinose, a lead (II) based column (Aminex HPX-87P, Bio-Rad) with two Micro-Guard Deashing (Bio-Rad) precolumns operated at 85 °C with 0.6 mL/min ultrapure water as eluent. Fungal biomass concentration (dry weight) was determined at the end of the cultivation by washing the pellet or mycelial biomass with deionized water followed by drying at 70 °C for 24 h before weight analysis. Limit™ digital Vernier caliper (resolution 0.01 mm) was used to measure the pellet size (diameter). Fermentation media viscosity was measured using a Brookfield digital viscometer-model DV-E (Chemical Instruments AB, Sweden).
All the results and values represented were the average of two independent experimental runs and reported intervals and error bars are ±2 standard deviations, unless otherwise specified. All the data were considered statistically significant at the 95% confidence interval with the P value < 0.05.
Rheological study
The rheological aspects of the N. intermedia fermentation in the straw hydrolysate media were analyzed based on the existing concept of the microbial fermentation process. It is estimated that the resistance to flow or viscosity in a fermentation media could either be constant or change during the fermentation process. The relationship between the shear stress τ (Pa) and the shear rate \( \frac{dv}{dy} \) or ý (s− 1) when the viscosity is constant is given by Newton's law as shown in eq. 1.
$$ \tau =-\upmu \frac{\mathrm{dv}}{\mathrm{dy}}=-\upmu \acute{\mathrm{y}}, $$
However, when the microorganism in the fermentation media grows up to an extent when the viscosity of the media is no longer constant, the fluid becomes non-Newtonian. The flow under this condition is pseudoplastic and it follows the relationship shown in eq. 2, where K is the fluid consistency index (Pa.sn) and n is a number less than one [26].
$$ \tau =\mathrm{K}{\acute{\mathrm{y}}}^n, $$
The viscosity under non-Newtonian condition is the apparent viscosity (μa) and can be expressed by the relationship shown in eq. 3.
$$ {\upmu}_{\mathrm{a}}=\mathrm{K}{\acute{\mathrm{y}}}^{\mathrm{n}-1}=\frac{\tau }{\acute{\mathrm{y}}}, $$
N. intermedia in the integrated ethanol process
Effect of lignocellulose pretreatment on fermentation
In order to facilitate the integrated model of first and second-generation ethanol processes, a specially designed fermentation media composed of dilute acid pretreated wheat straw hydrolysate and thin stillage, was used for the filamentous fungal cultivations (Section "Neurospora intermedia fermentation for the integration process"). From all the three hydrolysates used (H190, H195, and H201), the slurry obtained from the pretreatment at conditions 0.7% (w/v) acid conc. at 201 ± 4 °C for 7 min (i.e. H201), showed the highest ethanol and fungal biomass production in all the cultivations, both individually and in combination with thin stillage (Tables 2, 3 and 4). With the integration of thin stillage and with the inoculation using pellets, 0.188 ± 0.005 g ethanol /g dry substrate was obtained from enzymatically hydrolyzed straw slurry- H201 (3.5% total solids), with a reduced fermentation period to 24 h. However, the cultivation using only the hydrolysate slurry, resulted in an ethanol production as low as only 0.01 ± 0.08 g/g dry substrate straw. Similar results were also obtained with clear liquid hydrolysate (Table 2). With the filamentous mycelial forms as inoculum and with the integration of thin stillage to liquid hydrolysate, the ethanol production was increased from 0.054 ± 0.002 g/g substrate to about 0.23 ± 0.05 g/g dry substrate straw (Table 2) using enzymatically hydrolyzed straw slurry- H201 (7% total solids). The overall fermentation results suggest that while using the hydrolysate from different pretreatment conditions (H201, H195, H190), an improved fermentation process with high ethanol and fungal biomass production was observed only with the integrated media using thin stillage (Tables 2, 3 and 4). An improved growth of fungus in the mild pretreated wheat straw slurry (H201), where only 0.7% (w/v) acid concentration has been used, indicated the strong influence of acid loading on the subsequent fungal fermentation process. However, the effect of the dilute acid pretreatment on the ethanol fermentation was considerably reduced when using the integrated media using thin stillage. At various cultivation conditions using substrate hydrolysate with 3.5% total solids, an increase in ethanol yield by about 345, 394 and 544% for the liquid hydrolysate and 3561, 2293%; and 4213% for the slurry (solid and liquid) hydrolysate of H201, H195, and H190 respectively, was obtained while integrating thin stillage as a nutrient supplement for the fermentation media (Tables 2, 3 and 4).
Table 2 Fermentation profile by N. intermedia on integrated fermentation substrate using wheat straw pretreated with dilute phosphoric acid (H201- acid concentration of 0.7% (w/v), duration of 7 min, and temperature of 201 ± 4 °C) and thin stillage (THS) mixture
Table 4 Fermentation profile by N. intermedia on integrated fermentation substrate using wheat straw pretreated with dilute phosphoric acid (H190- acid concentration of 1.75% (w/v), duration of 10 min, and temperature of 190 ± 2 °C) and thin stillage (THS) mixture
N. intermedia growth in the form of pellets achieved in the liquid wheat straw hydrolysate showed improved ethanol and inhibitor tolerance (Tables 2, 3, 4 and 5). Fermentation using N. intermedia pellets in the liquid straw hydrolysate (WSL) at varying substrates solid loading (7 and 3.5%) resulted in up-to 31% increase in the ethanol yield, with an improved glucose assimilation by the pellets (up-to 82% reduction in initial glucose) as opposed to filamentous forms (up-to 51% reduction in initial glucose), under similar culture conditions (Table 2). Considering the ethanol productivity, the concentration was always higher in the integrated media than only the hydrolyzed media. A possible explanation for this fact is the presence of sufficient nutrients in thin stillage needed for the fungi to produce ethanol [27]. Though the fungal biomass concentration was higher in fermentation slurry (substrate hydrolysate) containing higher initial total solids especially for hydrolysate H201 (Table 2), the trend was not observed for hydrolysates H195, and H190. This could be attributed to the presence of higher amount of fermentation inhibitors, especially acetic acid at higher solid loading conditions (Table 1). Similar observations on the production of fungal biomass and ethanol using acid-pretreated wheat straw slurry were obtained in a previous study while using zygomycetes strains of filamentous fungus, Rhizopus sp.; however the biomass yields obtained was only up to 0.34 g biomass/g consumed monomeric sugars and acetic acid [28].
Table 5 Fermentation profile of N. intermedia in the presence of inhibitors. Mycelial growth in semi-synthetic potato dextrose media at pH 3.5 represents pellets and at pH 5.5 represents filamentous forms, at varying inhibitor concentrations as observed in wheat straw-hydrolysate at different cultivation conditions
Effect of inhibitors and acetic acid assimilation
The growth of fungal pellets in presence of acetic acid, HMF and furfural inhibitors (in liquid semi-synthetic media), at different concentrations, had resulted in increase in the ethanol production by as low as 11% to as high as 45%, compared to filamentous forms at similar growth conditions (Table 5). In this study, considering the relative inhibitor concentrations as represented in the initial slurry or liquid hydrolysate, the major detrimental effect on the fungal growth was observed with acetic acid as compared to other inhibitors (Table 5). The effect of acetic acid inhibition was however reduced by maintaining it in its dissociated form, unavailable for cell-membrane diffusion [29, 30]. This was achieved by a custom-made neutralization step where the extracellular media pH was increased to pH above 8.0 ± 0.5 (almost double the pKa value of acetic acid) using CaCl2 (100 mM) and then decreasing the pH to 3.5 ± 0.3 or 5.5 ± 0.2, using 1 M HCl, prior to pellets or filamentous inoculum, respectively. The results showed improved acetic acid assimilation by the fungal cells, with the decrease in its concentration by about 36 to 48%. However, the fungal biomass and ethanol yields decreased considerably with the increase in acetic acid concentration as compared to other fermentation inhibitors (Table 5). The presence of acetic acid in the fermentation media generally leads to a significant decrease in the maximum cell biomass concentration in most cultivations [30, 31].
Rheological aspects of N. intermedia growth and scale-up
The nature of the relationship between the shear stress and the shear rate in a fermentation media determines whether the media would be described as Newtonian or non-Newtonian (eqs. 1 and 2). Most non-viscous fermentation media is usually Newtonian at the beginning of the fermentation process, with the media viscosity being constant until the concentration of the biomass exceeds a threshold value. However, the viscosity would vary considerably, with the increase in the fungal biomass concentration [26]. In general, the viscosity influences oxygen transfer rate into the fermentation medium, which in turn influences the fungal growth. The initial media viscosity was measured to be 89.2 ± 0.7 cP, 102.1 ± 0.4 cP, and 98.3 ± 0.9 cP for the hydrolysate slurry H201, H195, and H190, respectively. When hydrolyzed second generation (wheat straw) substrate H201 was combined with first generation substrate (thin stillage), the viscosity of the combination would depend on how the substrates are mixed, together with the total solid content of the media. Fig. 2 represents the effect of viscosity of the fermentation media (such as wheat straw slurry hydrolysate (H201); 1:1 mixture of slurry hydrolysate (H201) and thin stillage; and the clear liquid supernatant of hydrolysate H201), on the fungal biomass concentration. It was observed that the viscosity of the fermentation media played a critical role in the fungal growth [32], where higher the viscosity, the less the oxygen transfer with reduced biomass yield. Fermentation experiments carried out using 7% (w/v) hydrolysate H195 and H190 in different media combinations of the integrated media hence showed no fungal growth attributing to its high viscosity (Tables 3 and 4). However, when the initial cultivation volume of the fermentation media was reduced by half, improved fungal growth was observed until 120 h of fermentation, possibly due to the increased oxygen transfer into the media [33, 34]. The integrated media using 1:1 mixture of 7% w/v (total solid) slurry hydrolysate (H201) and thin stillage with half initial volume, had resulted in a maximum of 18.0 ± 2.8 g/L of fungal biomass, pointing out the obligate aerobic nature of N. intermedia. This hence implies the significance of reduced media viscosity and adequate oxygen transfer into bioreactors, for an efficient fermentation using N. intermedia in the integrated first and second generation ethanol production process model.
The rheological effect of fermentation media on edible fungal biomass growth pattern. Figure represents wheat straw slurry hydrolysate (H201) - at solid hydrolysate loading of 7% hydrolysate (−▲−); at 1:1 mixture of 7% total solid hydrolysate and thin stillage (··■··); at 1:1 mixture of 7% total solid hydrolysate and thin stillage mixture with half initial volume (··♦··), and at clear hydrolysate of 7% solid loading (·−●−·); with each marker points representing 24 h of fermentation period.
Scale-up of the fermentation experiments using the optimum cultivation media showing the maximum fungal growth and ethanol production was carried out in a 4.5 L bench scale airlift reactor at an aeration rate of 1.4 vvm (as described in section "N. intermedia pellets fermentation in airlift reactor"). Integrated media containing 1:1 mixture of thin stillage and 7% w/v slurry hydrolysate (H201) was fed to the airlift reactor in batches. The media viscosity decreased during the stationary phase (24 h fermentation) to 28.1 ± 2.1 cP from the initial 90.2 ± 3.8 cP, which was similar to what was observed for the shake flask experiment (Fig. 2). However, the viscosity of the media increased to 209.8 ± 3.7 cP during the growth phase (48 h cultivations) and then decreased to 180.7 ± 1.8 cP during the stationary phase (72 h cultivations). Cultivations in the airlift under this condition resulted in the production of 18.2 ± 1.3 g/L of fungal biomass (72 h cultivations). However, operating the airlift reactor at 1.4 vvm aeration rates caused foaming and evaporation of the fermentation medium, which was controlled by applying intermitted mixing strategies.
Fungal growth morphology and product formation
The previous study has shown that the fungal growth morphology significantly influences the product formation rate, with pellet morphology favoring more ethanol production and filamentous growth favoring more biomass formation [19]. Additionally, different fungi growth morphologies have their advantages and drawbacks from an overall process perspective [35]. However, the oxygen uptake rate for fungi growing as pellets was much higher than that of the filamentous form, which implies that better aeration efficiency is obtained during fungi pellet growth [19]. The results from the fermentation experiments using either pellets or filamentous biomass as inoculum had indicated that the biomass growth could not occur as pellets in the integrated fermentation media (1:1 mixture of thin stillage and straw hydrolysate) in all the cultivation conditions using different substrates hydrolysates (Tables 2, 3 and 4). A possible reason for this is the high viscosity of the media (between 35 and 100 cP), which in turn leads to less oxygen transfer into the fermentation media [36]. Hence, filamentous morphology was the most common biomass form for the integrated media using the hydrolysate with 7% total solid content. Nevertheless, considering biomass growth as pellets (as found in liquid hydrolysate), it was observed that the viscosity of the fermentation media remains within a constant range, independent of the biomass growth rate (or concentrations) as compared to the growth in form of mycelial filaments as found in the fermentation using the integrated media (Fig. 2).
Integrating lignocelluloses (wheat straw) to existing first generation (1G) wheat-based ethanol facilities could possibly reduce its current dependence on the food grains (wheat), in addition to reducing the investment cost and risk associated with the 2G lignocellulose-to-ethanol process [3, 4, 37]. The ascomycetes filamentous fungus N. intermedia, capable of utilizing pentoses [2, 38], and traditionally used for the preparation of the indigenous Indonesian food oncom [39], was used in the present integration model. The fungal biomass obtained could effectively be used as an animal feed component or enrich the DDGS quality at the 1G-ethanol plants or be considered as a new valuable by-product [11]. However, the practical aspects of fermentation such as fluid rheology, the effect of fermentation inhibitors and the fungal growth morphology, highly affect the biomass growth and ethanol production. This study hence describes for the first time, various process aspects of N. intermedia fermentation on wheat-based integrated first and second-generation ethanol substrate. The previous challenges with 2G lignocellulose (wheat straw)-to-ethanol process while using N. intermedia [18, 37] could also be effectively addressed with the current integration model. Thin stillage, a process-waste stream at 1G-ethnaol facility, has also been valorized effectively using the integration model. Considering the fermentation inhibitors, the presence of fermentation inhibitors (mainly HMF, furfural and acetic acid) from the pretreatment process had posed severe challenges in N. intermedia growth during previous studies [18, 37]. Hence, in this study, the addition of fungal pellets capable of an improved fermentation and inhibitor tolerance [17, 19] was used as the starting inoculum for the cultivation.
Ethanol and biomass optimization for the integrated process
Higher ethanol and fungal biomass production are always beneficial for increasing the profitability of the integrated process model at the ethanol industries. The current results suggest that the feedstock (from both first and second-generation process) integration model greatly influences the optimal ethanol and biomass production. Integration after the solid removal from the lignocellulose hydrolysate (Fig. 1 b) would result in higher ethanol production as seen in the case of integrated thin stillage and clear hydrolysate (for example H201) media as compared to that of the integrated whole slurry hydrolysate and thin stillage media (Tables 2, 3 and 4). However, higher biomass production was most favored while using the whole slurry of the lignocellulose hydrolysate (Fig. 1 a) for the integrated fermentation media (Table 2, 3 and 4). The use of lower solid loading also facilitates the removal of suspended particles from the slurry after enzyme hydrolysis, allowing an easy separation of the fungal biomass. Hence, a trade-off between the fungal biomass and ethanol production clearly exists in the integration model, which would greatly depend on the prevailing market conditions. Nevertheless, from a process standpoint, the integration before solid removal is beneficial for minimizing the associated energy and investment /operation cost associated with the process steps, for example, centrifugation. This could also minimize the investment cost for an ethanol facility, considering that for a typical 100,000 m3 ethanol facility, the centrifuge accounts for 18% of the total fermentation investment cost [40]. However, a thorough techno-economic analysis is required to optimize the actual integration model for further developments at a larger scale.
The use of integrated media with wheat straw (dilute acid pretreated and hydrolyzed) and thin stillage (from first generation ethanol facility), overcomes the challenges previously faced by the filamentous fungi fermentation on wheat straw. The use of N. intermedia mycelial pellets as fermentation inoculum resulted in an improved ethanol and inhibitor tolerance, with acetic assimilation by about 36 to 48% (decreasing initial acid concentration). Overall rheological observations coupled with the high biomass yields at lower initial solid loading conditions, points out the significance of adequate oxygen transfer into the bioreactors, as the most critical factor for filamentous fungal growth in the proposed integrated ethanol process.
REN21 PS: Renewables 2015 global status report. In.: REN21 Secretariat Paris, France; 2015.
Lennartsson PR, Erlandsson P, Taherzadeh MJ. Integration of the first and second generation bioethanol processes and the importance of by-products. Bioresour Technol. 2014;165(0):3–8.
Dias MOS, Junqueira TL, Cavalett O, Cunha MP, Jesus CDF, Rossell CEV, Maciel Filho R, Bonomi A. Integrated versus stand-alone second generation ethanol production from sugarcane bagasse and trash. Bioresour Technol. 2012;103(1):152–61.
Erdei B. Development of integrated cellulose- and starch-based ethanol production and process design for improved xylose conversion [doctoral thesis]. Lund: Lund university; 2013.
Macrelli S, Mogensen J, Zacchi G. Techno-economic evaluation of 2(nd)generation bioethanol production from sugar cane bagasse and leaves integrated with the sugar-based ethanol process. Biotechnology for biofuels. 2012;5:22.
Dias MOS, da Cunha MP, Maciel Filho R, Bonomi A, Jesus CDF, Rossell CEV. Simulation of integrated first and second generation bioethanol production from sugarcane: comparison between different biomass pretreatment methods. J Ind Microbiol Biotechnol. 2011;38(8):955–66.
Bob Flach, Sabine Lieberz, Marcela Rondon, Barry Williams, Wilson C: global agricultural information network (GAIN), United States Department of Agriculture Foreign Agricultural Service https://gain.fas.usda.gov/Recent%20GAIN%20Publications/Biofuels%20Annual_The%20Hague_EU-28_6-29-2016.pdf. Accessed Aug 2017.
Joelsson E, Erdei B, Galbe M, Wallberg O. Techno-economic evaluation of integrated first- and second-generation ethanol production from grain and straw. Biotechnology for biofuels. 2016;9(1):1.
Rajendran K, Rajoli S, Taherzadeh MJ. Techno-economic analysis of integrating first and second-generation ethanol production using filamentous fungi: an industrial case study. Energies. 2016;9(5):359.
Macrelli S, Mogensen J, Zacchi G. Techno-economic evaluation of 2nd generation bioethanol production from sugar cane bagasse and leaves integrated with the sugar-based ethanol process. Biotechnology for biofuels. 2012;5:22.
Nair RB, Lundin M, Brandberg T, Lennartsson PR, Taherzadeh MJ. Dilute phosphoric acid pretreatment of wheat bran for enzymatic hydrolysis and subsequent ethanol production by edible fungi Neurospora intermedia. Ind Crop Prod. 2015;69:314–23.
Taherzadeh MJ, Fox M, Hjorth H, Edebo L. Production of mycelium biomass and ethanol from paper pulp sulfite liquor by Rhizopus oryzae. Bioresour Technol. 2003;88(3):167–77.
Nair RB, Taherzadeh MJ. Valorization of sugar-to-ethanol process waste vinasse: a novel biorefinery approach using edible ascomycetes filamentous fungi. Bioresour Technol. 2016;221:469–76.
Millati R, Edebo L, Taherzadeh MJ. Performance of Rhizopus, Rhizomucor, and Mucor in ethanol production from glucose, xylose, and wood hydrolyzates. Enzym Microb Technol. 2005;36(2–3):294–300.
Ferreira JA, Lennartsson PR, Taherzadeh MJ. Production of ethanol and biomass from thin stillage using food-grade zygomycetes and ascomycetes filamentous fungi. Energies. 2014;7(6):3872–85.
Bourdichon F, Casaregola S, Farrokh C, Frisvad JC, Gerds ML, Hammes WP, Harnett J, Huys G, Laulund S, Ouwehand A, et al. Food fermentations: microorganisms with technological beneficial use. Int J Food Microbiol. 2012;154(3):87–97.
Nair RB, Lennartsson PR, Taherzadeh MJ. Mycelial pellet formation by edible ascomycete filamentous fungi. Neurospora intermedia AMB Express. 2016;6(1):1–10.
Nair RB, Lundin M, Lennartsson PR, Taherzadeh MJ. Optimizing dilute phosphoric acid pretreatment of wheat straw in the laboratory and in a demonstration plant for ethanol and edible fungal biomass production using Neurospora intermedia. J Chem Technol Biotechnol. 2017;92(6):1256–65.
Osadolor OA, Nair RB, Lennartsson PR, Taherzadeh MJ. Empirical and experimental determination of the kinetics of pellet growth in filamentous fungi: a case study using Neurospora intermedia. Biochem Eng J. 2017;124:115–21.
Sluiter A, Hames B, Hyman D, Payne C, Ruiz R, Scarlata C, Sluiter J, Templeton D, Wolfe J. Detemination of total solids in biomass and total dissolved solids in liquid process samples. Colorado: NREL; 2008.
Sluiter A, Hames B, Ruiz R, Scarlata C, Sluiter J, Templeton D, Crocker D. Determination of structural carbohydrates and lignin in Biomass. Colorado: NREL; 2011.
Sluiter A, Sluiter J. Determination of starch in solid biomass samples by HPLC. Colorado: NREL; 2008.
Hames B, Ruiz R, Scarlata C, Sluiter A, Sluiter J, Templeton D. Preparation of samples for compositional analysis. Colorado: NREL; 2008.
Sluiter A, Ruiz R, Scarlata C, Sluiter J, Templeton D. Determination of extractives in Biomass. Colorado: NREL; 2008.
Sluiter A, Hames B, Ruiz R, Scarlata C, Sluiter J, Templeton D. Determination of ash in Biomass. Colorado: NREL; 2008.
Doran PM. Chapter 7 - fluid flow. In: Bioprocess engineering principles. Academic press; 2013. p. 201–54.
Ferreira JA, Lennartsson PR, Taherzadeh MJ. Production of ethanol and biomass from thin stillage by Neurospora intermedia: a pilot study for process diversification. Eng Life Sci. 2015;15(8):751–9.
FazeliNejad S, Ferreira JA, Brandberg T, Lennartsson PR, Taherzadeh MJ. Fungal protein and ethanol from lignocelluloses using Rhizopus pellets under simultaneous saccharification, filtration and fermentation (SSFF). Biofuel Research Journal. 2016;3(1):372–8.
Taherzadeh MJ, Niklasson C, Liden G. Acetic acid - friend or foe in anaerobic batch conversion of glucose to ethanol by Saccharomyces cerevisiae? Chem Engg Sci. 1997;15:2653–9.
Casey E, Sedlak M, Ho NWY, Mosier NS. Effect of acetic acid and pH on the cofermentation of glucose and xylose to ethanol by a genetically engineered strain of Saccharomyces cerevisiae. FEMS Yeast Res. 2010;10(4):385–93.
Narendranath NV, Thomas KC, Ingledew WM. Effects of acetic acid and lactic acid on the growth of Saccharomyces cerevisiae in a minimal medium. J Ind Microbiol Biotechnol. 2001;26(3):171–7.
Garcia-Ochoa F, Gomez E. Bioreactor scale-up and oxygen transfer rate in microbial processes: an overview. Biotechnol Adv. 2009;27(2):153–76.
Maier U, Büchs J. Characterisation of the gas–liquid mass transfer in shaking bioreactors. Biochem Eng J. 2001;7(2):99–106.
Maier U, Losen M, Büchs J. Advances in understanding and modeling the gas–liquid mass transfer in shake flasks. Biochem Eng J. 2004;17(3):155–67.
Zhang J, Zhang J. The filamentous fungal pellet and forces driving its formation. Crit Rev Biotechnol. 2016;36(6):1066–77.
Oostra J, Le Comte EP, Van den Heuvel JC, Tramper J, Rinzema A. Intra-particle oxygen diffusion limitation in solid-state fermentation. Biotechnol Bioeng. 2001;75(1):13–24.
Nair RB, Kabir MM, Lennartsson PR, Taherzadeh MJ, Horváth IS. Integrated process for ethanol, biogas, and edible filamentous Fungi-based animal feed production from dilute phosphoric acid-pretreated wheat straw. Appl Biochem Biotechnol. 2017;184:1–15.
Ferreira JA, Lennartsson PR, Edebo L, Taherzadeh MJ. Zygomycetes-based biorefinery: present status and future prospects. Bioresour Technol. 2013;135:523–32.
Ho CC. Identity and characteristics of Neurospora intermedia responsible for oncom fermentation in Indonesia. Food Microbiol. 1986;3(2):115–32.
Maiorella BL, Blanch HW, Wilke CR, Wyman CE. Economic evaluation of alternative ethanol fermentation processes. Biotechnol Bioeng. 2009;104:419–43.
This work was financially supported by the Swedish Energy Agency and the Swedish Research Council Formas.
The supplementary details of this study are available in the Swedish research-data repository, DiVA http://hb.diva-portal.org/smash/record.jsf?pid=diva2%3A1128276&dswid=-2387
Swedish Centre for Resource Recovery, University of Borås, 50190, Borås, SE, Sweden
Ramkumar B. Nair, Osagie A. Osadolor, Vamsi K. Ravula, Patrik R. Lennartsson & Mohammad J. Taherzadeh
Mycorena AB, Stena Center 1 A, 41292, Gothenburg, SE, Sweden
Ramkumar B. Nair
Osagie A. Osadolor
Vamsi K. Ravula
Patrik R. Lennartsson
Mohammad J. Taherzadeh
RBN conceived and designed the experiments; VKR performed the experiments; RBN and OAO analyzed the data and wrote the manuscript; PRL and MJT supervised the experiments and revised the manuscript. All authors read and approved the final manuscript.
Correspondence to Ramkumar B. Nair.
All authors declare the consent to publish the manuscript with no direct or indirect conflict of interests.
Nair, R.B., Osadolor, O.A., Ravula, V.K. et al. Lignocellulose integration to 1G-ethanol process using filamentous fungi: fermentation prospects of edible strain of Neurospora intermedia. BMC Biotechnol 18, 49 (2018). https://doi.org/10.1186/s12896-018-0444-z
Lignocelluloses
Edible filamentous fungi
Neurospora intermedia | CommonCrawl |
barak manos
logonos.com
Mathematics 41.2k 41.2k 77 gold badges4444 silver badges109109 bronze badges
Code Review 1.2k 1.2k 11 gold badge66 silver badges1818 bronze badges
MathOverflow 580 580 22 silver badges1414 bronze badges
109 Is there a "positive" definition for irrational numbers?
87 Quickly find whether a value is present in a C array?
79 What is a real-world metaphor for irrational numbers?
65 Every Real number is expressible in terms of differences of two transcendentals
65 How to determine without calculator which is bigger, $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ or $\left(\frac{1}{3}\right)^{\frac{1}{2}}$
57 python JSON object must be str, bytes or bytearray, not 'dict
54 How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis?
syntactic-analysis
4 What is the general term for "greater than or equal to" and "less than or equal to"? Feb 1 '15
3 "If you don't mind me asking" or "If you don't mind my asking"? [duplicate] Feb 21 '16
1 Construct a "the more x the more y" compound statement Sep 16 '14
0 Exceptions that allow the use of "will" after "if" [duplicate] Sep 16 '14 | CommonCrawl |
\begin{definition}[Definition:Smallest Natural Number]
Let $S \subseteq \N$ be a subset of the natural numbers $\N$.
The '''smallest''' element $m$ of $S$ is defined as:
:$\forall n \in S: m \le n$
That is, it is the minimal element of $S$ under the usual ordering.
\end{definition} | ProofWiki |
Mixed effects of remittances on child education
José R. Bucheli ORCID: orcid.org/0000-0002-3311-80861,
Alok K. Bohara1 &
Matías Fontenla1
IZA Journal of Development and Migration volume 8, Article number: 10 (2018) Cite this article
We exploit the size of the 2010 Ecuadorian Census to estimate the effect of remittances on secondary school enrollment across four key dimensions: gender, household wealth, rural vs. urban, and family migration status. Using a bivariate probit model that accounts for both endogeneity and non-linearity issues, we find both positive and negative effects of remittances on the likelihood of schooling. The strongest positive effects are for poorer, urban males, while the negative effects are for rural females. For children in wealthier households, the effects of remittances are either negative or non-significant. This suggests that the positive income effects of remittances may be offset by the negative effects of a missing parent due to migration, more visible in wealthier families where financial constraints may not be as binding. We find further support for this by estimating the effects of remittances conditional on migration status. Our results show positive effects on schooling for non-migrant households that receive remittances and no effects for children living in households where at least one parent has migrated. The sharp contrasts within and across groups, while using the same data and econometric specifications, help explain the lack of consensus in the literature.
International remittances continue to be a major source of income in developing countries. In Ecuador, remittances as a share of GDP neared 7% for 2005–2007, roughly matching the revenue from oil, the country's number one export. These large financial inflows have the potential to benefit poorer households by increasing income, educational attainment, and promoting health.Footnote 1 However, as Amuedo-Dorantes (2014) has noted, the effects of remittances may be heavily determined by the idiosyncrasies of each country. To address potential heterogeneity in our findings, we first develop a theoretical model that accounts for some of these differences and then test it by partitioning our data into population sub-groups.
The effects of remittances on children's schooling are of particular interest, as human capital accumulation may break the intergenerational transmission of poverty through higher future income, especially in the case of Ecuador where large labor returns to schooling have been found (Bertoli et al. 2011). The majority of the evidence in the literature supports the existence of potentially opposing effects of migration and remittances on education (Amuedo-Dorantes et al. 2010; Bargain and Boutin 2015; Hu 2012; Koska et al. 2013). Some studies find evidence for a higher likelihood of schooling in the presence of migration (Shrestha 2017; Theoharides: Manila to Malaysia, Quezon to Qatar: international migration and its effects on origin-country human capital, forthcoming) and remittances (Alcaraz et al. 2012; Calero et al. 2009; Göbel 2013). They argue that the positive effects may be driven by the additional income, wage premiums for migrants, contribution to household capital accumulation, and higher propensities of migrant families to invest in education. Bouoiyour and Miftah (2016) conclude that with remittances, children in Morocco are less likely to drop out of school and delay their entry into the labor market and that these improvements are especially notorious among girls. With higher income from remittances, Coon (2016) finds that Bolivian children work fewer hours, potentially leading to improvements in human capital. Even in terms of quality, Salas (2014) finds that remittances increase the likelihood of investing in sending children to private schools.
On the other hand, while some find that migration might raise parental academic aspirations for their children (Böhme 2015), others find detrimental effects of migration on education (Amuedo-Dorantes and Pozo 2010; Bouoiyour and Miftah 2015; McKenzie and Rapoport 2011). These studies argue that the negative effect could be driven by children having to compensate for the missing parent, by joining the labor force, or by taking over domestic responsibilities. This argument is also supported by Cortes (2015), who finds robust evidence that the mothers' migration, in contrast to fathers', has a negative effect on the educational outcomes of Filipino children. In terms of the effect of remittances, research continues to find instances in which remittances have no significant effect on schooling (Bargain and Boutin 2015; Nepal 2016; Pilarova and Kandakov 2017), or even where they might hinder child schooling, as households discount the value of education needed to thrive abroad (Davis and Brazil 2016).
This paper develops a theoretical model where receiving remittances may relax the household financial constraints, but migration may also change the household members' labor participation and education decisions. Although remittances may increase the likelihood of schooling among children, the effect of migration on schooling may be negative. We empirically test this relationship via a bivariate probit model that accounts for both endogeneity and non-linearity issues. We exploit the size and coverage of the 2010 Ecuadorian Population and Housing Census, which allows us to look at specific groups that may not be well represented in smaller samples.
In this way, we add to the work on Ecuador of Calero et al. (2009) and Göbel (2013), who use a sample of households collected in 2005–2006 to find a positive effect of remittances on education in Ecuador. However, due to data limitations acknowledged by the authors, the in-depth exploration of population sub-groups is not possible. In contrast, in this paper, we are able to partition the data into four key dimensions: gender, wealth, rural vs. urban location, and nuclear family migration status. While using the same data and econometric specifications, our findings vary greatly across these subgroups, and it is this variation that seems to explain the heterogeneity of results in the literature.
We find both positive and negative effects of remittances on education, depending on the particular group being studied. First, our findings reveal large gender inequalities, as boys benefit from remittances while the effect for girls is generally negative or insignificant. Second, urban children gain more than their rural counterparts, supporting the existence of regional inequalities. Also, while remittances relax households' budget constraints, the impact on education varies strongly by income level, where school attendance increases the most for poorer children. In contrast, for wealthier households, the effects of remittances on education are either negative or insignificant. This suggests that the positive income effects of remittances may be offset by the negative effects of a missing parent due to migration, more visible in wealthier families where financial constraints may not be binding. We find further support for this by estimating the effects of remittances conditional on migration status. Our results show positive effects on schooling for non-migrant households that receive remittances and no effects for children living in households where at least one parent has migrated.
During the latter part of the 1990s, the Ecuadorian economy suffered one of its most severe crises with a staggering real GDP contraction of 28% for 1999, when measured in US dollars. Roughly one third of the population fell below the poverty line, and people living in extreme poverty doubled (Acosta et al. 2006). The crisis was characterized by business failures, increased unemployment, the official dollarization of the economy, the freezing of bank deposits, increasing public debt, and a large drop in health and education indicators.
This period of economic turmoil had deep social repercussions, especially in the form of the unprecedented emigration of millions of Ecuadorians to the USA and Europe. Close to 20% of the economically active population left the country in the early 2000s, and almost a quarter of the households had one or both parents migrate (Camacho and Hernández 2008). Camacho and Hernández (2008) also report that, on average, every migrant left behind two children under the age of 18. As a result, many homes became mono parental, children were left under the care of extended kin, or older siblings became the head of household.
The massive migration to North America and Europe produced a large flow of remittances back to Ecuador. Between 1990 and 2002, the volume of remittances entering the country increased by 30% annually. After a short slowdown in 2003, the rate recovered to an annual increase of 22% until 2007. Since 1999, the volume of remittances has been higher than that of foreign direct investment (FDI), and in 2007 alone, remittances were 16 times higher than FDI (Quintana et al. 2014). Between 2005 and 2007, remittances as a share of GDP neared 7%, surpassing the share of GDP generated by the value-added tax, and roughly equal to the share of petroleum, Ecuador's number one export.
Similar to income, the distribution of remittances across Ecuador is highly unequal. Olivié et al. (2009) indicate that in 2006, the bottom two income quintiles of the population received less than 6% of the total volume of remittances, while the wealthiest 20% received over 34%. Further, the poorest 40% received remittances for an average period of 4.4 years, while the wealthiest 20% received them for almost seven years. Stated differently, poor children who start receiving remittances during the first grade of primary school may stop receiving them before they are able to complete the fifth grade.
In terms of how remittances are being used by recipients, the United Nations Population Fund reported that roughly 90% of remittances received are used for current expenditures (UNFPA and FLACSO 2008). Education and human capital formation expenditures account for 18% of the total.
Theoretical framework
To relate the probability of a child attending school to remittance reception and migration, we develop a utility maximization model with two adults a and b, and a child c, who is too young to make her own decisions. The household's utility depends on consumption and on the child's education, e c . The child can only use her time for work or education. Time is normalized to 1, and parents decide how it is allocated. The household's utility is given by
$$U=U\left(x,e_{c}\right), $$
where x represents the consumption bundle and e c is the time the child dedicates to education. Consequently, (1−e c ) is the time the child spends working.
Each period, the household receives remittances r and maximizes their utility by choosing a consumption level as well as the study and work time allocation for the child. The household's budget constraint is given by
$$ x=w_{a}+w_{b}+\left(1-e_{c}\right)w_{c}+r, $$
where w stands for wages. Notice that receiving remittances is not conditional on either parent migrating, as our data section below confirms. Remittances can also originate from relatives such as grandparents, friends, or from a biological parent not part of the household. With this, we can maximize the household's utility through the use of the Lagrangian and its first-order conditions:
$$\begin{array}{*{20}l} \mathcal{L} & =U(x,e_{c})-\lambda\left[x-w_{a}-w_{b}-r-\left(1-e_{c}\right)w_{c}\right] \end{array} $$
$$\begin{array}{*{20}l} \frac{{\partial\mathcal{L}}}{\partial x} & =\frac{\partial U}{\partial x}-\lambda=0 \end{array} $$
$$\begin{array}{*{20}l} \frac{\partial\mathcal{L}}{\partial e_{c}} & =\frac{\partial U}{\partial e_{c}}-\lambda w_{c}=0 \end{array} $$
$$\begin{array}{*{20}l} \frac{\partial\mathcal{L}}{\partial\lambda} & =x-w_{a}-w_{b}-r-\left(1-e_{c}\right)w_{c}=0. \end{array} $$
These lead to the optimal condition
$$ \frac{\partial U}{\partial e_{c}}=\frac{\partial U}{\partial x}w_{c}. $$
The left-hand side is the change in the household's utility from a change in the child's education, which we label θ. The right-hand side denotes the change in utility as the level of consumption changes, times the wage the child would earn if she worked. We label this change in utility as ϕ. The Marshallian demand curve for education is then given by
$$ e_{c}=1-\frac{\phi}{\theta}\left[x_{b}-w_{b}-w_{a}-r\right]. $$
Taking the partial derivative of education with respect to remittances, we obtain
$$ \frac{\partial e_{c}}{\partial r}=\frac{\phi}{\theta}. $$
The sign and magnitude of the change in education with a change in remittances is ambiguous, as it depends on the trade-off between the benefits of education and the household's consumption. Also, if member a of the household migrates, then w a =0, but r potentially increases. Thus, the trade-off between lost local parental wages vs. increased remittances makes the sign of Eq. (8) unclear. In addition, the trade-offs may vary across urban and rural settings, as imperfect labor markets, usually more pervasive in rural settings, may affect demand for child labor differently. Further, the relationship may also differ by income levels, age, and gender of the child.
Given the above discussion, we remain agnostic about the expected signs in our empirical results, as they may vary by population subgroups.
Our data comes from the 2010 Ecuadorian Population and Housing Census conducted by Ecuador's National Institute of Statistics and Census (INEC). The census collected information on schooling, remittances, household, and demographic characteristics for Ecuador's entire population. Household remittance reception was determined by whether any member received money from relatives or friends living abroad during 2010. Since the census was conducted in November 2010, there was a period of 11 months in which respondents could have received remittances intermittently, regularly, or just once. The data does not include information on frequency or amount of reception; thus, the results are interpreted as the average effect on all households that received remittances. To identify households with migrant family members, the survey asked whether any individuals who resided in the household during the 2001 census had moved to another country and had not returned permanently. Follow-up questions inquired about migrants' age, destination, and purpose, as well as year of emigration.
While there is variation in the effect of remittances across gender and income groups, the largest differences appear to be on the rural/urban dimension. Figure 1 presents the schooling rates for urban and rural regions by remittance reception status and age. Note that before the age of 11, there is little difference in the schooling rates across groups and enrollment rates are close to 100%. However, starting at age 12, the schooling rates rapidly diverge, where urban children that receive remittances have the highest schooling rates, and rural children that do not receive remittances are the ones least likely to attend school. Even when Ecuadorian laws make it mandatory for children to attend school until the age of 14, Fig. 1 shows that this requirement has little impact on the rate at which enrollment rates decrease. Rather, it is the primary/secondary school jump that creates the discontinuity. While the slope of enrollment rates with changes in age is practically horizontal in primary school, it is clearly negative between the ages of 12 and 17 across groups. About 95% of 12-year-olds attend secondary school, but this proportion quickly drops and diverges to around 85% for urban remittance receivers and 60% among rural children who who do not receive remittances. At least in part, these differences may be explained by the increasing opportunity costs that arise from delaying the child's entry into the labor force. As children age, they become more capable of contributing to their households' income and taking over domestic responsibilities that often times make them drop out of school. The rapid decline in enrollment rates may be also explained by availability of secondary schools, especially in rural areas. While primary schools are common across Ecuador, the density of secondary schools is significantly lower, and secondary schools are often located in urban centers. Additional transportation costs added to an increased opportunity cost may push the marginal cost of education above its marginal benefit and force children out of school.
Schooling rate by rural/urban region and remittance reception status
Due to the above discussion, our study focuses on the effect of remittances on the school enrollment of children who are between 12 and 17 years old. Although some children may graduate from school after the age of 17, restricting the bound ensures exclusion of non-schooling effects. Our final sample includes 1.7 million individuals.
The endogenous variable of interest, remittances, is instrumented via four variables. First, in line with the literature (Acosta 2011; Coon 2016; Davis and Brazil 2016), we connect an individual's likelihood of receiving remittances to historical migration networks. The expectation is that children who live in areas more prone to international migration are more likely to receive remittances. Note that migration of a nuclear family member is not a necessary condition to remittance reception, as relatives and friends commonly send income from abroad to one or more nuclear family units. In fact, Table 1 reports that only 32.5% of children who receive remittances have an immediate relative living overseas. To estimate historical migration networks, we use the 2001 census and calculate the proportion of migrants out of the total canton population.Footnote 2 This ensures that the historical migration patterns are not affected by our 2010 migration variable.
Second, we use migrants' characteristics to determine the probability of remitting without being directly related to the likelihood of education. We use migrants' age as an instrumental variable because it is potentially exogenous to socio-economic conditions in Ecuador, thus not affecting schooling decisions but having an effect on the probability to remit. We use a dummy variable as the instrument that indicates whether the migrants' ages at the time of survey were between 20 and 50 years old. In this way, we account for the higher probability of a migrant working and sending remittances if they are part of the working-age population. The identification strategy requires that variation in remittance reception as a result of the migrants' age is not directly related to education. As a matter of fact, our data shows that children with a migrant in this age group were 20 points more likely to receive remittances, while their probability of education only increased marginally.
Third, following Antman (2011), we capture the main destination countries for Ecuadorian migrants by including dummies that control for migration to either the USA and Canada or Europe. These variables capture the economic conditions by destination and consequently the differing probabilities of remitting. The rationale is that general economic conditions in destination countries determine the likelihood of remittances without directly affecting school enrollment rates at the origin. More detailed information on migrants' destination would have allowed to include time-varying instruments such as unemployment and GDP per capita (see, e.g., Böhme 2015). However, Table 1 shows that individuals who receive remittances are over 10 points more likely to have a migrant relative in the USA, Canada, or Europe than those who do not receive remittances. Thus, we use destination to explain in part the different probabilities of remitting.
To further validate our instruments, Table 6 in the Appendix presents the results of a correlation analysis between the instruments and the dependent variables. The coefficients indicate a very weak association between school enrollment and the instruments and a much stronger relationship with remittances. We also run a simple probit model to assess the joint likelihood of the instruments in predicting schooling and remittance reception, and report the coefficients and model summaries in Appendix: Table 7. We find that our instruments have a low predictive power for education and a high predictive power for remittances. The likelihood ratio χ2 test is significant in both cases, but the value for the remittances model is 30 times that of the school enrollment model. Similarly, McFadden's pseudo R2 for the remittances regression is 0.171 while it equals 0.003 for school enrollment. In line with McFadden (1977), who describes a pseudo R2 value between 0.2 and 0.4 as an "excellent fit," we conclude that the inclusion of instruments offers a considerable larger improvement for remittances than for school enrollment over their individual intercept models.
Our empirical model specifications control for province fixed effects to net out any potential local unobserved externalities that affect both recipient and non-recipient households, like quality of education and availability of schools. We also control for the following child, parent, and household characteristics: age, gender, number of children younger than five in the household, parents' highest level of education, location (urban/rural), ethnicity, presence of a disability, number of migrants, and wealth. As the the census does not collect information on actual income figures, nor its subgroup remittances, wealth is proxied by an index of 20 equally weighted variables that contain information on access to basic services and technologies, as well as materials, services, and housing conditions.Footnote 3 Although the use of a wealth index is a common control in studying the effect of remittances on household outcomes (e.g.,Acosta 2011; Antón 2010; Dustmann and Okatenko 2014), there is a potential for endogeneity if wealth is not assessed through pre-remittance reception data. Households who have received remittances for some time may be more likely to have moved up in the wealth distribution. To minimize this risk, we use variables that capture long-term socioeconomic status, like access to utilities and dwelling quality, rather than short-term measures, like income. The use of construction quality and asset ownership also allows us to use variables that are more responsive to past wealth than current flows of remittances. Furthermore, when we partition the sample to analyze effects within wealth groups, we use terciles as it is less likely that remittances would have caused households to cross the 33 and 66% thresholds to reach the next group.
Table 1 presents the descriptive statistics for the children, parent, and household variables by remittance reception status. Secondary school enrollment rate is 83%, with the remittance receiving group being 5.7% higher. We see that 7.5% of individuals between 12 and 17 reside in a remittance receiving household, meaning that over 130 thousand secondary school-aged children receive income from abroad. Not surprisingly, individuals who receive remittances are more likely to live in a migrant household and to have more relatives living outside the country. In terms of the migrants' main destinations, 40% of children who receive remittances see their relatives moving to the USA or Canada, while over 50% migrate to Europe. Table 1 additionally shows that families in the remittances group are relatively wealthier and more educated, with a lower proportion of recipients living in urban areas. In terms of ethnicity, mestizos and whites are over-represented among remittance recipients whereas individuals who identify as Afro-descendants, Montubios, or Indigenous are under-represented. The remainder of this paper estimates remittance marginal effects on schooling across sub-groups while addressing the potential endogeneity of remittances.
Econometric framework
To estimate the effect of remittances on child schooling given by Eq. (8), and because the response variable education, e c , is binary, we could use a probit model as a function of remittance reception, r, and a vector γ of child, parent, and household characteristics of the following form:
$$ e_{ck}=1\left[\alpha r_{k}+\gamma_{ck}\beta_{s}+\mu_{eck}>0\right] $$
To estimate this model, we would have to make two assumptions that could yield inconsistent and biased estimates. First, we would have to assume that all the differences between recipient and non-recipient households are explained by the characteristics in γ. However, remittances are a consequence of migration, and if migration has an effect on education in addition to its effect through remittances, the error term in (9) would suffer from omitted variable bias (Mckenzie 2006). Second, we would have to assume that the child schooling decisions are not correlated with the decision of a migrant to send back remittances. In fact, if schooling, migration, and remittance decisions are correlated, then we would run into a simultaneous causality problem. To address this potential endogeneity, we use a bivariate probit model to account for the presence of r as a binary endogenous variable (Roodman 2011) that equals one if the household receives remittances and zero otherwise. In Section 6.3, we verify our results by repeating the analysis on households without migration and reach a similar conclusion. The empirical counterpart of Eq. (7) is given by the following recursive model:
$$\begin{array}{*{20}l} e_{ck} & =1\left[\alpha r_{k}+\gamma_{ck}\beta_{s}+\epsilon_{eck}>0\right] \end{array} $$
$$\begin{array}{*{20}l} r_{k} & =1\left[\gamma_{ck}\beta_{r}+z_{r}\beta_{z}+\epsilon_{rck}>0\right] \end{array} $$
$$\begin{array}{*{20}l} \epsilon & =\left(\epsilon_{eck},\epsilon_{rck}\right)'\sim \text{Normal}~\left(0,\sum\right)\\ \sum & =\left[\begin{array}{cc} 1 & \rho\\ \rho & 1 \end{array}\right], \end{array} $$
where the subscripts c and k indicate child and household, respectively. α is the counterpart of Eq. (8). The vector z r includes observable instrumental variables for r k such that E(ε eck |z r =0). We could estimate this model through an IV-probit model by endogenizing remittances, r k , as a continuous variable. This would approximate (11) with a linear probability model and (10) with a standard probit. However, the IV-probit framework does not yield consistent estimates as it does not respect the non-linearity of the first stage, a procedure that has been called a "forbidden regression" by Wooldridge (2010). A more appropriate method respects the binary nature of r k to guarantee consistent and efficient parameters (Arendt and Larsen 2006). Thus, the preferred model to estimate (10) and (11) is a bivariate probit model that estimates remittances and education with ε e and ε r jointly normally distributed. We treat r k as a predetermined regressor in a SUR framework as the maximization of the likelihood function still generates consistent parameters (Wooldridge 2010).
Table 8 in the Appendix reports the bivariate probit first-stage results for the effect of our instruments on remittances. The instrument coefficients are consistently significant and positive across all specifications. We confirm the existence of an endogenous relationship between remittance reception and schooling via Wald's tests for ρ=0, where we reject the null hypothesis of no error correlation. We use specification three for all two-stage specifications, as it includes all instruments and controls, and its estimates provide a lower bound for the effect of remittances on school enrollment.
While the bivariate probit model is preferred, Table 2 also reports the estimates obtained through the standard probit and IV-probit models for comparative purposes.Footnote 4 We present marginal effects, clustered standard errors, and number of observations for each model. The first row presents the effects of remittance receipt on child education for the whole sample. According to our bivariate probit results, the overall effect of receiving remittances increases the probability of school enrollment by 2.6 percentage points, relative to children that do not receive remittances. Notice that when we do not correct for the endogeneity of remittances in the standard probit model, all results are downward biased. When we split the sample by gender, we see that the benefits tend to be higher for boys than for girls. While males' probability of being enrolled in secondary school increases by 3.4 percentage points, females' probability increases by only 1.3%, less than half of the probability for boys. In absolute terms though, this translates into an additional 26,400 males and 10,000 females attending secondary school. This positive effect may be explained by a reduced need for recipient households to send children to work, as remittances relax their budget constraints. As we further divide the population, we will find larger and more interesting effects and differences across groups.
Wealth inequalities
Table 3 further inquires into these heterogeneous results by partitioning the data into wealth terciles.Footnote 5 For the poorest tercile, the overall probability of attending school is 4.3 points higher than for their non-recipient counterparts. A smaller effect is found for children in the middle wealth group, and in sharp contrast, children in the top tercile seem to not be affected by remittances. This suggests that the impact of remittances depends on income levels and that levels in budget constraints may matter a great deal. In particular, the income effect of remittances may be stronger for poorer households with smaller budgets. In contrast, additional income may not offset the potential negative impact of a missing parent for wealthier households.
Table 3 also shows large gender differences for the poor, where remittances generally benefit boys more than girls. Males benefit the most the lower their wealth tercile, with a 7.8 point increase in schooling probability for the bottom third and a non-significant effect for the top category. Alternatively, effects on females seem to be mostly statistically and economically negligible. Only the middle tercile benefits from a mild improvement in the order of a 1.7 point increase in schooling probability. For both males and females, the effects for wealthier children remain insignificant in both magnitude and statistical significance.
Table 4 further narrows our view by splitting the population across rural and urban sub-samples, gender, and wealth levels. Our results indicate that there are significant differences in the effects of remittances on education conditional on rural-urban location. Overall, urban children in the bottom two wealth terciles seem to benefit the most, while their rural counterparts have either smaller or negative effects. We again find that the largest positive effects occur to those relatively more financially constrained, while either negative or not significant effects are seen for the wealthier group.
Table 4 Rural-urban effects of remittances on secondary school enrollment
The rural pooled sample seems to suggest that there is no effect of remittances on schooling across wealth groups. However, when we divide the observations into males and females, we observe that girls are driving the lack of effect. Boys who live in rural areas are better off if they receive remittances, with individuals in the bottom wealth tercile being the most affected by a 7.1 point higher likelihood of attending school. Unfortunately, these benefits do not transfer over to rural girls, with either no or negative effects on schooling probability. For more disadvantaged children, remittances might be the decisive factor that enables them to invest in human capital formation. However, these results may be attenuated by the parents' higher marginal utility for having an additional household member working, either to earn additional income or to do household work.
Turning to the urban sub-sample, Table 4 indicates that the gender differences are reversed for the most disadvantaged group as girls seem to be the most favored. In fact, remittances seem to increase the likelihood of secondary school enrollment for females by 13 points, the largest positive effect in our study. Even when looking at the pooled data, we see an overall improvement among the lowest wealth categories. The effects become non-significant for wealthier individuals, as this group's choice of education may be less constrained by their income level.
Remittance vs. migration
In this section, we disentangle the potentially positive effect of additional income via remittances vs. the potentially negative impact of a missing family member due to migration, hinted at in the previous sections and discussed in the literature (e.g., McKenzie and Rapoport 2011). To this end, we follow a similar approach to Amuedo-Dorantes and Pozo (2010) and estimate the effect of remittances conditional on migration status. The first group is composed of families where at least one parent has migrated, while the second group is limited to families in which no nuclear member has migrated. The latter could be families with children from previous relationships, or that receive remittances from grandparents, other relatives, and friends. These specific households are common in Ecuador, as over 65% of the children in our sample who receive remittances live in non-migrant households.
Unfortunately, the census data does not contain information on the senders nor the amount and frequency of remittances. Thus, we cannot assume that migrant and non-migrant households who receive remittances are strictly comparable. We are also unable to treat both groups as such after controlling for observables, because if these characteristics explained the systematic difference between the two, then we would observe similar migration and remitting behaviors. Migrant households that receive remittances are different from non-migrant households in difficult-to-capture areas such as drive, aptitudes, perceived returns from education, and concerns for their children (Mckenzie 2006). We therefore take the results in this section with caution. Still, if we find a negative effect among migrant households, and a positive effect for non-migrants, it suggests the existence of the two opposing effects. On the one hand, an increase in the schooling probability of children who live in non-migrant households would be driven by the income effect from remittances. On the other hand, this positive effect could be outweighed by the missing-parent effect among migrant households.
Table 5 presents the marginal effects of remittances on schooling probability for secondary school-aged rural and urban children. Notice that the effects of remittances are largely driven not only by migration status but also by gender and rural/urban location. Regardless of location, the effect of remittances among children in migrant households is non-significant. In sharp contrast, for non-migrant households, the effects are positive and significant, regardless of gender or location. These results support the existence of the two opposing forces. It seems that remittances benefit children only when they have not been directly affected by migration.
Table 5 By migration status: effects of remittances on secondary school enrollment
In the context of our Marshallian demand for education presented in Eq. (7), these results can be explained in terms of θ and ϕ (Eq. (6)). The positive effect of remittances observed among non-migrant households seems to be driven by a higher perceived utility from education relative to the additional utility obtained from an increase in the household consumption level. Instead, the lack of an effect of remittances on migrant households appears to be determined by the additional utility from consumption counteracting that of education.
We exploit the full dimension of the 2010 Ecuadorian Population and Housing Census database to identify the effects of remittances on children's education. Even though this topic has been previously explored, the heterogeneity of results across studies warrants a new look. Using the same data and econometric specifications, we find both positive and negative effects across different subgroups. In addition to gender, socioeconomic status, and urban/rural inequalities, our proposed model indicates that the variation in the results is mainly driven by two effects. First, the additional income received in the form of remittances relaxes the households' budget constraints, which increases the probability of investing in children's human capital. Second, the evidence indicates that the remittances income effect may be offset by the absenteeism effect from migrant relatives. Our results show that the difference in the effect of remittances between migrant households and non-migrant households is at least five percentage points. It is possible that under migration, the likelihood of children going to secondary school decreases, as they take on household responsibilities or are encouraged to work. Our evidence suggests that the magnitude of the effect depends on the group being considered. This highlights the importance of using large data sets, so that different sub-samples can be evaluated.
In terms of policy recommendations, the evidence suggests that the effect of remittances depends on the level of inclusion of the population sub-groups. Our findings suggest that girls living in rural areas are the most disadvantaged. Targeting policies toward them would would not only contribute to human capital formation but also contribute to the empowerment of these traditionally excluded groups.
Table 6 Correlations between dependent variables and instrumental variables
Table 7 Probit regressions on instruments
Table 8 Bivariate probit results: probit coefficients for school enrollment and remittance reception
Table 9 Proportion of children who are enrolled in school and who receive remittances
See Rapoport and Docquier (2006) for a comprehensive review.
Cantons are second-level administrative divisions, below provinces. They are the equivalent of a county in the USA.
Table 1 Descriptive statistics by household remittance reception status
Namely, access to dwelling, water source, plumbing, sewage, electricity, kitchen space, sanitation facilities, drinking water, landline telephone, cell phone, Internet access, computer access, and cable television, as well as roof, walls, and floor materials and condition.
We also test the consistency of our results with a Tobit model and find no substantial differences.
Table 2 Marginal effects of remittances (1/0) on secondary school enrollment (1/0)
Table 9 in the Appendix provides school enrollment rates and remittance reception by wealth tercile and rural-urban location.
Table 3 By wealth terciles: effects of remittances on secondary school enrollment
Acosta, A, López S, Villamar D (2006) La Migración en el Ecuador: Oportunidades y Amenazas. Universidad Andina Simón Bolívar/Corporación Editora Nacional, Quito, Ecuador.
Acosta, P (2011) School attendance, child labour, and remittances from international migration in El Salvador. J Dev Stud 47(6):913–36.
Alcaraz, C, Chiquiar D, Salcedo A (2012) Remittances, schooling, and child labor in Mexico. J Dev Econ 97(1):156–65.
Amuedo-Dorantes, C (2014) The good and the bad in remittance flows. IZA World of Labor. https://doi.org/10.15185/izawol.97.
Amuedo-Dorantes, C, Georges A, Pozo S (2010) Migration, remittances, and children's schooling in Haiti. Ann Am Acad Polit Soc Sci 630(1):224–44.
Amuedo-Dorantes, C, Pozo S (2010) Accounting for remittance and migration effects on children's schooling. World Dev 38(12):1747–59.
Antman, FM (2011) The intergenerational effects of paternal migration on schooling and work: what can we learn from children's time allocation?J Dev Econ 96:200–8.
Antón, JI (2010) The impact of remittances on nutritional status of children in Ecuador. Int Migr Rev 44(2):269–99.
Arendt, JN, Larsen HA (2006) Probit models with dummy endogenous regressors. University of Southern Denmark Business and Economics Discussion Paper 4:1–30.
Bargain, O, Boutin D (2015) Remittance effects on child labour: evidence from Burkina Faso. J Dev Stud 51(7):922–38.
Bertoli, S, Fernández-Huertas Moraga J, Ortega F (2011) Immigration policies and the Ecuadorian exodus. World Bank Econ Rev 25(1):57–76.
Böhme, MH (2015) Migration and educational aspirations—another channel of brain gain?IZA J Migr 4(12):1–24.
Bouoiyour, J, Miftah A (2015) Migration, remittances and educational levels of household members left behind: evidence from rural Mexico. Eur J Comp Econ 12(1):21–40.
Bouoiyour, J, Miftah A (2016) The impact of remittances on children's human capital accumulation: evidence from Morocco. J Int Dev 28(2):266–80.
Calero, C, Bedi AS, Sparrow R (2009) Remittances, liquidity constraints and human capital investments in Ecuador. World Dev 37(6):1143–54.
Camacho, G, Hernández K (2008) Niñez y Migración en el Ecuador: Diagnóstico de la Situación. UNICEF, CEPLAES, INNFA, Quito, Ecuador.
Coon, M (2016) Remittances and child labor in Bolivia. IZA J Migr 5(1):1–26.
Cortes, P (2015) The feminization of international migration and its effects on the children left behind: evidence from the Philippines. World Dev 65:62–78.
Davis, J, Brazil N (2016) Disentangling fathers' absences from household remittances in international migration: the case of educational attainment in Guatemala. Int J Educ Dev 50(1-11):1–11.
Dustmann, C, Okatenko A (2014) Out-migration, wealth constraints, and the quality of local amenities. J Dev Econ 110:52–63.
Göbel, K (2013) Remittances, expenditure patterns, and gender: parametric and semiparametric evidence from Ecuador. IZA J Migr 2(1):1–19.
Hu, F (2012) Migration, remittances, and children's high school attendance: the case of rural China. Int J Educ Dev 32(3):401–11.
Koska, OA, Saygin PÖ, Çağatay S, Artal-Tur A (2013) International migration, remittances, and the human capital formation of Egyptian children. Int Rev Econ Financ 28:38–50.
McFadden, D (1977) Quantitative methods for analyzing travel behaviour of individuals: some recent developments. Cowles Foundation for Research in Economics, Yale University, New Haven. No. 474.
Mckenzie, DJ (2006) International Migration, Remittances & Brain Drain, Chapter 4: beyond remittances: the effects of migration on Mexican households. The World Bank and Palgrave Macmillan:123–148.
McKenzie, DJ, Rapoport H (2011) Can migration reduce educational attainment? Evidence from Mexico. J Popul Econ 24(4):1331–58.
Nepal, AK (2016) The impact of international remittances on child outcomes and household eexpenditure in Nepal. J Dev Stud 52(6):838–53.
Olivié, I, Ponce J, Onofa M (2009) Remesas, Pobreza y Desigualdad: El Caso de Ecuador. Real Instituto Elcano, Madrid.
Pilarova, T, Kandakov A (2017) The impact of remittances on school attendance: the evidence from the Republic of Moldova. Int J Educ Dev 55:11–6.
Quintana, L, Mendoza MÁ, Correa R (2014) Regiones y Economía en el Ecuador: Crecimiento, Industria, Migración y Empleo, Volume 7 of Análisis Regional. Abya Yala, Ecuador.
Rapoport, H, Docquier F (2006) The economics of migrants' remittances. Handb Econ Giving Altruism Reciprocity 2:1135–98.
Roodman, D (2011) Fitting fully observed recursive mixed-process models with cmp. Stata J 11(2):159–206.
Salas, VB (2014) International remittances and human capital formation. World Dev 59:224–37.
Shrestha, SA (2017) No man left behind: effects of emigration prospects on educational and labour outcomes of non-migrants. The Economic Journal 127(600):495–521.
UNFPA and FLACSO (2008) Ecuador: La Migración Internacional en Cifras. Fondo de Población de las Naciones Unidas (UNFPA)/ Facultad Latinoamericana de Ciencias Sociales (FLACSO), Quito Ecuador.
Wooldridge, JM (2010) Econometric analysis of cross section and panel data. 2nd ed. MIT Press, Cambridge.
We would like to thank the participants at the International Economic Association conference in Mexico City, the Southwestern Social Science Association, seminar participants at the University of New Mexico, Melissa Binder, editor David Lam, and an anonymous referee for the suggestions that greatly improved the manuscript.
There is no funding to declare.
The datasets used and analyzed in the current study are available from the corresponding author or the website of the Ecuadorian National Institute of Statistics and Censuses.
Department of Economics, University of New Mexico, Albuquerque, USA
José R. Bucheli
, Alok K. Bohara
& Matías Fontenla
Search for José R. Bucheli in:
Search for Alok K. Bohara in:
Search for Matías Fontenla in:
Correspondence to José R. Bucheli.
The IZA Journal of Development and Migration is committed to the IZA Guiding Principles of Research Integrity. The authors declare that they have observed these principles.
Bucheli, J.R., Bohara, A.K. & Fontenla, M. Mixed effects of remittances on child education. IZA J Develop Migration 8, 10 (2018) doi:10.1186/s40176-017-0118-y
International migration | CommonCrawl |
THEORY DEPARTMENT
Research Codes
Neutral Transport
Magnetohydrodynamics
Basic Plasma Physics
Turbulence & Transport
Energetic Particles
Helio & Astro Plasmas
A peeling-ballooning eigenfunction calculated using M3D-C$^1$. The red and blue show the perturbed pressure, and the magenta curve the location of the last closed flux surface of the plasma equilibrium. The green triangles show the finite element mesh, which has resolution packed near the edge of the plasma to efficiently resolve the eigenfunction. The peeling-ballooning instability leads to Edge Localized Modes (ELMs) in tokamaks.
The following research codes are actively maintained:
XGC
M3D-C$^1$
COILOPT
STELLOPT
Gkeyll
HMHD
RBQ1D
DEGAS 2
Publications and Data Management website: website.
XGC: the X-point Gyrokinetic Code for Transport in Tokamaks
Near the edge of tokamak plasmas, strong particle and energy sources and sinks (e.g. radiation, contact with a material wall,...) drive the plasma away from thermal equilibrium, thus invalidating the assumptions underpinning fluid theories. The XGC, gyrokinetic, particle-in-cell (PIC) suite-of-codes, XGC1, XGCa and XGC0, were developed to provide a comprehensive description of kinetic transport phenomena in this complicated region; including: heating and cooling, radiation losses, neutral particle recycling and impurity transport.
A particle's "state" is described by the position, ${\bf x}$, and velocity, ${\bf v}$; constituting a a six-dimensional phase space. Gyrokinetic codes average over the very-fast, "gyromotion" of charged particles in strong magnetic fields, ${\bf B}$, and phase space becomes five-dimensional, ${\bf X} \equiv ({\bf x},v_\parallel,\mu)$, where ${\bf x}$ is the position of the "guiding center", $v_\parallel$ is the velocity parallel to ${\bf B}$, and $\mu$ is the "magnetic moment". The density of particles is given by a distribution function, $f({\bf X},t)$; the evolution of which, including collisons, is formally described by the "Vlasov-Maxwell" equation: $f$ evolves along "characteristics", which are the dynamical trajectories of the guiding centers [1], \begin{eqnarray} \dot {\bf x} & = & \left[ v_\parallel {\bf b} + v_\parallel^2 \nabla B \times {\bf b} + {\bf B} \times ( \mu \nabla B - {\bf E}) / B^2 \right] / D,\\ \dot v_\parallel & = & - ( {\bf B} + v_\parallel \nabla B \times {\bf b} ) \cdot ( \mu \nabla B - {\bf E}), \end{eqnarray} where ${\bf E}$ is the electric field, ${\bf b}={\bf B}/|B|$, and $D \equiv 1 + v_\parallel {\bf b} \cdot \nabla \times {\bf b}/B$ ensures the conservation of phase-space volume (Liouville theorem). The non-thermal-equilibrium demands that gyrokinetic codes must evolve the full distribution function, by applying classical "full-f" [2,3] and noise-reducing, "total-f" techniques [4]. (This is in contrast to so-called "$\delta f$" methods, which evolve only a small perturbation to an assumed-static, usually-Maxwellian distribution.) As full-f codes, XGC can include heat and torque input, radiation cooling; and neutral particle recycling [5].
Multiple particle-species (e.g. ions and electrons, ions and impurities) are included; and XGC uses a field-alligned, unstructured mesh in cylindrical coordinates, and so can easily accommodate the irregular magnetic fields in the plasma edge (e.g. the "X-point", "separatrix"). XGC calculates transport in the entire plasma volume; from the "closed-flux-surface", good-confinement region (near the magnetic axis) to the "scrape-off layer" (where magnetic fieldlines intersect the wall and confinement is lost). Collisions between ions, electrons and impurities are evaluated using either (i) a linear, Monte-Carlo operator [6] for test-particles, and the Hirshman-Sigmar operator [7] for field-particle collisions; or (ii) a fully-nonlinear, Fokker-Planck-Landau collision operator [8,9]. XGC codes efficiently exploit massively-parallel computing architecture.
[1] Robert G. Littlejohn, Phys. Fluids 28, 2015 (1985)
[2] C.S. Chang, Seunghoe Ku & H. Weitzner, Phys. Plasmas 11, 2649 (2004)
[3] S. Ku, C.S. Chang & P.H. Diamond, Nucl. Fusion 49, 115021 (2009)
[4] S. Ku, R. Hager et al., J. Comp. Phys. 315, 467 (2016)
[5] D.P. Stotler, C.S. Chang et al., J. Nucl. Mater. 438 S1275 (2013)
[6] Allen H. Boozer & Gioietta Kuo‐Petravic, Phys. Fluids 24, 851 (1981)
[7] S.P. Hirshman & D.J. Sigmar, Nucl. Fusion 21, 1079 (1981)
[8] E. S. Yoon & C.S. Chang, Phys. Plasmas 21, 032503 (2014)
[9] Robert Hager, E.S. Yoon et al., J. Comp. Phys. 315 644 (2016)
[10] Robert Hager & C.S. Chang, Phys. Plasmas 23, 042503 (2016)
[11] D.J. Battaglia, K.H. Burrell et al., Phys. Plasmas 21, 072508 (2014)
[#h34: C-S. Chang, R. Hager, S-H. Ku, 2016-08-05]
Gkeyll: Energy-conserving, discontinuous, high-order discretizations for gyro-kinetic simulations
Discontinuities at cell boundaries are allowed and used to compute a "numerical flux" needed to update the solution. Shown is a DG fit (in the least-square sense) of $x^4+\sin(5x)$ onto constant (left), linear (middle) and quadratic (right) basis functions.
Fusion energy gain in tokamaks depends sensitively on the plasma edge [1,2]; but, because of open- and closed-fieldlines, interaction with divertor plates, neutral particles, and large electromagnetic fluctuations, the edge region is, understandably, difficult to treat analytically. Large-scale, kinetic, numerical simulations are required. The "GKEYLL" code [3] is a flexible, robust, powerful, algorithm that provides numerical calculations of gyrokinetic turbulence, which importantly preserves the conservation laws of gyrokinetics.
Gyrokinetics [4] describes how a distribution of particles, described by a density-distribution function, $f(t,{\bf z})$, evolves in time, $t$; where ${\bf z}=({\bf x},{\bf v})$ describes the position and velocity, i.e. a point in "phase space", of the guiding-center. Elegant theoretical and numerical descriptions of this dynamical system exploit the "Hamiltonian properties", i.e. $\partial f / \partial t + \{f,H\} = 0$, where $H({\bf z})$ is the Hamiltonian (i.e. "energy", e.g. for a Vlasov system, $H(x, v, t)$ $=$ $mv^2/2$ $+$ $q \, \phi(x,t)$, where $\phi$ is the electro-static potential) and $\{g,f\}$ is the Poisson bracket operator [5]. For reliable simulations, numerical discretizations must preserve so-called "quadratic invariants", $\int H \{ f,H\} \, d{\bf z} $ $=$ $ \int f \{ f,H\} \, d{\bf z} = 0$.
Discontinuous Galerkin (DG) algorithms represent the "state-of-art" discretizations of hyperbolic, partial-differential equations [6]. DG combines the key advantages of finite-elements (e.g. low phase error, high accuracy, flexible geometries) with those of finite-volumes (e.g. up-winding, guaranteed positivity/monotonicity, . . . ), and makes efficient use of parallel computing architectures. DG is inherently super-convergent: e.g., whereas finite-volume methods interpolate $p$ points to get $p$-th order accuracy, DG methods in contrast interpolate $p$ points to get $(2p−1)$-th order accuracy, a significant advantage for $p>1$! Use of DG schemes may lead to significant advances towards a production-quality, edge gyrokinetic simulation software with reasonable computational costs.
[1] A. Loarte, et al. Nucl. Fusion 47, S203 (2007)
[2] P.C. Stangeby, The Plasma Boundary of Magnetic Fusion Devices, Institute of Physics Publishing (2000).
[3] A. Hakim, 26th IAEA Fusion Energy Conference (2016)
[4] E.A. Frieman & Liu Chen, Phys. Fluids 25, 502 (1982)
[5] John R. Cary & Alain J. Brizard, Rev. Mod. Phys. 81, 693 (2009)
[6] Bernardo Cockburn & Chi-Wang Shu, J. Comput. Phys. 141, 199 (1998)
[#h44: A. Hakim, 2016-08-05]
NOVA and NOVA-K
Hot beam ion orbits in the Tokamak Fusion Test Reactor (TFTF) shot #103101 [2,4]. Shown (click to enlarge) are passing (a) and trapped (b) orbits represented in both $\psi,R$ and $Z,R$ planes at a time when TAEs were observed.
NOVA is a suite of codes including the linear ideal eigenmode solver to find the solutions of ideal magnetohydrodynamic (MHD) system of equations [1], including such effects as plasma compressibility and realistic tokamak geometry. The kinetic post-processor of the suite, NOVA-K [2], analyses those solutions to find their stability properties. NOVA-K evaluates the wave particle interaction of the eigenmodes such as Toroidal Alfvén Eigenmodes (TAEs) or Reversed Shear Alfvén Eigenmodes (RSAEs) by employing the quadratic form with the perturbed distribution function coming from the drift kinetic equations [2,4]. The hot particle growth rate of ideal MHD eigenmode is expressed via the equation \begin{eqnarray} \frac{\gamma_{h}}{\omega_{AE}}=\frac{\Im\delta W_{k}}{2\delta K}, \end{eqnarray} where $\omega_{AE}$ is Alfvén eigenmode frequency, $\delta W_{k}$ is the potential energy of the mode, and $\delta K$ is the inertial energy. In computations of the hot ion contribution to the growth rate NOVA-K makes use of the constant of motion to represent the hot particle drift orbits. Example of such representation is shown in the figure.
NOVA is routinely used for AE structure computations and comparisons with the experimentally observed instabilities [5,6]. The main limitations of NOVA code are caused by neglecting thermal ion finite Larmor radius (FLR), toroidal rotation, and drift effects for the eigenmode computations. Therefore it can not describe accurately radiative damping for example. Finite element methods are used in radial direction and Fourier harmonics are used in poloidal and toroidal directions.
NOVA-K is able to predict various kinetic growth and damping rates perturbatively, such as the phase space gradient drive from energetic particles, continuum damping, radiative damping, ion/electron Landau damping and trapped electron collisional damping.
More information can be found on Dr. N. N. Gorelenkov's NOVA page.
[1] C. Z. Cheng, M. C. Chance, Phys. Fluids 29, 3695 (1986)
[2] C. Z. Cheng, Phys. Reports 211, 1 (1992)
[3] G. Y. Fu, C. Z. Cheng, K. L. Wong, Phys. Fluids B 5, 4040 (1993)
[4] N. N. Gorelenkov, C. Z. Cheng, G. Y. Fu, Phys. Plasmas 6, 2802 (1999)
[5] M. A. Van Zeeland, G. J. Kramer et al., Phys. Rev. Lett. 97, 135001 (2006)
[6] G. J. Kramer, R. Nazikian et al., Phys. Plasmas 13, 056104 (2006)
[#h45: N. N. Gorelenkov, 2018-07-05]
The guiding center code Orbit, using guiding center equations first derived in White and Chance [1] and more completely in The Theory of Toroidally Confined Plasmas [2], can use equal arc, PEST, or any other straight field line coordinates $\psi_p, \theta, \zeta$, which along with the parallel velocity and energy completely specify the particle position and velocity. The guiding center equations depend only on the magnitude of the $B$ field and functions $I$ and $g$ with $\vec{B} = g\nabla \zeta + I\nabla \theta + \delta \nabla \psi_p$. In simplest form, in an axisymmetric configuration, in straight field line coordinates, and without field perturbation or magnetic field ripple, the equations are \begin{eqnarray} \dot{\theta} = \frac{ \rho_{\parallel}B^2}{D} (1 - \rho_{\parallel}g^{\prime}) + \frac{g}{D}\left[ (\mu + \rho_{\parallel}^2B)\frac{\partial B}{\partial\psi_p} + \frac{\partial \Phi}{\partial\psi_p}\right], \label{tdot2} \end{eqnarray} \begin{eqnarray} \dot{\psi_p} = -\frac{g}{D}\left[(\mu +\rho_{\parallel}^2B)\frac{\partial B}{\partial\theta} + \frac{\partial \Phi}{\partial\theta}\right], \label{psdot2} \end{eqnarray} \begin{eqnarray} \dot{\rho_{\parallel}} = -\frac{(1 -\rho_{\parallel}g^{\prime})}{D} \left[(\mu +\rho_{\parallel}^2B)\frac{\partial B}{\partial\theta} + \frac{\partial\Phi}{\partial\theta}\right], \label{rdot2} \end{eqnarray} \begin{eqnarray} \dot{\zeta} = \frac{ \rho_{\parallel}B^2}{D} (q +\rho_{\parallel}I^{\prime}_{\psi_p}) - \frac{I}{D} \left[(\mu +\rho_{\parallel}^2B)\frac{\partial B}{\partial\psi_p} + \frac{\partial \Phi}{\partial\psi_p}\right], \label{zdot2} \end{eqnarray} with $ D = gq + I + \rho_\parallel(g I'_{\psi_p} - I g'_{\psi_p})$. The terms in $\partial \Phi/\partial\psi_p$, $\partial \Phi/\partial\theta$, $\partial \Phi/\partial\zeta$ are easily recognized as describing $\vec{E}\times \vec{B}$ drift.
Orbit can read numerical equilibria developed by TRANSP or other routines, particle distributions produced by TRANSP, and mode eigenfuncitons produced by NOVA.
The code uses a fourth order Runge Cutta integration routine. It is divided into a main program Orbit.F, which is essentially a heavily commented name list and a set of switches for choosing the type of run, the diagnostics, the data storage, and output.
The code has been used extensively since its inception at PPPL, General Atomics, RFX (Padova) and Kiev for the analysis of mode-induced particle loss for fishbones and TAE modes, induced ripple loss, the modification of particle distributions, local particle transport, etc.
To obtain a copy take all files from /u/ftp/pub/white/Orbit. To submit a job modify the script batch, modify Orbit.F to choose the run desired, type make, and type qsub -q sque batch
[1] R. B. White & M. S. Chance, Phys. Fluids 27, 2455 (1984)
[1] R. B. White, The theory of Toroidally Confined Plasmas, Imperial College Press, 3rd. ed. (2014)
[#h46: R. B. White, 2018-07-05]
GTS: Gyrokinetic Tokamak Simulation Code
Snapshot of a GTS ion-temperature-gradient instability simulation showing the field-line-following mesh and the quasi-2D structure of the electrostatic potential in the presence of microturbulence. Notice the fine structure on the two poloidal planes perpendicular to the toroidal direction, while the potential along the field lines changes very little as you go around the torus. (Image generated from a GTS simulation by Kwan-Liu Ma and his group at the University of California, Davis.)
The Gyrokinetic Tokamak Simulation (GTS) code is a full-geometry, massively parallel $\delta f$ particle-in-cell code [1,2] developed at PPPL to simulate plasma turbulence and transport in practical fusion experiments. The GTS code solves the modern gyrokinetic equation in conservative form [3]: \begin{equation} \frac{\partial f_a}{\partial t}+\frac{1}{B^{*}}\nabla_{\bf Z}\cdot (\dot{{\bf Z}}B^{*}f_a)=\sum\limits_{b} C[f_a, f_b]. \end{equation} for a gyro-center distribution function $f ({\bf Z}, t)$ in 5-dimension phase space ${\bf Z}$, along with the gyrokinetic Poisson equation and Ampere's law for potentials using a $\delta f$ method.
GTS features high robustness at treating globally consistent, shaped cross-section tokamaks; in particular, the highly challenging spherical tokamak geometry such as NSTX and its upgrade NSTX-U. GTS simulations directly import plasma profiles of temperature, density and toroidal rotation from the TRANSP experimental database, along with the related numerical MHD equilibria, including perturbed equilibria. General magnetic coordinates and a field-line-following mesh are employed [1]. The particle gyro-center motion is calculated by Lagrangian equations in the flux coordinates, which allows for accurate particle orbit integration thanks to the separation between fast parallel motion and slow perpendicular drifts. The field-line-following mesh accounts for the nature of the quasi-2D mode structure of drift-wave turbulence in toroidal systems, and hence offers a highly efficient spatial resolution for strongly anisotropic fluctuations in fusion plasmas. Fully-kinetic electron physics is included. In particular, both trapped and untrapped electrons are included in the non-adiabatic response. GTS solves the field equations in configuration space for the turbulence potentials using finite element method on unstructured mesh, which is carried out by PETSc. The real space, global field solver, in principle, retains all toroidal modes from $(m,n) = (0,0)$ all the way up to a limit which is set by grid resolution, and therefore retains full-channel nonlinear energy couplings.
One remarkable feature in GTS, which distinguishes it from the other $\delta f$ particle simulations, is that the weight equations satisfy the incompressibility condition in extended phase space $({\bf Z}, w)$ [4]. Satisfying the incompressibility is actually required in order to correctly solve the $\delta f$ kinetic equation using simulation markers whose distribution function $F({\bf Z}, w, t)$ is advanced along the marker trajectory in the extended phase space according to $F({\bf Z},w,t)=const.$.
In GTS, Coulomb collisions between like particles are implemented via a linearized Fokker-Plank operator with particle, momentum and energy conservation. Electron-ion collisions are simulated by the Lorentz operator. By modeling the same gyrokinetic-Poisson system, GTS is extended to performing global neoclassical simulation in additional to traditional turbulence simulation. More importantly, GTS now is able to do a global gyrokinetic simulation with self-consistent turbulence and neoclassical dynamics coupled together. This remarkable capability is shown to lead to significant new features regarding nonlinear turbulence dynamics, impacting a number of important transport issues in tokamak plasmas. In particular, this capability is critical for the proposed study of nonlinear NTM physics. For example, it allows to calculate bootstrap current in the presence of turbulence [5, 6], which plays a key role for NTM evolution. Currently, GTS has been extended to simulating finite beta physics. This includes low-n shear-Alfvén modes, current driven tearing modes, kinetic ballooning modes and micro-tearing modes.
A state-of-the-art electromagnetic algorithm has been developed and implemented into GTS [7,8] with the goal to achieve a robust, global electromagnetic simulation capability to attack the highly challenging electron transport problem in high-$\beta$ NSTX-U plasmas and be used as a first-principles-based module for integrated whole device modeling of turbulence/neoclassical/MHD physics.
On the front of physics study, GTS simulations have been applied to wide experiments for various problems. Recent applications include discovering new turbulence sources responsible for plasma transport and understanding underlying physics behind the confinement scaling in spherical tokamak experiments [4,9], validating the physics of turbulence-driven plasma flow and first-principles-based model prediction of intrinsic rotation profile against profile against experiments [10], and plasma self-generated non-inductive current in turbulent fusion plasmas [5].
GTS has three levels of parallelism: a one-dimensional domain decomposition in the toroidal direction, dividing both the grid and the particles, a particle distribution within each domain, which further divides the particles between processors, and a loop-level multi-threading method. The domain decomposition and the particle distribution are implemented with MPI, while the loop-level multi-threading is implemented with OpenMP directives.
[1] W. X. Wang, Z. Lin et al., Phys. Plasmas 13, 092505 (2006)
[2] W. X. Wang, P. H. Diamond et al., Phys. Plasmas 17, 072511 (2010)
[3] A. J. Brizard & T. S. Hahm, Rev. Mod. Phys. 79, 421 (2007)
[4] W. X. Wang, S. Ethier et al., Phys. Plasmas 22, 102509 (2015)
[5] W. X. Wang et al., Proc. 24th IAEA Fusion Energy Conference, (2012), TH/P7-14
[6] T. S. Hahm, Nucl. Fusion 53 104026 (2013)
[7] E. A. Startsev & W. W. Lee, Phys. Plasmas 21, 022505 (2014)
[8] E. A. Startsev et al., Paper BM9.00002, APS-DPP Conference, San Jose, CA (2016)
[9] W. X. Wang, S. Ethier et al., Nucl. Fusion 55, 122001 (2015)
[10] B. A. Grierson, W. X. Wang et al., Phys. Rev. Lett. 118, 015002 (2017)
[#h47: W. Wang, 2018-07-11]
HMHD: A 3D Extended MHD Code
A snapshot of plasmoid-mediated turbulent reconnection simulation showing the parallel current density and samples of magnetic field lines.
HMHD is a massively parallel, general purpose 3D extended MHD code that solves the fluid equations of particle density $n$ and momentum density $n\boldsymbol{u}$: \[ \partial_{t}n+\nabla\cdot\left(n\boldsymbol{u}\right)=0, \] \[ \partial_{t}\left(n\boldsymbol{u}\right)+\nabla\cdot\left(n\boldsymbol{uu}\right)=-\nabla\left(p_{i}+p_{e}\right)+\boldsymbol{J}\times\boldsymbol{B}-\nabla\cdot\boldsymbol{\Pi}+\boldsymbol{F}, \] where $\boldsymbol{J}=\nabla\times\boldsymbol{B}$ is the electric current, $p_{e}$ and $p_{i}$ are the electron and ion pressures, $\boldsymbol{\Pi}$ is the viscous stress tensor, and $\boldsymbol{F}$ is an external force. The magnetic field ${\bf B}$ is stepped by the Faraday's law \[ \partial_{t}\boldsymbol{B}=-\nabla\times\boldsymbol{E}, \] where the electric field $\boldsymbol{E}$ is determined by a generalized Ohm's law that incorporates the Hall term and the electron pressure term in the following form: \[ \boldsymbol{E}=-\boldsymbol{u}\times\boldsymbol{B}+d_{i}\frac{\boldsymbol{J}\times\boldsymbol{B}-\nabla p_{e}}{n}+\eta\boldsymbol{J}, \] with $\eta$ the resistivity and $d_{i}$ the ion skin depth. The set of equations is completed by additional equations for electron and ion pressures $p_{e}$ and $p_{i}$, where several options are available. In the simplest level an isothermal equation of state $p_{i}=p_{e}=nT$ is assumed; in a sophisticated level, the ion and electron pressure can be evolved individually with viscous and resistive heating, anisotropic thermal conduction, and thermal exchange between the two species; various additional options are available between these two levels. HMHD is flexible to switch on and off various effects in the governing equations.
HMHD uses a single Cartesian grid, with the capability of variable grid spacing. The numerical algorithm [1] approximates spatial derivatives by finite differences with a five-point stencil in each direction. The time-stepping scheme has several options including a second-order accurate trapezoidal leapfrog method as well as three-stage or four-stage strong stability preserving Runge-Kutta methods [2,3]. HMHD is written in Fortran 90 and parallelized with MPI for domain decomposition, augmented with OpenMP for multi-threading in each domain. HMHD has been employed to carry out large-scale 2D simulations plasmoid-mediated reconnection in resistive MHD [4,5,6] and Hall MHD [7,8]. It has been used to carry out 3D self-generated turbulent reconnection simulations [9,10].
[1] P. N. Guzdar, J. F. Drake et al., Phys. Fluids B 5, 3712 (1993)
[2] S. Gottlieb, C.-W. Shu & E. Tadmor, SIAM Review 43, 89 (2001)
[3] R. J. Spiteri & S. J. Ruuth, SIAM J. Numer. Anal. 40, 469 (2002)
[4] Y.-M. Huang & A. Bhattacharjee, Phys. Plasmas 17, 062104 (2010)
[5] Y.-M. Huang & A. Bhattacharjee, Phys. Rev. Lett. 109, 265002 (2012)
[6] Y.-M. Huang, L. Comisso & A. Bhattacharjee, Astrophys. J. 849, 75 (2017)
[7] Y.-M. Huang, A. Bhattacharjee & B. P. Sullivan, Phys. Plasmas 18, 072109 (2011)
[8] J. Ng, Y.-M. Huang et al., Phys. Plasmas 22, 112104 (2015)
[9] Y.-M. Huang & A. Bhattacharjee, Astrophys. J. 818, 20 (2016)
[10] D. Hu, A. Bhattacharjee & Y.-M. Huang, Phys. Plasmas 25, 062305 (2018)
[#h48: Y-M. Huang, 2018-07-11]
FOCUS: Flexible Optimized Coils Using Space curves
Modular coils for a conventional rotating elliptical stellarator. The color on the internal plasma boundary indicates the strength of mean curvature.
Finding an easy-to-build coils set has been a critical issue for stellarator design for decades. The construction of the coils is only one component of modern fusion experiments; but, realizing that it is the currents in the coils that produce the "magnetic bottle" that confines the plasma, it is easy to understand that designing and accurately constructing suitable coils is paramount.
Conventional approaches simplify this problem by assuming that coils are lying on a defined toroidal "winding" surface [1]. The Flexible Optimized Coils using Space Curves (FOCUS) code [2] represents coils as one-dimensional curves embedded in three-dimensional space. A curve is described directly, and completely generally, in Cartesian coordinates as ${\bf x}(t) = x(t) \, {\bf i} + y(t) \, {\bf j} + z(t) \, {\bf k}$. The coil parameters, ${\bf X}$, are then varied to minimize a target function consisting of multiple objective functions, \begin{equation} \chi^2({\bf X}) \equiv w_{B}\int_S \frac{1}{2} \left ( { \bf B} \cdot {\bf n} \right )^2 ds \ + \ w_{\Psi}\int_0^{2\pi} \frac{1}{2} \left( \Psi_\zeta \; - \; \Psi_o \right)^2 d\zeta \ + \ w_{L} \; \sum_{i=1}^{N_C} L_i + \cdots \end{equation} These objective functions cover both "physics" requirements and "engineering" constraints, such as the normal magnetic field, the toroidal flux, the resonant magnetic harmonics, coil length, coil-to-coil separation and coil-to-plasma separation.
With analytically calculated derivatives, FOCUS computes the gradient and Hessian with respect to coil parameters fast and accurately. It allows FOCUS to employ numerous powerful optimization algorithms, like the Newton method [3]. The figure (click to enlarge) shows using FOCUS to design modular coils for a rotating elliptical stellerator. FOCUS is also applied to analyze the error field sensitivity to coil deviations [4], vary the shape of plasma surface in order to simplify the coil geometry [5] and design non-axisymmetric RMP coils for tokamaks.
[1] P. Merkel, Nucl. Fusion, 27, 867 (1987)
[2] Caoxiang Zhu, Stuart R. Hudson et al., Nucl. Fusion, 58, 016008 (2017)
[3] Caoxiang Zhu, Stuart R. Hudson et al., Plasma Phys. Control. Fusion 60, 065008 (2018)
[5] S. R. Hudson, C. Zhu et al., Phys. Lett. A, in press
[#h49: C. Zhu, 祝曹祥, 2016-08-05]
(a) Mode evolution for different levels of collisionality featuring intermitency and steady saturation within RBQ1D depending on the effective collisionality and (b) scaling of saturation amplitude as a function of collisionality within RBQ1D. Calculations are done for the global TAE shown in panel (c).
The interaction between fast ions and Alfvénic eigenmodes has proved to be numerically expensive to be modeled in realistic large tokamak configurations such as ITER. Therefore, it is motivating to explore reduced, numerically efficient models such as the Resonance Broadened Quasilinear model, used to build the code in its one-dimensional version (RBQ1D). The code is capable of modeling the fast ion distribution function in the direction of the canonical toroidal momentum while self-consistently evolving the amplitude of interacting Alfvénic modes. The theoretical approach is based on the model proposed by Berk et al [1] to address the resonant energetic particle interaction in both regimes of isolated and overlapping modes. RBQ1D is written by using the same structure of the conventional quasilinear equations for the fast ion distribution function but with the resonant delta function broadened primarily in the radial direction. The model was designed to reproduce the expected saturation level for non-overlapping modes from analytic theory. In the model, the nonlinear trapping (bounce) frequency is the fundamental variable for the dynamics in the vicinity at a resonance. The diffusion equation of RBQ1D, derived using action and angle variables, is [2,3] \begin{eqnarray} \frac{\partial f}{\partial t}=\pi\sum_{l,M}\frac{\partial}{\partial P_{\varphi}}C_{M}^{2}\mathcal{E}^{2}\frac{G_{m^{\prime}l}^{*}G_{ml}}{\left|\partial\Omega_{l}/\partial P_{\varphi}\right|_{res}}\mathcal{F}_{lM}\frac{\partial}{\partial P_{\varphi}}f_{lM}+\nu_{eff}^{3}\sum_{l,M}\frac{\partial^{2}}{\partial P_{\varphi}^{2}}\left(f_{lM}-f_{0}\right), \end{eqnarray} and the amplitude evolution satisfies \begin{eqnarray} C_{M}\left(t\right)\sim e^{\left(\gamma_{L}+\gamma_{d}\right)t}\Rightarrow\frac{dC_{M}^{2}}{dt}=2\left(\gamma_{L}+\gamma_{d}\right)C_{M}^{2}. \end{eqnarray} RBQ1D employs a finite-difference scheme used for numerical integration of the distribution function. It recovers several scenarios for amplitude evolution, such as pulsations, intermittency and quasi-steady saturation, see figure (a). The code is interfaced with linear ideal MHD code NOVA, which provides eigenstructures, and the stability code NOVA-K, which provides damping rates and wave-particle interaction matrices for resonances in 3D constant of motion space. RBQ1D employs an iterative procedure to account for mode structure non-uniformities within the resonant island [4]. RBQ1D has been subject to rigorous verification exercises [4]. Both the wave-particle resonant diffusion and Coulomb collisional scattering diffusion operator are thoroughly verified against analytical expressions in limiting cases. In addition, the collisional scattering frequency dependence of the modes saturation level is similar to the value expected by the analytical theory, as shown in figure (b).
[1] H. L. Berk, B. N. Breizman et al., Nucl. Fusion 35, 1661 (1995)
[2] V. N. Duarte, Ph.D. Thesis, U. São Paulo (2017)
[3] N. Gorelenkov, V. Duarte et al., Nucl. Fusion 58, 082016 (2018)
[4] N. N. Gorelenkov, Invited Talk, APS-DPP (2018)
[#h50: V. Duarte, 2018-09-25]
HYbrid and MHD simulation code (HYM)
(a-b) Representative co-helicity spheromak merging simulation results, magnetic field lines are shown; final configuration corresponds to a 3D Taylor eigenstate. (c-d) Hybrid simulations of FRC spin-up and instability for non-symmetric end-shortening boundary conditions, magnetic field lines and plasma density are shown.
The nonlinear 3-D simulation code (HYM) has been originally developed at PPPL to carry out investigations of the macroscopic stability properties of FRCs [1,2]. The HYM code has also been used to study spheromak merging [3], and the excitation of sub-cyclotron frequency waves by the beam ions in the National Spherical Torus Experiment (NSTX) [4,5]. In the HYM code, three different physical models have been implemented: (a) a 3-D nonlinear resistive MHD or Hall-MHD model; (b) a 3-D nonlinear hybrid scheme with fluid electrons and particle ions; and (c) a 3-D nonlinear hybrid MHD/particle model where a fluid description is used to represent the thermal background plasma, and a kinetic (particle) description is used for the low-density energetic beam ions. The nonlinear delta-f method has been implemented in order to reduce numerical noise in the particle simulations. The capabilities of the HYM code also include an option to switch from the delta-f method to the regular particle-in-cell (PIC) method in the highly nonlinear regime. An MPI-based, parallel version of the HYM code has been developed to run on distributed-memory parallel computers. For production-size MHD runs, very good parallel scaling has been obtained for up to 1000 processors at the NERSC Computer Center. The HYM code has been validated against FRC experiments [6], SSX spheromak merging experiments [3], and NSTX and NSTX-U experimental results related to stability of sub-cyclotron frequency Alfven eigenmodes [4,5].
The HYM code is unique in that it employs the delta-f particle simulation method and a full-ion-orbit description in a toroidal geometry. Second-order, time-centered, explicit scheme is used for time stepping, with smaller time steps for field equations (subcycling). Fourth-order finite difference and cylindrical geometry are used to advance fields and apply boundary conditions, while a 3D Cartesian grid is used for the particle pushing and gathering of fast ion density and current density. Typically, the total energy is conserved within 10% of the wave energy, provided that the numerical resolution is sufficient for the mode of interest. Both 3D hybrid simulations of spheromak merging and 3D simulations of the FRC compression require use of a full-f simulation scheme, and therefore a large number of simulation particles. The HYM code has been modified in order to allow simulations with up to several billions of simulation particles.
The initial equilibrium used in the HYM code is calculated using a Grad-Shafranov solver. The equilibrium solver allows the computation of MHD equilibria including the effects of toroidal flows [1]. In addition, the MPI version of the Grad-Shafranov solver has been developed for kinetic equilibria with a non-Maxwellian and anisotropic ion distribution function [7].
The ability to choose between different physical models implemented in the HYM code facilitates the study of a variety of physical effects for a wide range of magnetic configurations. Thus, the numerical simulations have been performed for both oblate and prolate field-reversed configurations, with elongation in the range E=0.5-12, in both kinetic and MHD-like regimes; in support of co-helicity and counter-helicity spheromak merging experiments; for rotating magnetic field (RMF) FRC studies; and investigation of the effects of neutral beam injection (NBI) ions on FRC stability. In addition, hybrid simulations using the HYM code have predicted for the first time, destabilization of the Global Alfven Eigenmodes (GAEs) by the energetic beam ions in the National Spherical Torus Experiment (NSTX) [8], subsequently confirmed both by the analytical calculations and the experimental observations, and suggested a new energy channeling mechanism explaining flattening of the electron temperature profiles at high beam power in NSTX [4].
[1] E. V. Belova, S. C. Jardin et al., Phys. Plasmas 7, 4996 (2000)
[2] E. V. Belova, R. C. Davidson et al., Phys. Plasmas 11, 2523 (2004)
[3] C. Myers, E. V. Belova et al., Phys. Plasmas 18, 112512 (2011)
[4] E. V. Belova, N. N. Gorelenkov et al., Phys. Rev. Lett. 115, 015001 (2015)
[5] E.D. Fredrickson, E. Belova et al., Phys. Rev. Lett. 118, 265001 (2017)
[6] S. P. Gerhardt, E. Belova et al., Phys. Plasmas 13, 112508 (2006)
[7] E. V. Belova, N. N. Gorelenkov & C.Z. Cheng, Phys. Plasmas 10, 3240 (2003)
[8] E. V. Belova, N. N. Gorelenkov et al., "Numerical Study of Instabilities Driven by Energetic Neutral Beam Ions in NSTX", Proceedings of the 30th EPS Conference on Contr. Fusion and Plasma Phys., (2003) ECA Vol. 27A, P-3.102.
[#h51: E. Belova, 2016-08-05]
Slices through the three-dimensional atomic (vertical) and molecular (horizontal) deuterium profiles in a simulation of data from the NSTX Edge Neutral Density Diagnostic.
Neutral atoms and molecules in fusion plasmas are of interest for multiple reasons. First, neutral particles are produced via the interactions of the plasma as it flows along open field lines to surrounding material surfaces. Unconfined by the magnetic field, the atoms and molecules provide a channel for heat transport across the field lines and also serve as a source of plasma particles via ionization. Second, atoms that penetrate well past the last closed flux surface can charge exchange with plasma ions to generate high energy neutrals that can strike the vacuum vessel wall, sputtering impurities into the plasma and possibly damaging the wall. Third, the most common means of fueling plasmas is with an external puff of gas. Finally, the light emitted by the neutral atoms and molecules in all of the above processes can be monitored and used as the basis for diagnostics.
Kinetic models of neutral particle transport are based on the Boltzmann equation. For the simple case of a single ``background'' species and a single binary collision process, this is: \begin{eqnarray*} \frac{\partial f({\bf r}, {\bf v}, t)}{\partial t} & + & {\bf v} \cdot \nabla_{\bf r} f({\bf r}, {\bf v}, t) \\ & = & \int d{\bf v}^{\prime} \, d{\bf V}^{\prime} \, d{\bf V} \sigma( {\bf v}^{\prime}, {\bf V}^{\prime}; {\bf v}, {\bf V}) | {\bf v}^{\prime} - {\bf V}^{\prime} | f({\bf v}^{\prime}) f_{b}({\bf V}^{\prime}) \\ & - & \int d{\bf v}^{\prime} \, d{\bf V}^{\prime} \, d{\bf V} \sigma( {\bf v}, {\bf V}; {\bf v}^{\prime}, {\bf V^{\prime}}) | {\bf v} - {\bf V} | f({\bf v}) f_{b}({\bf V}), \label{BE} \end{eqnarray*} where $f({\bf r}, {\bf v}, t)$ and $f_{b}({\bf r}, {\bf V}, t)$ are the neutral and background distribution functions, respectively, and $\sigma$ is the differential cross section for the collision process. The first (second) integral on the right-hand side represents scattering into (out of) the velocity $v$.
DEGAS 2 [1], like its predecessor, DEGAS [2], uses the Monte Carlo approach to integrating the Boltzmann equation, allowing the treatment of complex geometries, atomic physics, and wall interactions. DEGAS 2 is written in a ``macro-enhanced'' version of FORTRAN via the FWEB library, providing an object oriented capability and simplifying tedious tasks, such as dynamic memory allocation and the reading and written of self-describing binary files. As a result, the code is extremely flexible and can be readily adapted to problems seemingly far removed from tokamak divertor physics, e.g., its use in simulating the diffusive evaporation of lithium in NSTX [3] and LTX [4]. DEGAS 2 has been extensively verified, as is documented in its User's Manual [5]. Experimental validation has been largely centered on the Gas Puff Imaging (GPI) technique for visualizing plasma turbulence in the tokamak edge. The validation against deuterium gas puff data from NSTX is described in the paper by B. Cao et al. [6] Analogous work with both deuterium and helium has been carried out on Alcator C-Mod. A related application of DEGAS 2 is in the interpretation of data from the Edge Neutral Density Diagnostic on NSTX [7] and NSTX-U. DEGAS 2 has been applied to many other devices, including JT-60U [8], ADITYA [9], and FRC experiments at Tri-Alpha Energy [10]. Neutral transport codes are frequently coupled to plasma simulation codes to allow a self-consistent plasma-neutral solution to be computed. Initially, DEGAS 2 was coupled to UEDGE [11]. More recently, DEGAS 2 has been coupled to the drift-kinetic XGC0 [12], and has been used in the development and testing of the simplified built-in neutral transport module in XGC1 [13]. A related project is a DEGAS 2-based synthetic GPI diagnostic for XGC1 [14].
[1] D. P. Stotler & C. F. F. Karney, Contrib. Plasma Phys. 34, 392 (1994)
[2] D. Heifetz, D. Post et al., J. Comp. Phys. 46, 309 (1982)
[3] D. P. Stotler et al., J. Nucl. Mater. 415, S1058 (2011)
[4] J. C. Schmitt et al., J. Nucl. Mater. 438, S1096 (2013)
[5] DEGAS 2 Home Page
[6] B. Cao et al., Fusion Sci. Tech. 64, 29 (2013)
[7] D. P. Stotler et al., Phys. Plasmas 22, 082506 (2015)
[8] H. Takenaga et al., Nucl. Fusion 41, 1777 (2001)
[9] R. Dey et al., Nucl. Fusion 57, 086003 (2017)
[10] E. M. Granstedt et al., Presented at 60th Annual Meeting of the APS Division of Plasma Physics
[11] D. P. Stotler et al., Contrib. Plasma Phys. 40, 221 (2000)[
[12] D. P. Stotler et al., Comput. Sci. Disc. 6, 015006 (2013)
[13] D. P. Stotler et al., Nucl. Fusion 57, 086028 (2017)
[14] D. P. Stotler et al., Nucl. Mater. Energy 19, 113 (2019)
[#h52: D. Stotler, 2016-08-05]
M3D-C$^1$ Extended MHD code
The accurate calculation of the equilibrium, stability and dynamical evolution of magnetically-confined plasma is fundamental for fusion research. The most suitable, macroscopic model to address some of the most critical challenges confronting tokamak plasmas is given by the extended-magnetohydrodynamic (MHD) equations, which describe plasmas as electrically conducting fluids of ions and electrons. The M3D-C$^1$ code [?] solves the fluid equations: for example, the "single-fluid" model, in which the ions and electrons are considered to have the same fluid velocity and temperature, the dynamical equations for the particle number density $n$, the fluid velocity $\vec{u}$, the total pressure $p$ are \begin{eqnarray} \frac{\partial n}{\partial t} + \nabla \cdot (n \vec{u}) & = & 0 \\ n m_i \left( \frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u} \right) & = & \vec{J} \times \vec{B} - \nabla p \color{blue}{ - \nabla \cdot \Pi + \vec{F}} \\ \frac{\partial p}{\partial t} + \vec{u} \cdot \nabla p + \Gamma p \nabla \cdot \vec{u} & = & \color{blue}{(\Gamma - 1) \left[Q - \nabla \cdot \vec{q} + \eta J^2 - \vec{u} \cdot \vec{F} - \Pi : \nabla u \right]} \end{eqnarray} together with a "generalized Ohm's law", $\vec{E} = - \vec{u} \times \vec{B} \color{blue}{ + \eta \vec{J}}$; and with a reduced set of Maxwell's equations for the electrical-current density, $\vec{J} = \nabla \times \vec{B} / \mu_0$, and for the time evolution of the magnetic field, $\partial_t \vec{B} = - \nabla \times \vec{E}$. The manifestly divergence-free magnetic field, $\vec{B}$, and the fluid velocity, $\vec{u}$, are represented using stream functions and potentials, $\vec{B} = \nabla \psi \times \nabla \varphi + F \nabla \varphi$ and $\vec{u} = R^2 \nabla U \times \nabla \varphi + R^2 \omega \nabla \varphi + R^{-2} \nabla_\perp \chi$.
The "ideal-MHD" model is described by the above equations with the terms in blue set to zero. M3D-C$^1$ is also capable of computing the "two-fluid model", which accommodates differences in the ion and electron fluid velocities: the generalized Ohm's law is augmented with $\color{red}{( \vec{J} \times \vec{B} - \nabla p_e - \nabla \cdot \Pi_e + \vec{F}_e ) / n_e}$, additional terms must be added to $\partial_t p$, and an equation for the "electron pressure", $p_e$, must be included.
Physically-meaningful, "reduced" models provide accurate solutions in certain physical limits and are obtained, at a fraction of the computational cost, by restricting the scalar fields that are evolved: e.g., the two-field, reduced model is obtained by only evolving $\psi$ and $U$, and the "four-field", reduced model by only evolving $\psi$, $U$, $F$, and $\omega$.
To obtain accurate solutions efficiently, over a broad range of temporal and spatial scales, M3D-C1 employs advanced numerical methods, such as: high-order, $C^1$, finite elements, an unstructured geometrical mesh, fully-implicit and semi-implicit time integration, physics-based preconditioning, domain-decomposition parallelization, and the use of scalable, parallel, sparse, linear algebra solvers.
M3D-C$^1$ is used to model numerous tokamak phenomena, including: edge localized modes (ELMs) [1]; sawtooth cycles [2]; and vertical displacement events (VDEs), resistive wall modes (RWMs), and perturbed (i.e. three-dimensional, 3D) equilibria [3].
[1] N.M. Ferraro, S.C. Jardin & P.B. Snyder, Phys. Plasmas 17, 102508 (2010)
[2] S.C. Jardin, N. Ferraro et al., Com. Sci. Disc. 5, 014002 (2012)
[3] N.M. Ferraro, T.E. Evans et al., Nucl. Fusion 53, 073042 (2013)
[#h4: N. Ferraro, 2016-05-23]
Stepped Pressure Equilibrium Code (SPEC)
Building on the theoretical foundations of Bruno & Laurence [1], that three-dimensional, (3D) magnetohydrodynamic (MHD) equilibria with "stepped"-pressure profiles are well-defined and guaranteed to exist, whereas 3D equilibria with integrable magnetic-fields and smooth pressure (or with non-integrable fields and continuous-but-fractal pressure) are pathological [2], Dr. S.R. Hudson wrote the "Stepped Pressure Equilibrium Code", (SPEC) [3]. SPEC finds minimal-plasma-energy states, subject to the constraints of conserved helicity and fluxes in a collection, $i=1, N_V$, of nested sub-volumes, ${\cal R}_i$, by extremizing the multi-region, relaxed-MHD (MRxMHD), energy functional, ${\cal F}$, introduced by Dr. M.J. Hole, Dr. S.R. Hudson & Prof. R.L. Dewar [4,5], \begin{eqnarray} {\cal F} \equiv \sum_{i=1}^{N_V} \left\{ \int_{{\cal R}_i} \! \left( \frac{p}{\gamma-1} + \frac{B^2}{2} \right) dv - \frac{\mu_i}{2} \left( \int_{{\cal R}_i} \!\! {\bf A}\cdot{\bf B} \, dv - H_i \right) \right\}. \end{eqnarray} Relaxation is allowed in each ${\cal R}_i$, so unphysical, parallel currents are avoided and magnetic reconnection is allowed; and the ideal-MHD constraints are enforced at selected "ideal interfaces", ${\cal I}_i$, on which ${\bf B}\cdot{\bf n}=0$. The Euler-Lagrange equations derived from $\delta {\cal F}=0$ are: $\nabla \times {\bf B} = \mu_i {\bf B}$ in each ${\cal R}_i$; and continuity of total pressure, $[[p+B^2/2]]=0$, across each ${\cal I}_i$, so that non-trivial, stepped pressure profiles may be sustained. If $N_V=1$, MRxMHD equilibria reduce to so-called "Taylor states" [6]; and as $N_V \rightarrow \infty$, MRxMHD equilibria approach ideal-MHD equilibria [7]. Discontinuous solutions are admitted. The figure (click to enlarge) shows an $N_V=4$ equilibrium with magnetic islands, chaotic fieldlines and a non-trivial pressure.
[1] Oscar P. Bruno & Peter Laurence, Commun. Pure Appl. Math. 49, 717 (1996)
[2] H. Grad, Phys. Fluids 10, 137 (1967)
[3] S.R. Hudson, R.L. Dewar et al., Phys. Plasmas 19, 112502 (2012)
[4] M.J. Hole, S.R. Hudson & R.L. Dewar, J. Plas. Physics 72, 1167 (2006)
[5] S.R. Hudson, M.J. Hole & R.L. Dewar, Phys. Plasmas 14, 052505 (2007)
[6] J.B. Taylor, Rev. Modern Phys. 58, 741 (1986)
[7] G.R. Dennis, S.R. Hudson et al.,Phys. Plasmas 20, 032509 (2013)
[#h5: S.R. Hudson, 2016-05-23]
Designing Stellarator Coils with the COILOPT++, Coil-Optimization Code
Concept design for a quasi-axisymmetric stellarator fusion reactor with the modular coils straightened and spaced for ease of access. (Figure courtesy of Tom Brown.)
Electrical currents flowing inside magnetically-confined plasmas cannot produce the magnetic field required for the confinement of the plasma itself. Quite aside from understanding the physics of plasmas, the task of designing external, current-carrying coils that produce the confining magnetic field, ${\bf B}_C$, remains a fundamental problem; particularly for the geometrically-complicated, non-axisymmetric, "stellarator" class of confinement device [1]. The coils are subject to severe, engineering constraints: the coils must be "buildable", and at a reasonable cost; and the coils must be supported against the forces they exert upon each other. For reactor maintenance, the coils must allow access to internal structures, such as the vacuum vessel, and must allow room for diagnostics; many, closely-packed coils that give precise control over the external magnetic field might not be satisfactory.
The COILOPT code [2] and its successor, COILOPT++ [3], vary the geometrical degrees-of-freedom, ${\bf x}$, of a discrete set of coils to minimize a physics+engineering, "cost-function", $\chi({\bf x})$, defined as the surface integral over a given, "target", plasma boundary, ${\cal S}$, of the squared normal component of the "error" field, \begin{eqnarray} \chi^2 \equiv \oint_{\cal S} \left( \delta {\bf B} \cdot {\bf n} \right)^2 da + \mbox{engineering constraints}, \end{eqnarray} where $\delta {\bf B} \equiv {\bf B}_C - {\bf B}_P$ is the difference between the externally-produced magnetic field (as computed using the Biot-Savart law [5]) and the magnetic field, ${\bf B}_P$, produced by the plasma currents (determined by an equilibrium calculation). Using mathematical optimization algorithms, by finding an ${\bf x}$ that minimizes $\chi^2$ we find a coil configuration that minimizes the total, normal magnetic field at the boundary; thereby producing a "good flux surface".
A new design methodology [3] has been developed for "modular" coils for so-called "quasi-axisymmetric" stellarators (stellarators for which the field strength appears axisymmetric in appropriate coordinates), leading to coils that are both simpler to construct and that allow easier access [4]. By straightening the coils on the outboard side of the device, additional space is created for insertion and removal of toroidal vessel segments, blanket modules, and so forth. COILOPT++ employs a "spline+straight" representation, with fast, parallel optimization algorithms (e.g. the Levenberg-Marquardt algorithm [6], differential evolution [7], ...), to quickly generate coil designs that produce the target equilibrium. COILOPT++ has allowed design of modular coils for a moderate-aspect-ratio, quasi-axisymmetric, stellarator reactor configuration, an example of which is shown in the figure (click to enlarge).
[1] Lyman Spitzer Jr., Phys. Fluids 1, 253 (1958)
[2] Dennis J. Strickler, Lee A. Berry & Steven P. Hirshman, Fusion Sci. & Technol. 41, 107 (2002)
[3] J.A. Breslau, N. Pomphrey et al., in preparation (2016)
[4] George H. Neilson, David A. Gates et al., IEEE Trans. Plasma Sci. 42, 489 (2014)
[5] Wikipedia, Biot-Savart Law
[6] Wikipedia, Levenberg–Marquardt Algorithm
[7] Wikipedia, Differential Evolution
[#h12: J. Breslau, 2016-05-23]
STELLOPT: Stellarator Optimization and Equilibrium Reconstruction Code
Design of the National Compact Stellarator eXperiment (NCSX) optimized by STELLOPT. The shape of three-dimensional nested flux surfaces is optimized to have good MHD stability, plasma confinement and turbulence transport. External non-planar coils with relatively simple geometries, large coil-plasma space and coil-coil separation are also obtained.
One of the defining characteristics of the "stellarator" class of magnetic confinement device [1] is that the confining magnetic field is, for the most part, generated by external, current-carrying coils; stellarators are consequently more stable than their axisymmetric, "tokamak" cousins, for which an essential component of the confining field is produced by internal plasma currents. Stellarators have historically had degraded confinement, as compared to tokamaks, and require more-complicated, "three-dimensional" geometry; however, by exploiting the three-dimensional shaping of the plasma boundary (which in turn determines the global, plasma equilibrium), stellarators may be designed to provide optimized plasma confinement. This is certainly easier said-than-done: the plasma equilibrium is a nonlinear function of the boundary, and the particle and heat transport are nonlinear functions of the plasma equilibrium!
STELLOPT [2,3] is a versatile, optimization code that constructs suitable, magnetohydrodynamic equilibrium states via minimization of a "cost-function", $\chi^2$, that quantifies how "attractive" an equilibrium is; for example, $\chi^2$ quantifies the stability of the plasma to small perturbations (including "ballooning" stability, "kink" stability) and the particle and heat transport (including "neoclassical" transport, "turbulent" transport, "energetic particle" confinement). The independent, degrees-of-freedom, ${\bf x}$, describe the plasma boundary, which is input for the VMEC equilibrium code [4]. The boundary shape is varied (using either a Levenberg-Marquardt [5], Differential-Evolution [6] or Particle-Swarm [7] algorithms) to find minima of $\chi^2$; thus constructing an optimal, plasma equilibrium with both satisfactory stability and confinement properties.
A recent extension of STELLOPT enables "equilibrium reconstruction" [8,9]. By minimizing $\chi^2({\bf x})$ $\equiv$ $\sum_{i} [ y_i - f_i({\bf x}) ]^2 / \sigma_i^2$, where the $y_i$ and the $f_i({\bf x})$ are, respectively, the experimental measurements and calculated "synthetic diagnostics" (i.e. numerical calculations that mimic signals measured by magnetic sensors) of Thomson scattering, charge exchange, interferometry, Faraday rotation, motional Stark effect, and electro-cyclotron emission reflectometry, to name just a few, STELLOPT solves the "inverse" problem of inferring the plasma state given the experimental measurements. (The $\sigma_i$ are user-adjustable "weights", which may reflect experimental uncertainties.) Equilibrium reconstruction is invaluable for understanding the properties of confined plasmas in present-day experimental devices, such as the DIIID device [10].
[2] S.P. Hirshman, D.A. Spong et al., Phys. Rev. Lett. 80, 528 (1998)
[3] D.A. Spong, S.P. Hirshman et al., Nucl. Fusion 40, 563 (2000)
[4] S.P. Hirshman, J.C. Whitson, Phys. Fluids 26, 3553 (1983)
[5] Donald W. Marquardt, SIAM J. Appl. Math. 11, 431 (1963)
[6] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Reading, Addison-Wesley, 1989)
[7] Riccardo Poli, James Kennedy & Tim Blackwell, Swarm Intell. 1, 33 (2007)
[8] S. Lazerson & I.T. Chapman, Plasma Phys. Control. Fusion 55, 084004 (2013)
[9] J.C. Schmitt, J. Bialek et al., Rev. Sci. Instrum. 85, 11E817 (2014)
[10] S. Lazerson S and the DIII-D Team, Nucl. Fusion 55, 1 (2015)
[#h35: S. Lazerson, 2016-08-18]
© 2023 PPPL Theory Department. All rights reserved.
Research & Review Seminars
Princeton, NJ 08543-0451 U.S.A
Follow PPPL | CommonCrawl |
How to calculate radius of a spherical surface having four circles touching one another?
There are four circles having radii $r_1, r_2, r_3 $ and $r_4$ touching one another on a spherical surface of radius $R$ (as shown in the picture below, four colored circles touching one another at 6 points on the sphere). Any help to find out the radius $R$ in terms of $r_1, r_2, r_3 $ and $r_4$?
Assume that $r_1, r_2, r_3, r_4<R$
circles surfaces
Harish Chandra Rajpoot
Harish Chandra RajpootHarish Chandra Rajpoot
(Since you didn't specify, I'm assuming that the distances $r_1, ...$ are distances on the surface, not in $\mathbb{R}^3$. Sorry if this turns out to be useless!)
Draw the lines between the centers of the circles. This divides the sphere into four spherical triangles -- making it a "spherical tetrahedron", if you will. We know the side lengths of the triangles: the sum of the radii of the circles. For instance, one triangle has side lengths $\{r_1+r_2, r_1+r_3, r_2+r_3\}$.
Then we can find the area $\Delta$ of each triangle. To do so we find the angles using the spherical law of cosines, and then the spherical triangle area formula (see this answer).
$$a = \frac{r_1+r_2}{R}$$ $$C = \arccos\left(\frac{\cos c-\cos a \cos b}{\sin a \sin b}\right)$$ $$\Delta = R^2 (A+B+C-\pi)$$ By substituting these values in for the four sides, twelve angles, and four triangles, and requiring that their areas sum to $4\pi R^2$, we can build an equation for $R$. This can provide the value of having an equation in only one variable to solve -- but I worry that it will be far from solvable by anything other than numerical methods.
Alex MeiburgAlex Meiburg
Sorry, this is not a complete answer, but I don't have enough reputation to simply comment. As a starting point I think you need some constraints, for example, it appears from the image that each circle touches all three other circles - I will assume that this a requirement. Also, as long as it is not necessary that $r1,r2,r3,r4 < R$, then perhaps it would simplify things to consider the special case of $r1 = R$, and then $r2,r3,r4$ reside in a hemisphere (assuming that all circles must touch three other circles).
apoapo
Not the answer you're looking for? Browse other questions tagged circles surfaces or ask your own question.
Area of a Spherical Triangle from Side Lengths
finding one circles radius so that it tangentially touches two other set circles
Twelve identical circles touching one another on the surface of a sphere
Apollonian gasket
Prove that $\frac{1}{r_1}-\frac{1}{r_2}=\frac{2}{a}+\frac{2}{b}+\frac{2}{c}$
A problem on four kissing circles (Descartes Theorem)
Find the center of a third externally tangent circle
Radius of the smaller circle
How long is this line making a loop? | CommonCrawl |
A circle is inscribed in quadrilateral $ABCD$, tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q$. Given that $AP=19$, $PB=26$, $CQ=37$, and $QD=23$, find the square of the radius of the circle.
Call the center of the circle $O$. By drawing the lines from $O$ tangent to the sides and from $O$ to the vertices of the quadrilateral, four pairs of congruent right triangles are formed.
Thus, $\angle{AOP}+\angle{POB}+\angle{COQ}+\angle{QOD}=180$, or $(\arctan(\tfrac{19}{r})+\arctan(\tfrac{26}{r}))+(\arctan(\tfrac{37}{r})+\arctan(\tfrac{23}{r}))=180$.
Take the $\tan$ of both sides and use the identity for $\tan(A+B)$ to get\[\tan(\arctan(\tfrac{19}{r})+\arctan(\tfrac{26}{r}))+\tan(\arctan(\tfrac{37}{r})+\arctan(\tfrac{23}{r}))=n\cdot0=0.\]
Use the identity for $\tan(A+B)$ again to get\[\frac{\tfrac{45}{r}}{1-19\cdot\tfrac{26}{r^2}}+\frac{\tfrac{60}{r}}{1-37\cdot\tfrac{23}{r^2}}=0.\]
Solving gives $r^2=\boxed{647}$. | Math Dataset |
Active transport driven by ATP hydrolysis
Free energy stored in the activated carrier ATP can be used to drive the transport of an ion or small molecule against a gradient - i.e., transport from a region of low to high concentration.
Membrane-embedded transporter proteins can function as machines because binding and catalytic events are coupled to conformational changes.
The cycle below is a simplified, schematic model of a transporter whose ATP hydrolysis step is coupled to "eversion" - transition between inward-facing and outward-facing conformations.
The alpha helices in membrane proteins achieve eversion in roughly the same manner as you might re-adjust the orientations of a group of irregular pens and pencils held in your hand.
Basics of the Cycle
This cycle will pump molecule A from inside to outside a cell or compartment in typical conditions (relatively high ATP concentration).
Like all cycles, this one can be reversed with sufficiently large [A] outside and sufficiently small [ATP]. Thus, free energy stored in the gradient of A can be used to synthesize ATP: see below.
As in the case of gradient-driven transport, the pump itself is actually a passive element or catalyst for the cycle. The free energy is fully supplied by ATP.
Note that the model as shown is a simplified version that omits some possible states and connections among states - e.g., ATP cannot bind the empty receptor in our model.
Additional states and connections could reduce the efficiency of pumping. See discussion in Hill's book.
The presence of additional/states connections would have to be verified on a system-by-system basis.
Understanding how the cycle is driven
A simple but powerful qualitative analysis starts by considering the condition of equilibrium. Equilibrium describes a (hypothetical) very large set or ensemble of identical systems of which - on average - an equal number are executing the forward and reverse of every process. Thus, for example, between steps 2 and 3, if there are $N_{23}$ systems in which ATP binds every second, there is an equal number ($N_{32} = N_{23}$) for which ATP unbinds.
In the perfectly balanced state of equilibrium, there will be equal numbers of systems performing full clockwise and full counter-clockwise cycles per unit time. Thus, although individual cycles may pump A or synthesize ATP, there is no net pumping or synthesis/hydrolysis of ATP.
Because the balance of equilibrium is so perfect, however, it can be disturbed at almost any point in the cycle. Most obviously, adding an excess of ATP will cause more 2-to-3 transitions, which will cause 3-to-4 transitions and so on, leading to counter-clockwise cycling. It is the binding process that provides directionality. Binding events can be considered the "handles" used to drive a cycle in a given direction.
Importantly, the cycle can also be driven by imbalancing the detailed equilibrium at any point in the cycle:
Excess A added inside will also drive the cycle counter-clockwise.
Excess A added outside will drive the cycle clockwise.
Excess ADP will also drive the cycle clockwise.
A quantitative model: Simulation and analysis
We will employ a chemical-kinetics model, which is formulated solely in terms of state populations and rate constants for transitions among states, assumed to obey mass-action behavior.
To keep the model simple, we will make the following assumptions:
We assume ATP hydrolysis to ADP is a unimolecular event -- i.e., no phosphate (Pi) is released. This does not affect the key conclusions.
$\kon$ will be the on-rate (see Notation) for both A and nucleotide binding -- for transitions from state 1 to 2, from 2 to 3, from 6 to 5, and from 5 to 4.
$\koff$ will be the off-rate for all unbinding of A or nucleotide (ATP, ADP) -- reversals of the transitions noted above.
$\kconf$ will be the rate for conformational transitions in both directions in the apo transporter (no A or nucleotide) -- transitions between states 1 and 6.
The coupled process of conformational change and ATP conversion to ADP will be governed by the rates $\ktd$ and $\kdt$ for the processes 3 to 4 and 4 to 3, respectively.
We will assume that the outside and inside volumes are the same -- so that the numbers of A, ATP, and ADP molecules tell us the concentrations in a simple way.
The model is now fully specified. For example, the differential equation governing the population of state 1 (denoted [1]) is gievn by
In words, the equation means that the population of the unbound state 1 decreases due to binding of A with rate $\kon$ and due to conformational transitions to state 6 with rate $\kconf$; it increases due to unbinding from state 2 with rate $\koff$ and due to conformational transitions from state 6 with rate $\kconf$.
Simulating the model
If we start a simulation with all A molecules on the inside and equal numbers of ATP and ADP (which is far from equilibrium), the system equilibrates to a state with essentially all the A molecules pumped outside. A molecules are pumped because the initial condition of equal ATP and ADP concentrations is far from equilibrium: ADP is greatly favored and the system moves toward increased ADP concentration even at the (lesser) cost of pushing A beyond its apparent equilibrium of equal inside and outside concentrations. The data below show the behavior due to a single transporter.
On the other hand, if we start from a very different non-equilibrium condition where no ATP is present and all the A is outside, then ATP will be sythesized (provided there is a sufficiently large amount of A present). The data below show the system evolution when multiple transporters are present.
These simulations were performed using BioNetGen, a rule-based platform for kinetic modeling. The brief source code for the model (a .bngl file) can be downloaded by right-clicking here.
Analyzing the model
Although molecular bio-machines tend to operate out of equilibrium, an equilibrium analysis is the simplest and clearest reference point from which to understand their behavior. In equilibrium, every process (i.e., arrows in the model) will be in balance with its reverse process. In terms of the simple mass-action kinetics we are employing (see above for rates), this means
Solving the full set of equations (by going around the cycle and eliminating one of the numbered-state concentrations at a time) yields the result
Despite its simple appearance, this is actually a constrained equilibrium, as we will see below.
We need to carefully determine the ratio $ \ktd / \kdt $. We can do this by considering any equilibrium situation because the ratio is not a property of the various rates involved, but rather of the overall "stoichiometry" of the machine (i.e., of the fact that one ATP is required to pump one A molecule). We can imagine that our transporter is present in a membrane where there is also a simple channel that allows A to flow in or out without impediment. In this case, A must equilibrate to equal inside and outside concentrations. But also, ATP and ADP will be unconstrained and hence will relax to their "natural" equilibrium -- i.e., the equilibrium they would reach in solution if they were not coupled to any other process. This equilibrium is governed by the ratio of solution ("sol") rates:
which is the same as the equilibrium constant by definition.
To finish our analysis, we substitute (5) into the full-system equilibrium condition (4) for the special case $\conceq{A}_{\mathrm{out}} / \conceq{A}_{\mathrm{in}} = 1$, finding
We thus conclude
which is a result for the particular choices we made for rate constants. Had we chosen different rates for the various processes, a different relation would have been found in terms of the various rates. Note that (7) does not say that the rates in and out of solution are the same: it only says their ratio is the same.
A Constrained Equilibrium
Let us re-write the equilibrium condition (4) using the constraint (7):
This is actually a constrained equilibrium, assuming that the only way A is transported -- and the only way ATP and ADP interconvert -- is through our transporter. Specifically, the concentrations of A generally will not reach their "natural" equilibrium (equal inside and outside concentrations) for arbitrary initial conditions of ATP and ADP. This point is illustrated in the simulation examples above. The equilibrium is constrained by the fact that one ATP is converted to ADP for each A pumped from inside to outside (or the reverse).
Furthermore, as you can show in an excercise, the constrained equilibrium (8) does not depend on our simple rate choices. The same result holds for any rate choices in our cycle which allow detailed balance to be satisfied in equilibrium.
Another exercise: Write an equation for the equilibrium values of all species based on an arbitrary initial condition using a single variable. Ignore molecules that might remain in the transporter. [Hint: Consider the consequences for all species of a single A molecule transported by this model.]
B. Alberts et al., "Molecular Biology of the Cell," Garland Science (many editions available).
T.L. Hill, "Free Energy Transduction and Biochemical Cycle Kinetics," (Dover, 2005). Absolutely the book on cycles. Describes the effects of including additional states and transitions.
C. Hallam and R. Whittam, "The Role of Sodium Ions in ATP Formation by the Sodium Pump," Proc. R. Soc. Lond. B (1977) 198:109-128, doi: 10.1098/rspb.1977.0088
‹ Antiporters (Exchanger or Counter-transporter) up Stoichiometric Effects in Transport › | CommonCrawl |
\begin{document}
\title{Accelerating Large Scale Real-Time GNN Inference using Channel Pruning}
\author{Hongkuan Zhou} \author{Ajitesh Srivastava} \author{Hanqing Zeng} \affiliation{
\institution{University of Southern California}
\city{Los Angeles}
\country{USA} } \email{{hongkuaz,ajiteshs,zengh}@usc.edu}
\author{Rajgopal Kannan} \affiliation{
\institution{US Army Research Lab}
\city{Los Angeles}
\country{USA} } \email{[email protected]}
\author{Viktor Prasanna} \affiliation{
\institution{University of Southern California}
\city{Los Angeles}
\country{USA} } \email{[email protected]}
\begin{abstract}
Graph Neural Networks (GNNs) are proven to be powerful models to generate node embedding for downstream applications. However, due to the high computation complexity of GNN inference, it is hard to deploy GNNs for large-scale or real-time applications. In this paper, we propose to accelerate GNN inference by pruning the dimensions in each layer with negligible accuracy loss. Our pruning framework uses a novel LASSO regression formulation for GNNs to identify feature dimensions (channels) that have high influence on the output activation.
We identify two inference scenarios and design pruning schemes based on their computation and memory usage for each.
To further reduce the inference complexity, we effectively store and reuse hidden features of visited nodes, which significantly reduces the number of supporting nodes needed to compute the target embedding.
We evaluate the proposed method with the node classification problem on five popular datasets and a real-time spam detection application. We demonstrate that the pruned GNN models greatly reduce computation and memory usage with little accuracy loss.
For full inference, the proposed method achieves an average of $3.27\times$ speedup with only $0.002$ drop in F1-Micro on GPU. For batched inference, the proposed method achieves an average of $6.67\times$ speedup with only $0.003$ drop in F1-Micro on CPU.
To the best of our knowledge, we are the first to accelerate large scale real-time GNN inference through channel pruning. \end{abstract}
\maketitle
\begingroup\small\noindent\raggedright\textbf{PVLDB Reference Format:}\\ \vldbauthors. \vldbtitle. PVLDB, \vldbvolume(\vldbissue): \vldbpages, \vldbyear.\\ \href{https://doi.org/\vldbdoi}{doi:\vldbdoi} \endgroup \begingroup \renewcommand\thefootnote{}\footnote{\noindent This work is licensed under the Creative Commons BY-NC-ND 4.0 International License. Visit \url{https://creativecommons.org/licenses/by-nc-nd/4.0/} to view a copy of this license. For any use beyond those covered by this license, obtain permission by emailing \href{mailto:[email protected]}{[email protected]}. Copyright is held by the owner/author(s). Publication rights licensed to the VLDB Endowment. \\ \raggedright Proceedings of the VLDB Endowment, Vol. \vldbvolume, No. \vldbissue\ ISSN 2150-8097. \\ \href{https://doi.org/\vldbdoi}{doi:\vldbdoi} \\ }\addtocounter{footnote}{-1}\endgroup
\ifdefempty{\vldbavailabilityurl}{}{
\begingroup\small\noindent\raggedright\textbf{PVLDB Availability Tag:}\\ The source code of this research paper has been made publicly available at \url{\vldbavailabilityurl}. \endgroup }
\section{Introduction} \label{sec: intro}
Recently, Graph Neural Networks (GNNs) have attracted the attention of many AI researchers due to the high expressive power and generalizability of graphs in many applications. The node embedding generated from GNNs outperforms other graph representation learning methods when fed into downstream applications like node classification, edge prediction, and graph classification. Table \ref{tab: gnn application} shows some popular applications of GNNs on various size graphs with different latency requirements. The knowledge graphs used in few-shot learning could only contain around one hundred of nodes and hundreds of edges, while the social network graphs could have billions of nodes and trillions of edges. Most of these GNN applications are latency sensitive at inference. For example, the applications related to Computer Vision need to perform streaming real-time inference on the data captured by the cameras. The applications related to fraud and spam detection need to identify malicious posts and transactions as fast as possible to avoid the property loss of the victim users. In addition to latency, some vision applications that utilize GNNs \cite{qi20173d} need to perform inference on edge devices with limited computing power and memory, such as self-driving cars with 3D-cameras and radars.
\begin{figure}
\caption{Accuracy and throughput of full inference on the Reddit dataset on GPU.}
\label{fig: acct}
\end{figure}
Compared with traditional graph analytics algorithms, GNNs have high computation cost as one node needs to gather and aggregate feature vectors from all the neighbors in its receptive field to compute a forward pass. To accelerate the training of GNNs, many works \cite{graphsage,graphsaint,fastgcn,s-gcn} adopt stochastic node sampling techniques to reduce the number of supporting neighbors. \textcolor{reb}{GraphNorm \cite{cai2020graphnorm} normalizes the node attributes to speedup the convergence.} As a result, GNNs training scales well with graph size. It only takes seconds to minutes to train on a graph with millions of nodes. However, many GNN applications struggle at inference when deployed to production environment. Performing the full forward pass with all the neighbors at inference leads to high memory usage and latency. The node sampling techniques, when applied to inference, struggle to maintain high accuracy on every sample. In consequence, GNN applications either turn to traditional graph analytics algorithms with lower complexity, or rely on obsolete (not updated recently) embedding. For example, Youtube \cite{halcrow2020grale} turns to label propagation to detect abusive videos. Pinterest \cite{pinterest} has to use obsolete embedding generated with the MapReduce framework in an offline process. Taobao \cite{liuheterogeneous} runs the GNN based malicious account detection daily, instead of immediately after one transaction pops. Even with the compromise of offline inference, GNN inference is still expensive on large graphs. It is reported that a cluster with 378 computing nodes still needs one day to generate embedding for 3 billion nodes \cite{pinterest}. In addition, GraphBERT \cite{zhang2020graph} shows that pre-trained GNN models could be directly (or with light fine-tuning) transferred to address new tasks, which makes accelerating GNN inference more important.
\begin{table}[t!]
\centering
\caption{\textcolor{reb}{GNN Applications with their conventional graph sizes (in number of nodes) and latency requirement.}}
\begin{tabular}{>{\centering\arraybackslash}m{2.0in}|c|c}
Applications & Nodes & Lat.\\
\toprule
\multicolumn{3}{c}{\textbf{Knowledge Graph}}\\
\midrule
Few-shot image classification \cite{garcia2018fewshot,gidaris2019generating} & \textcolor{reb}{$10^2-10^3$} & ms \\
\midrule
Relation extraction and reasoning \cite{schlichtkrull2018modeling} & \textcolor{reb}{$10^3-10^6$} & ms-s \\
\midrule
\multicolumn{3}{c}{\textbf{Image Graph}}\\
\midrule
Point cloud segmentation \cite{wang2019dynamic,qi2017pointnet} & \textcolor{reb}{$10^3-10^6$} & ms\\
\midrule
\multicolumn{3}{c}{\textbf{Spatio-Temporal Graph}}\\
\midrule
Traffic prediction \cite{guo2019attention} & \textcolor{reb}{$10^3-10^6$} & s\\
\midrule
Action recognition \cite{yan2018spatial,qi2018learning} & \textcolor{reb}{$10^2-10^3$} & ms\\
\midrule
\multicolumn{3}{c}{\textbf{Social Network Graph}}\\
\midrule
Recommending system \cite{pinterest,zhang2019star,fanmetapath} & \textcolor{reb}{$10^6-10^9$} & ms\\
\midrule
Spam detection \cite{xianyu,wang2019semi,liuheterogeneous} & \textcolor{reb}{$10^6-10^9$} & ms\\
\end{tabular}
\label{tab: gnn application} \end{table}
Although it has not caught much attention of researchers, accelerating GNN inference is as important as accelerating GNN training. Based on these GNN applications, we define two inference scenarios -- full inference where the target nodes are all the nodes (or a large portion of nodes, i.e., the test set) in the graph, and batched inference where the target nodes are a few nodes. Full inference applies to GNN applications that operate on small to medium size graphs, or perform offline inference on large graphs. Batched inference applies to GNN applications that have strict requirements on latency, or need to be executed on edge devices such as embedded systems and FPGAs. Full inference performs forward propagation on all the nodes in the graph, while batched inference only propagates from the selected supporting nodes of the target nodes. For batched inference, the number of supporting nodes grows exponentially with the number of GNN layers, which is referred to as the ``neighbor explosion'' problem. In this work, we propose to accelerate GNN inference by reducing the input feature dimensions in each GNN layer and reusing the hidden features for visited nodes. Our pruning framework works on most GNN architectures and can significantly improve their inference throughput with little or no loss in accuracy. The main contributions of this work are
\begin{itemize}
\item We develop a novel LASSO regression formulation to prune input channels for GNN layers, which outperforms random and greedy pruning methods.
\item We design different pruning schemes for full inference and batched inference addressing their computation complexity and memory usage.
\item We develop a novel technique to store and reuse the hidden features of visited nodes for batched inference, which mitigates the ``neighbor explosion'' problem.
\item We evaluate the performance of the pruned models on five popular datasets \textcolor{reb}{and a real-time spam detection application}. The pruned GNN models greatly reduce the complexity and memory usage with negligible accuracy loss. For full/batched inference, the pruned models reduce the computation to $0.19\times$/$0.10\times$ and memory requirements to $0.43\times$/$0.18\times$ with only $0.002$/$0.003$ F1-Micro drop on average. The pruned models achieve an average of $3.27\times$/$6.67\times$ speedup for full/batched inference on GPU/CPU. \end{itemize}
\section{Background}
\begin{figure*}
\caption{Illustration of one pruned GNN layer. The $\times$ operator denotes sparse or dense matrix multiplication while the shaded areas denote the pruned channels. The blue areas in the weight matrices $\bm{W}_k^{(i)}$ and the output features $\bm{h}'^{(i)}$ before activation show the pruned channels in the next GNN layer. The red areas in weight matrices $\bm{W}_k^{(i)}$ and the input features $\bm{h}^{(i-1)}$ show the pruned channels in this GNN layer.}
\label{fig:prune}
\end{figure*}
\subsection{Graph Neural Networks}
For a graph $G(V,E)$ where each node $v\in V$ has node attributes $\bm{h}(v)\in\mathbb{R}^f$, GNNs iteratively gather and aggregate information from neighbors to compute node embedding. Denote the matrix of all the output features $\bm{h}^{(i)}(v)$ stacked horizontally in layer-$i$ by $\bm{h}^{(i)}$. Let $\bm{\tilde{A}}$ be the normalized adjacency matrix. In general, the output features $\bm{h}^{(i)}$ of layer-$i$ is computed by
\begin{equation}
\bm{h}^{(i)}=\sigma\left(\mathop{\Vert}_{k=K'}^K\bm{\tilde{A}}^k\bm{h}^{(i-1)}\bm{W}_k^{(i)}\right)
\label{eq: forward} \end{equation} where $\Vert$ denotes the horizontal concatenation operation. $\bm{W}_k^{(i)}$ is the learnable weight matrix of order $k$ in layer-$i$. And $\sigma(\cdot)$ denotes the ReLU activation. We stack multiple layers and let the input of the first layer $\bm{h}^{(0)}=\bm{h}$ to compute the node embedding. For $K'=K=1$, Equation \ref{eq: forward} shows the forward propagation of vanilla Graph Convolutional Network \cite{kipfgcn}. For $K'=0,K=1$, Equation \ref{eq: forward} is the GraphSAGE \cite{graphsage} architecture. For $K'=0,K>1$, Equation \ref{eq: forward} is the MixHop \cite{mixhop} architecture. For other variants of GNNs \cite{gat,jumpingknowledge,gin}, Equation \ref{eq: forward} could be adapted by adding residue connections or alternating the normalized adjacency matrix.
\subsection{Case Study: GraphSAGE Inference} \label{sec: case}
We perform a case study to analyze the complexity and memory usage for both inference scenarios on the widely used GraphSAGE architecture. We choose to analyze the GraphSAGE architecture as it achieves top tier accuracy with relatively high throughput (see Figure \ref{fig: acct}). For the GraphSAGE architecture, $K'=0,K=1$ and the adjacency matrix $\bm{A}$ is normalized by $\tilde{\bm{A}}=\bm{D}^{-1}\bm{A}$ where $\bm{D}$ is the diagonal degree matrix.
\subsubsection{Full Inference}
To perform full inference that computes node embedding for all the nodes in the graph, we \textcolor{reb}{batch the node-wise aggregation and compute sparse-dense matrix multiplication $\bm{\tilde{A}}\cdot\bm{h}^{(i-1)}$.}
Denote the input and output feature dimensions of the weight matrices $\bm{W}_{k}^{(i)}$ by $f_{k}^{\text{in}(i)}$ and $f_{k}^{\text{out}(i)}$ (all input feature dimensions are equal in each layer). Let $|V|$ be the number of nodes in the graph. Assume the average degree of the whole graph is $d$. The average complexity per node $C^{(i)}_{\text{full}}$ and total memory consumption $M^{(i)}_{\text{full}}$ of full inference are
\begin{equation}
\begin{aligned}
C^{(i)}_{\text{full}}&=\mathcal{O}\left(d\min (f_{1}^{\text{in}(i)},f_1^{\text{out}(i)})+\sum_{k=0}^1f_{k}^{\text{in}(i)}f_{k}^{\text{out}(i)}\right)\\
M^{(i)}_{\text{full}}&=|V|\left(f_{0}^{\text{in}(i)}+f_{0}^{\text{out}(i)}+f_{1}^{\text{out}(i)}\right)+\sum_{k=0}^1f_{k}^{\text{in}(i)}f_{k}^{\text{out}(i)}\\
\end{aligned} \end{equation} As the output features of all nodes are computed in every layer, the computation and memory consumption distribute evenly in each layer. \textcolor{reb}{Each branch in one layer also contributes to a non-negligible portion of the computation and memory usage.}
\subsubsection{Batched Inference}
For batched inference, the GraphSAGE architecture aggregates from $L$-hop neighbors. Denote the set of target nodes to infer by $V_t$. In layer-$i$, the average number of supporting nodes is $|V_t|\sum_{l=0}^{L-i+1}d^l$, \textcolor{reb}{which leads to the average complexity per node dominated by the complexity in the last layer}
\begin{equation}
C_{\text{batched}}=\sum_{i=1}^LC_{\text{batched}}^{(i)}=\sum_{i=1}^L\sum_{l=0}^{L-i}d^lC_{\text{full}}^{(i)}=\mathcal{O}(d^{L-1}C_{\text{full}}^{(1)})
\label{eq: c_batched} \end{equation} \textcolor{reb}{Similarly, the memory consumption also peaks in the first layer with the most supporting neighbors.}
\subsection{Related Work} \label{sec: relatedwork}
There are many existing works on channel pruning in Deep Neural Networks. \textcolor{reb}{ The works \cite{he2017channel,10.5555/3157096.3157329} prune the channels in the convolution layer by applying penalized regression on the input channels. ThiNet \cite{thinet} prunes the channels based on statistics from the next layer. The work \cite{ye2018rethinking} forces some channels to freeze during the training and remove them at inference. } \textcolor{reb}{Unlike performing inference on texts or images where each instance is independent with the others, inference of nodes depends on the graph structure and attributes of other supporting nodes. The computation pattern is also different for different inference scenarios. These two challenges make it hard to directly apply the existing channel pruning techniques on GNNs.} Recently, several works \cite{sgc,frasca2020sign} have tried to accelerate training and inference of GNNs by removing the nonlinearity of internal layers and pre-computing the feature aggregation ($\bm{A}^k\bm{h}$). PPRGo \cite{PPRGo} accelerates inference by performing less aggregation as in training. These methods require pre-processing on either the node attributes or the adjacency matrix, which do not apply to evolving graphs. TinyGNN \cite{tinygnn} speeds up inference by training a shallow student GNN supervised by a teacher GNN. \textcolor{reb}{Recently, several works have tried to accelerate the full batch propagation in Equation \ref{eq: forward} through matrix partitioning \cite{graphsaint-ipdps19}, node re-ordering \cite{fullinffpga}, and runtime scheduling \cite{9139807}. Others have developed hardware accelerators \cite{9065592,Zeng_2020} and in-memory processors \cite{8327035}. These hardware-specific optimization techniques do not address the basic problem -- high computation complexity of GNNs.} On the other hand, although not aiming at rapid inference, some works \cite{zhang2018graph,Xu2020Dynamically} propose to prune the edges to reduce the noise aggregated from neighbors. However, they are limited to knowledge graphs with specific inference queries.
In contrast, we propose a general method to reduce the inference complexity by directly pruning the input channels. \textcolor{reb}{ Our method works for all types of graphs and most GNN architectures. Combined with edge pruning methods and architecture simplification methods, our method has the potential to further speed up the inference. In addition, our pruning method does not incur extra sparse operations. The dimensions of the matrix operations are also lower, which makes it easier to design hardware accelerators and in-memory processors. }
\section{Approach}
In GNNs, channels refer to column vectors in the hidden features matrix $\bm{h}^{(i)}$. We propose to solve the channel pruning problem by applying LASSO regression \cite{tibshirani1996regression} directly on the input channels. For a pre-trained GNN model, we prune the channels reversely from the output layer to the input layer. Figure \ref{fig:prune} shows one pruned GNN layer with multiple branches. In this section, we first introduce the formulation and optimization of channel pruning in a single layer. Then, we discuss the pruning schemes for full inference and batched inference. For batched inference, we further propose a novel technique that stores the hidden features for visited nodes and aggregates directly from them during inference.
\subsection{Single Branch Pruning}
To prune the channels, we aim to generate the same output features in a branch before activation $\bm{h}'^{(i)}$ with fewer input dimensions. We focus on $\bm{h}'^{(i)}$ instead of $\bm{h}^{(i)}$ to keep all operations linear. For branch $k$ ($K'\leq k\leq K$) in layer-$i$, let $c_k^{(i)}=f_k^{\text{in}(i)}$ be the number of channels in the original GNN. We formulate the channel pruning problem with budget $\eta_k^{(i)}$ as the following optimization problem
\begin{equation}
\begin{aligned}
&\argmin_{\hat{\bm\beta}_k^{(i)},\widehat{\bm{W}}_k^{(i)}}\left\|\bm{Y}_k^{(i)}-\tilde{\bm{A}}^k\bm{h}^{(i-1)}\odot\hat{\bm\beta}_k^{(i)}\widehat{\bm{W}}_k^{(i)}\right\|_2^2\\
&\text{subject to }\left\|\hat{\bm\beta}_k^{(i)}\right\|_0\leq\eta_k^{(i)}c_k^{(i)}
\end{aligned} \end{equation}
where $\bm{Y}_k^{(i)}=\tilde{\bm{A}}^k\bm{h}^{(i-1)}\bm{W}_k^{(i)}$ is the target output features. $\left\|\cdot\right\|_2$ is the L2-norm and $\left\|\cdot\right\|_0$ is the L0-norm measuring the number of non-zero elements. $\hat{\bm\beta}_k^{(i)}\in\mathbb{R}^{c_k^{(i)}}$ is the coefficient vector acting as masks for each channel. $\odot$ denotes element-wise multiplication on each row of the matrix. If $\hat{\bm\beta}_k^{(i)}(j)=0$, then the $j^\text{th}$ channel of the input features $\bm{h}^{(i-1)}$ can be removed. To solve the optimization problem, we first relax the L0-norm to L1-norm and add a penalty term with penalty factor $\lambda$.
\begin{equation}
\begin{aligned}
&\argmin_{\hat{\bm\beta}_k^{(i)},\widehat{\bm{W}}_k^{(i)}}\left\|\bm{Y}_k^{(i)}-\tilde{\bm{A}}^k\bm{h}^{(i-1)}\odot\hat{\bm\beta}_k^{(i)}\widehat{\bm{W}}_k^{(i)}\right\|_2^2+\lambda\left\|\hat{\bm\beta}_k^{(i)}\right\|_1\\
\end{aligned} \end{equation}
We separate the optimization of $\hat{\bm\beta}_k^{(i)}$ and $\widehat{\bm{W}}_k^{(i)}$ into two sub-problems and optimize on the sub-problems iteratively to find the global minimum. Initially, $\widehat{\bm{W}}_k^{(i)}=\bm{W}_k^{(i)}$ and $\hat{\bm\beta}_k^{(i)}=\mathbbm{1}$. We optimize both sub-problems on the hidden features of the training nodes. We use the training graph \textcolor{reb}{as the normalized adjacency matrix during optimization to avoid information leak}.
\subsubsection{Optimization on $\hat{\bm\beta}_k^{(i)}$}
To optimize $\hat{\bm\beta}_k^{(i)}$, $\widehat{\bm{W}}_k^{(i)}$ is fixed. The problem becomes a classic LASSO regression problem with ``large $n$, small $p$''. We solve the LASSO regression with Stochastic Gradient Descent (SGD). Due to the L1-norm term in the constraint, some mask values in the solution of $\hat{\bm\beta}_k^{(i)}$ would shrink to zero, leading to the removal of the corresponding channels.
\begin{equation}
\begin{aligned}
&\argmin_{\hat{\bm\beta}_k^{(i)}}\left\|\bm{Y}_k^{(i)}-\bm{Z}_k^{(i)}\odot\hat{\bm\beta}_k^{(i)}\right\|_2^2+\lambda\left\|\hat{\bm\beta}_k^{(i)}\right\|_1\\
\end{aligned} \end{equation} where $\bm{Z}_k^{(i)}=\tilde{\bm{A}}^k\bm{h}^{(i-1)}\widehat{\bm{W}}_k^{(i)}$.
\subsubsection{Optimization on $\widehat{\bm{W}}_k^{(i)}$} To optimize $\widehat{\bm{W}}_k^{(i)}$, $\hat{\bm\beta}_k^{(i)}$ is fixed. The problem becomes a quadratic programming problem
\begin{equation}
\argmin_{\widehat{\bm{W}}_k^{(i)}}\left\|\bm{Y}_k^{(i)}-\bm{X}_k^{(i)}\widehat{\bm{W}}_k^{(i)}\right\|_2^2 \end{equation} where $\bm{X}_k^{(i)}=\tilde{\bm{A}}^k\bm{h}^{(i-1)}\odot\hat{\bm\beta}_k^{(i)}$. The closed-form least square solution is given by $\widehat{\bm{W}}_k^{(i)}=(\bm{X}_k^{(i)\top}\bm{X}_k^{(i)})^{-1}\bm{X}_k^{(i)\top}\bm{Y}_k^{(i)}$.
\subsection{Single Layer Pruning} \label{sec: single layer pruning}
To prune the input channels for one layer with multiple branches, we need to ensure each branch shares the same pruned channels in $\bm{h}^{(i-1)}$ so that these channels could be removed in the output of the previous layer. We enforce that the same channels are pruned in each branch by applying a shared coefficient vector $\bm{\hat{\beta}}^{(i)}$ and jointly optimizing on all the branches.
\begin{equation}
\begin{aligned}
&\argmin_{\hat{\bm\beta}^{(i)},\widehat{\bm{W}}_k^{(i)}}\sum_{k=K'}^{K}\left\|\bm{Y}_k^{(i)}-\tilde{\bm{A}}^k\bm{h}^{(i-1)}\odot\hat{\bm\beta}^{(i)}\widehat{\bm{W}}_k^{(i)}\right\|_2^2+\lambda\left\|\hat{\bm\beta}^{(i)}\right\|_1\\
\end{aligned} \end{equation} The sub-problem of $\bm{\hat{\beta}}^{(i)}$ forms a generalized LASSO optimization problem.
\begin{equation}
\begin{aligned}
&\argmin_{\hat{\bm\beta}^{(i)}}\left\|\bm{Y}^{(i)}-g\left(\bm{h}^{(i-1)}\odot\hat{\bm\beta}^{(i)}\right)\right\|_2^2+\lambda\left\|\hat{\bm\beta}^{(i)}\right\|_1\\
\end{aligned} \end{equation} where $\bm{Y}^{(i)}=\Vert_{k=K'}^K\bm{Y}_k^{(i)}$ stacked horizontally. And the generalized function $g(\bm{h}'^{(i-1)})=\Vert_{k=K'}^{K}\tilde{\bm{A}}^k\bm{h}'^{(i-1)}\widehat{\bm{W}}_k^{(i)}$ stacked horizontally. However, as $\bm{Y}^{(i)}$ is also concatenated horizontally, we can rewrite the LASSO optimization problem by substituting the original observations $\bm{h}^{(i-1)}$ with $\tilde{\bm{A}}^k\bm{h}^{(i-1)}$ concatenated vertically. Then, the sub-problem of $\bm{\hat{\beta}}^{(i)}$ falls back to a classic LASSO regression with $(K-K')$ times the observations in the single branch pruning and the same number of predictors.
\textcolor{reb}{Our pruning method also works for other GNN architectures. For GNN architectures with averaging instead of concatenating the output features in each branch, the generalized function becomes $g(\bm{h}^{(i-1)})=\sum_{k=K'}^K\tilde{\bm{A}}^k\bm{h}^{(i-1)}\widehat{\bm{W}}_k^{(i)}/(K-K'+1)$, which can be optimized by concatenating the observations horizontally. For multi-head attention based architecture, we can prune the layers by treating each attention head as a branch.}
\begin{figure}
\caption{Illustration of forward propagation on $L+1$-layer GraphSAGE architecture with stored hidden features. $v^{(i)}$ denotes the supporting nodes for the branch $k=0$. $u_{unv}^{(i)},u_{vis}^{(i)}$ denote the unvisited and visited supporting nodes for the branch $k=1$. The hidden features of the visited nodes $u_{vis}^{(i)}$ are obtained directly from the stored hidden features.}
\label{fig: batched}
\end{figure}
\subsection{End-to-End GNN Pruning}
We design the pruning schemes for different inference scenarios based on the computation complexity and memory consumption in the case study in Section \ref{sec: case}. The channel pruning in layer-$i$ not only ignores some input channels in $\bm{h}^{(i-1)}$, but also leads to the reduction of output channels in the weight matrices $\bm{W}_k^{(i)}$ of the previous layer. Thus, when pruning the whole network, we prune reversely from the output layer to the input layer. Dense layers are treated as GNN layers with $K'=K=0$.
\subsubsection{Pruned Full Inference} \label{sec: prunedfullinf}
For full inference, we simply prune each layer with a constant budget $\eta$ except the input layer as the computation and memory distribute evenly in each layer. We do not remove any dimensions of the raw node attributes in layer-$0$. \textcolor{reb}{ The complexity per node and memory usage of the pruned models range in $(\eta^2,\eta)$ and $(\eta,1)$, compared with the original model. For other GNN architectures that follow similar forward propagation in Equation \ref{eq: forward} like JK \cite{jumpingknowledge} and SIGN \cite{frasca2020sign}, our pruning method could be directly applied with constant budget.}
\subsubsection{Pruned Batched Inference} \label{sec: prunedbatchinf}
The major challenge in batched inference is the ``neighbor explosion'' problem where the number of supporting nodes grows exponentially as the network goes deeper. We need to visit the node attributes for an exponential amount of nodes to compute the embedding for one target node. Therefore, we focus on reducing the computation and memory usage \textcolor{reb}{in the first layer by reducing the channels in the first layer and the second layer. In the first layer, we focus on the branches that have more neighbors than others.} For the GraphSAGE architecture with pruning budget $\eta$, we prune the $k=1$ branch in layer-$1$ and the whole layer-$2$ with budget $\eta$, which reduces the dominant terms in the computation and memory usage by $\eta$.
In addition to channel pruning, we store the hidden features $\bm{h}^{(i)}$ of visited nodes in the middle layers. Their neighbors, when aggregating from them, directly aggregate from the stored hidden features, instead of iteratively looking at farther neighbors. Figure \ref{fig: batched} shows the supporting nodes in each layer with stored hidden features. Ideally, if we store the hidden features for all visited nodes, the batched inference would have exactly the same complexity as full inference (i.e., $d=1$ in Equation \ref{eq: c_batched}). However, indexing and storing the hidden features incur extra data transfer which increases the latency. On evolving graphs, out-dated hidden features also affect accuracy. \textcolor{reb}{The portion of hidden features to store in each batch could be dynamically determined by the task-specific target latency and accuracy. Applications with high latency tolerance could potentially save more hidden features to increase throughput. For out-dated hidden features, we can set a threshold and discard them when the accuracy drop reaches the threshold. In practice, we find storing the hidden features for the root nodes at inference is a good balance point for the datasets we use. In addition, the root nodes usually have the most up-to-date hidden features in batched inference.}
\subsubsection{Detailed Optimization Procedure} \label{sec: detailopt}
In the experiment, we perform one iteration on each sub-problem instead of multiple iterations \cite{he2017channel}. For the sub-problem of $\widehat{\bm{W}}$, instead of the least square solution, we also apply SGD as the size of $\bm{X}$ could be large. We partition the matrix $\tilde{\bm{A}}^k\bm{h}^{(i-1)}$ and $\bm{X}_k^{(i)}$ row-wise to form mini-batches. Define one epoch as performing SGD on the whole matrix once. To optimize the whole problem, we first optimize several epochs on the sub-problem of $\hat{\bm\beta}$. At the end of each epoch, we slightly increase the penalty factor $\lambda$ until \textcolor{reb}{pruning budget is met or over-penalized (all values in $\bm\beta$ are decreasing)}. Note that as the mask values converge to zero, some mask values may be exactly zero while the others are close to zero. We clip the masks with small values to zero according to the pruning budget to make sure the corresponding channels are completely removed. Then, we optimize the sub-problem for $\widehat{\bm{W}}_k^{(i)}$ until converge. The final weights of the pruned layer are obtained by applying the mask $\hat{\bm\beta}_k^{(i)}$ to the weights $\widehat{\bm{W}}_k^{(i)}$.
\section{Experiments}
\begin{table}[t!]
\centering
\caption{Dataset statistics. The Attr. column shows the dimension of the node attributes. (s) in Classes denotes multi-class single-label classification problem while (m) denotes multi-class multi-label classification problem. The Test\% column shows the percentage of test nodes.}
\begin{tabular}{r|c@{\hspace{1.2ex}}c@{\hspace{1.2ex}}c@{\hspace{1.2ex}}c@{\hspace{1.2ex}}c}
Dataset & Nodes & Edges & Attr. & Classes & Test\%\\
\toprule
Flickr & 89,250 & 899,756 & 500 & 7(s) & 25\%\\
Arxiv & 169,343 & 1,166,243 & 128 & 40(s) & 29\%\\
Reddit & 232,965 & 11,606,919 & 602 & 41(s) & 24\%\\
Yelp & 716,847 & 6,977,410 & 300 & 100(m) & 10\%\\
Products & 2,449,029 & 61,859,140 & 100 & 47(s) & 88 \%\\
\midrule
\textcolor{reb}{YelpCHI} & \textcolor{reb}{67,395} & \textcolor{reb}{287,619} & \textcolor{reb}{769} & \textcolor{reb}{2(s)} & \textcolor{reb}{23\%}\\
\end{tabular}
\label{tab: dataset} \end{table}
\begin{table*}
\setlength{\tabcolsep}{1.35mm}
\fontsize{9}{9}\selectfont
\centering
\caption{\textcolor{reb}{Pruned full inference results on GPU. The $\ast$ nodes in the plots denote the results of the reference models (no pruning).}}
\begin{tabular}{c|cccc|cccc|cccc|cccc}
& & \multicolumn{2}{c}{Flickr} & & & \multicolumn{2}{c}{Arxiv} & & & \multicolumn{2}{c}{Reddit} & & & \multicolumn{2}{c}{Yelp}\\
\toprule
Budget & - & 2$\times$ & 4$\times$ & 8$\times$ & - & 2$\times$ & 4$\times$ & 8$\times$ & - & 2$\times$ & 4$\times$ & 8$\times$ & - & 2$\times$ & 4$\times$ & 8$\times$\\
\midrule
\multirow{1}{*}{F1-Micro} & 0.511 & \textcolor{reb}{0.517} & \textcolor{reb}{0.520} & \textcolor{reb}{0.517} & 0.716 & 0.712 & \textcolor{reb}{0.710} & \textcolor{reb}{0.706} & 0.966 & 0.966 & \textcolor{reb}{0.964} & \textcolor{reb}{0.959} & 0.654 & \textcolor{reb}{0.654} & \textcolor{reb}{0.651} & \textcolor{reb}{0.640}\\
\midrule
\#kMACs/node & 545 & 211 & 94 & 48 & 1242 & 360 & 115 & 40 & 317 & 172 & 112 & 85 & 1490 & 485 & 180 & 77\\
\midrule
Mem. (MB) & 531 & 269 & 221 & 199 & 1997 & 1002 & 505 & 257 & 852 & 738 & 681 & 652 & 8459 & 4256 & 2155 & 1225\\
\midrule
Thpt. (mN/s) & 2.69 & 5.28 & 9.11 & 15.13 & 1.11 & 1.98 & 3.79 & 6.72 & 2.47 & 4.30 & 7.17 & 10.35 & 0.90 & 1.95 & 3.05 & 4.40\\
\midrule
Thpt. Impr. & - & 1.96$\times$ & 3.39$\times$ & 5.63$\times$ & - & 1.79$\times$ & 3.42$\times$ & 6.07$\times$ & - & 1.74$\times$ & 2.90$\times$ & 4.19$\times$ & - & 2.16$\times$ & 3.38$\times$ & 4.82$\times$ \\
\midrule
\begin{tabular}{c}F1mic-Thpt\end{tabular} &
\multicolumn{4}{c|}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e9.pdf}
\end{tabular}
} &
\multicolumn{4}{c|}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e10.pdf}
\end{tabular}
} &
\multicolumn{4}{c|}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e11.pdf}
\end{tabular}
} &
\multicolumn{4}{c}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e12.pdf}
\end{tabular}
}
\end{tabular}
\label{tab: full} \end{table*}
We evaluate the performance of the proposed method with the node classification problem on five popular datasets: 1. \textbf{Flickr} \cite{graphsaint} classifying the types of user-uploaded images, 2. \textbf{Arxiv} \cite{hu2020open} classifying subject areas of Arxiv CS papers, 3. \textbf{Reddit} \cite{graphsage} classifying communities of Reddit posts, 4. \textbf{Yelp} \cite{graphsaint} classifying types of businesses on Yelp, 5. \textbf{Products} \cite{hu2020open} classifying categories of products on Amazon. \textcolor{reb}{For batched inference, we also evaluate with a real world spam detection application on the YelpCHI\cite{yelpchi} dataset that identifies spam reviews on Yelp.} We adopt supervised and inductive settings on all datasets.
For the models to prune, we use the 2-layer GraphSAGE \cite{graphsage} architecture (Equation \ref{eq: forward} with $K'=0,K=1$) with the common hidden feature size $256,512,128,512,512$ on the five \textcolor{reb}{node classification} datasets, respectively. \textcolor{reb}{On the YelpCHI dataset, we use $128$ as the hidden feature size.} We use the standard single floating point precision for both the original models and the pruned models. To obtain trained models to prune, we adopt the sub-graph based training technique from GraphSAINT \cite{graphsaint} with the random walk sampler. For each dataset, we prune with three global budgets $\eta=0.5,0.25,0.125$ and obtain three pruned models ($2\times,4\times,8\times$) in different sizes. We choose 1024 as the batch size and use the ADAM optimizer for SGD in the two sub-problems. After pruning, we re-train the pruned models until convergence.
To test the speedup of the pruned models, we measure the throughput and latency of full inference on the first four datasets with GPU, and batched inference on all five datasets with CPU and GPU. For batched inference, we form batches randomly from the nodes in the test set until all the nodes in the test set are covered. All accuracy (F1-Micro) results are for the test nodes only. The pruning framework is implemented using PyTorch and Python3. \textcolor{reb}{We run all experiments on a machine with 64-core ThreadRipper 2990WX CPU with 256GB of DDR4 RAM, and a single NVIDIA RTX A6000 GPU with 48GB of GDDR6 RAM.} All the accuracy results are averages of three runs. For batched inference, we limit the number of hop-2 neighbors to be 32.
\subsection{Performance of Single Layer Pruning}
\begin{figure}
\caption{Loss and F1-Micro curves under different numbers of pruned channels in layer-$2$ on the Reddit dataset. \textcolor{reb}{We also show the percentage of $\bm\beta$ that shrinks to zero for the LASSO pruning method in the left figure.}}
\label{fig: loss-acc}
\end{figure}
We compare the proposed pruning method (LASSO) with pruning the channels with small L1-norm in the corresponding weight matrix (Max Res.) and randomly pruning the channels (Random). Figure \ref{fig: loss-acc} shows the loss and F1-Micro curves under different numbers of pruned channels in both branches of layer-$2$ on the Reddit dataset. We apply layer-wise re-training for all three pruning methods. The proposed pruning method clearly outperforms other pruning methods, especially when the number of pruned channels is more than $30\%$.
\subsection{Full Inference}
\begin{table*}
\setlength{\tabcolsep}{1.1mm}
\centering
\caption{\textcolor{reb}{Pruned batched inferences results on CPU (batch size=512). The second rows in each metric show the results with stored hidden features. The $\ast$ nodes in the plots denote the results of the reference models (no pruning, w/o store).}}
\fontsize{9.3}{9.3}\selectfont
\begin{tabular}{c|cccc|cccc|cccc|cccc}
& & \multicolumn{2}{c}{Arxiv} & & & \multicolumn{2}{c}{Reddit} & & & \multicolumn{2}{c}{Yelp} & & & \multicolumn{2}{c}{Products}\\
\toprule
Budget & - & 2$\times$ & 4$\times$ & 8$\times$ & - & 2$\times$ & 4$\times$ & 8$\times$ & - & 2$\times$ & 4$\times$ & 8$\times$ & - & 2$\times$ & 4$\times$ & 8$\times$\\
\midrule
F1-Micro & 0.714 & 0.710 & 0.709 & 0.707 & 0.966 & 0.966 & 0.964 & 0.955 & 0.654 & 0.654 & 0.652 & 0.646 & 0.792 & 0.791 & 0.785 & \textcolor{reb}{0.764}\\
w/ store & 0.714 & 0.710 & 0.709 & 0.707 & 0.966 & 0.966 & 0.964 & 0.954 & 0.653 & 0.653 & 0.652 & 0.646 & 0.792 & 0.792 & 0.786 & \textcolor{reb}{0.766}\\
\midrule
\#kMACs/node & 3135 & 1620 & 846 & 395 & 17665 & 7409 & 3288 & 1052 & 7870 & 3696 & 1650 & 840 & 3952 & 2044 & 1090 & 520\\
w/ store & 2118 & 1096 & 577 & 286 & 6225 & 2627 & 1171 & 381 & 3908 & 1888 & 895 & 485 & 1590 & 827 & 446 & 240\\
\midrule
Mem. (MB) & 85 & 49 & 30 & 14 & 3086 & 1551 & 790 & 409 & 225 & 122 & 53 & 26 & 96 & 65 & 49 & 28\\
w/store & 72 & 42 & 23 & 10 & 1431 & 568 & 288 & 147 & 165 & 92 & 39 & 19 & 70 & 37 & 21 & 10\\
\midrule
Lat. (ms) & 27 & 20 & 17 & 13 & 411 & 217 & 128 & 85 & 56 & 38 & 25 & 16 & 120 & 58 & 48 & 35\\
w/ store & 15 & 8 & 6 & 5 & 101 & 56 & 34 & 24 & 22 & 14 & 10 & 7 & 47 & 30 & 27 & 20\\
\midrule
Lat. Impr. & - & 1.33$\times$ & 1.54$\times$ & 2.12$\times$ & - & 1.90$\times$ & 3.22$\times$ & 4.86$\times$ & - & 1.46$\times$ & 2.20$\times$ & 3.59$\times$ & - & 2.09$\times$ & 2.50$\times$ & 3.42$\times$\\
w/ store & 1.82$\times$ & 3.24$\times$ & 4.18$\times$ & 5.29$\times$ & 4.09$\times$ & 7.35$\times$ & 12.26$\times$ & 17.04$\times$ & 2.50$\times$ & 3.84$\times$ & 5.84$\times$ & 8.08$\times$ & 2.56$\times$ & 3.99$\times$ & 4.40$\times$ & 6.02$\times$ \\
\midrule
\begin{tabular}{c}F1mic-Lat.\\\textcolor{blue}{w/o store}\\\textcolor{red}{w/ store}\end{tabular} &
\multicolumn{4}{c|}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e13.pdf}
\end{tabular}
} &
\multicolumn{4}{c|}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e6.pdf}
\end{tabular}
} &
\multicolumn{4}{c|}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e7.pdf}
\end{tabular}
} &
\multicolumn{4}{c}{
\begin{tabular}{c}
\includegraphics{EXTERNAL_TIKZ_FOLDER/e8.pdf}
\end{tabular}
}
\end{tabular}
\label{tab: batched} \end{table*}
Table \ref{tab: full} shows the results for full inference on GPU. The computation complexity is measured in thousands of Multiplication-and-ACcumulation operations per node (\#kMACs/node). The throughput is measured in thousand of target nodes computed per second (kN/s) or million of target nodes computed per second (mN/s). For memory usage, we adopt in-place point-wise operations without storing the intermediate values as we only need to compute forward propagation at inference. The latency is the GPU execution time of a complete forward propagation. The throughput and memory usage are calculated for all the nodes in the graphs. In the F1mic-Thpt. row, the x and y axes are the throughput (in mN/s) and F1-micro. On the Flickr dataset, the pruned models achieve higher F1-Micro than the original models, possibly due to better convergence of smaller models. The reduction in computation and memory usage depends on the dimension of the input node attributes. We achieve close to $\eta^2$ reduction in computation complexity and $\eta$ reduction in memory usage on the Arxiv and Yelp dataset with small input node attributes dimensions. We achieve an average of $3.27\times$ speedup on GPU with less than 0.006 drop in F1-Micro for all datasets with $4\times$ pruned models. On the Flickr, Arxiv and Reddit datasets, the $8\times$ pruned models still achieve similar accuracy as the original models. The pruned models for full inference reduce the latency on small datasets to meet the requirements for real-time applications and increase the throughput for large datasets. The pruned models also make it possible to run full batch inference of small graphs on edge devices with limited memory. \textcolor{reb}{We observe consistent GPU utilization of around 50\% for models with different pruning budgets.}
\textcolor{reb} {
On the Flickr, Arxiv, Reddit, and Yelp dataset, our pruning method takes 2.35, 4.34, 6.35, and 32.15 seconds in pruning and 1.36, 6.38, 10.02, and 346.21 seconds in re-training.
Due to the small number of parameters in the pruned models, the re-training of the pruned models takes less time than training the original models. }
\subsection{Batched Inference} \label{sec: batchedinf}
Table \ref{tab: batched} shows the results for batched inference on CPU. We calculate the memory usage by the amount of memory needed to compute the forward path of one batch. The attributes and stored hidden features of the supporting nodes in each batch are fed into CPU from DDR4 (peak bandwidth 68GB/s) and GPU from GDDR6 (peak bandwidth 768GB/s) memory. In the F1mic-Thpt. row, the x and y axes are the throughput (in kN/s) and F1-micro. We achieve $\eta$ reduction in computation complexity and memory usage on all datasets for batched inference without stored hidden features. We store the hidden features of training and validation nodes, and the root nodes in each batch of inference. The storing of hidden features further reduces an average of $33\%$ of supporting nodes in layer-$1$. We reduce the memory usage from 85-3086MB to 10-147MB, which makes it possible to perform inference on edge devices like mobiles. The memory usage also reflects an upper bound of the amount of input node attributes needed to perform one batch of forward propagation. On all five datasets, the pruned models with stored hidden features achieve less than 30ms (up to $17\times$ improvement) latency on CPU with less than 0.012 F1-Micro drop. The pruned models with stored hidden feature meet the requirements of most real-time applications on CPU. \textcolor{reb}{On GPU, our $4\times$ models achieve $4,16,4,8$ms latency without stored hidden features and $4,6,3,6$ms latency with stored hidden features on the Arxiv, Reddit, Yelp, and Products datasets, respectively.} Compared with full inference, batched inference requires less memory and computation for a small number of target nodes. For latency-sensitive or large scale applications, batched inference provides a light-weight and low-latency solution. \textcolor{reb}{We observe 100\% CPU utilization on all models, and 20\% to 50\% GPU utilization on GPU depending on the model size.}
\begin{figure}
\caption{\textcolor{reb}{(a). Latency curves under different batch sizes on the Reddit dataset on CPU. (b). Maximum extra latency and accuracy drop curves with different percentages of stored hidden features.}}
\label{fig: batchsize}
\end{figure}
Figure \ref{fig: batchsize}.a shows the latency under different batch sizes on the Reddit dataset on CPU. \textcolor{reb}{The latency grows linearly with the batch size, which shows that our pruning method accommodates applications with different inference batch sizes.} \textcolor{reb}{Figure \ref{fig: batchsize}.b shows the trade-off between storing hidden features and extra latency and drop in F1-Micro. Note that the extra latency is mostly caused by the storing of the hidden features, which can be done offline. }
\subsubsection{Spam Detection Application} \label{sec: yelpchi}
\textcolor{reb}{To evaluate the performance of the proposed pruning technique on real-time applications, we over-sample the YelpCHI \cite{yelpchi} dataset 400 times to create a graph with 27 million nodes which has similar scale as the Yelp website. The nodes in the graph represent reviews for restaurants and hotels in Chicago and are attached with timestamps. The task is to identify spam reviews from the posted reviews between October 2011 and October 2012. We adopt the strategy to perform inference on the emerging reviews every 30 minutes and re-train the model every month. The $1\times$(reference), $2\times$, $4\times$, and $8\times$ models achieve 0.873, 0.871, 0.866, and 0.865 accuracy on the test set. Figure \ref{fig: yelpchi} shows the accuracy and maximum latency of each day in the first month. For inference with stored hidden features, the first few days have higher latency due to the indexing and storing of the hidden features. However, the latency is still lower than inference without stored hidden features, even in the first few days.}
\begin{figure}
\caption{\textcolor{reb}{Accuracy and maximum latency of each day in the first month on the YelpCHI dataset. We only show the accuracy without stored hidden features because the accuracy with stored hidden features is very close. }}
\label{fig: yelpchi}
\end{figure}
\subsection{Comparison with Other GNNs}
We compare the throughput and accuracy of full inference with GAT \cite{gat}, SIGN \cite{frasca2020sign}, Jumping Knowledge Network (JK) \cite{jumpingknowledge}, GraphSAGE \cite{graphsage}, PPRGo \cite{PPRGo} \footnote{We tune the parameters of PPRGo to fit the supervised learning setting.}, GCN \cite{kipfgcn}, SGC \cite{sgc}, and \textcolor{reb}{TinyGNN \cite{tinygnn}}. We use a similar two-layer (or equivalent of two-hop propagation) architecture on all baselines \textcolor{reb}{except a one-layer student network supervised by a two-layer teacher network for TinyGNN}. Figure \ref{fig: acct} shows the inference throughput and accuracy of various GNN architectures on the Reddit dataset on GPU. Our $4\times$ model achieves top-tier accuracy, comparable with GAT, SIGN and GraphSAGE, but with significant improvement in throughput ($6.96\times,4.74\times,2.59\times$ with GAT, SIGN, and GraphSAGE, respectively).
\newcommand{\mcrot}[4]{\multicolumn{#1}{#2}{\rlap{\rotatebox{#3}{#4}~}}} \begin{table}[t]
\centering
\begin{tabular}{c|c@{\hspace{-1ex}}c@{\hspace{2ex}}c@{\hspace{1ex}}c}
& & Pre-Proc. & F1-micro & \#kMACs/node\\
\toprule
\multirow{5}{*}{\rotatebox[origin=c]{90}{\parbox{1cm}{Full Inf.}}} & \multirow{2}{*}{SGC} & - & \multirow{2}{*}{0.949} & 146 \\
& & \ding{51} & & 25 \\
\cline{2-5}
& \multirow{2}{*}{SIGN(2,0,0)} & - & \multirow{2}{*}{0.966} & 978 \\
& & \ding{51} & & 858 \\
\cline{2-5}
& PPRGo & - & 0.937 & 148\\
\cline{2-5}
& \textcolor{reb}{TinyGNN} & \textcolor{reb}{-} & \textcolor{reb}{0.957} & \textcolor{reb}{273} \\
\cline{2-5}
& ours-4$\times$ & - & 0.964 & 112 \\
\toprule
\multirow{3}{*}{\rotatebox[origin=c]{90}{\parbox{1cm}{Batched Inf.}}}& MLP-2 & - & 0.702 & 120\\
\cline{2-5}
& ours-4$\times$ w/o& - & 0.964 & 3288 \\
& ours-4$\times$ w/& - & 0.964 & 1171 \\
\end{tabular}
\caption{Comparison of accuracy and per node computation for full inference and batched inference on the Reddit dataset. \textcolor{reb}{The w/ and w/o in batched inference denotes with and without stored hidden features.}}
\label{tab: comp} \end{table}
\subsubsection{Computation Comparison with Simplified GNNs} We compare the accuracy and per node computation on the Reddit dataset of our 4$\times$ pruning models with SGC, SIGN with $(r,s,t)=(2,0,0)$, PPRGo with two-pass inference, \textcolor{reb}{TinyGNN with a 1-layer PAM student network supervised by a 2-layer teacher network,} and 2-layer MLP with 128 hidden features (MLP-2). \textcolor{reb}{Table \ref{tab: comp} shows the result of the comparison with other simplified GNNs.} The pre-processing for both SGC and SIGN is to twice compute feature propagation ($\tilde{\bm{A}}^2\cdot\bm{h}^{(0)}$) for 120 kMACs/node. If any graph structure or node attributes change, the pre-processing needs to be re-computed. For SGC, if the input node features are pre-processed, there is only one MLP layer transforming the aggregate features to class probabilities, leading to the lowest computation. SIGN has the highest per node computation as the numbers of hidden units in the feedforward layers are high (460 for GNN layers and 675 for the classification layer). For full inference, our pruned model achieves higher accuracy than SGC and TinyGNN, and comparable accuracy to SIGN with less computation. For batched inference, our pruned models achieve remarkably higher accuracy than MLP.
\section{Conclusion}
We presented a novel method of pruning the input channels to accelerate large scale and real-time GNN inference. We formulated the GNN pruning problem as a LASSO optimization problem to select from the input channels to approximate the output. We developed different pruning schemes according to the computation complexity and memory usage in different inference scenarios. We designed a unique technique for batched inference to further reduce computation by storing and reusing the hidden features. We conducted experiments on real-world datasets to demonstrate that the pruned models greatly reduce computation and memory usage while still maintaining high accuracy. We showed the improvement on latency and throughput of using the pruned models on CPU and GPU. The light-weight pruned models are attractive to energy-efficient devices like mobile processors and FPGA, as well as applications like real-time recommendation and fraud detection on social networks.
\begin{acks}
This work is supported by the National Science Foundation (NSF) Research Fund of SPX: Collaborative Research: FASTLEAP: FPGA based compact Deep Learning Platform (Number CCF-1919289). Any opinions, findings and conclusions or recommendations expressed in this work are those of the authors and do not necessarily reflect the views of NSF. \end{acks}
\end{document} | arXiv |
\begin{definition}[Definition:Self-Map]
Let $S$ be a set.
A '''self-map on $S$''' is a mapping from $S$ to itself:
:$f: S \to S$
\end{definition} | ProofWiki |
\begin{document}
\title{Methods for Efficient Unfolding of Colored Petri Nets}
Colored Petri nets offer a compact and user friendly representation of traditional Petri nets also known as P/T nets. Colored Petri nets with finite color ranges can be unfolded into traditional P/T nets. However, this unfolding may produce exponentially larger P/T nets. We present two novel formal techniques based on static analyses for reducing the size of unfolded colored Petri nets. The first method identifies colors that behave equivalent in the colored Petri net and groups them into equivalence classes representing the colors, producing a smaller quotiented colored Petri net. The second method analyses which colors can ever exist in any place and excludes colors that can never be present in a given place. Both methods show great promise individually, but even more when combined. The combined method is able to reduce the size of multiple colored Petri nets from the Model Checking Contest compared to state of the art unfolders MCC, Spike and ITS-Tools, while still remaining competitive in terms of unfolding time. Lastly, we show the effect of the smaller unfolded nets by verifying queries from the 2020 Model Checking Contest using the unfolded nets from each tool. \end{abstract}
\begin{abstract} Colored Petri nets offer a compact and user friendly representation of the traditional Place/Transition (P/T) nets and colored nets with finite color ranges can be unfolded into the underlying P/T nets, however, at the expense of an exponential explosion in size. We present two novel techniques based on static analysis in order to reduce the size of unfolded colored nets. The first method identifies colors that behave equivalently and groups them into equivalence classes, potentially reducing the number of used colors. The second method overapproximates the sets of colors that can appear in places and excludes colors that can never be present in a given place. Both methods are complementary and the combined approach allows us to significantly reduce the size of multiple colored Petri nets from the Model Checking Contest benchmark. We compare the performance of our unfolder with state-of-the-art techniques implemented in the tools MCC, Spike and ITS-Tools, and while our approach is competitive w.r.t. unfolding time, it also outperforms the existing approaches both in the size of unfolded nets as well as in the number of answered model checking queries from the 2021 Model Checking Contest. \end{abstract}
When modelling large systems it is important to verify that the system is working as intended. Since many systems are under constant change it is necessary to automate the verification process. One of the most widely used of such verifiable models are Petri nets introduced by Carl A. Petri in 1962~\cite{CarlPetriNets}.
Petri nets are most often seen in a classical Place/Transition (P/T) form and thus most verification techniques have been developed for P/T nets~\cite{MurataPaper}. However, P/T nets often become too large and incomprehensible for humans to read. Therefore, colored Petri nets (CPN)~\cite{CPNJensen} have been introduced which are compressed versions of Petri nets, that employ different techniques from programming such as types, variables and logical expressions.
In CPNs each place is assigned a color domain and each token in that place has a color from the color domain. Arcs have expressions that define what colored tokens to consume and produce and transitions have guard expressions that restrict when a transition may fire.
A CPN can be translated into a P/T net if every color domain is finite through a process called \textit{unfolding}, which allows the use of efficient verification methods already developed for P/T nets. When unfolding a CPN each place is unfolded into a new place for each color that a token can take in the place; a naive approach is to create a new place for each color in the color domain of the place. Transitions are unfolded such that each binding of variables that satisfies the guard is unfolded into a new transition in the unfolded net.
The size of an unfolded net can be exponentially larger than the colored net and therefore requires certain optimizations and analyses to unfold in realistic time and memory. Several types of analyses have been proposed that consider especially transition guards and arc expressions~\cite{MCCUnfolder, PatternPaper, IDDUnfolder}.
However, even with the optimizations there still exists CPNs that cannot be unfolded. As an example, the largest instances of the nets \textit{FamilyReunion}~\cite{FamilyReunionNet, FamilyReunionPaper} and \textit{DrinkVendingMachine}~\cite{DrinkVendingMachineNet, DrinkVendingMachinePaper} from the Model Checking Contest~\cite{mcc:2020} have yet to be unfolded. This is the case since the color domains and number of possible bindings are far too large to be unfolded in realistic time and memory with current techniques. Even if the net is unfolded it may still be too large to verify due to the exponential increase in size.
We propose two new methods for statically analysing a CPN to reduce the size of the unfolded P/T net. The first method called \textit{color quotienting} uses the fact that sometimes multiple colors behave the same throughout the colored net. If such colors exist in the net we can create equivalence classes that represents the colors with similar behaviour. As such, we can reduce the amount of colors that we need to consider when unfolding, because we now only have to consider the equivalence class of colors instead of each color individually.
The second method called \textit{color approximation} detects which colors may actually be present in any given place s.t. we only unfold places for the colors that can exist. This method also allows for invalidating bindings that are dependent on unreachable colors thus reducing the amount of transitions that are unfolded.
\paragraph{Related work:} Several unique approaches for unfolding CPNs effectively have been proposed. In~\cite{PatternPaper} Heiner et al. analyses the arc- and guard expressions to reduce the amount of bindings that need to be considered for a given transition by collecting \textit{patterns}. The pattern analysis is implemented in the tool Snoopy~\cite{SnoopyTool}. The color approximation method captures the reductions of the pattern analysis. In~\cite{IDDUnfolder} the same authors present a technique for representing and using patterns utilizing Interval Decision Diagrams. This technique is used in the tools Snoopy~\cite{SnoopyTool}, MARCIE~\cite{MARCIETool} and Spike~\cite{SpikeTool}. It proved to be generally faster than the method presented in~\cite{PatternPaper}.
In~\cite{MCCUnfolder} (MCC) Dal Zilio describes a method which he calls \textit{stable places}. A stable place is a place that never changes from the initial marking, i.e. every time a token is consumed from this place an equivalent token is added to the place. For a place to be stable it has to hold that for each transition the place is connected to, it is connected by both an input and output arc that have equivalent arc expressions. As such, it is only a syntactical check, meaning no further analysis is done. This method is especially efficient on the net BART from the Model Checking Contest~\cite{mcc:2020}. However, this method is not very general and does not find places that deviate even a little from the initial marking. The color approximation method is a more general form of the stable places and is able to capture these deviations from the initial marking. In~\cite{MCCUnfolder} an additional method called \textit{Component Analysis} is presented where it is detected that a net consists of a number of copies of the same component. MCC is used in the TINA toolchain~\cite{TINAtool} and to our knowledge in the latest release of the LoLA tool~\cite{LoLA2}.
GreatSPN~\cite{GSPNPaper} is another tool for unfolding CPNs, however in~\cite{MCCUnfolder} it is demonstrated that MCC was able to greatly outperform GreatSPN and as such we omit GreatSPN from later experiments.
ITS-Tools~\cite{ITSToolPaper} has an integrated unfolding engine. The tool uses a technique we refer to as \textit{variable symmetry identification}, in which it is analyzed whether variables $x$ and $y$ are actually permutable in a binding. This allows for invalidation of bindings that are equivalent by constraining $x \leq y$. Furthermore, they use stable places during the binding and they apply analysis to choose the binding order of parameters to simplify false guards as soon as possible. After unfolding ITS-Tools applies some post-unfolding reductions that removes orphan places and transitions and removes behaviourally equivalent transitions should they make them~\cite{YannCorrespondence}.
In \cite{YannSymmetryDetection} Thierry-Mieg et al. present a technique for automatic detection of symmetries in high level Petri nets used to construct symbolic reachability graphs in the GreapSPN tool. This detection of symmetries is very reminiscent of the color quotienting method presented in this paper, although the color quotienting method is used for unfolding the colored Petri net instead of symbolic model checking.
In~\cite{KlostergaardThesis} Klostergaard presents the unfolding method implemented in \textit{verifypn} revision 226 (untimed engine of TAPAAL~\cite{TAPAALTool, verifypnPaper}), which is the base of our implementation. The implementation is efficient, but since it is implemented as the naive approach mentioned earlier, there are several nets which it cannot unfold such as BART.
Both the color quotienting method and the color approximation method are advanced static analyses techniques and all of the above mentioned techniques, except symmetric variables and component analysis, are captured by color approximation and color quotienting.
\subsection{Bibliographical Remark} Parts of this thesis are inspired or derived from the work in our 9th semester project~\cite{ninthsemester}. Specifically, the Abstract and Section~\ref{sec:Introduction} are modified and extended versions from the previous work where especially the related work has been extended to cover more related tools. Section \ref{sec:Preliminaries} has been modified s.t. we instead define bisimulation and define isomorphism as a special case of bisimulation. Section \ref{Sec:IntegerCPN} uses the same syntax and semantics as described in~\cite{ninthsemester} but is described more informally in this work. Section \ref{Sec:ColorApprox} is a revised and improved version of the main contribution in~\cite{ninthsemester}. Specifically, we have updated the color expansion to be a function instead of a transition relation, in order to use fixed point theory. The rest of the section is adjusted accordingly. Furthermore, we optimize the color approximation implementation presented in~\cite{ninthsemester}. Lastly, Section~\ref{sec:FutureWork} includes similar future work as in the previous work, but we extend it to consider new data structures and the new methods presented in this project.
\section{Introduction} \label{sec:Introduction}
Petri nets~\cite{CarlPetriNets}, also known as P/T nets, are a powerful modelling formalism supported by a rich family of verification techniques~\cite{MurataPaper}. However, P/T nets often become too large and incomprehensible for humans to read. Therefore, colored Petri nets (CPN)~\cite{CPNJensen} were introduced to allow for high level modelling of distributed systems. In CPNs, each place is assigned a color domain and each token in that place has a color from its domain. Arcs have expressions that define what colored tokens to consume or produce, and transitions have guard expressions that restrict transition enabledness.
A CPN can be translated into an equivalent P/T net, provided that every color domain is finite, through a process called \textit{unfolding}. This allows us to use efficient verification tools already developed for P/T nets. When unfolding a CPN, each place is unfolded into a new place for each color that a token can take in that place; a naive approach is to create a new place for each color in the color domain of the place. Transitions are unfolded such that each binding of variables to colors, satisfying the guard, is unfolded into a new transition copy in the unfolded net. The size of an unfolded net can be exponentially larger than the colored net and the unfolding process therefore requires optimizations in order to finish in realistic time and memory. Several types of improvements were proposed that analyse transition guards and arc expressions~\cite{MCCUnfolder, PatternPaper, IDDUnfolder}. However, even with these optimizations, there still exist CPNs that cannot be unfolded using the existing tools. As an example, the largest instances of the nets \textit{FamilyReunion}~\cite{FamilyReunionNet, FamilyReunionPaper} and \textit{DrinkVendingMachine}~\cite{DrinkVendingMachineNet, DrinkVendingMachinePaper} from the Model Checking Contest~\cite{mcc:2021} have not yet been unfolded in the competition setup.
We propose two novel methods for statically analysing a CPN to reduce the size of the unfolded P/T net. The first method called \textit{color quotienting} uses the fact that sometimes tokens with different colors can be indistinguishable in the sense that they generate bisimilar behaviour. If such colors exist in the net, we can create equivalence classes that represent the colors with similar behaviour. As such, we can reduce the amount of colors that we need to consider when unfolding.
The second method called \textit{color approximation} overapproximates which colors can possibly be present in any given place such that we only unfold places for the colors that can exist. This method also allows for invalidating bindings that are dependent on unreachable colors, thus reducing the amount of transitions that are unfolded.
Our two methods are implemented in the model checker TAPAAL~\cite{TAPAALTool, verifypnPaper} and an extensive experimental evaluation shows convincing performance compared to the state-of-the-art tools for CPN unfolding.
\paragraph{Related work.}
Heiner et al.~\cite{PatternPaper} analyse the arc and guard expressions to reduce the amount of bindings by collecting \textit{patterns}. The pattern analysis is implemented in the tool Snoopy~\cite{SnoopyTool} and our color approximation method further extends this method.
In~\cite{IDDUnfolder} the same authors present a technique for representing the patterns as Interval Decision Diagrams. This technique is used in the tools Snoopy~\cite{SnoopyTool}, MARCIE~\cite{MARCIETool} and Spike~\cite{SpikeTool} and performs better compared to~\cite{PatternPaper}; it also allows to unfold a superset of colored nets compared to the format adopted by the Model Checking Contest benchmark~\cite{mcc:2021}.
In~\cite{MCCUnfolder} Dal-Zilio describes a method (part of the unfolder MCC) called \textit{stable places}. A stable place is a place that never changes from the initial marking, i.e. every time a token is consumed from this place an equivalent token of the same color is added to the place.
This method is especially efficient on the net BART from the Model Checking Contest~\cite{mcc:2021}, however, it does not detect places that deviate even a little from the initial marking. Our color approximation method includes a more general form of the stable places.
In the unfolder MCC~\cite{MCCUnfolder}, a \textit{component analysis} is introduced and it detects if a net consists of a number of copies of the same component. MCC is used in the TINA toolchain~\cite{TINAtool} and to our knowledge in the latest release of the LoLA tool~\cite{LoLA2}. GreatSPN~\cite{GSPNPaper} is another tool for unfolding CPNs, however, in~\cite{MCCUnfolder} it is demonstrated that MCC is able to greatly outperform GreatSPN and as such we omit GreatSPN from later experiments.
ITS-Tools~\cite{ITSToolPaper} has an integrated unfolding engine. The tool uses a technique of \textit{variable symmetry identification}, in which it is analyzed whether variables $x$ and $y$ are permutable in a binding. Furthermore, they use stable places during the binding and they apply analysis to choose the binding order of parameters to simplify false guards as soon as possible. After unfolding, ITS-Tools applies further post-unfolding reductions that remove orphan places/transitions and behaviourally equivalent transitions~\cite{YannCorrespondence}. Our implementation includes a variant of the symmetric variables reduction as well. In \cite{YannSymmetryDetection} Thierry-Mieg et al. present a technique for automatic detection of symmetries in high level Petri nets used to construct symbolic reachability graphs in the GreatSPN tool. This detection of symmetries is reminiscent of the color quotienting method presented in this paper, although our color quotienting method is used for unfolding the colored Petri net instead of symbolic model checking.
In~\cite{KlostergaardThesis} Klostergaard presents a simple unfolding method implemented in TAPAAL~\cite{TAPAALTool, verifypnPaper}, which is the base of our implementation. The implementation is efficient
but there are several nets which it cannot unfold. Both unfolding methods introduced in this paper are advanced static analyses techniques and we observe that the above mentioned techniques, except symmetric variables and component analysis, are captured by color approximation and/or color quotienting.
This paper is an extended version of the conference paper~\cite{BJPST:RP:2021} with full proofs, complete definitions and additional examples and last but not least a substantial reimplementation of the methods in the tool TAPAAL with improved experimential results compared to~\cite{BJPST:RP:2021}.
\section{Preliminaries} \label{sec:Preliminaries} Let $\mathbb{N}^{> 0}$ be the set of positive integers and $\mathbb{N}^{0}$ the set of nonnegative integers. A Labeled Transition System (LTS) is a triple $(Q,Act, \xrightarrow{})$ where $Q$ is a set of states, $Act$ is a finite, nonempty set of actions, and $\xrightarrow{} \subseteq Q \times Act \times Q$ is the transition relation. We write $s \xrightarrow{a}$ if there is $s' \in Q$ such that $s \xrightarrow{a} s'$ and $s \not\xrightarrow{a}$ if there is no such $s'$. A binary relation $R$ over the set of states of an LTS is a \emph{bisimulation} iff for every $(s_1,s_2) \in R$ and $a \in Act$ it holds that if $s_1 \xrightarrow{a} s_1'$ then there is a transition $s_2 \xrightarrow{a} s_2'$ such that $(s_1',s_2') \in R$, and if $s_2 \xrightarrow{a} s_2'$ then there is a transition $s_1 \xrightarrow{a} s_1'$ such that $(s_1',s_2') \in R$. Two states $s$ and $s'$ are \emph{bisimilar}, written $s \sim s'$, iff there is a bisimulation $R$ such that $(s,s') \in R$.
A finite \emph{multiset} over some nonempty set $A$ is a collection of elements from $A$ where each element occurs in the multiset a finite amount of times; a multiset $S$ over a set $A$ can be identified with a function $S: A \xrightarrow{} \mathbb{N}^{0}$ where $S(a)$ is the number of occurrences of element $a \in A$ in the multiset $S$. We shall represent multisets by a formal sum
$\sum_{a \in A} S(a)'(a)$
such that e.g. $1'(x) + 2'(y)$ stands for a multiset containing one element $x$ and two elements $y$. We assume the standard multiset operations of membership ($\in$), inclusion ($\subseteq$), equality ($=$), union ($\uplus$), subtraction ($\setminus$) and by $|S|$ we denote the cardinality of $S$ (including the repetition of elements). By $\mathcal{S}(A)$ we denote the set of all multisets over the set $A$.
Finally, we also define the function $\textit{set}$
as a way of reducing multisets of colors to sets of colors given by
$ \textit{set}(S) \defeq \{a \ | \ a \in S\}$
where $\textit{set} (S)$ is the set of all colors with at least one occurrence in $S$.
\subsection{Colored Petri Nets} \label{sec:CPN} Colored Petri nets (CPN) are an extension of traditional P/T nets introduced by Kurt Jensen~\cite{CPNJensen} in 1981. In CPNs, places are associated with color domains where colors represent the values of tokens. Arc expressions describe what colors to consume and add to places depending on a given binding (assignment of variables to colors). Transitions may contain guards restricting which bindings are valid.
There exist several different definitions of CPNs from the powerful version defined in~\cite{JensenCPNBook2} that includes the ML language for describing arcs expressions and guards to less powerful ones such as the one used in the Model Checking Contest~\cite{mcc:2021}. We shall first give an abstract definition of a CPN.
\begin{definition} A colored Petri net is a tuple $\mathcal{N} = \CPN{}$ where \begin{enumerate}
\item $P$ is a finite set of places,
\item $T$ is a finite set of transitions such that $P \cap T = \emptyset$,
\item $\mathbb{C}$ is a nonempty set of colors,
\item $\mathbb{B}$ is a nonempty set of bindings,
\item $C : P \xrightarrow{} 2^\mathbb{C} \setminus \emptyset$ is a place color type function,
\item $G: T \times \mathbb{B} \xrightarrow{} \{\mathit{true}, \mathit{false}\}$ is a guard evaluation function,
\item $W: ((P \times T) \cup (T \times P)) \times \mathbb{B} \xrightarrow{} \mathcal{S}(\mathbb{C})$ is an arc evaluation function such that $\textit{set}(W((p,t),b)) \subseteq C(p)$ and $\textit{set}(W((t,p),b)) \subseteq C(p)$ for all $p \in P$, $t \in T$ and $b \in \mathbb{B}$,
\item $W_I : P \times T \xrightarrow{} \mathbb{N}^{> 0} \cup \{\infty\}$ is an inhibitor arc weight function, and
\item $M_0$ is the initial marking where a marking $M$ is a function $M : P \xrightarrow{} \mathcal{S}(\mathbb{C})$ such that $\textit{set} (M(p)) \subseteq C(p)$ for all $p \in P$. \end{enumerate} \end{definition}
In this definition, we assume an abstract set of bindings $\mathbb{B}$, representing the concrete configurations a net can be in. For example, colored Petri nets often use variables on the arcs and in the guards, and these variables can be assigned concrete colors. Such a variable assignment is then referred to as a binding. Notice that $G$ and $W$ are semantic functions which are in different variants of CPN defined by a concrete syntax. These functions take as an argument a binding and for this binding return either whether the guard is true or false, or in case of $W$ the function returns the multiset of tokens that should be consumed/produced when a transition is fired.
The set of all markings on a CPN $\mathcal{N}$ is denoted by $\mathbb{M}(\mathcal{N})$. In order to avoid the use of partial functions, we allow $W((p,t),b) = W((t,p),b) = \emptyset$ and $W_I(p,t) = \infty$,
meaning that if the arc evaluation function returns the empty multiset then the arc has no effect on transition firing and if the inhibitor arc function returns infinity then it never inhibits the connected transition.
Let $\mathcal{N} = \CPN{}$ be a fixed CPN for the rest of this section.
Let $B(t) \defeq \{ b \in \mathbb{B} \ | \ G(t,b) = \textit{true} \}$ be the set of all bindings that satisfy the guard of transition $t \in T$.
Let $\ell : T \xrightarrow{} Act$ be a transition labeling function. The semantics of a CPN $\mathcal{N}$ is defined as an LTS $L(\mathcal{N}) = (\mathbb{M}(\mathcal{N}),Act,\xrightarrow{})$ where
$\mathbb{M}(\mathcal{N})$ is the set of states defined as all markings on $\mathcal{N}$,
$Act$ is the set of actions, and
$M \xrightarrow{a} M'$ iff there exists $t \in T$ where $\ell(t) = a$ and there is $b \in B(t)$ such that
\begin{align*}
&W((p,t),b) \subseteq M(p) \text{ and }
W_I(p,t) > |M(p)| \text{ for all } p \in P , \text{ and}\\
&M'(p) = (M(p) \setminus W((p,t),b)) \uplus W((t,p),b) \text{ for all } p \in P.
\end{align*}
We denote the firing of a transition $t \in T$ in marking $M$ reaching $M'$ as $M \xrightarrow{t} M'$. Let $\xrightarrow{} = \Bcup{t \in T}\xrightarrow{t}$ and let $\xrightarrow{}^*$ be the reflexive and transitive closure of $\xrightarrow{}$.
\begin{remark} \label{remark:Finite} To reason about model checking of CPNs, we need to have a finite representation of colored nets that can be passed as an input to an algorithm. One way to enforce such a representation is to assume that all color domains are finite and the semantic functions $C$, $G$, $W$ and $W_I$ are effectively computable. \end{remark}
Finally, let us define the notion of postset and preset of $p \in P$
as $\postset{p} = \{ t \in T \mid \exists b \in \mathbb{B}.\ W((p,t),b) \neq \emptyset \}$ and $\preset{p} = \{ t \in T \mid \exists b \in \mathbb{B}.\ W((t,p),b) \neq \emptyset \}$. Similarly, for a transition $t \in T$ we define $\postset{t} = \{ p \in P \mid \exists b \in \mathbb{B}.\ W((t,p),b) \neq \emptyset \}$
and $\preset{t} = \{ p \in P \mid \exists b \in \mathbb{B}.\ W((p,t),b) \neq \emptyset \}$. We also define the preset of inhibitor arcs as $\presetstar{t} = \{ p \in P \ | \ W_I(p,t) \neq \infty \}$.
\subsection{P/T Nets} A Place/Transition (P/T) net is a CPN $\mathcal{N} = \CPN{}$ with one color $\mathbb{C} = \{ \bullet \}$ and only one binding $\mathbb{B} = \{b_\upepsilon\}$ such that every guard evaluates to true i.e. $G(t,b_\upepsilon) = \textit{true}$ for all $t\in T$ and every arc evaluates to a multiset over $\{\bullet\}$ i.e. $W((p,t),b_\upepsilon) \in \mathcal{S}(\{ \bullet \})$ and $W((t,p),b_\upepsilon) \in \mathcal{S}(\{ \bullet \})$ for all $p \in P$ and $t \in T$.
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw (209,103.75) .. controls (209,93.95) and (217.06,86) .. (227,86) .. controls (236.94,86) and (245,93.95) .. (245,103.75) .. controls (245,113.55) and (236.94,121.5) .. (227,121.5) .. controls (217.06,121.5) and (209,113.55) .. (209,103.75) -- cycle ;
\draw (245,107.5) -- (326.3,147.18) ; \draw [shift={(329,148.5)}, rotate = 206.02] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (329,135) -- (337,135) -- (337,168.5) -- (329,168.5) -- cycle ;
\draw (419,150.75) .. controls (419,140.95) and (427.06,133) .. (437,133) .. controls (446.94,133) and (455,140.95) .. (455,150.75) .. controls (455,160.55) and (446.94,168.5) .. (437,168.5) .. controls (427.06,168.5) and (419,160.55) .. (419,150.75) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (217.5,103.75) .. controls (217.5,102.23) and (218.73,101) .. (220.25,101) .. controls (221.77,101) and (223,102.23) .. (223,103.75) .. controls (223,105.27) and (221.77,106.5) .. (220.25,106.5) .. controls (218.73,106.5) and (217.5,105.27) .. (217.5,103.75) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (229.5,103.75) .. controls (229.5,102.23) and (230.73,101) .. (232.25,101) .. controls (233.77,101) and (235,102.23) .. (235,103.75) .. controls (235,105.27) and (233.77,106.5) .. (232.25,106.5) .. controls (230.73,106.5) and (229.5,105.27) .. (229.5,103.75) -- cycle ;
\draw (209,190.75) .. controls (209,180.95) and (217.06,173) .. (227,173) .. controls (236.94,173) and (245,180.95) .. (245,190.75) .. controls (245,200.55) and (236.94,208.5) .. (227,208.5) .. controls (217.06,208.5) and (209,200.55) .. (209,190.75) -- cycle ;
\draw (246,189.75) -- (326.25,154.7) ; \draw [shift={(329,153.5)}, rotate = 516.4100000000001] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (224.5,190.75) .. controls (224.5,189.23) and (225.73,188) .. (227.25,188) .. controls (228.77,188) and (230,189.23) .. (230,190.75) .. controls (230,192.27) and (228.77,193.5) .. (227.25,193.5) .. controls (225.73,193.5) and (224.5,192.27) .. (224.5,190.75) -- cycle ;
\draw (335,146.5) .. controls (369.74,120.44) and (396.1,129.78) .. (416.77,143.94) ; \draw [shift={(419,145.5)}, rotate = 215.54] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (343.06,156.72) .. controls (352,161.92) and (358.77,164.71) .. (369,167.5) .. controls (380,170.5) and (402,168.5) .. (420,156.5) ; \draw [shift={(341,155.5)}, rotate = 30.96] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (0, 0) circle [x radius= 3.35, y radius= 3.35] ;
\draw (186,101.4) node [anchor=north west][inner sep=0.75pt] {$p_{1}$};
\draw (428,172.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}$};
\draw (327,112.4) node [anchor=north west][inner sep=0.75pt] {$t$};
\draw (186,188.4) node [anchor=north west][inner sep=0.75pt] {$p_{2}$};
\draw (276,105.4) node [anchor=north west][inner sep=0.75pt] {$2'( \bullet )$};
\end{tikzpicture}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw (152,149.4) node [anchor=north west][inner sep=0.75pt] [font=\large] {$p_{1} :2'( \bullet ) \ +\ p_{2} :1'( \bullet )$};
\draw (334,141.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\xrightarrow{t}$};
\draw (379,148.4) node [anchor=north west][inner sep=0.75pt] [font=\large] {$p_{3} :1'( \bullet )$};
\end{tikzpicture}
\subsection{Integer Colored Petri Nets} \label{Sec:IntegerCPN} An integer CPN (as used for example in the Model Checking Contest~\cite{mcc:2021}) is a CPN $$\mathcal{N} = \CPN{}$$ where all colors are integer products i.e. $\mathbb{C} = \Bcup{k \geq 1}(\mathbb{N}^0)^{k}$. We use interval ranges to describe sets of colors such that a tuple of ranges $([a_1,b_1],...,[a_k,b_k])$ where $a_i,b_i \in \mathbb{N}^0$ for $i, 1 \leq i \leq k$, describes the set of colors
$\{(c_1,...,c_k) \ | \ a_i \leq c_i \leq b_i \text{ for all } 1 \leq i \leq k \}$. If the interval upper-bound is smaller than the lower-bound, the interval range denotes the empty set and by $[a]$ we denote the singleton interval $[a,a]$. As an example, consider the place color type of some place $p$ as $C(p) = \llrr{([1,2],[6,7])}$ describing the set of colors $\{(1,6),(1,7),(2,6),(2,7)\}$. For notational convenience, we sometimes omit the semantic paranthesis and simply write $([1,2],[6,7])$ instead of $\llrr{([1,2],[6,7])}$.
We use the set of variables $\mathcal{V}=\{x_1,...,x_n\}$ to represent colors. Variables can be present on arcs and in guards. A binding $b: \mathcal{V} \xrightarrow{} \mathbb{C}$ assigns colors to variables. We write $b \equiv \langle x_1=c_1,...,x_n=c_n \rangle$ for a binding where $b(x_i)=c_i$ for all $i$, $1 \leq i \leq n$. We now introduce the syntax of arc/guard expressions and its intuitive semantics by an example.
Figure~\ref{fig:CPNexample} shows an integer CPN where places (circles) are associated with ranges. The initial marking contains five tokens (two of color $0$ and three of color $2$) in $p_1$ and two tokens of color $5$ in place $p_2$. There is a guard on transition $t$ (rectangle) that compares $x$ with the integer $1$ and restricts the valid bindings. We can see that the arc from $t$ to $p_3$ creates a product of the integers $x$ and $y$, where the value of $x$ is decremented by one. We assume that all ranges are cyclic, meaning that the predecessor of $0$ in the color set $A$ is $2$. Figure~\ref{fig:CPNexample} also shows an example of transition firing.
Markings are written as formal sums showing how many tokens of what colors are in the different places. The transition $t$ can fire only once, as the inhibitor arc (for unlabelled inhibitor arcs we assume the default weight 1) from place $p_3$ to transition $t$ inhibits the second transition firing.
\begin{figure}
\caption{Integer CPN and transition firing under the binding $\langle x=0,y=5\rangle $}
\caption{Unfolding of CPN from Figure~\ref{fig:CPNexample} }
\caption{A CPN and its Unfolding to a P/T net}
\label{fig:CPNexample}
\label{fig:naiveUnfolding}
\end{figure}
We shall now present the formal syntax of arc expressions and guards. In integer CPNs, each arc $(P \times T) \cup (T \times P)$ excluding inhibitor arcs is assigned an arc expression $\alpha$ given by the syntax: \begin{align*}
&\alpha ::= n'(\tau_1,...,\tau_k) \ |\ \alpha_1 \pm \alpha_2\ | \ n \cdot \alpha \\
&\tau ::= c \ | \ x \ | \ x \pm s\ \end{align*}
where $c \in \mathbb{C}$, $x \in \mathcal{V}$, $s \in \mathbb{N}^{> 0}$, $n \in \mathbb{N}^{> 0}$ and $\pm ::= + \ | \ -$. The $\alpha$ expressions allow us to make tuples consisting of $\tau$ expressions that can be combined using multiset union ($+$) and subtraction ($-$), or multiplied by $n$ (creating $n$ copies of the tuple). The expressions of type $\tau$ are called simple arc expressions. The semantics of arc expressions is straightforward and demonstrated by the following example. \begin{example} Let $a$ be an arc annotated by the arc expression $1'(x-1) + 1'(y + 1) + 1'(z)$ and let $b_1 = \lanrang{x=3,y=3,z=1}$ and $b_2 = \lanrang{x=1,y=2,z=2}$ be bindings with integer range $([1,3])$ over the variables $x, y$ and $z$. The CPN semantics of the arc $a$ is defined as the multiset where $W(a,b_1) = 1'(2) + 2'(1)$ since the colors are cyclic in nature such that $3+1 = 1$ and $W(a,b_2) = 2'(3) + 1'(2)$ because $1-1 = 3$. \end{example}
Guards in integer CPNs are expressed by the following syntax: \begin{equation*}
\gamma ::= \mathit{true}\ |\ \mathit{false}\ |\ \neg\gamma\ |\ \gamma_1 \wedge \gamma_2\ |\ \gamma_1 \vee \gamma_2\ |\ \alpha_1 = \alpha_2\ |\ \alpha_1 \neq \alpha_2\ |\ \tau_1 \bowtie \tau_2 \ \end{equation*}
where $\bowtie \ ::= <\ |\ \leq\ |\ >\ |\ \geq\ |\ =\ |\ \neq$ and where $\alpha_1$ and $\alpha_2$ are arc expressions and $\tau_1$ and $\tau_2$ are simple arc expressions. The semantics of guards (their evaluation to true or false in a given binding) is also straightforward and demonstrated by an example. \begin{example} Let $g = (x > 2 \wedge y = 2) \vee z + 2 = 3$ be a guard on a transition $t$ and let $b_1 = \lanrang{x=3,y=3,z=1}$ and $b_2 = \lanrang{x=1,y=2,z=2}$ be bindings with range $([1,3])$ over the variables $x, y$ and $z$. Then $G(t,b_1) = \textit{true}$ and $G(t,b_2) = \textit{false}$. \end{example}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw (150.5,214.25) .. controls (150.22,204.17) and (158.17,196) .. (168.25,196) .. controls (178.33,196) and (186.72,204.17) .. (187,214.25) .. controls (187.28,224.33) and (179.33,232.5) .. (169.25,232.5) .. controls (159.17,232.5) and (150.78,224.33) .. (150.5,214.25) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ] (440.5,177.25) .. controls (440.22,167.17) and (448.17,159) .. (458.25,159) .. controls (468.33,159) and (476.72,167.17) .. (477,177.25) .. controls (477.28,187.33) and (469.33,195.5) .. (459.25,195.5) .. controls (449.17,195.5) and (440.78,187.33) .. (440.5,177.25) -- cycle ;
\draw (150.5,135.25) .. controls (150.22,125.17) and (158.17,117) .. (168.25,117) .. controls (178.33,117) and (186.72,125.17) .. (187,135.25) .. controls (187.28,145.33) and (179.33,153.5) .. (169.25,153.5) .. controls (159.17,153.5) and (150.78,145.33) .. (150.5,135.25) -- cycle ;
\draw [color={rgb, 255:red, 0; green, 0; blue, 0 } ,draw opacity=1 ][fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (313.99,160.59) -- (313.69,194.18) -- (303.7,194.09) -- (304,160.5) -- cycle ;
\draw (187,135.25) -- (300.13,169.63) ; \draw [shift={(303,170.5)}, rotate = 196.9] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (188,214.25) -- (300.1,184.27) ; \draw [shift={(303,183.5)}, rotate = 525.03] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (308.84,177.34) .. controls (346.24,148.1) and (404.61,148.47) .. (439.87,170.15) ; \draw [shift={(442,171.5)}, rotate = 213.31] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (320.23,185.66) .. controls (361.76,209.18) and (404.78,201.16) .. (443,184.5) ; \draw [shift={(317.69,184.18)}, rotate = 30.9] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (0, 0) circle [x radius= 3.35, y radius= 3.35] ;
\draw (346,135.4) node [anchor=north west][inner sep=0.75pt] {$1'( x-1,y)$};
\draw (224,127.4) node [anchor=north west][inner sep=0.75pt] {$1'( x)$};
\draw (224,203.4) node [anchor=north west][inner sep=0.75pt] {$1'( y)$};
\draw (304,201.4) node [anchor=north west][inner sep=0.75pt] {$t$};
\draw (290,141.4) node [anchor=north west][inner sep=0.75pt] {$x< 1$};
\draw (111,114) node [anchor=north west][inner sep=0.75pt] [align=left] {[$A$]};
\draw (113,194) node [anchor=north west][inner sep=0.75pt] [align=left] {[$B$]};
\draw (444,134) node [anchor=north west][inner sep=0.75pt] [align=left] {[$AB$]};
\draw (112,133.4) node [anchor=north west][inner sep=0.75pt] {$ \begin{array}{l} 2'( 0)\\ 3'( 2) \end{array}$};
\draw (112,213.4) node [anchor=north west][inner sep=0.75pt] {$2'( 5)$};
\draw (497,124) node [anchor=north west][inner sep=0.75pt] [align=left] {\textbf{Declarations:}\\color set $A=([0,2])$\\color set $B=([4,5])$\\color set $AB=A \times B$ \\variable $x: A$\\variable $y: B$};
\draw (162,161.4) node [anchor=north west][inner sep=0.75pt] {$p_{1}$};
\draw (162,241.4) node [anchor=north west][inner sep=0.75pt] {$p_{2}$};
\draw (451,204.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}$};
\draw (372,205.4) node [anchor=north west][inner sep=0.75pt] {$$};
\end{tikzpicture}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw (16,158.4) node [anchor=north west][inner sep=0.75pt] [font=\large] {$p_{1} :( 2'( 0) +3'( 2)) \ +\ p_{2} :2'( 5)$};
\draw (268,151.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\xrightarrow{t}$};
\draw (313,158.4) node [anchor=north west][inner sep=0.75pt] [font=\large] {$p_{1} :( 1'( 0) +3'( 2)) \ +\ p_{2} :1'( 5) \ +\ p_{3} :( 1'(( 2,5)))$}; \draw (255,113.4) node [anchor=north west][inner sep=0.75pt] [font=\large] {$\text{binding: } \langle x=0,y=5\rangle $};
\end{tikzpicture}
\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
\draw (176,70) .. controls (176,60.61) and (183.61,53) .. (193,53) .. controls (202.39,53) and (210,60.61) .. (210,70) .. controls (210,79.39) and (202.39,87) .. (193,87) .. controls (183.61,87) and (176,79.39) .. (176,70) -- cycle ;
\draw (226,70) .. controls (226,60.61) and (233.61,53) .. (243,53) .. controls (252.39,53) and (260,60.61) .. (260,70) .. controls (260,79.39) and (252.39,87) .. (243,87) .. controls (233.61,87) and (226,79.39) .. (226,70) -- cycle ;
\draw (276,70) .. controls (276,60.61) and (283.61,53) .. (293,53) .. controls (302.39,53) and (310,60.61) .. (310,70) .. controls (310,79.39) and (302.39,87) .. (293,87) .. controls (283.61,87) and (276,79.39) .. (276,70) -- cycle ;
\draw (278,260) .. controls (278,250.61) and (285.61,243) .. (295,243) .. controls (304.39,243) and (312,250.61) .. (312,260) .. controls (312,269.39) and (304.39,277) .. (295,277) .. controls (285.61,277) and (278,269.39) .. (278,260) -- cycle ;
\draw (340,260) .. controls (340,250.61) and (347.61,243) .. (357,243) .. controls (366.39,243) and (374,250.61) .. (374,260) .. controls (374,269.39) and (366.39,277) .. (357,277) .. controls (347.61,277) and (340,269.39) .. (340,260) -- cycle ;
\draw (213,261) .. controls (213,251.61) and (220.61,244) .. (230,244) .. controls (239.39,244) and (247,251.61) .. (247,261) .. controls (247,270.39) and (239.39,278) .. (230,278) .. controls (220.61,278) and (213,270.39) .. (213,261) -- cycle ;
\draw (310,210) .. controls (310,200.61) and (317.61,193) .. (327,193) .. controls (336.39,193) and (344,200.61) .. (344,210) .. controls (344,219.39) and (336.39,227) .. (327,227) .. controls (317.61,227) and (310,219.39) .. (310,210) -- cycle ;
\draw (401,70) .. controls (401,60.61) and (408.61,53) .. (418,53) .. controls (427.39,53) and (435,60.61) .. (435,70) .. controls (435,79.39) and (427.39,87) .. (418,87) .. controls (408.61,87) and (401,79.39) .. (401,70) -- cycle ;
\draw (453,71) .. controls (453,61.61) and (460.61,54) .. (470,54) .. controls (479.39,54) and (487,61.61) .. (487,71) .. controls (487,80.39) and (479.39,88) .. (470,88) .. controls (460.61,88) and (453,80.39) .. (453,71) -- cycle ;
\draw (404,260) .. controls (404,250.61) and (411.61,243) .. (421,243) .. controls (430.39,243) and (438,250.61) .. (438,260) .. controls (438,269.39) and (430.39,277) .. (421,277) .. controls (411.61,277) and (404,269.39) .. (404,260) -- cycle ;
\draw (213,210) .. controls (213,200.61) and (220.61,193) .. (230,193) .. controls (239.39,193) and (247,200.61) .. (247,210) .. controls (247,219.39) and (239.39,227) .. (230,227) .. controls (220.61,227) and (213,219.39) .. (213,210) -- cycle ;
\draw (402,210) .. controls (402,200.61) and (409.61,193) .. (419,193) .. controls (428.39,193) and (436,200.61) .. (436,210) .. controls (436,219.39) and (428.39,227) .. (419,227) .. controls (409.61,227) and (402,219.39) .. (402,210) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (273,142) -- (305,142) -- (305,148.5) -- (273,148.5) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (347,142) -- (379,142) -- (379,148.5) -- (347,148.5) -- cycle ;
\draw (206,82.5) -- (283.57,138.74) ; \draw [shift={(286,140.5)}, rotate = 215.94] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (206,82.5) -- (358.19,139.45) ; \draw [shift={(361,140.5)}, rotate = 200.52] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (402,78.5) -- (295.61,139.02) ; \draw [shift={(293,140.5)}, rotate = 330.37] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (457,84.5) -- (367.56,138.94) ; \draw [shift={(365,140.5)}, rotate = 328.66999999999996] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (289,145.25) -- (243.05,194.31) ; \draw [shift={(241,196.5)}, rotate = 313.12] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (363,145.25) -- (406,193.27) ; \draw [shift={(408,195.5)}, rotate = 228.15] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (184.5,69.75) .. controls (184.5,68.23) and (185.73,67) .. (187.25,67) .. controls (188.77,67) and (190,68.23) .. (190,69.75) .. controls (190,71.27) and (188.77,72.5) .. (187.25,72.5) .. controls (185.73,72.5) and (184.5,71.27) .. (184.5,69.75) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (196.5,69.75) .. controls (196.5,68.23) and (197.73,67) .. (199.25,67) .. controls (200.77,67) and (202,68.23) .. (202,69.75) .. controls (202,71.27) and (200.77,72.5) .. (199.25,72.5) .. controls (197.73,72.5) and (196.5,71.27) .. (196.5,69.75) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (296.5,66.75) .. controls (296.5,65.23) and (297.73,64) .. (299.25,64) .. controls (300.77,64) and (302,65.23) .. (302,66.75) .. controls (302,68.27) and (300.77,69.5) .. (299.25,69.5) .. controls (297.73,69.5) and (296.5,68.27) .. (296.5,66.75) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (284.5,67) .. controls (284.5,65.48) and (285.73,64.25) .. (287.25,64.25) .. controls (288.77,64.25) and (290,65.48) .. (290,67) .. controls (290,68.52) and (288.77,69.75) .. (287.25,69.75) .. controls (285.73,69.75) and (284.5,68.52) .. (284.5,67) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (290.5,76.75) .. controls (290.5,75.23) and (291.73,74) .. (293.25,74) .. controls (294.77,74) and (296,75.23) .. (296,76.75) .. controls (296,78.27) and (294.77,79.5) .. (293.25,79.5) .. controls (291.73,79.5) and (290.5,78.27) .. (290.5,76.75) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (461.5,70.75) .. controls (461.5,69.23) and (462.73,68) .. (464.25,68) .. controls (465.77,68) and (467,69.23) .. (467,70.75) .. controls (467,72.27) and (465.77,73.5) .. (464.25,73.5) .. controls (462.73,73.5) and (461.5,72.27) .. (461.5,70.75) -- cycle ;
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (473.5,70.75) .. controls (473.5,69.23) and (474.73,68) .. (476.25,68) .. controls (477.77,68) and (479,69.23) .. (479,70.75) .. controls (479,72.27) and (477.77,73.5) .. (476.25,73.5) .. controls (474.73,73.5) and (473.5,72.27) .. (473.5,70.75) -- cycle ;
\draw (289,145.25) .. controls (284.18,170.34) and (290.52,184.49) .. (310.74,198.03) ; \draw [shift={(313,199.5)}, rotate = 212.47] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (363,145.25) .. controls (371.6,174.86) and (358.3,186.45) .. (342.28,197.88) ; \draw [shift={(340,199.5)}, rotate = 324.78] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (8.93,-4.29) -- (0,0) -- (8.93,4.29) -- cycle ;
\draw (320,195.5) .. controls (320,182.96) and (316.28,159.24) .. (298.04,153.1) ; \draw [shift={(296,152.5)}, rotate = 194.04] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (0, 0) circle [x radius= 3.35, y radius= 3.35] ;
\draw (332,195.5) .. controls (332.96,171.5) and (338.53,161.32) .. (354.9,153.47) ; \draw [shift={(357,152.5)}, rotate = 336.04] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (0, 0) circle [x radius= 3.35, y radius= 3.35] ;
\draw (177,29.4) node [anchor=north west][inner sep=0.75pt] {$p_{1}( 0)$};
\draw (227,29.4) node [anchor=north west][inner sep=0.75pt] {$p_{1}( 1)$};
\draw (277,29.4) node [anchor=north west][inner sep=0.75pt] {$p_{1}( 2)$};
\draw (308,228.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}(\mathbf{sum})$};
\draw (400,29.4) node [anchor=north west][inner sep=0.75pt] {$p_{2}( 4)$};
\draw (453,29.4) node [anchor=north west][inner sep=0.75pt] {$p_{2}( 5)$};
\draw (263,278.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}(( 0,4))$};
\draw (331,278.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}(( 0,5))$};
\draw (197,278.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}(( 1,4))$};
\draw (395,277.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}(( 1,5))$};
\draw (201,227.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}(( 2,4))$};
\draw (392,227.4) node [anchor=north west][inner sep=0.75pt] {$p_{3}(( 2,5))$};
\draw (154,137.4) node [anchor=north west][inner sep=0.75pt] {$t( \langle x=0,y=4\rangle )$};
\draw (396,137.4) node [anchor=north west][inner sep=0.75pt] {$t( \langle x=0,y=5\rangle )$};
\end{tikzpicture}
The Model Checking Contest~\cite{mcc:2021} further includes color types called dots and cyclic enumerations which are excluded from these definitions as these can be trivially translated to tuples of integer ranges. The color type dot, $\{\bullet\}$, is represented by the color domain $([1])$ and a cyclic enumeration with elements $\{e_1,e_2,...,e_n\}$ is encoded as the integer range $([1,n])$ corresponding to the indices of the cyclic enumeration. Furthermore, the contest uses the \textit{.all} expression, which creates one of each color in the color domain. For example if $A = ([0,2])$ then $A.all = 1'(1) + 1'(2) + 1'(3)$.
All examples in this paper are expressed in integer CPN syntax.
\subsection{Unfolding of CPNs}
CPNs with finite color domains can be \textit{unfolded} into an equivalent P/T net~\cite{JensenCPNBook}. Each place $p$ is unfolded into $|C(p)|$ places, a transition is made for each legal binding and we translate the multiset of colors on the arc to a multiset over $\bullet$. We now provide a formal definition of unfolding in our syntax, following the approach from~\cite{KlostergaardThesis,LattePaper} that also consider inhibitor arcs.
For each place connected to an inhibitor arc, we create a fresh summation place that contains the sum of tokens across the rest of the unfolded places. The summation places are created to ensure that inhibitor arcs work correctly after unfolding.
\begin{definition}[Unfolding]\label{Def:NaiveUnfolding} Let $\mathcal{N} = \CPN{}$ be a colored Petri net. The unfolded P/T net $\mathcal{N}^u = \CPN{u}$ is given by \begin{enumerate}
\item $P^u = \{p(c) \ | \ p \in P \wedge c \in C(p)\} \cup \{ p(\textit{\textbf{sum}}) \ | \ t \in T, p \in \presetstar{t}\},$
\item $T^u = \bigcup_{t \in T}\bigcup_{b \in B(t)} t(b),$
\item $\mathbb{C}^u = \{\bullet\}$,
\item $\mathbb{B}^u = \{b_\upepsilon\}$,
\item $C^u(p(c)) = \{\bullet\}$ for all $p(c)\in P^u$,
\item $G^u(t(b),b_\upepsilon) = \textit{true}$ for all $t(b) \in T^u$,
\item $W^u((p(c),t(b)),b_\upepsilon) = W((p,t), b)(c)'(\bullet) \text{ and }
W^u((t(b),p(c)),b_\upepsilon) = \\ W((t,p), b)(c)'(\bullet)$ for all \ $p(c) \in P^u$ and \ $t(b) \in T^u$, \textit{and} \\
$W^u((p(\textbf{sum}),t(b)),b_\upepsilon) = |W((p,t), b)|'(\bullet) \text{ and } W^u((t(b),p(\textbf{sum})),b_\upepsilon)
= \\ |W((t,p), b)|'(\bullet)$ for all $p(\textbf{sum}) \in P^u$ and $t(b) \in T^u$,
\item $W_I^u(p(\textbf{sum}),t(b)) = W_I(p,t)$ for all $p(\textbf{sum}) \in P^u$ and $t(b) \in T^u$, and
\item $M_{0}^u(p(c)) = M_0(p)(c)'(\bullet)$ for all $p(c) \in P^u$ \textit{ and }\\
$M_{0}^u(p(\textbf{sum})) = |M_0(p)|'(\bullet)$ for all $p(\textbf{sum}) \in P^u$ \end{enumerate}
where $p(\textbf{sum})$ denotes the sum of all tokens regardless of the color for the place $p$. \end{definition}
Consider the CPN in Figure~\ref{fig:CPNexample}. The unfolded version of this can be seen in Figure~\ref{fig:naiveUnfolding}. We notice that each place of the CPN is unfolded to a fresh new place for every color in the color type of the place as well as a \textbf{\textit{sum}} place for $p_3$. Additionally, the transition is unfolded to a new transition for each legal binding.
The theorem showing that the unfolded net is bisimilar to the original CPN was proved in~\cite{KlostergaardThesis,LattePaper}; we only added a small optimization on the summation places.
\begin{theorem}[\cite{KlostergaardThesis,LattePaper}] \label{theorem:BisimilarUnfolding} Given a colored Petri net $\mathcal{N} = \CPN{}$ and the unfolded P/T net $\mathcal{N}^u = \CPN{u}$, it holds that $M_0 \sim M_0^u$ with the labeling function $\ell(t(b)) = t$ for all $t(b) \in T^u$. \end{theorem}
\section{Color Quotienting} \label{Sec:Partitioning} Unfolding a CPN without any further analysis will often lead to many unnecessary places and transitions. We shall now present our first technique that allows to group equivalently behaving colors into equivalence classes in order to reduce the number of colors and hence also to reduce the size of the unfolded net.
As an example consider the CPN in Figure~\ref{fig:UnstablePartition}, the unfolded version of this net adds five places for both $p_1$ and $p_2$. However, we see that in $p_1$ all colors greater than or equal to $3$ behave exactly the same throughout the net and can thus be represented by a single color.
We can thus \textit{quotient} the CPN by \textit{partitioning} the color domain of each place into a number of \textit{equivalence classes} of colors such that the colors behaving equivalently are represented by the same equivalence class. Using this approach, we can construct a bisimilar CPN seen in Figure~\ref{fig:partitionUnfolding} where the color $3$ now represents all colors greater than or equal to $3$.
Such a reduction in the number of colors is possible to include already during the design of a CPN model, however, the models may look less intuitive for human modeller or the nets can be auto-generated and hence contain redundant/equivalent colors as observed in the benchmark of CPN models from the annual Model Checking Contest benchmark~\cite{mcc:2021}.
\begin{figure}
\caption{Example CPN}
\caption{Quotiented net from Figure~\ref{fig:UnstablePartition}}
\caption{Example of an unstable partition $\delta$ and markings showing why it is unstable}
\caption{Example of stable partition $\delta'$}
\caption{Quotienting example}
\label{fig:UnstablePartition}
\label{fig:partitionUnfolding}
\label{fig:UnstablePartitionFiring}
\label{fig:StablePartition}
\end{figure}
We thus introduce \textit{color partition} on places where all colors with similar behaviour in a given place are grouped into an \textit{equivalence class}, denoted by $\theta$. For the rest of this section, let us assume a fixed CPN $\mathcal{N} = \CPN{}$.
A partition $\delta$ is a function $\delta : P \xrightarrow{} 2^{2^\mathbb{C}} \setminus \emptyset$ that for a place $p$ returns the equivalence classes of $C(p)$ such that $(\Bcup{\theta \in \delta(p)} \theta) = C(p)$ and $\theta_1 \cap \theta_2 = \emptyset$ for all $\theta_1,\theta_2 \in \delta (p)$ where $\theta_1 \neq \theta_2$.
\begin{definition}\label{def:dequiv} Given a partition $\delta$ and markings $M$ and $M'$, we write $M(p) \dequiv M'(p)$ for a $p \in P$ iff for all $\theta \in \delta(p)$ it holds that
$\Bsum{c \in \theta}M(p)(c) = \textstyle \sum_{c \in \theta}M'(p)(c)$.
We write $M \dequiv M'$ iff $M(p) \dequiv M'(p)$ for all $p \in P$. A partition $\delta$ is \emph{stable} if the relation $\dequiv$ on markings induced by $\delta$ is a bisimulation. \end{definition}
Consider the CPN in Figure~\ref{fig:UnstablePartition}. The partition shown in the Figure~\ref{fig:UnstablePartitionFiring} is not stable as demonstrated by the transition firing from $M_1$ and $M_2$ to $M_1'$ resp. $M_2'$ where $M_1 \dequiv M_2$ but $M_1' \centernot{\dequiv} M_2'$. Figure~\ref{fig:StablePartition} shows an example of a stable partition (here we describe the partition with integer ranges in the same manner as in integer CPNs).
We now show how a CPN can be quotiented using a stable partition. First, we define the notion of binding equivalence under a partition. \begin{definition}\label{Def:PartBindingEquiv} Given a partition $\delta$, a transition $t \in T$ and bindings $b,b' \in B(t)$, we write $b \dtequiv b'$ iff for all $p \in \preset{t}$ and for all $ \theta \in \delta(p)$ it holds that \begin{equation*}
\Bsum{c \in \theta}W((p,t),b)(c) = \Bsum{c \in \theta}W((p,t),b')(c) \end{equation*} and for all $p \in \postset{t}$ and for all $\theta \in \delta(p)$ it holds that \begin{equation*}
\Bsum{c \in \theta}W((t,p),b)(c) = \Bsum{c \in \theta}W((t,p),b')(c). \end{equation*} \end{definition}
We can now define classes of equivalent bindings given a partition $\delta$ which are bindings that have the same behaviour for a given transition, formally
$B^\delta(t) \defeq \{[b]_t \ | \ b \in B(t) \} \text{ where } [b]_t = \{b' \ | \ b' \dtequiv b\}$.
For a given stable partition, we now construct a quotiented CPN where the set of colors are the equivalence classes of the stable partition and
the set of bindings are the equivalence classes of bindings. As such, we rewrite the arc and guard evaluation functions to instead consider an equivalence class of bindings, which is possible since each binding in the equivalence class behaves equivalently.
\begin{definition} \label{def:partitionedCPN} Let $\mathcal{N} = \CPN{}$ be a CPN and $\delta$ a stable partition of $\mathcal{N}$. The quotiented CPN $\mathcal{N}^\delta = (P, T,\mathbb{C}^\delta,\mathbb{B}^\delta, C^\delta, G^\delta, W^\delta, W_I^\delta, \Mdelta_0)$ is defined by \begin{enumerate}
\item $\mathbb{C}^\delta = \Bcup{p \in P} \delta(p)$
\item $\mathbb{B}^\delta = \biguplus_{t \in T} B^\delta(t)$.
\item \label{item:guards}
$G^\delta(t, [b]_t) = G(t, b)$ for all $t \in T$ and $[b]_t \in B(t)$,
\item $C^\delta (p) = \delta (p)$ for all $p \in P$,
\item \label{item:arcs}
$W^\delta((p,t),[b]_t) = S \text{ where } S(\theta) = \Bsum{c \in \theta}W((p,t),b)(c) \text{ for all } \theta \in \delta(p)$ and\\
$W^\delta((t,p),[b]_t) = S \text{ where } S(\theta) = \Bsum{c \in \theta}W((t,p),b)(c) \text{ for all } \theta \in \delta(p)$ \\ for all $p \in P$, $t \in T$ and $[b]_t \in \mathbb{B}^\delta$,
\item \label{item:inhibarcs}
$W_I^\delta(p,t)$ = $W_I(p,t)$ \textit{ for all $p \in P$ and $t \in T$, and }
\item \label{item:initialMarking} $M^\delta_0 (p) = S \text{ where } S(\theta) = \Bsum{c \in \theta}M_0(p)(c)$ for all $p \in P$ and $\theta \in \delta (p)$. \end{enumerate} \end{definition}
We can now present our main correctness theorem, stating that the original and quotiented colored nets are bisimilar.
\begin{theorem} Let $\mathcal{N} = \CPN{}$ be a CPN, $\delta$ a stable partition and $\mathcal{N}^\delta = \CPN{\delta}$ the quotiented CPN. Then $M_0 \sim M^\delta_0$. \end{theorem} \begin{proof} We show that
$R = \{(M,M^\delta) \ | \ \Bsum{c \in \theta}M(p)(c) = M^\delta(p)(\theta) \text{ for all } p \in P \text{ and all } \theta \in \delta (p) \}$ is a bisimulation relation. We first notice that $(M_0,M^\delta_0) \in R$ by Item \ref{item:initialMarking} in Definition \ref{def:partitionedCPN}.
Assume that $(M,M^\delta) \in R$ and $t \in T$ such that $M \xrightarrow{t} M'$ under binding $b \in B(t)$, we want to show that $M^\delta \xrightarrow{t} M^{\delta\prime}$ under binding $[b]_t \in B(t)$ such that $(M',M^{\delta\prime}) \in R$. As such, we need to prove the following: \begin{align*}
\bm{(a)} \quad &W^\delta((p,t),[b]_t) \subseteq M^\delta(p) \text{ for all } p \in P \\
\bm{(b)} \quad &W_I^\delta(p,t) > |M^\delta (p)| \text{ for all } p \in P\\
\bm{(c)} \quad &(M',M^{\delta\prime}) \in R \text{ where } M^{\delta\prime}(p) = (M^\delta (p) \setminus W^\delta((p,t),[b]_t)) \uplus W^\delta((t,p),[b]_t)\\ &\text{for all } p \in P \ . \end{align*}
$\bm{(a)}$ We start by showing $W^\delta((p,t),[b]_t) \subseteq M^\delta(p)$ for all $p \in P$. First, because $(M,M^\delta) \in R$, we know that \begin{equation} \label{eq:markingsInRelation}
\Bsum{c \in \theta}M(p)(c) = M^\delta(p)(\theta) \text{ for all } p \in P \text{ and all } \theta \in \delta (p). \end{equation} Since $W((p,t),b) \subseteq M(p)$ we know for all $c \in C(p)$ that $ W((p,t),b))(c) \leq M(p)(c)$ which implies that \begin{equation} \label{eq:inputArcsTheorem}
\Bsum{c \in \theta}W((p,t),b))(c) \leq \Bsum{c \in \theta}M(p)(c) \end{equation} for all $\theta \in \delta (p)$. We now show that $W^\delta((p,t),[b]_t) \subseteq M^\delta(p)$ for all $p \in P$ i.e. $W^\delta((p,t),[b]_t)(\theta) \leq M^\delta(p)(\theta)$ for all $\theta \in \delta (p)$:
\begin{center} \begin{tabularx}{\textwidth}{lXr}
$W^\delta((p,t),[b]_t)(\theta)$ & = & substitute by Def. \ref{def:partitionedCPN} Item \ref{item:arcs} \\
$\Bsum{c \in \theta}W((p,t),b))(c)$& $\leq$ &by Equation (\ref{eq:inputArcsTheorem}) \\
$\Bsum{c \in \theta}M(p)(c)$ & = & by Equation (\ref{eq:markingsInRelation}) \\
$M^\delta(p)(\theta)$ \ . & & \\ \end{tabularx} \end{center}
$\bm{(b)}$ Next we show $W_I^\delta((p,t) > |M^\delta(p)|$. We know that \begin{equation} \label{Eq:inhibArcs}
W_I(p,t) > |M(p)| \end{equation} by definition of CPN semantics since $M \xrightarrow{t} M'$. We then observe that for all $p \in P$, it holds \begin{center} \begin{tabularx}{\textwidth}{lXr}
$W^\delta_I(p,t)$ & = &substitute by Def. \ref{def:partitionedCPN} Item \ref{item:inhibarcs} \\
$W_I(p,t)$& > &by Equation (\ref{Eq:inhibArcs})\\
$|M(p)|$ & = & multiset definition\\
$\Bsum{c \in C(p)}M(p)(c)$ &= & since $(\Bcup{\theta \in \delta(p)} \theta) = C(p)$\\
$\Bsum{\theta \in \delta (p)}\Bsum{c \in \theta}M(p)(c)$ & = & by Equation (\ref{eq:markingsInRelation}) \\
$\Bsum{\theta \in \delta(p)}M^\delta(p)(\theta)$ & = & multiset definition\\
$|M^\delta(p)|$ \ .& &\\ \end{tabularx} \end{center}
$\bm{(c)}$ Lastly, we show that $(M',M^{\delta\prime}) \in R$. Assume $p \in P$, $b \in B(t)$ and equivalence class $[b]_t$. We know that $M'(p) = (M(p) \setminus W((p,t),b)) \uplus W((t,p),b)$ and $M^{\delta\prime}(p) = (M^\delta(p) \setminus W^\delta((p,t),[b]_t)) \uplus W^\delta((t,p),[b]_t)$ and we need to show that $\Bsum{c \in \theta}M'(p)(c) = M^{\delta\prime}(p)(\theta)$ for all $\theta \in \delta(p)$:
\begin{center} \begin{tabularx}{\textwidth}{lXr}
$\Bsum{c\in \theta} M'(p)(c)$ & = & by def. of CPN semantics \\
$\Bsum{c \in \theta} (M(p) \setminus W((p,t),b) \uplus W((t,p),b))(c)$ & = & \specialcell{substitute by multiset definitions\\ and by enabledness of $t$}\\
\specialcell{$\Bsum{c \in \theta}M(p)(c) - \Bsum{c \in \theta}W((p,t),b)(c)$\\$ + \Bsum{c \in \theta}W((t,p),b)(c)$} & = &substitute by Def. \ref{def:partitionedCPN} Item \ref{item:arcs}\\
\specialcell{$\Bsum{c \in \theta}M(p)(c) - W^\delta((p,t),[b]_t)(\theta)$\\ $+ W^\delta((t,p),[b]_t)(\theta) $} &= &since $(M,M^\delta) \in R$\\
\specialcell{$M^\delta(p)(\theta) - W^\delta((p,t),[b]_t)(\theta)$ \\$+ W^\delta((t,p),[b]_t)(\theta) $} & = & by definition of CPN semantics \\
$M^{\delta\prime}(p)(\theta)$ \ . & & \\ \end{tabularx} \end{center}
We then have to show that the same is the case for the opposite direction such that assume $(M,M^\delta) \in R$ and $t \in T$ such that $M^\delta \xrightarrow{t} M^{\delta\prime}$, we want to show that $M \xrightarrow{t} M'$
for some $b \in B(t)$ such that $(M',M^{\delta\prime}) \in R$. As such, we want to show that: \begin{align*}
\bm{(d)} \quad &W((p,t),b) \subseteq M(p) \text{ for all } p \in P \\
\bm{(e)} \quad &W_I(p,t) > |M(p)| \text{ for all } p \in P\\
\bm{(f)} \quad &(M',M^{\delta\prime}) \in R \text{ where } M'(p) = (M(p) \setminus W((p,t),b)) \uplus W((t,p),b) \\ &\text{for all } p \in P \ . \end{align*}
We first notice that $\bm{(e)}$ and $\bm{(f)}$ can be showed by the same argumentation as $\bm{(b)}$ and $\bm{(c)}$. For the case $\bm{(d)}$, we show that $W((p,t),b) \subseteq M(p) \text{ for all } p \in P$. From $\bm{(a)}$ we know that $W^\delta((p,t),[b]_t) \subseteq M^\delta(p)$ which implies $\Bsum{c \in \theta}W((p,t),b))(c) \leq \Bsum{c \in \theta}M(p)(c)$ for all $\theta \in \delta(p)$ and $p \in P$.
Hence observe that there exists a marking $M_1$ such that $\Bsum{c \in \theta}M_1(p)(c) = M^\delta(p)(\theta)$ and where $\Bsum{c \in \theta}W((p,t),b)(c) \leq \Bsum{c \in \theta}M_1(p)(c)$ for all $\theta \in \delta(p)$ and $p \in P$. Clearly $t$ is enabled in $M_1$ since $W((p,t),b) \subseteq M_1(p)$ for all $p \in P$ by the multiset definition of $\subseteq$ and we know that the inhibitor arcs do not inhibit the transition by $\bm{(e)}$.
We then want to show that $M_1 \dequiv M$, i.e. $\Bsum{c \in \theta}M_1(p)(c) = \Bsum{c \in \theta}M(p)(c)$ for all $\theta \in \delta(p)$ and $p \in P$. Since $(M,M^\delta) \in R$ we know that $\Bsum{c \in \theta}M(p)(c) = M^\delta(p)(\theta) = \Bsum{c \in \theta}M_1(p)(c)$ for all $\theta \in \delta(p)$ and $p \in P$ and thus $M_1 \dequiv M$. Since $\delta$ is stable we know that $t$ is enabled in $M$.
Thus we know that the opposite direction also holds meaning that $R$ is a bisimulation.
\end{proof}
\subsection{Computing Stable Partitions} Our main challenge is how to efficiently compute a stable partition in order to apply the quotienting technique. To do so, we first define a partition refinement.
\begin{definition} Given two partitions $\delta$ and $\delta'$ we write
$ \delta \geq \delta'$ if for all $p \in P$ and all $\theta' \in \delta'(p)$ there exists $\theta \in \delta (p)$ such that $\theta' \subseteq \theta$.
Additionally, we write $\delta > \delta'$ if $\delta \geq \delta'$ and $\delta' \neq \delta$. \end{definition}
\begin{comment} As an example, consider partitions $\delta$ and $\delta'$ from Figure \ref{fig:UnstablePartition} and Figure \ref{fig:StablePartition}, then $\delta > \delta'$ since for all $\theta' \in \delta'(p)$ there exists $\theta \in \delta(p)$ such that $\theta' \subseteq \theta$ for $p \in \{p_1,p_2\}$ and $\delta \neq \delta'$. \end{comment}
Note that for any finite CPN as assumed in Remark~\ref{remark:Finite},
the refinement relation $>$ is well-founded as for any $\delta > \delta'$ the partition $\delta'$ has strictly more equivalence classes for at least one place $p \in P$.
We now define also the union of two partitions as the smallest partition that has both of the partitions as refinements.
\begin{definition} Given two partitions $\delta_1$, $\delta_2$ and $p \in P$, let $\xleftrightarrow{}$ be a relation over $\delta_1(p) \cup \delta_2(p)$ such that
$ \theta \xleftrightarrow{} \theta' \textit{ iff } \theta \cap \theta' \neq \emptyset$ where $\theta, \theta' \in \delta_1(p) \cup \delta_2(p)$.
Let $\xleftrightarrow{}^*$ be the reflexive, transitive closure of $\xleftrightarrow{}$ and let $[\theta] \defeq \Bcup{\theta' \in \delta_1(p) \cup \delta_2(p), \theta \xleftrightarrow{}^* \theta'} \theta'$ where $\theta \in \delta_1(p) \cup \delta_2(p)$. Finally, we define the partition union operator $\sqcup$ by
$(\delta_1 \sqcup \delta_2)(p) = \Bcup{\theta \in \delta_1(p) \cup \delta_2(p)} \{[\theta]\} \text{ for all } p \in P$.
\end{definition}
For example, assume some place $p$ such that $C(p) = \{([1,5])\}$ and partitions $\delta_1$ and $\delta_2$ such that $\delta_1(p) = \{([1,2]),([3,4]),([5])\}$ and $\delta_2(p) = \{([1]),([2,3]),([4]),([5])\}$ then $(\delta_1 \sqcup \delta_2)(p) = \{([1,4]),([5])\}$.
\begin{lemma} \label{lemma:SmallestUnion} \label{lemma:PartitionUnion} Let $\delta_1$ and $\delta_2$ be two partitions. Then (i) $\delta_1 \sqcup \delta_2 \geq \delta_1$ and $\delta_1 \sqcup \delta_2 \geq \delta_2$, and (ii) if $\delta_1$ and $\delta_2$ are stable partitions then so is $\delta_1 \sqcup \delta_2$. \end{lemma} \begin{proof} For the first part of the claim, from definition of partition union, we see that $[\theta]$ is the union of any $\theta'$ that overlaps with $\theta$ and $\delta_1 \sqcup \delta_2$ just collects all such unions for every $\theta$. As such, it is trivial that for any $\theta \in \delta_1 (p)$ there exists $\theta' \in \delta_1 (p) \sqcup \delta_2 (p)$ such that $\theta \subseteq \theta'$ for all $p \in P$, in other words $ \delta_1 \sqcup \delta_2 \geq \delta_1$. The same is the case for $\delta_2$.
For the second part of the claim, let $\delta = \delta_1 \sqcup \delta_2$. Assume $M$ and $M'$ such that $M \dequiv M'$ i.e. for all $p \in P$ and all $[\theta ]\in \delta (p)$ it holds that $\Bsum{c \in [\theta]} M(p)(c) = \Bsum{c \in [\theta]} M'(p)(c)$.
We want to show that $\delta$ is stable, i.e that $\dequivparam{}$ is a bisimulation relation, while assuming that $\dequivparam{1}$ and $\dequivparam{2}$ are bisimulation relations. Assume a fixed $p \in P$. From definition of partition union, we get that for every fixed $[\theta] \in \delta (p)$ and for all $c,c' \in [\theta]$ there exist $c_1,...,c_k \in [\theta]$ such that $c,c_1 \in \theta_1$, $c_1,c_2 \in \theta_2$, $c_2,c_3 \in \theta_3$, \ldots, $c_k,c' \in \theta_k$, where $\theta_i \in \delta_1 (p) \cup \delta_2 (p)$ for all $i, 1 \leq i \leq k$. Let us define markings $M_i$, for $1 \leq i \leq k$, such that $M_i(p)(c_i) = \Bsum{c \in [\theta]} M(p)(c)$ and $M_i(p)(c) = 0$ if $c \neq c_i$ and $c \in [\theta]$, otherwise $M_i(p)(c) = M(p)(c)$. In other words, in order to obtain $M_i$, we replace in $M$ all colors in the equivalence class $[\theta]$ with the color $c_i$ and obtain the chain $M(p) \dequivparam{j_1} M_1(p) \dequivparam{j_2} M_2(p) \dequivparam{j_3} ... \dequivparam{j_k} M_k(p)$ where $j_i \in \{1,2\}$, i.e. $M_i$ and $M_{i+1}$ are related either by $\dequivparam{1}$ or $\dequivparam{2}$, both of them being bisimulation relations. We can also observe that $M(p) \dequiv M_k(p)$.
We can then repeat this process for all other equivalence classes in $\delta(p)$ in order to conclude that $M(p) \dequiv M'(p)$.
The same process can then be applied to all $p \in P$, implying that
$M \dequiv M'$. Hence there is a chain of markings as above starting from $M$ and ending in $M'$. Since $\delta_1$ and $\delta_2$ are stable, meaning that both $\dequivparam{1}$ and $\dequivparam{2}$ are bisimulation relations, this implies that every transition from $M$ can be (by transitivity) matched by a transition from $M'$ and vice versa, implying that $\dequivparam{}$ is a bisimulation relation and $\delta$ is so stable.
\end{proof}
The lemma above implies the existence of a unique maximum stable partition. \begin{theorem}
There is a unique maximum stable partition $\delta$ such that $\delta \geq \delta'$ for all stable partitions $\delta'$. \end{theorem} \begin{proof} We prove this by contradiction. Assume two maximum stable partitions $\delta_1$ and $\delta_2$ where $\delta_1 \neq \delta_2$. By Lemma \ref{lemma:PartitionUnion} (ii) we know that $\delta_1 \sqcup \delta_2$ is stable and by Lemma \ref{lemma:SmallestUnion} (i) we know that $\delta_1 \sqcup \delta_2 \geq \delta_1$ and $\delta_1 \sqcup \delta_2 \geq \delta_2$. Thus $\delta_1$ and $\delta_2$ cannot both be maximum stable partitions. \end{proof}
In order to provide an algorithm for computing a stable partition, we define the maximum arc size for a given CPN $\mathcal{N}$ as the function
$\textit{max}(\mathcal{N}) = \max_{p \in P, t\in T,b \in \mathbb{B}}(|W((p,t),b)|, |W((t,p),b)|)$.
The set of all markings smaller than the \textit{max} arc size over $\mathcal{N}$ is defined by
$\mathbb{M}^{bounded}(\mathcal{N}) = \{ M \in \mathbb{M}(\mathcal{N}) \ | \ |M(p)| \leq \textit{max}(\mathcal{N}) \text{ for all } p \in P \}$.
As such, $\mathbb{M}^{bounded}(\mathcal{N})$ is a finite set of all bounded markings of $\mathcal{N}$ with cardinality less than or equal to $\textit{max}(\mathcal{N})$.
In order to compute stable partitions we need to show some properties for markings in $\mathbb{M}^{bounded}(\mathcal{N})$. The following lemma shows that if there are two non-bisimilar markings that break the fact that the relation $\dequiv$ is a bisimulation, then there are also two markings that are bounded and that also break the bisimulation property.
\begin{lemma}\label{lemma:BoundedMarkings} Let $\mathcal{N}$ be a CPN and $\delta$ a partition. Then for all $t \in T$ it holds that \begin{enumerate}
\item[$\bm{(a)}$] if there exist $M_1,M_2 \in \mathbb{M}(\mathcal{N})$ such that $M_1 \dequiv M_2$, $M_1 \xrightarrow{t} \text{ and } \ M_2 \centernot{\xrightarrow{t}}$ then there exist $M_3,M_4 \in \mathbb{M}^{bounded}$ such that $M_3 \dequiv M_4$, $M_3 \xrightarrow{t} \text{ and } \ M_4 \centernot{\xrightarrow{t}}$, and
\item[$\bm{(b)}$] if there exist $M_1,M_2 \in \mathbb{M}(\mathcal{N})$ where $M_1 \dequiv M_2$ and there exists $M_1' \in \mathbb{M}(\mathcal{N})$ such that $M_1 \xrightarrow{t} M_1'$ and for all $M_2' \in \mathbb{M}(\mathcal{N})$ where $M_2 \xrightarrow{t} M_2'$ it holds that $M_1' \centernot{\dequiv} M_2'$ then there exists $M_3,M_4 \in \mathbb{M}^{bounded}(\mathcal{N})$ where $M_3 \dequiv M_4$ and there exists $M_3' \in \mathbb{M}^{bounded}(\mathcal{N})$ such that $M_3 \xrightarrow{t} M_3'$ and for all $M_4' \in \mathbb{M}^{bounded}(\mathcal{N})$ where $M_4 \xrightarrow{t} M_4'$ it holds that $M_3' \centernot{\dequiv} M_4'$. \end{enumerate} \end{lemma} \begin{proof}
Recall that $\textit{max}(\mathcal{N})$ is defined as the largest cardinality of all arc multisets in $\mathcal{N}$, i.e. $|W((p,t),b)| \leq \textit{max}(\mathcal{N})$ for all $p \in P$ and $b \in B(t)$.
\begin{enumerate}
\item[$\bm{(a)}$] Let $M_1,M_2 \in \mathbb{M}(\mathcal{N})$ such that $M_1 \dequiv M_2$, $M_1 \xrightarrow{t} \text{ and } \ M_2 \centernot{\xrightarrow{t}}$. We construct a marking $M_3$ such that $M_3(p) = W((p,t),b)$ for all $p \in P$ and some $b \in B(t)$. It clearly follows that $|M_3(p)| \leq \textit{max}(\mathcal{N})$ for all $p \in P$, hence $M_3 \in \mathbb{M}^{bounded}(\mathcal{N})$. We see that $M_1(p) = M_3(p) \uplus \overline{M}_3(p)$ since $M_3(p) \subseteq M_1(p)$ for all $p \in P$ where $\overline{M}_3(p)$ describes the remaining tokens in $M_1(p)$ that are not in $M_3(p)$.
We know that no inhibitor arc can be the reason that $M_2$ is not enabled because $M_1 \dequiv M_2$. We also know that $M_2(p) \centernot{\subseteq} W((p,t),b)$ for at least one $p \in P$ for all $b \in B(t)$.
We pick a marking $M_4$ where $M_4 \dequiv M_3$, $M_4(p) \centernot{\subseteq} W((p,t),b)$ for at least one $p \in P$ and $M_2(p) = M_4(p) \uplus \overline{M}_4(p)$ for all $p \in P$ such that $\overline{M}_4 \dequiv \overline{M}_3$. Notice that $M_4 \in \mathbb{M}^{bounded}(\mathcal{N})$. We know that $M_4$ exists because $M_2 \dequiv M_1$ and $M_2(p) \centernot{\subseteq} W((p,t),b)$ meaning that $M_4(p) \uplus \overline{M}_4(p) \centernot{\subseteq} W((p,t),b)$ and thus $M_4(p) \centernot{\subseteq} W((p,t),b)$ for some $p \in P$.
\item[$\bm{(b)}$] Let $M_1,M_2 \in \mathbb{M}(\mathcal{N})$ such that $M_1 \dequiv M_2$ and $M_1' \in \mathbb{M}(\mathcal{N})$ such that $M_1 \xrightarrow{t} M_1'$ while for all $M_2' \in \mathbb{M}(\mathcal{N})$ where $M_2 \xrightarrow{t} M_2'$ we know that $M_1' \centernot{\dequiv} M_2'$. We construct a marking $M_3$ exactly as before such that $M_3(p) = W((p,t),b)$ for all $p \in P$ and some $b \in B(t)$ and $M_1(p) = M_3(p) \uplus \overline{M}_3(p)$ for all $p \in P$.
We then pick a marking $M_4$ such that $M_4 \dequiv M_3$
and $M_2(p) = M_4(p) \uplus \overline{M}_4(p)$ for all $p \in P$ and $b \in B(t)$ such that $\overline{M}_4 \dequiv \overline{M}_3$. We know that $M_4 \in \mathbb{M}^{bounded}(\mathcal{N})$ since $M_4 \dequiv M_3$.
Let $M_3' \in \mathbb{M}^{bounded}(\mathcal{N})$ such that $M_3 \xrightarrow{t} M_3'$, which is possible because $M_3(p) \subseteq M_1(p)$ for all $p \in P$ such that no inhibitor arc can inhibit $M_3$. For the sake of contradiction now assume there exists a marking $M_4' \in \mathbb{M}^{bounded}(\mathcal{N})$ such that $M_4 \xrightarrow{t} M_4'$ and $M_3' \dequiv M_4'$. Then notice that we can let $M_1'(p) = M_3'(p) \uplus \overline{M}_3(p)$ where $M_1 \xrightarrow{t} M_1'$ since $M_3(p) \subseteq M_1(p)$ for all $p \in P$ and let $M_2'(p) = M_4'(p) \uplus \overline{M_4}(p)$ where $M_2 \xrightarrow{t} M_2'$ since $M_4(p) \subseteq M_2(p)$ for all $p \in P$. But since $M_3'(p) \dequiv M_4'(p)$ and $\overline{M}_3(p) \dequiv \overline{M}_4(p)$ it means that $M_3'(p) \uplus \overline{M}_3(p) \dequiv M_4'(p) \uplus \overline{M_4}(p)$ for all $p \in P$, i.e. $M_1' \dequiv M_2'$. However, this contradicts the conditions of $\bm{(b)}$, and as such $M_3' \dequiv M_4'$ cannot hold.
\end{enumerate} \end{proof}
\begin{algorithm}[!t] \SetAlgoLined \textbf{Input}: $\mathcal{N} = \CPN{}$\\ \textbf{Output}: Stable partition $\delta$\\ let $\delta(p) := \{C(p)\}$ for all $p \in P$\\ \For{$t \in T$}{
\While{$\exists M_1,M_2 \in \mathbb{M}^{bounded}(\mathcal{N}).M_1 \dequiv M_2 \wedge M_1 \centernot{\xrightarrow{t}} \wedge M_2 \xrightarrow{t} $}
{
pick $\delta' < \delta$ such that $M_1 \centernot{\dequivprime} M_2$ \\
$\delta := \delta'$\\
}
} let $\mathcal{Q} := P$ \ //Waiting list of places\\ \While{$\mathcal{Q} \neq \emptyset$}{
let $p \in \mathcal{Q}$; $\mathcal{Q} := \mathcal{Q} \setminus \{p\}$\\
\For{$t \in \preset{p}$}{
\If{$\exists M_1, M_2 \in \mathbb{M}^{bounded}(\mathcal{N}).M_1 \dequiv M_2.\exists M_1' \in \mathbb{M}^{bounded}(\mathcal{N}). M_1 \xrightarrow{t} M_1' \wedge \forall M_2' \in \mathbb{M}^{bounded}(\mathcal{N}).M_2 \xrightarrow{t} M_2' \wedge M_1'(p) \centernot{\dequiv} M_2'(p)$}
{
pick $\delta' < \delta$ such that $M_1\centernot{\dequivprime} M_2$
and $\delta'(p') = \delta(p')$ for all $p' \in P \setminus \preset{t}$ \\
$\mathcal{Q} := \mathcal{Q} \cup \{p' \ | \ \delta'(p') \neq \delta(p')\}$ \\
$\delta := \delta'$
}
} } \Return $\delta$ \caption{$\textit{Stabilize}(\mathcal{N})$} \label{Algo:StablePartition} \end{algorithm}
Algorithm \ref{Algo:StablePartition} now gives a procedure for computing a stable partition over a given CPN. It starts with an initial partition where every color in the color domain is in the same equivalence class for each place. The algorithm is then split into two parts. The first part from line 4 to 9 creates an initial partition applying the guard restrictions to the input places of the transitions. The second part from line 11 to 20 back propagates the guard restrictions throughout the net such that only colors that behave the same are quotiented together. Depending on the choices in lines 6 and 15, the algorithm may return the maximum stable partition, however in the practical implementation this is not guaranteed due to an approximation of the guard/arc expression analysis.
\begin{theorem} Given a CPN $\mathcal{N}$, the algorithm $\textit{Stabilize}(\mathcal{N})$ terminates and returns a stable partition of $\mathcal{N}$. \end{theorem} \begin{proof} We first prove that $\textit{Stabilize} (\mathcal{N})$ terminates. Notice that each iteration produces a new $\delta$ according to the $>$ operator, and since the operator is well-founded we know that the algorithm terminates. We then show that for $\delta = \textit{Stabilize} (\mathcal{N})$, $\delta$ is a stable partition of $\mathcal{N}$. Recall, a partition $\delta$ is stable iff for any markings $M_1 \dequiv M_2$ whenever $M_1 \xrightarrow{t} M_1'$ for some $t$ and $M_1'$ then $M_2 \xrightarrow{t} M_2'$ for some $M_2'$ such that $M_1' \dequiv M_2'$. We prove this by contradiction. Assume that $\delta$ is not a stable partition. As such there must exists markings $M_1,M_2 \in \mathbb{M}(\mathcal{N})$ such that $M_1 \dequiv M_2$ and there exists a marking $M_1' \in \mathbb{M}(\mathcal{N})$ such that $M_1 \xrightarrow{t} M_1'$ for some transition $t$ where for all $M_2' \in \mathbb{M}(\mathcal{N})$ such that $M_2 \xrightarrow{t} M_2'$ then $M_1' \centernot{\dequiv} M_2'$. This is exactly the property stated in the if statement on line 17 and we know from Lemma \ref{lemma:BoundedMarkings} that if the property is satisfied with two markings from $\mathbb{M}(\mathcal{N})$ then there exists two markings from $\mathbb{M}^{bounded}(\mathcal{N})$ that also satisfy the property.
\end{proof}
\subsection{Stable Partition Algorithm for Integer CPNs} The $\textit{Stabilize}$ computation presented in Algorithm \ref{Algo:StablePartition} can be used to find a stable partition for any finite CPN. However, implementation-wise it is inefficient to represent every color in a given equivalence class individually. Hence, for integer CPN we represent an equivalence class as a tuple of ranges.
As an example of computing stable partitions with Algorithm~\ref{Algo:StablePartition}, consider the integer CPN in Figure~\ref{fig:GuardPartition}. Table~\ref{tab:exampleTable} shows the different stages that $\delta$ undergoes in order to become stable. In iteration 0, the guard restrictions from the first for-loop are applied, followed by the iterations of the main while-loop. In our implementation, we do not iterate through every bounded marking and we instead (for efficiency reasons) statically analyze the places, arcs and guards in order to partition the color sets. For example, in iteration number 1, we consider the place $p_3$ and we can see that the colors in the range $[1,3]$ must be distinguished from the color $4$. This partitioning propagates back to the place $p_1$ as firing the transition $t_1$ moves tokens from $p_1$ to $p_3$ without changing its color.
\begin{tabular}{l|c|c|c|c|l}
Iteration & \multicolumn{1}{c|}{$p_1$} & \multicolumn{1}{c|}{$p_2$} & \multicolumn{1}{c|}{$p_3$} & \multicolumn{1}{c|}{$p_4$} & \multicolumn{1}{c}{$\mathcal{Q}$} \\ \hline
0 & $\{([1,4])\}$ & $\{([1,4])\}$ & $\{([1,3]),([4])\}$ & \begin{tabular}[c]{@{}l@{}}$\{([1,4],[1]), ([1,4],[2]), ([1,4],[3,4])\}$\end{tabular} & $\{p_1,p_2,p_3,p_4\}$ \\ \hline 1, $p=p_3$ & $\{([1,3]),([4])\}$ & - & - & - & $\{p_1,p_2,p_4\}$ \\ \hline 2, $p=p_4$ & - & \specialcelltwo{$\{([1]),$ $([2]), ([3,4])\}$} & - & - & $\{p_1,p_2\}$ \\ \hline 3, $p=p_2$ & - & - & - & \begin{tabular}[c]{@{}l@{}}$\{([1,4],[1]), ([1,4],[2]),$ \\ $([1,4],[3]), ([1,4],[4])\}$\end{tabular} & $\{p_1,p_4\}$ \\ \hline 4, $p=p_4$ & - & \specialcelltwo{$\{([1]),$ $([2]), ([3]), ([4])\}$} & - & - & $\{p_1,p_2\}$ \\ \hline 5, $p=p_2$ & - & - & - & - & $\{p_1\}$ \\ \hline 6, $p=p_1$ & - & - & - & - & $\{\}$ \\ \hline
\end{tabular}
\begin{comment} As an example of what happens in the following iterations consider iteration~1. In this iteration we pick $p_3$ as our place. We then look at the input places for $t_1$ i.e. $p_1$ and $p_2$. We see that there exist markings $M_1 = p_1: 1'(4) + p_2: 1'(4)$ and $M_2 = p_1: 1'(3) + p_2: 1'(4)$ where $M_1 \dequiv M_2$. If we fire $t_1$ from the markings $M_1 \xrightarrow{t_1} M_1'$ and $M_2 \xrightarrow{t_1} M_2'$ then $M_1' = p_3: 2'(4)$ and $M_2' = p_3: 2'(3)$ where $M_1'(p_3) \centernot{\dequiv} M_2'(p_3)$ since $3$ and $4$ are not in the same equivalence class for $\delta (p_3)$. Therefore, we let $\delta( p_{1}) =\{([1,3]),([4])\}$ such that $M_1 \centernot{\dequiv} M_2$.
Since the marking for $p_3$ is not dependent on the value of $y$, we do not modify $\delta (p_2)$ in this iteration. After all iterations are done a stable partition is computed. \end{comment}
\begin{figure}
\caption{Example CPN}
\label{fig:GuardPartition}
\label{tab:exampleTable}
\label{tab:cfpTable}
\end{figure}
\section{Color Approximation}\label{Sec:ColorApprox} We now introduce another technique for safely overapproximating what colors can be present in each place of a CPN. Let $\mathcal{N} = \CPN{}$ be a fixed CPN for the rest of this section.
A \emph{color approximation} is a function $\alpha : P \xrightarrow{} 2^{\mathbb{C}}$ where $\alpha (p)$ approximates the possible colors in place $p \in P$ such that $\alpha (p) \subseteq C(p)$. Let $\mathbb{A}$ be the set of all color approximations.
For a marking $M$ and color approximation $\alpha$, we write $M \subseteq \alpha$ iff $\textit{set}(M(p)) \subseteq \alpha(p)$ for all $p \in P$. A \emph{color expansion} is a function $E : \mathbb{A} \xrightarrow{} \mathbb{A}$ defined by \begin{align*}
E(\alpha)(p) = \left \{\begin{array}{lll}
&\alpha(p) \cup \textit{set}(W((t,p),b)) \textit{ }&\text{if } \exists t \in T.\exists b \in B(t).
\\ &&\textit{set}(W((q,t),b)) \subseteq \alpha(q) \text{ for all } q \in P
\\
&\alpha (p) \textit{ }& \text{otherwise.}
\end{array}\right. \end{align*}
A color expansion iteratively expands the possible colors that exist in each place but without keeping a count of how many tokens of each color are present (if a color is present in a place, we assume that arbitrary many copies of the color are present). The expansion function obviously preserves the following property.
\begin{lemma}\label{Lemma:Monotonicity}
Let $\alpha$ be a color approximation then $\alpha(p) \subseteq E(\alpha)(p)$ for all $p \in P$. \end{lemma}
Let $\alpha_0$ be the initial approximation such that $\alpha_0(p) \defeq \textit{set}(M_0(p))$ for all $p \in P$.
Since $E$ is a monotonic function on a complete lattice, we can compute its minimum fixed point and formulate the following key lemma.
\begin{lemma} \label{alphatheorem} Let $\alpha$ be a minimum fixed point of $E$ such that $\alpha_0(p) \subseteq \alpha(p)$ for all $p\in P$. If $M_0 \xrightarrow{}^* M$ then $M \subseteq \alpha$. \end{lemma} \begin{proof} By induction on $k$ we prove if $M_0 \xrightarrow{}^k M$ then $M \subseteq \alpha$. \textit{Base step.} If $k=0$, we know that $M_0 \subseteq \alpha$ by the assumption of the lemma.
\textit{Induction step.} Let $M_0 \xrightarrow{}^k M \xrightarrow{t} M'$ by some transition $t$ with some binding $b \in B(t)$. We want to show that $M' \subseteq \alpha$. By induction hypothesis we know that $M \subseteq \alpha$. If $M \xrightarrow{t} M'$ for some $b \in B(t)$, then $M'(p) = (M(p) \setminus W((p,t),b)) \uplus W((t,p),b)$ for all $p \in P$. Since $E(\alpha)$ is a fixed point then $\alpha(p) = \alpha(p) \cup \textit{set}(W((t,p),b))$ for transition $t$ under binding $b$ for all $p \in P$ i.e. $\textit{set}(W((q,t),b)) \subseteq \alpha(q)$ for all $q \in P$. Thus we get $M' \subseteq \alpha$. \end{proof}
Given a color approximation $\alpha$ satisfying the preconditions of Lemma~\ref{alphatheorem}, we can now construct a reduced CPN $\mathcal{N}^\alpha = (P, T, \mathbb{C}, \mathbb{B}, C^\alpha, G, W, W_I, M_0 )$ where $C^\alpha(p) = \alpha(p)$ for all $p \in P$. The net $\mathcal{N}^\alpha$ can hence have possibly smaller set of colors in its color domains and it satisfies the following theorem.
\begin{theorem}
The reachable fragments from the initial marking $M_0$ of the LTSs generated by $\mathcal{N}$ and $\mathcal{N}^\alpha$ are isomorphic. \end{theorem} \begin{proof} By Lemma~\ref{alphatheorem} we know that $M \subseteq \alpha$ for any marking $M \in \mathbb{M}(\mathcal{N})$ reachable from $M_0$. Since the reachable fragments of $\mathcal{N}$ and $\mathcal{N}^\alpha$ are exactly the reachable markings, we know that the reachable fragments are isomorphic. \end{proof}
\subsection{Computing Color Approximation on Integer CPNs} As with color quotienting, representing each color individually becomes inefficient.
We thus employ integer ranges to represent color approximations.
Consider the approximation $\alpha$ where $\alpha(p) = \{(1,2),(2,2),(3,2),(5,6),(5,7)\}$ are possible colors (pairs of integers) in the place $p$; this can be more compactly represented as a set of tuples of ranges $\{([1,3],[2]),([5],[6,7])\}$.
However, computing the minimum fixed point of $E$ using ranges is not as trivial as using complete color sets. To do so, we need to compute new ranges depending on arcs and guards. We demonstrate this on the CPN in Figure \ref{fig:GuardPartition}. Table~\ref{tab:cfpTable} shows the computation of the minimum fixed point of $E$, starting from the initial approximation $\alpha_0$. For example, in iteration number 5, we check if firing transition $t_1$ can produce any additional tokens to the places $p_3$ and $p_4$. Clearly, there is no change to the possible token colors in $p_3$ as $\alpha(p_1)$ did not change, however the addition of the integer range $[1]$ to $\alpha(p_2)$ in the previous iteration now allows us to produce a new token color $(1,1)$ into $p_4$ and hence we add the singleton range $([1],[1])$ to $\alpha(p_4)$. Due to the guard $y \geq 3$ on the transition $t_2$, we know that the added token $(1,1)$ does not generate any further behaviour and hence we reached a fixed point.
\section{Experiments} We implemented the quotienting method from Section~\ref{Sec:Partitioning} as well as the color approximation method from Section \ref{Sec:ColorApprox} in C++ as an extension to the verification engine \textit{verifypn}~\cite{verifypnPaper} from the TAPAAL toolchain~\cite{TAPAALTool}.
\begin{comment} The variable symmetry technique works by identifying whether two variables of a binding are permutable to avoid creating equivalent bindings. For example, assume variables $x,y,z$ with range $[1,3]$ and an arc expression $1'(x) + 1'(y) + 1'(z)$. Assuming that there are no restrictions on these variables from guards or arcs there are $3^3 = 27$ possible bindings. However, if the variables are detected as symmetric bindings such as $\langle x = 1, y = 1, z = 2 \rangle$ and $\langle x = 2, y = 1, z = 1 \rangle$ are equivalent and we only need to consider $10$ of the $27$ bindings. \end{comment}
\begin{comment} In Figure \ref{Fig:flowchart} a flowchart diagram of the process of unfolding and verifying a CPN can be seen. We parse the CPN, process the net with the different static analysis techniques and then use the information gained to unfold the CPN into a P/T net which can then be verified. Since the different static analysis techniques are independent any of the techniques can be disabled e.g. it is possible to only do color quotienting and color approximation and skip detection of symmetric variables. For more details on the verification process refer to~\cite{tapaal_structural_reductions}. \begin{figure}
\caption{Flowchart of a CPN through verifypn}
\label{Fig:flowchart}
\end{figure} \end{comment}
\begin{comment} \subsection{Correctness of our implementation} We check the correctness of our implementation on the colored nets and queries provided in the 2021 Model Checking Contest~\cite{mcc:2021}. The correctness is tested by verifying 16 queries in each of the following categories: ReachabilityCardinality, ReachabilityFireability, CTLCardinality, CTLFireability, LTLCardinality and LTLFireability per net for 213 colored nets (of which we unfold 207). All together we test 20448 queries of which we provide an answer for 16368 queries.
We compare query answers to the Oracle database, which is a database of agreed upon answers by tools from the Model Checking Contest~\cite{oracle-database}. For the above mentioned categories there are 10096 answers where we answer consistently on all of them.
\subsection{TAPAAL GUI Support} For the development of our methods and implementation we made use of the recently developed and soon to be released colored GUI of TAPAAL~\cite{TAPAALGUI}. The GUI supports the integer CPNs described in Section \ref{Sec:IntegerCPN} and has helped us refining and debugging our implementation. \end{comment}
We perform experiments by comparing several different approaches; the quotienting approach (method A), the color approximation approach (method B) and the combination of both (method A+B) against \begin{itemize} \item the unfolder MCC~\cite{MCCUnfolder} (used also by TINA~\cite{TINAtool} and LoLA~\cite{LoLA2}), \item ITS-Tools unfolder~\cite{ITSToolPaper}, \item the Spike unfolder~\cite{SpikeTool} (also used by MARCIE~\cite{MARCIETool} and Snoopy~\cite{SnoopyTool}), \item and \textit{verifypn} TAPAAL unfolder with methods A and B disabled, referred to as Tapaal. \end{itemize}
We compare the tools on the complete set of CPN nets and queries from
2021 Model Checking Contest~\cite{mcc:2021}\footnote{We omit the \emph{UtilityControlRoom} family of models as they contain operators not supported by ITS-Tools.}. The experiments are conducted on a compute cluster running Linux version 5.4.0, where each experiment is conducted on a AMD Epyc 7642 processor with a 15 GB memory limit and 5 minute timeout. To reduce noise in the experiments, we read and write models to a RAM-disk (not included in the 15 GB). A repeatability package is available in~\cite{Repeatabilitypackage}.
We conduct two series of experiments: in Section~\ref{sec:unf} we study the unfolding performance of the tools; i.e. how fast and how many models can be unfolded by each tool and how large are the unfolded models, and in Section~\ref{sec:query} we study the impact of this unfolding on the ability to answer the queries from the Model Checking Contest 2021.
\subsection{Unfolding Experiments} \label{sec:unf} \begin{comment} \subsection{Methodology} \subsubsection{Size}
To measure the size differences we use the size ratio defined as $\dfrac{size_1}{size_2}$ where the size for a given net is $|P| + |T|$ where $|P|$ is the number of places and $|T|$ is the number of transitions. Since we work with division we make the following rules: $\dfrac{size_1}{NaN} = 0$ and $\dfrac{NaN}{size_2} = 1000$ where $NaN$ is the size value for a net that has not been unfolded. Lastly, if neither can unfold the value will be $1$. \subsubsection{Time} When discussing unfolding time, we only measure pure unfolding time in seconds (and post-processing reduction time in the case of ITS), which means we exclude time used on reading, parsing, outputting etc. We make a rule that if the unfolding did not complete, the unfolding time will be set to $1000$ seconds. \subsubsection{Cactus Plots} We present the results for unfolding time and size using \textit{cactus plots}. A cactus plot is a graph for which each method has their values sorted from lowest to highest, i.e. the first element is the one with the lowest value for a given method. As such, two points with the same x-value for different methods may not be for the same net. \subsubsection{Options} For method A we set the timeout to 5 seconds. For method B we set the granularity constant to 250 and let it go down to 5 after 10 seconds. \end{comment}
Table~\ref{tab:unfoldedNets} shows for each of the unfolders the number of unfolded nets within the memory/time limit. The last column shows the total number of unfolded nets by all tools combined, and we notice that the combination of methods A$+$B allows us to unfold a superset of models unfolded by the other tools. Our method A$+$B can unfold 3 nets that no other tool can unfold; DrinkVendingMachine48, 76, 98. This is directly attributed to method A.
\begin{table}[t] \centering
\newcolumntype{x}[1]{>{\centering\arraybackslash\hspace{0pt}}p{#1}}
\begin{tabular}{l|x{0.9cm}x{1.1cm}x{0.9cm}x{0.9cm}x{0.9cm}x{0.9cm}x{0.9cm}|x{0.9cm}}
& Spike & Tapaal & ITS-Tools & A & B & MCC & A$+$B & Total\\ \hline Unfolded nets & 170 & 192 & 193 & 195 & 200 & 200 & 203 & 203 \end{tabular}
\caption{Number of unfolded nets for each unfolder} \label{tab:unfoldedNets}
\end{table}
The comparison of the sizes (total number of transitions and places) of unfolded nets is done by plotting the ratios between the size produced by our A+B method and the competing unfolder.
Figure~\ref{Fig:Sizeratio90best} shows the size ratios where at least one comparison is not equal to 1. We see that our method has a size ratio always smaller or equal to 1 (no other method unfolds any of the nets to a smaller size compared to our method A+B) and the size ratio is strictly smaller than 1 for 144 colored nets (out of 213 nets in the database). Moreover, we can reduce 45 nets by at least one order of magnitute, compared to all other unfolders.
\begin{figure}
\caption{Size-ratios (nondecreasingly ordered)}
\caption{Worst 80 unfolding times}
\caption{Unfolding size and unfolding time comparison}
\label{Fig:Sizeratio90best}
\label{Fig:time80best}
\end{figure}
\begin{comment} In Table~\ref{tab:medianRatios} the median size ratio for each type of model is presented. We see that method A+B is better on over half of the models, and is able to reduce the size to a fraction of what the other tools can for many nets. However, we again see that ITS-Tools slightly outperforms method A+B on a few models due to the post-reductions. \setlength{\tabcolsep}{8pt} \begin{table}[h] \centering
\begin{tabular}{l|ccc}
\multicolumn{1}{c|}{Name} & A+B/ITS & A+B/MCC & A+B/Spike \\ \hline DrinkVendingMachine & \textless{}0.001 & \textless{}0.001 & - \\ GlobalResAllocation & 0.003 & \textless{}0.001 & \textless{}0.001 \\ Referendum & 0.021 & 0.021 & 0.021 \\ Airplane & 0.023 & 0.018 & 0.018 \\ Bart & 0.051 & 0.032 & - \\ PermAdmissibility & 0.074 & 0.045 & 0.045 \\ DotAndBoxes & 0.281 & 0.207 & 0.207 \\ CSRepetition & 0.463 & 0.463 & 0.463 \\ Sudoku-COL-A & 0.576 & 0.576 & - \\ Sudoku-COL-B & 0.632 & 0.632 & - \\ FamilyReunion & 0.686 & 0.685 & 0.686 \\ QuasiCertifProtocol & 0.947 & 0.947 & 0.947 \\ PolyORBNT & 1 & 0.446 & 0.446 \\ PolyORBLF & 1 & 0.557 & 0.557 \\ LamportFastMutEx & 1 & 0.807 & 0.807 \\ BridgeAndVehicles & 1 & 1 & 1 \\ DatabaseWithMutEx & 1 & 1 & 1 \\ Philisophers & 1 & 1 & 1 \\ PhilisophersDyn & 1 & 1 & 1 \\ TokenRing & 1 & 1 & 1 \\ SharedMemory & 1.007 & 1 & 1 \\ SafeBus & 1.009 & 0.997 & 0.997 \\ NeoElection & 1.04 & 0.057 & 0.077 \\ Peterson & 1.047 & 1 & 1 \\ VehicularWifi & 2.234 & 0.174 & - \end{tabular}
\caption{Size-ratios on the median models for all tools. A '-' indicates that the median net was not unfolded by the tool} \label{tab:medianRatios} \end{table}
One important property of method A$+$B is the ability to have constant size scaling on certain nets. This means that no matter how many extra colors are added to the color domains, the unfolded net will be the same size. As such, we can go from an increase in size with each new instance to no increase at all. The nets where this is the case can be seen in Table~ \ref{tab:constantSizeScaling}.
\begin{table}[] \centering
\begin{tabular}{l|llll}
& A+B & ITS & MCC & Spike \\ \hline AirplaneLD-COL-0010 & 55 & 145 & 177 & 177 \\ AirplaneLD-COL-0020 & 55 & 265 & 327 & 327 \\ AirplaneLD-COL-0050 & 55 & 625 & 777 & 777 \\ AirplaneLD-COL-0100 & 55 & 1225 & 1527 & 1527 \\ AirplaneLD-COL-0200 & 55 & 2425 & 3027 & 3027 \\ AirplaneLD-COL-0500 & 55 & 6025 & 7527 & 7527 \\ AirplaneLD-COL-1000 & 55 & 12025 & 15027 & 15027 \\ AirplaneLD-COL-2000 & 55 & 24025 & 30027 & 30027 \\ AirplaneLD-COL-4000 & 55 & 48025 & 60027 & 60027 \\ BART-COL-002 & 447 & 668 & 1410 & - \\ BART-COL-005 & 447 & 1670 & 3117 & - \\ BART-COL-010 & 447 & 3340 & 5962 & - \\ BART-COL-020 & 447 & 6680 & 11652 & - \\ BART-COL-030 & 447 & 12509 & 17342 & - \\ BART-COL-040 & 447 & 13360 & 23032 & - \\ BART-COL-050 & 447 & 16700 & 28722 & - \\ BART-COL-060 & 447 & 20040 & 34412 & - \\ DrinkVendingMachine-COL-02 & 22 & 76 & 96 & 96 \\ DrinkVendingMachine-COL-10 & 22 & 28780 & 111280 & 111280 \\ DrinkVendingMachine-COL-16 & 22 & 248352 & 1118752 & 1118752 \\ DrinkVendingMachine-COL-24 & 22 & 1685232 & 8309232 & - \\ DrinkVendingMachine-COL-48 & 22 & - & - & - \\ DrinkVendingMachine-COL-76 & 22 & - & - & - \\ DrinkVendingMachine-COL-98 & 22 & - & - & - \\ Referendum-COL-0010 & 7 & 52 & 52 & 52 \\ Referendum-COL-0015 & 7 & 77 & 77 & 77 \\ Referendum-COL-0020 & 7 & 102 & 102 & 102 \\ Referendum-COL-0050 & 7 & 252 & 252 & 252 \\ Referendum-COL-0100 & 7 & 502 & 502 & 502 \\ Referendum-COL-0200 & 7 & 1002 & 1002 & 1002 \\ Referendum-COL-0500 & 7 & 2502 & 2502 & 2502 \\ Referendum-COL-1000 & 7 & 5002 & 5002 & 5002 \end{tabular} \caption{Nets with constant size scaling. A '-' indicates that the net was not unfolded by the tool} \label{tab:constantSizeScaling} \end{table}
In general we see that method A+B has a large effect on the sizes of the unfolded nets, and reduces the size considerably compared to the other unfolders with a few exceptions compared to ITS. \end{comment}
As our method outperforms the state-of-the-art unfolders w.r.t. the size of the unfolded nets, the question is whether the overhead of the advanced static analysis does not kill the benefits. Fortunately, this is not the case as shown in Figure~\ref{Fig:time80best} where the 80 slowest unfolding times (independently sorted in nondecreasing order for each tool) are depicted. The plots show that our method has almost a magnitude reduction in running-time compared to ITS-Tools and MCC (which are close in performance), while Spike is significantly slower. ITS-Tools is generally fast on the nets that are unfolded in less than 10 seconds, however it becomes gradually slower and has problems unfolding the larger nets. Our unfolder and the MCC unfolder demonstrate a similar degradation trend in performance.
The overall conclusion is that our advanced analyses requires less overhead compared to other existing unfolders, while at the same time it significantly decreases the size of the unfolded nets.
\subsection{Query Verification Experiments} \label{sec:query} In these experiments, we examine which unfolding engine allows for most query answers verified on the unfolded nets. To allow for a fair comparison, we let each tool unfold and output the net to a PNML file. Regarding queries, both method A$+$B and ITS-Tools can already output the unfolded queries. For MCC, we implement our own translation from the colored queries to the unfolded queries for the given nets. For Spike, we were not able to construct a query unfolder that worked consistently and for this reason Spike is excluded from the query verification experiments.
Since we are testing the effect of the unfolding and not the verification engine, we use \textit{verifypn} (revision 507c8ee0) to verify the queries on the nets unfolded by the different unfolders.
There is a total of 20,448 queries to be answered. The results can be seen in Table~\ref{tab:queriesAnswered}.
\begin{table}[] \centering \begin{tabular}{lllllll} \hline \multicolumn{7}{c}{\textbf{Cardinality Queries}} \\ \hline
\multicolumn{1}{l|}{} & \multicolumn{2}{c|}{A+B} & \multicolumn{2}{c|}{MCC} & \multicolumn{2}{c}{ITS-Tools} \\ \hline
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{Solved} & \multicolumn{1}{c|}{\%} & \multicolumn{1}{l|}{Solved} & \multicolumn{1}{c|}{\%} & \multicolumn{1}{l|}{Solved} & \multicolumn{1}{c}{\%} \\ \hline
\multicolumn{1}{l|}{ReachabilityCardinality} & \multicolumn{1}{l|}{2909} & \multicolumn{1}{l|}{85.4} & \multicolumn{1}{l|}{2793} & \multicolumn{1}{l|}{82.0} & \multicolumn{1}{l|}{2759} & 81.0 \\ \hline
\multicolumn{1}{l|}{CTLCardinality} & \multicolumn{1}{l|}{2651} & \multicolumn{1}{l|}{77.8} & \multicolumn{1}{l|}{2490} & \multicolumn{1}{l|}{73.1} & \multicolumn{1}{l|}{2443} & 73.8 \\ \hline
\multicolumn{1}{l|}{LTLCardinality} & \multicolumn{1}{l|}{2952} & \multicolumn{1}{l|}{86.6} & \multicolumn{1}{l|}{2818} & \multicolumn{1}{l|}{82.7} & \multicolumn{1}{l|}{2785} & 81.7 \\ \hline
\multicolumn{1}{l|}{\textbf{Total}} & \multicolumn{1}{l|}{\textbf{8512}} & \multicolumn{1}{l|}{\textbf{83.3}} & \multicolumn{1}{l|}{\textbf{8101}} & \multicolumn{1}{l|}{\textbf{79.2}} & \multicolumn{1}{l|}{\textbf{7987}} & \textbf{78.1}\\ \hline \multicolumn{7}{c}{\textbf{Fireability Queries}} \\ \hline
\multicolumn{1}{l|}{ReachabilityFireability} & \multicolumn{1}{l|}{2567} & \multicolumn{1}{l|}{75.3} & \multicolumn{1}{l|}{2484} & \multicolumn{1}{l|}{72.9} & \multicolumn{1}{l|}{2513} & 73.7 \\ \hline
\multicolumn{1}{l|}{CTLFireability} & \multicolumn{1}{l|}{2047} & \multicolumn{1}{l|}{60.1} & \multicolumn{1}{l|}{1878} & \multicolumn{1}{l|}{55.1} & \multicolumn{1}{l|}{1695} & 49.7 \\ \hline
\multicolumn{1}{l|}{LTLFireability} & \multicolumn{1}{l|}{2798} & \multicolumn{1}{l|}{82.1} & \multicolumn{1}{l|}{2639} & \multicolumn{1}{l|}{77.4} & \multicolumn{1}{l|}{2520} & 73.9 \\ \hline
\multicolumn{1}{l|}{\textbf{Total}} & \multicolumn{1}{l|}{\textbf{7412}} & \multicolumn{1}{l|}{\textbf{72.5}} & \multicolumn{1}{l|}{\textbf{7001}} & \multicolumn{1}{l|}{\textbf{68.5}} & \multicolumn{1}{l|}{\textbf{6728}} & \textbf{65.8} \\ \hline \hline
\multicolumn{1}{l|}{\textbf{Total query answers}} & \multicolumn{1}{l|}{\textbf{15924}} & \multicolumn{1}{l|}{\textbf{77.9}} & \multicolumn{1}{l|}{\textbf{15102}} & \multicolumn{1}{l|}{\textbf{73.9}} & \multicolumn{1}{l|}{\textbf{14715}} & \textbf{72.0} \\ \hline \end{tabular}
\caption{Number of queries answered for the unfolded nets of each tool. The \% column describes how many percent of the total available queries in each category were answered. Each category counts $213 \cdot 16=3408$ queries.} \label{tab:queriesAnswered} \end{table}
We see that using the method A$+$B to unfold nets allows us to answer more queries in every category due to the generally smaller nets it unfolds to. In total, we are able to answer $4$ percentage points more queries using the unfolded nets of method A$+$B compared to using the unfolded nets of MCC and $5.9$ percentage points more compared to ITS-Tools.
\section{Conclusion} We presented two complementary methods for reducing the unfolding size of colored Petri nets (CPN). Both methods are proved correct and implemented in an open-source verification engine of the tool TAPAAL. Experimental results show a significant improvement in the size of unfolded nets, compared to state-of-the-art tools, without compromising the unfolding speed. The actual verification on the models and queries from the 2021 Model Checking Contest shows that our unfolding technique allows us to solve 4\% more queries compared to the second best competing tool. In future work, we plan to combine our approach with structural reduction techniques applied directly to the colored nets.
\paragraph{Acknowledgments.} We would like to thank Yann Thierry-Mieg for his answers and modifications to the ITS-Tools,
Silvano Dal Zilio for his answers/additions concerning the MCC unfolder
and Monika Heiner and Christian Rohr for their answers concerning the tools Snoopie, Marcie and Spike.
\sloppy
\end{document} | arXiv |
\begin{document}
\title{The geometry of marked contact Engel structures}
\vskip 1.truecm
\author{\textsc{Gianni Manno}\\
Dipartimento di Scienze Matematiche (DISMA),\\
Politecnico di Torino,\\
Corso Duca degli Abruzzi, 24,
10129 Torino, Italy\\
\textsf{[email protected]}\\[0.5cm]
\textsc{Pawe\l\ Nurowski}\\
Center for Theoretical Physics PAS\\
Al. Lotnik\'ow 32/46,
02-668 Warsaw, Poland\\
\textsf{[email protected]}\\[0.5cm]
\textsc{Katja Sagerschnig}\\
Center for Theoretical Physics PAS\\
Al. Lotnik\'ow 32/46,
02-668 Warsaw, Poland\\
\textsf{[email protected]}}
\date{\today}
\maketitle
\begin{abstract} A \emph{contact twisted cubic structure} $({\cal M},\mathcal{C},\gamma)$ is a $5$-dimensional manifold ${\cal M}$ together with a contact distribution $\mathcal{C}$ and a bundle of twisted cubics $\gamma\subset\mathbb{P}(\mathcal{C})$ compatible with the conformal symplectic form on $\mathcal{C}$. In Engel's classical work, the Lie algebra of the exceptional Lie group $\mathrm{G}_2$ was realized as the symmetry algebra of the most symmetrical contact twisted cubic structure; we thus refer to this one as the \emph{contact Engel structure}. In the present paper we equip the contact Engel structure with a smooth section $\sigma: {\cal M}\to\gamma$, which ``marks'' a point in each fibre $\gamma_x$.
We study the local geometry of the resulting structures $({\cal M},\mathcal{C},\gamma, \sigma)$, which we call \emph{marked contact Engel structures}. Equivalently, our study can be viewed as a study of foliations of ${\cal M}$ by curves whose tangent directions are everywhere contained in $\gamma$.
We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $\geq 6$, and we prove an analogue of the classical Kerr theorem from relativity.
\end{abstract}
\tableofcontents
\section{Introduction}
Consider a smooth $5$-dimensional manifold ${\cal M}^5$ together with a contact distribution, i.e., a rank $4$ subbundle $\mathcal{C}\subset T{\cal M}^5$ such that the Levi bracket \begin{equation}\label{Levi}\mathcal{L}:\Lambda^2\mathcal{C}\to T{\cal M}^5/\mathcal{C},\quad \xi_x\wedge\eta_x\mapsto [\xi,\eta]_x \ \mathrm{mod}\ \mathcal{C}_x\end{equation} is non-degenerate at each point $x\in {\cal M}$. Then $\mathcal{L}_x$ endows each fibre $\mathcal{C}_x$ with the structure of a conformal symplectic vector space. Locally, $\mathcal{C}$ is the kernel of a contact form, i.e., $\mathcal{C}=\mathrm{ker}(\alpha)$, where $\alpha\in\Omega^1({\cal M}^5)$ satisfies ${\rm d}\alpha\wedge{\rm d}\alpha\wedge\alpha\neq 0$, and the conformal symplectic structure on $\mathcal{C}_x$ is generated by ${\rm d}\alpha\vert_{{\mathcal{C}}_x}$.
Now suppose that each contact plane $\mathcal{C}_x$ is equipped with a cone $\hat{\gamma}_x\subset\mathcal{C}_x$ whose projectivization $\gamma_x\subset\mathbb{P}(\mathcal{C}_x)$ is the image of the map $$ \mathbb{R}\mathbb{P}^1\to\mathbb{P}(\mathcal{C}_x)\cong\mathbb{R}\mathbb{P}^3,\quad [t,s]\mapsto [t^3,t^2 s,t s^2,s^3]\,; $$ such a curve is called a twisted cubic curve (also, rational normal curve of degree three). Moreover, assume that $\hat{\gamma}_x$ is a Lagrangian in the sense that the tangent space at each non-zero point is a $2$-dimensional subspace of $\mathcal{C}_x$ on which the conformal symplectic form vanishes identically. Further suppose that $\gamma=\bigsqcup_{x\in\mathcal{M}^5}\gamma_x\to{\cal M}^5$ is a subbundle of $\mathbb{P}(\mathcal{C})\to{\cal M}^5$. Then $({\cal M}^5,\mathcal{C},\gamma)$ is called a \emph{contact twisted cubic structure}.
In 1893 Cartan and Engel, in the same journal but independent articles \cite{cartan, engel}, provided the first explicit realizations of the Lie algebra of the exceptional Lie group $\mathrm{G}_2$\footnote{In this paper $\mathrm{G}_2$ denotes a Lie group whose Lie algebra is the split real form of the smallest of the complex exceptional simple Lie algebras, see Section \ref{LieTheory}.}
as infinitesimal automorphisms of geometric structures on $5$-dimensional manifolds. One of these structures was the simplest maximally non-integrable rank two distribution, while the other was the simplest contact twisted cubic structure. (In other words, Cartan and Engel gave local coordinate descriptions of the geometric structures on the two $5$-dimensional homogeneous spaces $\mathrm{G}_2/\mathrm{P}_1$ and $\mathrm{G}_2/\mathrm{P}_2$ whose automorphism groups are precisely $\mathrm{G}_2$.) Engel's description of the invariant contact twisted cubic structure
was (up to a different choice of coordinates) as follows:
Let $(x^0,x^1,x^2,x^3,x^4)$ be local coordinates on an open subset $\mathcal{U}\subset \mathbb{R}^5$ and consider
co-frame $(\alpha^0,\alpha^1,\alpha^2,\alpha^3,\alpha^4),$ \begin{equation}\label{coframeintro}\alpha^0=\mathrm{d}x^0+x^1 \mathrm{d}x^4- 3 x^2 \mathrm{d}x^3,\quad \alpha^1={\rm d} x^1,\quad \alpha^2={\rm d} x^2,\quad \alpha^3={\rm d} x^3,\quad \alpha^4={\rm d} x^4,\end{equation} with dual frame
$(X_0,X_1,X_2,X_3,X_4)$, \begin{equation}\label{frameintro} \begin{aligned}X_0=\partial_{x^0},\quad X_1=\partial_{x^1}, \quad X_2=\partial_{x^2},\quad X_3=3x^2\partial_{x^0}+\partial_{x^3},\quad X_4=-x^1\partial_{x^0}+\partial_{x^4}. \end{aligned} \end{equation} Here $\alpha^0$ is a contact form and defines a contact distribution $\mathcal{C}=\mathrm{ker}(\alpha^0)$. Now consider the set of horizontal null vectors $$\hat{\gamma}=\{\,Y\in\mathcal{C}:\,g_1(Y,Y)=g_2(Y,Y)=g_3(Y,Y)=0\,\}$$
of the three degenerate metrics \footnote{We refer to the tensor fields $g_1, g_2, g_3\in\Gamma(\smash{\bigodot^2T^*\mathcal{U}})$ as metrics, although strictly speaking these are not metrics, since all three of them are indeed degenerate, and $g_1$ and $g_2$ are even degenerate when restricted to the distribution $\mathcal{C}$.} \begin{equation}\begin{aligned}\label{cone}
g_1=\alpha^1\alpha^3- (\alpha^2)^2,\quad
g_2= \alpha^2\alpha^4- (\alpha^3)^2,\quad
g_3= \alpha^2\alpha^3- \alpha^1 \alpha^4,
\end{aligned} \end{equation} where $\alpha^i\alpha^j=\tfrac{1}{2}(\alpha^i\otimes\alpha^j+\alpha^j\otimes\alpha^i)$. Then $Y\in\Gamma(\mathcal{C})$ takes values in $\hat{\gamma}$ if and only if is of the form $$Y=t^3 X_1+t^2 s X_2+ ts^2X_3+s^3X_4.$$
Hence the projectivization $\gamma_x\subset\mathbb{P}(\mathcal{C}_x)$ of $\hat{\gamma}_x$ is a twisted cubic curve, and it is straightforward to verify that $\hat{\gamma}_x\subset\mathcal{C}_x$ is Lagrangian.
We shall call the structure $(\mathcal{U},\mathcal{C},\gamma)$
the \emph{contact Engel structure} in view of Engel's classical work.\footnote{Contact Engel structures should not be confused with Engel distributions, sometimes also called Engel structures, which are maximally non-integrable rank $2$ distributions on $4$-dimensional manifolds.}
The contact Engel structure is the \emph{flat model} for contact twisted cubic structures in the following sense. One can show that a contact twisted cubic structure is the underlying structure of a certain type of Cartan geometry, more specifically parabolic geometry, see \cite{tanaka, book}. As such it admits a \emph{canonical Cartan connection}, which has in general \emph{nonzero curvature}. There is a unique, up to a local equivalence, contact twisted cubic structure
whose curvature vanishes identically. This is the one we call the contact Engel structure.
A specialization of contact twisted cubic structures can go \emph{independently} in other directions. For example, instead of imposing restrictions on the curvature
of a given contact twisted cubic structure, one can restrict its structure group by adding more structure.
The structure group of the corresponding enriched geometry must preserve this additional structure, and gets reduced. We will explain below that a natural choice for such a reduction is to add a section
$$\sigma:\mathcal{M}^5\to\gamma\subset \mathbb{P}(\mathcal{C})$$ of the bundle
$\,\mathbb{R}\mathbb{P}^1\to\gamma\to {\cal M}^5$ of twisted cubics to the geometric structure.
Since such a section $\sigma$ marks a point $*=\sigma(x)$ in each twisted cubic $\gamma_x$, $x\in \mathcal{M}^5$, we refer to the enriched structure $({\cal M}^5,\mathcal{C},\gamma,\sigma)$
as a \emph{marked contact twisted cubic structure}. If the underlying contact twisted cubic structure is flat, then the resulting structure will be called a \emph{marked contact Engel structure}.
One may think of a marked contact twisted cubic structure as a foliation of a contact twisted cubic structure by special horizontal curves.
Suppose we are given a marked contact twisted cubic structure $({\cal M}^5,\mathcal{C},\gamma,\sigma)$. For each $x\in {\cal M}^5$, the point $\sigma(x)\in\gamma_x$ corresponds to a direction $\ell^{\sigma}_x$ in the contact plane $\mathcal{C}_x$.
Therefore, the section $\sigma$ defines a rank one distribution $\ell^{\sigma}\subset T{\cal M}^5$ whose integral manifolds define a foliation of $\mathcal{M}^5$ by curves (a \emph{congruence}). Conversely, a congruence on ${\cal M}^5$ by curves whose tangent directions are everywhere contained in $\gamma\subset\mathbb{P}(\mathcal{C})$ uniquely determines a section $\sigma:\mathcal{M}^5\to\gamma$. Since $\gamma_x\subset\mathbb{P}(\mathcal{C}_x)$ is cut out by three polynomials, the congruences corresponding to sections $\sigma:{\cal M}^5\to\gamma$ can be also seen as null congruences.
\subsection{Context and motivation}
Before we outline the main results of this paper, a few words of motivation are in order:
It follows from the above brief description that the marked contact Engel structures, or their more general cousins, the marked contact twisted cubic structures, are \emph{special} contact twisted cubic structures. This places the area of our present study in the context of \emph{special geometries}, which are mostly developed in Riemannian geometry. For example, similarly to the addition of a section $\sigma$ to a contact twisted cubic structure $({\cal M}^5,{\mathcal C},\gamma)$, one can add an almost Hermitian structure $\mathcal{J}$ to an even-dimensional Riemannian manifold $({\cal M}^{2n},g)$. In this way one passes from the Riemmannian geometry $({\cal M}^{2n},g)$ to the \emph{special} Riemannian geometry (almost Hermitian geometry) $({\cal M}^{2n},g,\mathcal{J})$, as we are passing from $({\cal M}^5,{\mathcal C},\gamma)$ to the special geometry $({\cal M}^5,{\mathcal C},\gamma,\sigma)$.
The analogy between our marked contact Engel structures and special geometries is particularly striking if we replace Riemannian geometry by \emph{conformal Lorentzian geometry in 4-dimensions} $({\cal M}^4,[g])$. These are the geometries studied in General Relativity, when the related physics is concerned with massless particles only.
Of particular importance in General Relativity are \emph{null congruences}, i.e. \emph{foliations} of $({\cal M}^4,[g])$ \emph{by null curves}. Suppose that we have such a congruence on $({\cal M}^4,[g])$. Let $K\subset T{\cal M}^4$ be the null line subbundle such that any section $s: {\cal M}^4\to K$ be tangent to the congruence. Then we have a \emph{special} Lorentzian conformal geometry $({\cal M}^4,[g],K)$, which we call a \emph{null congruence structure}. One can study the local equivalence problem of such geometries, where two null congruence structures $({\cal M}^4_i,[g_i],K_i)$, $i=1,2$, are locally equivalent if and only if there exists a local diffeomorphism $\phi:{\cal M}^4_1\to {\cal M}^4_2$ such that $\phi^*(g_2)=f^2 g_1$, $\phi^*(K_2)=K_1$, with $f$ a non-vanishing function on ${\cal M}^4_1$. One very quickly establishes that there are locally non-equivalent null congruence structures even if both conformal structures are conformally flat. For example, if the curves of one null congruence are \emph{geodesics} (this is a conformally invariant property) and the curves of the other one are not, the two congruences are locally non-equivalent. Even if we have two null congruences such that both are weaved by geodesics, they are still in general \emph{not} locally equivalent. The next important conformally invariant property distinguishing locally non-equivalent structures is \emph{shearfreeness} \cite{Robinson}, see \cite{GHN, HJN, nurTraut, ArmanTwistor, Curtis} for more details. So here is our analogy:\\ \begin{center}
\begin{tabular}{|c|c|}
\hline
conformal spacetime &contact twisted cubic structure\\ $({\cal M}^4,[g])$ & $({\cal M}^5,{\mathcal C},\gamma)$\\
\hline
conformally flat spacetime & Engel structure\\
\hline
null congruence & marked contact twisted cubic \\structure $({\cal M}^4,[g],K)$& structure $({\cal M}^5,{\mathcal C},\gamma,\sigma)$\\
\hline
conformally flat null &marked contact \\ congruence structure & Engel structure\\
\hline
conformally flat null & integrable marked contact \\congruence structure & Engel structure \\ of geodesics & \\ \hline
conformally flat null &integrable marked contact \\ congruence structure & Engel structure \\
of shearfree geodesics& \\\hline
Kerr theorem&Kerr theorem for contact\\& Engel structures\\ \hline \end{tabular} \end{center}
The relevance of the integrability condition on marked contact Engel structure, which appears in the above Table, will be explained in Section \ref{SecKerr}. Here we only mention that in our analogy it is related to the celebrated Kerr theorem of General Relativity, see \cite{penroserindler2, Tafel}, which gives a construction of all null congruence structures of shearfree geodesics that can live in conformally flat spacetimes. This theorem is the origin of Penrose's \emph{twistor theory} \cite{RP}. The analogy described above shows that it has a well defined interesting counterpart for marked contact Engel structures.
\subsection{Structure and main results of the article}
Section \ref{sec_marked} introduces the notions of a contact twisted cubic structure, Engel structure, marked contact twisted cubic structure and marked contact Engel structure. First observations about these structures are presented. In particular, the so-called ``osculating filtration'' determined by a marked contact twisted cubic structure is introduced: This is a filtration of the contact bundle $\mathcal{C}$ by distributions $$\ell^{\sigma}\subset\mathcal{D}^{\sigma}\subset\mathcal{H}^{\sigma}\subset\mathcal{C},$$ with respective ranks $1$, $2$, $3$, $4$, where $\mathcal{D}^{\sigma}$ is a \emph{Legendrian} rank two distribution. It corresponds fibre-wise to the osculating sequence
of the twisted cubic $\gamma_x\subset\mathbb{P}(\mathcal{C}_x)$ at a point $\sigma(x)$.
We call a marked contact twisted cubic structure (respectively the section $\sigma$) \emph{integrable} if the distribution $\mathcal{D}^{\sigma}$ is integrable.
The core of the present paper is Section \ref{CartanEquiv}, where we apply Cartan's method of equivalence to study the local equivalence problem of marked contact Engel structures. Throughout this paper, we shall refer to the set of \emph{all} vector fields preserving a given marked contact Engel structure as \emph{the infinitesimal symmetry algebra}, or simply \emph{the symmetry algebra}, of the marked contact Engel structure. We shall denote by $\mathfrak{g}$ the Lie algebra of the exceptional Lie group $\mathrm{G}_2$.\footnote{We chose to denote the Lie algebra of the Lie group $\mathrm{G}_2$ by $\mathfrak{g}$ in order to avoid confusion with a certain grading component that is commonly denoted by $\mathfrak{g}_2$.}
\begin{itemize} \item We show that there exists a (locally unique) maximally symmetric model for marked contact Engel structures. Its symmetry algebra is isomorphic to the $9$-dimensional parabolic subalgebra $\mathfrak{p}_1$ of $\mathfrak{g}$ that may be realized as the stabilizer of a highest weight line in the $7$-dimensional irreducible representation of $\mathfrak{g}$ on $\mathbb{R}^{3,4}$ (Theorem \ref{main}).
\item We provide an explicit construction of a unique coframe (absolute parallelism) on a $9$-dimensional bundle naturally associated with any marked contact Engel structure (Proposition \ref{propo_coframe}). Differentiating this coframe yields a complete set of local invariants for marked contact Engel structures.
\item In particular, we obtain a filtration of differential conditions for marked contact Engel structures, where the first is the integrability condition described above, and the last is equivalent to flatness, i.e., to local equivalence with the aforementioned maximally symmetric model (Theorem \ref{main}).
\item We systematically use the filtration of invariant conditions to classify, up to local equivalence, all homogeneous marked contact Engel structures whose symmetry algebra is of dimension $\geq 6$. Our analysis shows that there are precisely two locally non-equivalent homogeneous marked contact Engel structures whose symmetry algebras are $8$-dimensional (they are $\mathfrak{sl}(3,\mathbb{R})$ and $\mathfrak{su}(1,2)$). Moreover, we provide differential conditions characterizing these sub-maximally symmetric marked contact Engel structures. We show that there are no homogeneous marked contact Engel structures with $7$-dimensional symmetry algebra, and that there are precisely two locally non-equivalent homogeneous marked contact Engel structures with $6$-dimensional symmetry algebra (one of them is semisimple and isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$). We provide examples of locally non-equivalent homogeneous marked contact Engel structures with $5$-dimensional symmetry algebra as well. These results are summarized in Theorem \ref{thmsummary}, see also Table \ref{table}.
\end{itemize}
Sections \ref{sec_123} and \ref{SecKerr} provide geometric interpretations of some of the invariant properties of contact Engel structures derived in Section \ref{CartanEquiv}. In particular, the central notion of integrability will be revisited.
In Section \ref{SecKerr} we prove an analogue of the Kerr Theorem (Theorem \ref{Kerr1}), which provides a construction method of all integrable marked contact Engel structures. We subsequently recast the result in terms of the double filtration for the exceptional Lie group $\mathrm{G}_2$: \begin{align}\label{doublefib}
\xymatrix{
&\mathrm{G}_2/\mathrm{P}_{1,2} \ar[dl]_{\pi_2} \ar[dr]^{\pi_1} & \\
\mathrm{G}_2/\mathrm{P}_2 & & \mathrm{G}_2/\mathrm{P}_1\ . } \end{align} Here $\mathrm{P}_1$ and $\mathrm{P}_2$ are the $9$-dimensional parabolic subgroups of $\mathrm{G}_2$ and $\mathrm{P}_{1,2}=\mathrm{P}_1\cap \mathrm{P}_2$ is the $8$-dimensional Borel subgroup of $\mathrm{G}_2$. The contact Engel structure is a local coordinate description of the $\mathrm{G}_2$-invariant structure on the $5$-dimensional space $\mathrm{G}_2/\mathrm{P}_2$.
The total space of the $\mathbb{R}\mathbb{P}^1$-bundle $\gamma\to \mathrm{G}_2/\mathrm{P}_2$ can be identified with the $6$-dimensional homogeneous space $\mathrm{G}_2/\mathrm{P}_{1,2}$. Marked contact Engel structures can be identified with local sections $\sigma$ of the first leg in the double fibration, $$\mathrm{G}_2/\mathrm{P}_2\supset{\cal U}\xrightarrow{\sigma} \sigma({\cal U})\subset\mathrm{G}_2/\mathrm{P}_{1,2}.$$ The image of such a section defines a hypersurface in $\mathrm{G}_2/\mathrm{P}_{1,2}$, which descends to a hypersurface in the second $5$-dimensional homogeneous space $\mathrm{G}_2/\mathrm{P}_1$ if and only if $\sigma$ is integrable. This yields a local one-to-one correspondence between integrable sections and generic hypersurfaces in $\mathrm{G}_2/\mathrm{P}_1$ (Corollary \ref{Kerr2} of Theorem \ref{Kerr1}). The correspondence is then used to describe the maximal and submaximal marked contact Engel structures; these correspond to the simplest hypersurfaces in $\mathrm{G}_2/\mathrm{P}_1$, namely, identifying $\mathrm{G}_2/\mathrm{P}_1$ with the projectivized null cone in $\mathbb{R}^{3,4}$, they correspond to intersections of the null cone with hyperplanes in $\mathbb{R}^{3,4}$.
Section \ref{sec_general} provides a first analysis of
general marked contact twisted cubic structures. Following the general framework due to Tanaka, see \cite{tanaka, Morimoto, zelenko}, they are viewed as particular types of filtered $G_0$-structures in this section. We compute the (algebraic) Tanaka prolongation associated with these structures, which implies the existence of a canonical coframe on a $9$-dimensional bundle associated with any marked contact twisted cubic structure in a natural manner. Finally, we investigate the question whether a normalization condition in the sense of \cite{CapCartan} can be found. We prove that this is not the case, and thereby provide an example of a structure where such a normalization condition does not exist.
\
\subsection{Conventions and Notation}\label{notation} Throughout the paper all of our objects are smooth, all of our considerations are local and it follows from the context which neighbourhoods are taken into account.
We use the notations
\begin{equation}\label{abbriv}E^1E^2\dots E^k=E^1\odot E^2\odot \dots \odot E^k=\smash{\tfrac{1}{k!}\sum_{\sigma\in S_k}(E^{\sigma 1}\otimes E^{\sigma 2}\otimes\dots\otimes E^{\sigma k})},\end{equation} where $S_k$ is the symmetric group of degree $k$, for the symmetrized tensor product.
For a general coframe $(\omega^i)$ we write $F_{\omega^i}$ for the derivatives with respect to the coframe, i.e., ${\rm d} F=\sum_{i} F_{\omega^i}\omega^i$. If we consider a coordinate coframe $({\rm d} x^i) $, we simply write $F_{x^i}$.
\section{Marked contact twisted cubic structures}\label{sec_marked} Marked contact twisted cubic structures are $5$-dimensional contact structures equipped with additional geometric structures, and we shall introduce these additional geometric structures in the following section. We shall start with purely pointwise considerations, that is, facts about Legendrian twisted cubics in a conformal symplectic vector space in Section \ref{algebra}. Then we will define and discuss general contact twisted cubic structures and marked contact twisted cubic structures on $5$-dimensional manifolds in Section \ref{contact_twisted_section}. We shall introduce the notion of an \emph{integrable} marked contact twisted cubic structure. Finally, we will focus on marked contact twisted cubic structures whose underlying contact twisted cubic structure is flat, which we call marked contact Engel structures, in Section \ref{LieTheory}.
\subsection{Preliminaries on Legendrian twisted cubics}\label{algebra}
We shall first collect some algebraic background. References are e.g. \cite{Harris, bu}.
The \emph{twisted cubic} (rational normal curve of degree three) $\gamma\subset \mathbb{R}\mathbb{P}^3$ is the image of the Veronese map \begin{equation}\textstyle{\mathbb{R}\mathbb{P}^1=\mathbb{P}(\mathbb{R}^2)\to\mathbb{P}(\smash{\bigodot^3\mathbb{R}^2})=\mathbb{R}\mathbb{P}^3,\quad [w]\mapsto [w\odot w\odot w].}\end{equation} In coordinates with respect to bases $(e_1,e_2)$ of $\mathbb{R}^2$ and $(E_1=e_1\odot e_1\odot e_1, E_2=3 e_1\odot e_1\odot e_2, E_3=3 e_1\odot e_2\odot e_2, E_4=e_2\odot e_2\odot e_2)$ of $\smash{\bigodot^3\mathbb{R}^2}$ it is of the form
$$\gamma=[s^3,s^2 t,s t^2,t^3].$$ Denoting by $(E^1, E^2, E^3, E^4)$ the dual basis, the twisted cubic is also given by the zero set of the three quadratic forms
\begin{equation}\label{3polynomials} g_1=E^1E^3-(E^2)^2,\quad g_2=E^2 E^4-(E^3)^2, \quad g_3=E^2E^3- E^1E^4.\end{equation}
With respect to the introduced bases, the irreducible representation \begin{equation}\label{rho} \textstyle{\rho:\mathrm{GL}(2,\mathbb{R})\to\mathrm{End}(\mathbb{R}^4)},\quad\mathbb{R}^4=\smash{\bigodot^3\mathbb{R}^2}, \end{equation} is of the form \begin{equation} \label{irrepres}
\begin{pmatrix}\alpha&\beta\\\rho&\delta\end{pmatrix} \mapsto \begin{pmatrix}\alpha^3&3\alpha^2\beta&3\alpha\beta^2&\beta^3\\ \alpha^2\rho&\alpha^2\delta+2\alpha\beta\rho&2\alpha\beta\delta+\beta^2\rho&\beta^2\delta\\ \alpha\rho^2&2\alpha\delta\rho+\beta\rho^2&\alpha\delta^2+2\beta\delta\rho&\beta\delta^2\\ \rho^3&3\delta\rho^2&3\delta^2\rho&\delta^3\end{pmatrix}.\end{equation} The tangent map at the identity of \eqref{rho} defines the irreducible Lie algebra representation $$\rho'=T_e\rho:\mathfrak{gl}(2,\mathbb{R})\to \mathrm{End}(\mathbb{R}^4).$$
One can check the following.
\begin{proposition}\label{subalg}
The subalgebra of $\mathrm{End}(\mathbb{R}^4)$ preserving $\gamma\subset\mathbb{P}(\mathbb{R}^4)$ is precisely $\rho'\left(\mathfrak{gl}(2,\mathbb{R})\right)$. \end{proposition}
The
decomposition
$\bigwedge^2(\smash{\bigodot^3\mathbb{R}^2})\cong \bigodot^4\mathbb{R}^2\oplus \mathbb{R}$
shows that there is a unique (up to scalars) skew-symmetric bilinear form on $\smash{\bigodot^3\mathbb{R}^2}$ preserved by the $\mathrm{GL}(2,\mathbb{R})$-action up to scalars. Explicitly, it is given by \begin{equation}\label{symp}\omega=E^1\wedge E^4 - 3 E^2\wedge E^3
.\end{equation}
In order to characterize the $\mathrm{GL}(2,\mathbb{R})$-invariant conformal class of the symplectic form \eqref{symp} in terms of the twisted cubic, we shall introduce some more terminology: Let $\omega$ be a symplectic form on $\mathbb{R}^4$ and let $[\omega]$ be the conformal class of all non-zero multiples of $\omega$. Recall that a maximal subspace $\mathbb{W}$ on which a symplectic form $\omega$ (and then any $\omega'\in[\omega]$) vanishes identically is called \emph{Lagrangian}. A twisted cubic $\gamma\subset\mathbb{P}(\mathbb{R}^4)$ is called \emph{Legendrian} with respect to $[\omega]$, see \cite{bu}, if the cone $$\hat{\gamma}=\{\,w\odot w\odot w : w\in\mathbb{R}^2\,\}\subset \mathbb{R}^4$$ is Lagrangian, i.e., the tangent space at each point $p$ of $\hat{\gamma}\setminus \{0\}$ is a Lagrangian subspace of $T_p\mathbb{R}^4\cong\mathbb{R}^4$.
\begin{proposition}\label{confsymp}
The
conformal symplectic structure $[\omega]$ generated by $\omega=E^1\wedge E^4 - 3 E^2\wedge E^3$
is the unique conformal symplectic structure
such that $\gamma=[s^3,s^2 t,s t^2,t^3]$ is Legendrian with respect to $[\omega]$. \end{proposition} \begin{proof} The tangent space to $\hat{\gamma}$ at a point \begin{align}\label{point}\hat{p}={s}^3E_1+{s}^2{t}E_2+{s}{t}^2E_3+{t}^3E_4\end{align} is spanned by \begin{align}\label{X,Y} X= 3{s}^2 E_1+ 2{s}{t} E_2+ {t}^2 E_3\quad \mathrm{and} \quad Y={s}^2 E_2+ 2{s}{t} E_3+ 3{t}^2 E_4. \end{align} Let $\omega=\tfrac12\omega_{ij}E^i\wedge E^j$ be a symplectic form, then \begin{align*} \omega(X,Y)= 3{s}^4 \omega_{12} + 6{s}^3{t} \omega_{13}+ 3{s}^2{t}^2 (3 \omega_{14}+ \omega_{23}) + 6{s}{t}^3\omega_{24}+ 3{t}^4\omega_{34}. \end{align*} Hence $\gamma$ is Legendrian with respect to $\omega$ if and only if \begin{align*}\omega_{14}=-\tfrac13\omega_{23}\end{align*} and, modulo antisymmetry, the remaining $\omega_{ij}$ vanish. This determines $\omega$ uniquely up to scale. \end{proof}
We now introduce some additional data. Namely, we suppose the twisted cubic is marked, that is, a point $p\in\gamma\subset\mathbb{P}(\mathbb{R}^4)$ is distinguished. The point $p$ corresponds to a line $\ell\subset\hat{\gamma}\subset\mathbb{R}^4$. Such a line is of the form $\ell=\mathrm{Span}(\{l\odot l\odot l :l\in L\})$ for a unique $1$-dimensional subspace $L \subset\mathbb{R}^2$. Clearly $\mathrm{GL}(2,\mathbb{R})$ acts transitively on $\gamma$ and we may choose our line $\ell$ to be spanned by the first basis vector $e_1\odot e_1\odot e_1$. The stabilizer subgroup \begin{equation}\label{borel} B:=\{A\in\mathrm{GL}(2,\mathbb{R}): \rho (A)(\ell)\subset\ell\}=\{A\in\mathrm{GL}(2,\mathbb{R}): A(L)\subset L\} \end{equation} is a Borel subgroup $B\subset \mathrm{GL}(2,\mathbb{R})$; in the presentation \eqref{irrepres}, it is given by those matrices for which $\gamma=0$. In particular, $B$ preserves a full filtration of $\mathbb{R}^4$. This immediately implies:
\begin{lemma}\label{lemfilt} A distinguished point $p\in\gamma$ determines a filtration by subspaces \begin{align} \ell\subset\mathrm{D}\subset \mathrm{H}\subset\mathbb{R}^4. \end{align} If $\gamma$ is Legendrian, then $\mathrm{D}$ is a Lagrangian subspace and $\mathrm{H}$ is the symplectic orthogonal to $\ell$. \end{lemma}
In terms of $\mathbb{R}^4=\smash{\bigodot^3\mathbb{R}^2}$, $\mathrm{D}=\mathrm{Span}(\{l\odot l\odot e$\,: $l\in L, \,e\in\mathbb{R}^2\})$, and $\mathrm{H}=\mathrm{Span}(\{l\odot e\odot f$\,: $l\in L, \,e,f\in\mathbb{R}^2\})$. Geometrically, $\mathrm{D}$ is the de-projectivized tangent line to $\gamma$ at $p$ and $\mathrm{H}$ is the de-projectivized osculating plane to $\gamma$ at $p$. Thus we refer to the above filtration as the \emph{osculating sequence} at $p$.
\begin{remark}\label{rem3metrics} We underline that we need all the three quadratic forms $g_1, g_2, g_3$ in \eqref{3polynomials} to define a twisted cubic $\gamma$. In fact, the common zero locus in $\mathbb{R}\mathbb{P}^3$ of any two of the quadric forms belonging to $\mathrm{Span}(g_1, g_2, g_3)$ gives a twisted cubic plus a line (the so called residual intersection, see \cite{Harris}). In the present paper we are interested in the case when this line is tangent to the twisted cubic. The point of tangency is the distinguished point $p\in\gamma$.
\end{remark}
\subsection{Definitions and descriptions of (marked) contact twisted cubic structures}\label{contact_twisted_section}
We are now in the position to define the central objects of this article. \begin{definition} \label{contact_twisted_def} A \emph{contact twisted cubic structure} on a $5$-dimensional smooth manifold ${\cal M}$ is a contact distribution $\mathcal{C}\subset T{\cal M}$ together with a sub-bundle $\gamma\subset \mathbb{P}(\mathcal{C})$ whose fibre $\gamma_x$ at each point $x\in {\cal M}$ is a Legendrian twisted cubic with respect to the conformal symplectic structure $\mathcal{L}_x$ on $\mathcal{C}_x$. An equivalence between contact twisted cubic structures $({\cal M},\mathcal{C},\gamma)$ and $(\widetilde{{\cal M}},\widetilde{\mathcal{C}},\widetilde{\gamma})$ is a diffeomorphism $f:{\cal M}\to\widetilde{{\cal M}}$ such that
$f_*(\mathcal{C}_x)=\widetilde{\mathcal{C}}_{f(x)}$ and $f_*(\gamma_x)=\widetilde{\gamma}_{f(x)}$ for all $x\in {\cal M}$. A self equivalence is called an automorphism, or a symmetry. \end{definition}
\begin{definition} A \emph{marked contact twisted cubic structure} is a contact twisted cubic structure equipped with a smooth section $\sigma$ of $\gamma\to {\cal M}$.
An equivalence between marked contact twisted cubic structures $({\cal M},\mathcal{C},\gamma,\sigma)$ and $(\widetilde{{\cal M}},\widetilde{\mathcal{C}},\widetilde{\gamma},\widetilde{\sigma})$ is an equivalence $f$ between the underlying contact twisted cubic structures $({\cal M},\mathcal{C},\gamma)$ and $(\widetilde{{\cal M}},\widetilde{\mathcal{C}},\widetilde{\gamma})$ such that $f_*(\sigma_x)=\widetilde{\sigma}_{f(x)}$ for all $x\in {\cal M}$. A self equivalence is called an automorphism, or a symmetry. \end{definition}
Throughout this paper we will use various, locally equivalent, viewpoints on (marked) contact twisted cubic structures, which we shall summarize in Propositions \ref{prop_twisted} and \ref{prop_punctured}. Yet another important description, in terms of \emph{adapted coframes}, shall be given in Section \ref{adaptedcoframes}.
Before stating the Propositions, we recall that the $5$-dimensional Heisenberg Lie algebra is the graded nilpotent Lie algebra $\mathfrak{m}=\mathfrak{m}_{-1}\oplus\mathfrak{m}_{-2}$, where $\mathfrak{m}_{-1}\cong\mathbb{R}^4$, $\mathfrak{m}_{-2}\cong\mathbb{R}$, and the only non-trivial component of the Lie bracket $[,]:\Lambda^2\mathfrak{m}_{-1}\to\mathfrak{m}_{-2}$ defines a non-degenerate skew-symmetric bilinear form.
It then follows from non-degeneracy of the Levi bracket \eqref{Levi} that the associated graded $\mathrm{gr}(T{\cal M})=\mathcal{C}\oplus T{\cal M}/\mathcal{C}$ of the contact structure $\mathcal{C}\subset T{\cal M}$ equipped with the Levi bracket $\mathcal{L}$ is a bundle of graded nilpotent Lie algebras modeled on the Heisenberg Lie algebra $\mathfrak{m}$.
It has an associated graded frame bundle $\mathcal{F}\to {\cal M}$ with structure group the grading preserving Lie algebra automorphisms $\mathrm{Aut}_{gr}(\mathfrak{m})\cong\mathrm{CSp}(2,\mathbb{R})$ of $\mathfrak{m}$; its fibre $\mathcal{F}_x$, at each point $x\in {\cal M}$, comprises all graded Lie algebra isomorphisms $\varphi:\mathrm{gr}(T_x{\cal M})\to\mathfrak{m}$.
\begin{proposition}\label{prop_twisted} A contact twisted cubic structure on a $5$-dimensional manifold ${\cal M}$, locally, admits the following locally equivalent descriptions: \begin{enumerate}
\item It is given by a contact distribution $\mathcal{C}\subset T{\cal M}$, an auxiliary rank $2$ bundle $\mathcal{E}\to {\cal M}$ and a vector bundle isomorphism \begin{equation}\label{Psi}\Psi: \textstyle{\bigodot^3\mathcal{E}}\to \mathcal{C}\end{equation} compatible in the sense that it pulls back the conformal symplectic structure $\mathcal{L}_x$ on $\mathcal{C}_x$ to the $\mathrm{GL}(\mathcal{E}_x)$-invariant one on $ \smash{\bigodot^3\mathcal{E}_x}$ for all $x\in {\cal M}$.
\item It is given by a reduction of the graded frame bundle $\mathcal{F}\to {\cal M}$ of a contact structure
to the structure group $\rho(\mathrm{GL}(2,\mathbb{R}))$ with respect to an irreducible representation $\rho:\mathrm{GL}(2,\mathbb{R})\to\mathrm{CSp}(2,\mathbb{R})$.
\item It is given by a contact distribution $\mathcal{C}=\mathrm{ker}(\alpha)$ on ${\cal M}$ and a reduction of the structure group of the frame bundle of $\mathcal{C}$ from $\mathrm{GL}(4,\mathbb{R})$ to the irreducible $\mathrm{GL}(2,\mathbb{R})\subset \mathrm{CSp}(\mathrm{d}\alpha)$. \end{enumerate} \end{proposition}
We only sketch the proof. Given an isomorphism \eqref{Psi},
the image of the map \begin{align*} \iota:\mathbb{P}^1=\mathbb{P}(\mathcal{E}_x)\to\mathbb{P}(\textstyle{\bigodot^3\mathcal{E}_x})\cong\mathbb{P}(\mathcal{C}_x),\quad [\lambda]\mapsto [\Psi(\lambda\odot\lambda\odot\lambda)], \end{align*}
is a twisted cubic $\gamma_x$. By the compatibility requirement of the conformal symplectic structures and Proposition \ref{confsymp}, the twisted cubic is Legendrian.
Conversely, given a sub-bundle $\gamma\subset \mathbb{P}(\mathcal{C})$ of twisted cubics, then in a neighbourhood of each point there exists a rank $2$ bundle $\mathcal{E}$ and a vector bundle isomorphism $\Psi: \smash{\bigodot^3\mathcal{E}\cong\mathcal{C}}$. The compatibility of the conformal symplectic structures follows from the fact that the twisted cubic is Legendrian and by Proposition \ref{confsymp}.
The equivalence between the first and the second description is explained in \cite{book}. The equivalence of the second and third follows from the fact that any graded Lie algebra automorphism of $\mathfrak{m}$ is uniquely determined by its restriction to $\mathfrak{m}_{-1}$.
\begin{remark} A contact twisted cubic structure is the natural contact analogue of an irreducible $\mathrm{GL}(2,\mathbb{R})$-structure in dimension four, as studied, for instance, in \cite{bryant, nurowski}. In particular, one could also call it an irreducible $\mathrm{GL}(2,\mathbb{R})$-contact structure.
Contact twisted cubic structures are also known as a $\mathrm{G}_2$-contact structure in the literature, since they are the underlying structures of regular, normal parabolic geometries of type $(\mathrm{G}_2,\mathrm{P}_2)$, see \cite{book}. \end{remark}
\begin{proposition}\label{prop_punctured} A marked contact twisted cubic structure, locally, admits the following locally equivalent descriptions:
\begin{enumerate} \item It is given by a contact distribution $\mathcal{C}\subset T{\cal M}$, an auxiliary rank $2$ bundle $\mathcal{E}\to {\cal M}$, a vector bundle isomorphism $\Psi:\smash{\bigodot^3\mathcal{E}}\to \mathcal{C}$ compatible with the conformal symplectic structures and, in addition, a line subbundle $L\subset \mathcal{E}.$
\item It is given by a reduction of structure group of the graded frame bundle $\mathcal{F}\to {\cal M}$ of a contact structure in dimension $5$ with respect to the restriction $$\rho:B\to\mathrm{CSp}(2,\mathbb{R})$$ of an irreducible $\mathrm{GL}(2,\mathbb{R})$-representation $\rho$ to the Borel subgroup $B\subset\mathrm{GL}(2,\mathbb{R})$. \item It is given by a contact twisted cubic structure equipped with a $\gamma$-congruence, that is, a foliation of ${\cal M}$ by curves whose tangent directions are everywhere contained in $\gamma\subset\mathbb{P}(\mathcal{C})$. \end{enumerate}
\end{proposition} In view of Proposition \ref{prop_twisted}, the equivalence of the first two descriptions is obvious. Concerning the last description, note that a section $\sigma: {\cal M}\to\gamma$ is the same as a rank $1$ distribution $\ell^{\sigma}\subset\hat{\gamma}\subset\mathcal{C},$ where $\hat{\gamma}\subset\mathcal{C}$ is the cone over $\gamma\subset\mathbb{P}(\mathcal{C})$. The integral manifolds of this line distribution define the $\gamma$-congruence. Conversely, one obtains $\ell^{\sigma}$ from the $\gamma$-congruence by considering the field of tangent directions to the curves.
By Lemma \ref{lemfilt}, we have the following ``osculating filtration''. \begin{proposition}\label{propfilt} A marked contact twisted cubic structure $({\cal M},\mathcal{C},\gamma,\sigma)$ is equipped with a flag of distributions \begin{align}\label{filtration} \ell^{\sigma}\subset\mathcal{D}^{\sigma}\subset \mathcal{H}^{\sigma}\subset\mathcal{C}\subset T{\cal M}, \end{align} where the rank $2$ distribution $\mathcal{D}^{\sigma}\subset\mathcal{C}$ is Legendrian (i.e., totally null with respect to the conformal symplectic structure on $\mathcal{C}$) and the rank $3$ distribution $\mathcal{H}^{\sigma}$ is the symplectic orthogonal to $\ell^{\sigma}$. \end{proposition}
\begin{definition} We call a marked contact twisted cubic structure \emph{integrable} if the distribution $\mathcal{D}^{\sigma}$ is integrable. In this case the section $\sigma:{\cal M}\to\gamma$ is called an integrable section. \end{definition}
\subsection{(Marked) contact Engel structures and the exceptional Lie group $\mathrm{G}_2$}
\label{LieTheory} As mentioned in the introduction, the most symmetric contact twisted cubic structure, that we refer to as the contact Engel structure, is intimately related to the exceptional Lie group $\mathrm{G}_2$. We shall explain this relationship in the following section. For further references see e.g. \cite{engel, yamaguchi, BryantCartan, book}.
Let $\mathrm{G}_2$ denote the connected Lie group with center $\mathbb{Z}_2$ whose Lie algebra $\mathfrak{g}$ is the split real form of
the smallest of the exceptional complex simple Lie algebras.
$\mathrm{G}_2$ can be defined as the stabilizer subgroup in $\mathrm{GL}(7,\mathbb{R})$ of a generic $3$-form $\Phi\in\Lambda^3(\mathbb{R}^7)^*$. It preserves a non-degenerate bilinear form $h\in \bigodot^2(\mathbb{R}^7)^*$ of signature $(4,3)$.
The Lie algebra $\mathfrak{g}$ of $\mathrm{G}_2$ has, up to conjugacy, three parabolic subalgebras: the maximal parabolic algebras $\mathfrak{p}_1,$ $\mathfrak{p}_2$ and the Borel subalgebra $\mathfrak{p}_{1,2}$. Corresponding parabolic subgroups of $\mathrm{G}_2$ can be realized as follows: $\mathrm{P}_1$ is the stabilizer of a null line in $\mathbb{R}^7$ with respect to the $\mathrm{G}_2$-invariant bilinear form $h$, $\mathrm{P}_2$ is the stabilizer of a totally null $2$-plane in $\mathbb{R}^7$ that inserts trivially into $\Phi$, and $\mathrm{P}_{1,2}=\mathrm{P}_1\cap \mathrm{P}_2$.
For a parabolic subgroup $P$ of a simple Lie group $\mathrm{G}$, let $G_+\subset P$ be the unipotent radical and $G_0=P/G_+$ the reductive Levi factor, so that $P=G_0\ltimes G_+$. Denote by $\mathfrak{g}_+$ and $\mathfrak{g}_0=\mathfrak{p}/\mathfrak{g}_+$ the corresponding Lie algebras. Via the adjoint action, $P$ preserves a filtration \begin{align}\label{para_filt}\mathfrak{g}=\mathfrak{g}^{-k}\supset \mathfrak{g}^{-k+1}\supset\cdots\supset\mathfrak{g}^0\supset\mathfrak{g}^1\supset\cdots\supset\mathfrak{g}^k,\end{align} where $\mathfrak{g}^1=\mathfrak{g}_+$, $\mathfrak{g}^j=[\mathfrak{g}^{j-1},\mathfrak{p}_+]$ for $j\geq 2$, $\mathfrak{g}^{j+1}=(\mathfrak{g}^{-j})^{\perp}$ for $j\leq -1$ (the complement is taken with respect to the Killing form) and, in particular, $\mathfrak{g}^0=\mathfrak{p}$. Any splitting $\mathfrak{g}_0\to\mathfrak{p}$ determines an identification of the filtered Lie algebra $\mathfrak{g}$ with its associated graded Lie algebra $$\mathrm{gr}(\mathfrak{g})=\mathfrak{g}_{-k}\oplus\cdots\oplus\mathfrak{g}_0\oplus\cdots\oplus\mathfrak{g}_{k}.$$
For complex simple Lie algebras (and their split-real forms) conjugacy classes of parabolic subalgebras are in on-to-one correspondence with subsets of simple roots (having fixed a Cartan subalgebra $\mathfrak{h}$ and a set of simple roots $\Delta^0$). The correspondence is given as follows: Recall that any root can be uniquely decomposed into a sum of simple roots $\alpha=\sum_{i}a_i \alpha_i$ where all coefficients $a_i$ (if non-zero) are integers of the same sign. For any subset $\Sigma\subset\Delta^0$ one now defines the $\Sigma$-height $\mathrm{ht}_{\Sigma}(\alpha)$ of a root to be
$\mathrm{ht}_{\Sigma}(\alpha)=\sum_{i:\alpha_i\in\Sigma}a_i.$ Then $$\mathfrak{p}=\mathfrak{h}\oplus_{\{\alpha:\mathrm{ht}_{\Sigma}(\alpha)\geq0\}}\mathfrak{g}_{\alpha}$$ is a parabolic subalgebra. In fact, these choices determine a grading: $\mathfrak{g}_0=\mathfrak{h}\oplus_{\{\alpha:\mathrm{ht}_{\Sigma}(\alpha)=0\}}\mathfrak{g}_{\alpha}$ is a Levi subalgebra and the remaining grading components are given by $\mathfrak{g}_{i}=\oplus_{\{\alpha:\mathrm{ht}_{\Sigma}(\alpha)=i\}}\mathfrak{g}_{\alpha}.$
In the $\mathrm{G}_2$ case we have two simple roots $\Delta^0=\{\alpha_1,\alpha_2\}$, and the parabolic subalgebras $\mathfrak{p}_1$, $\mathfrak{p}_2$ and $\mathfrak{p}_{1,2}$ correspond to the sets $\Sigma_1=\{\alpha_1\}$, $\Sigma_2=\{\alpha_2\}$ and $\Sigma=\Delta^0$.
In this paper we are particularly interested in the contact grading, corresponding to $\Sigma_2=\{\alpha_2\}$. Here we have $\mathfrak{g}_0\cong\mathfrak{gl}(2,\mathbb{R})$, $\mathfrak{g}_{-}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{-2}$ and $\mathfrak{g}_+=\mathfrak{g}_1\oplus\mathfrak{g}_2$ are dual with respect to the Killing form and isomorphic to the $5$-dimensional Heisenberg algebra. Moreover, the $\mathfrak{g}_0$-representation $\mathfrak{g}_{-1}$ is irreducible; hence $\mathfrak{g}_{-1}\cong \smash{\bigodot^3\mathbb{R}^2}$ as a representation of the semisimple part ${\mathfrak{g}_0}^{ss}\cong\mathfrak{sl}(2,\mathbb{R})$. \begin{equation}\label{cont_grad} \begin{tikzpicture}[scale=1,baseline=-5pt]
\node[black] at (-1.6,2.3)
{$\mathfrak{p}_2=\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2$};
\draw[black] (-1.4, -0.2) rectangle (1.4,0.2);
\draw[black] (-1.85, 0.6) rectangle (1.85,1.85);
\draw[black] ( 0 , 1.732) -- (0,0);
\filldraw[black] ( 0 , 1.732) circle (0.06);
\draw[black] (0, 0) -- (1.5, 0.866);
\filldraw[black] ( 1.5, 0.866) circle (0.06);
\draw[black] (0, 0) -- (0.5, 0.866);
\filldraw[black] ( 0.5, 0.866) circle (0.06);
\draw[black] (0, 0) -- (-0.5, 0.866);
\filldraw[black] ( -0.5, 0.866) circle (0.06);
\draw[black] (0, 0) -- (-1.5, 0.866);
\filldraw[black] ( -1.5, 0.866) circle (0.06);
\draw[black] (0, 0) -- (0.866, 0);
\filldraw[black] (0.866, 0) circle (0.06);
\draw[black] (0, 0) -- (1.5, -0.866);
\filldraw[black] ( 1.5, -0.866) circle (0.06);
\draw[black] ( 0 , -1.732) -- (0,0);
\filldraw ( 0 , -1.732) circle (0.06);
\draw[black] (0, 0) -- (-1.5, -0.866);
\filldraw ( -1.5, -0.866) circle (0.06);
\draw[black] (0, 0) -- (-0.5, -0.866);
\filldraw ( -0.5, -0.866) circle (0.06);
\draw[black] (0, 0) -- (0.5, -0.866);
\filldraw ( 0.5, -0.866) circle (0.06);
\draw[black] (0, 0) -- (-0.866, 0);
\filldraw[black] (-0.866, 0) circle (0.06);
\filldraw[color=black] (0,0) circle (0.1);
\draw[dashed](- 2, 1.2)--(2, 1.2);
\draw[dashed](- 2, 0.4)--(2, 0.4);
\draw[dashed](- 2, -1.2)--(2, -1.2);
\draw[dashed](- 2, -0.4)--(2, -0.4);
\draw (2.5,1.7) node {$\mathfrak{g}_{2}$}; \draw (2.5,-1.7) node {$\mathfrak{g}_{-2\,}$}; \draw (2.5,0.9) node {$\mathfrak{g}_{1}$}; \draw (2.5,-0.9) node {$\mathfrak{g}_{-1\,}$}; \draw (2.5,0) node {$\mathfrak{g}_{0}\ $};
\end{tikzpicture}
\end{equation}
The model for contact twisted cubic structures is the
homogeneous space $\mathrm{G}_2/\mathrm{P}_2$. \begin{proposition}\label{homog} The homogeneous space $\mathrm{G}_2/\mathrm{P}_2$ is naturally equipped with a $\mathrm{G}_2$-invariant contact twisted cubic structure. \end{proposition} \begin{proof} The tangent bundle of $\mathrm{G}_2/\mathrm{P}_2$ is the associated bundle \begin{align}\label{iso}T(\mathrm{G}_2/\mathrm{P}_2)=\mathrm{G}_2\times_{\mathrm{P}_2}(\mathfrak{g}/\mathfrak{p}_2),\end{align} where $\mathfrak{g}$ denotes the Lie algebra of $\mathrm{G}_2$. The identification is induced by the trivialization of the tangent bundle of the Lie group $\mathrm{G}_2$ by left-invariant vector fields. Using the Maurer-Cartan form $\omega_{G_2}\in\Omega^1(\mathrm{G}_2,\mathfrak{g}),$ $\omega_{G_2}(\xi_g)=T\lambda_{g^{-1}}\xi_g$, it can be written as $$\xi_{x}\mapsto [g, \omega_{G_2}(\xi_{x})+\mathfrak{p}_2],$$ where $g\in\mathrm{G}_2, x=g\mathrm{P}_2 $ and $\xi_x\in T_x(\mathrm{G}_2/\mathrm{P}_2) .$ The filtration \eqref{para_filt} induces a $\mathrm{P}_2$-invariant filtration $$\mathcal{C}_o=\mathfrak{g}^{-1}/\mathfrak{p}_2\subset\mathfrak{g}/\mathfrak{p}_2=T_o(\mathrm{G}_2/\mathrm{P}_2)$$ and, via the identification \eqref{iso}, a subbundle $\mathcal{C}\subset T(\mathrm{G}_2/\mathrm{P}_2)$ of codimension one. The Levi bracket $\mathcal{L}:\Lambda^2\mathcal{C}\to T{\cal M}/\mathcal{C}$ corresponds to the Lie bracket on $\mathfrak{g}_{-}=\mathrm{gr}(\mathfrak{g}/\mathfrak{p}_2)$. Since this is the $5$-dimensional Heisenberg Lie algebra, $\mathcal{C}$ is contact. Moreover, since the unipotent radical acts trivially on $\mathfrak{g}^{-1}/\mathfrak{p}_2$, the $\mathrm{P}_2$ action factors to a $G_0$ action on $\mathcal{C}_o=\mathfrak{g}^{-1}/\mathfrak{p}_2$.
The latter action is irreducible, and the orbit through a highest weight line defines a $G_0$-invariant Legendrian twisted cubic $\gamma_o\subset\mathbb{P}(\mathcal{C}_o)$.
\end{proof} \begin{definition} A contact twisted cubic structure is called \emph{flat}, or \emph{contact Engel structure}, if and only if it is locally equivalent to the $\mathrm{G}_2$-invariant structure on $\mathrm{G}_2/\mathrm{P}_2$. \end{definition}
\begin{remark}\label{remark6} It follows from the general theory, see \cite{book}, that there is an equivalence of categories between general contact twisted cubic structures and certain regular, normal parabolic geometries.
The Engel structure is the locally unique contact twisted cubic structure with infinitesimal symmetry algebra of maximal dimension, and it is characterized, up to local equivalence, by the vanishing of the harmonic part of the curvature of the canonically associated Cartan connection. The infinitesimal automorphisms of a general contact twisted cubic structure form a Lie algebra of dimension $\leq 14$. In fact, if the structure is non-flat, it is known that the symmetry algebra is of dimension $\leq 7,$ see \cite{gap}.
\end{remark}
\begin{proposition} \label{G2modG12} Let $\gamma\subset\mathbb{P}(\mathcal{C})$ be the $\mathrm{G}_2$-invariant contact twisted cubic structure on $\mathrm{G}_2/\mathrm{P}_2$. Then
$$\gamma=\mathrm{G}_2\times_{\mathrm{P}_2} {\mathrm{P}_2/\mathrm{P}_{1,2}}=\mathrm{G}_2/\mathrm{P}_{1,2} .$$
\end{proposition}
\begin{proof} The left $\mathrm{G}_2$ action on $\mathrm{G}_2/\mathrm{P}_2$ lifts to a $\mathrm{G}_2$ action on $\gamma$.
Consider the fibre $\gamma_o\subset\mathbb{P}(\mathfrak{g}^{-1}/\mathfrak{p}_2)$ over the origin $o=e\mathrm{P}_2$. Then the $\mathrm{G}_2$ action on $\gamma$ restricts to a $\mathrm{P}_2$ action on $\gamma_o$, which
factors to an action of $G_0=\mathrm{GL}(2,\mathbb{R})$, since the unipotent radical acts trivially. The latter action is transitive on $\gamma_o$ and the stabilizer of a point in $\gamma_o$ (which is a highest weight line in $\mathfrak{g}^{-1}/\mathfrak{p}_2$) is the Borel subgroup $B\subset \mathrm{GL}(2,\mathbb{R})$ as in \eqref{borel}. Then the stabilizer in $\mathrm{P}_2$ of the point is $B\ltimes\mathrm{exp}(\mathfrak{g}_+)$, which is the Borel subgroup $\mathrm{P}_{1,2}\subset\mathrm{G}_2$, and so $$\gamma=\mathrm{G}_2\times_{\mathrm{P}_2} {\gamma_o}=G_2\times_{\mathrm{P}_2} {\mathrm{P}_2/\mathrm{P}_{1,2}}=G_2/\mathrm{P}_{1,2} .$$ \end{proof}
\begin{definition}\label{contEngel} A \emph{marked contact Engel structure} is a marked contact twisted cubic structure whose underlying contact twisted cubic structure is flat. \end{definition}
\begin{remark} Also in the general, non-flat case, we can identify $\gamma$ with the so-called \emph{correspondence space} $\mathcal{G}\times_{\mathrm{P}_2} {\mathrm{P}_2/\mathrm{P}_{1,2}}=\mathcal{G}/\mathrm{P}_{1,2}$ by means of the associated canonical Cartan connection $\omega\in\Omega^1(\mathcal{G},\mathfrak{g})$.
\end{remark}
\section{Local invariants and homogeneous models of marked contact Engel structures via Cartan's equivalence method}\label{CartanEquiv}
In this section we apply Cartan's method of equivalence (see e.g. \cite{Olver} for an introduction to the general method) to the local equivalence problem of marked contact Engel structures. We derive a set of local differential invariants of marked contact Engel structures. These allow us, in particular, to characterize the maximal and submaximal symmetric models. We further obtain a tree of locally non-equivalent branches of marked contact Engel structures, and we derive the structure equations for the maximally symmetric homogeneous structures in (almost all) branches. In particular, this yields a complete classification of all homogeneous marked contact Engel structures with the symmetry algebra of dimension $\geq 6$ up to local equivalence.
\subsection{Adapted coframes}\label{adaptedcoframes}
In order to apply Cartan's method to the equivalence problem of marked contact Engel structures, we shall recast the problem in terms of adapted coframes. A (marked) contact twisted cubic structure on a manifold ${\cal M}$ defines a natural coframe bundle, and adapted coframes are the sections of these bundles.
\begin{definition}\label{adapted1}
Let $\gamma\subset\mathbb{P}(\mathcal{C})$ be a contact twisted cubic structure on $\mathcal{U}$.
A (local) coframe $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ on $\mathcal{U}$ is adapted to the contact twisted cubic structure $\gamma\subset\mathbb{P}(\mathcal{C})$ if in terms of this coframe $$\mathcal{C}=\mathrm{ker}(\omega^0)$$ and $\gamma\subset \mathbb{P}(\mathcal{C})$ is the projectivization of the set of all tangent vectors contained in $\mathcal{C}$ that are simultaneously null for the following three symmetric tensor fields
\begin{equation}\label{3metrics}
\begin{aligned}
g_1=\omega^1\omega^3- (\omega^2)^2,\quad
g_2= \omega^2\omega^4- (\omega^3)^2,\quad
g_3= \omega^2\omega^3-\omega^1\omega^4.
\end{aligned}
\end{equation}
\end{definition}
\begin{proposition} Two coframes $(\hat{\omega}^0,\hat{\omega}^1,\hat{\omega}^2,\hat{\omega}^3,\hat{\omega}^4)$ and $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ on $\mathcal{U}$ are adapted to the same contact twisted cubic structure if and only if \begin{align}\label{coframe2} \begin{pmatrix} \hat{\omega}^0\\ \hat{\omega}^1\\ \hat{\omega}^2\\ \hat{\omega}^3\\ \hat{\omega}^4\\
\end{pmatrix}= \begin{pmatrix} s_0& 0&0&0&0\\ s_1&{s_5}^3& 3{s_5}^2s_6&3 s_5{s_6}^2&{s_6}^3\\ s_2&{s_5}^2s_7& {s_5}(s_5 s_8+2 s_6s_7)& s_6(2s_5s_8+s_6s_7)&{s_6}^2s_8\\ s_3& s_5 {s_7}^2& s_7(2 s_5 s_8+s_6s_7) & s_8(s_5{s_8}+2s_6s_7)&s_6{s_8}^2\\ s_4& {s_7}^3& 3 {s_7}^2s_8&3s_7{s_8}^2&{s_8}^3 \end{pmatrix}
\begin{pmatrix}\omega^0\\ \omega^1\\ \omega^2\\ \omega^3\\ \omega^4\\ \end{pmatrix} \end{align} where $s_0, s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8$ are smooth functions on $\mathcal{U}$ such that the determinant $s_0(s_6s_7-s_5s_8)^6\neq 0$.
Two contact twisted cubic structures represented by coframes $(\omega^0,\dots, \omega^4)$ on $\mathcal{U}$ and $(\bar{\omega}^0,\dots,\bar{\omega}^4)$ on $\mathcal{V}$ are (locally) equivalent if and only if there exists a (local) diffeomorphism $f:\mathcal{U}\to \mathcal{V}$ such that $(f^*(\bar{\omega}^0),\dots, f^*(\bar{\omega}^4))$ is related to $(\omega^0,\dots,\omega^4)$ by a transformation matrix of the form as in \eqref{coframe2}. \end{proposition}
Note that the bottom right $4\times 4$ block in the transformation matrix from \eqref{coframe2} is $\mathrm{GL}(2,\mathbb{R})$ in the $4$-dimensional irreducible representation \eqref{irrepres}.
\begin{definition} Let $\sigma:\mathcal{U}\to\gamma\subset\mathbb{P}(\mathcal{C})$ be a marked contact twisted cubic structure on $\mathcal{U}$. A (local) coframe $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ is adapted to the marked contact twisted cubic structure $\sigma:U\to\gamma\subset\mathbb{P}(\mathcal{C})$ if it is adapted to the underlying contact twisted cubic structure as in Definition \ref{adapted1}
and moreover the line field $\ell^{\sigma}$ is given by \begin{align*} \ell^{\sigma} =\mathrm{ker} (\omega^0,\omega^1, \omega^2, \omega^3). \end{align*}
\end{definition}
\begin{proposition} Two coframes $(\hat{\omega}^0,\hat{\omega}^1,\hat{\omega}^2,\hat{\omega}^3,\hat{\omega}^4)$ and $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ on $\mathcal{U}$ are adapted to the same marked contact twisted cubic structure if and only if \begin{align}\label{equiv_punct} \begin{pmatrix} \hat{\omega}^0\\ \hat{\omega}^1\\ \hat{\omega}^2\\ \hat{\omega}^3\\ \hat{\omega}^4\\ \end{pmatrix}= \begin{pmatrix} s_0& 0&0&0&0\\ s_1&{s_5}^3& 0 & 0&0\\ s_2&{s_5}^2s_7& {s_5}s_5 s_8& 0 &0\\ s_3& s_5 {s_7}^2& 2s_7 s_5 s_8 & s_8s_5{s_8}& 0\\ s_4& {s_7}^3& 3 {s_7}^2s_8&3s_7{s_8}^2&{s_8}^3
\end{pmatrix}
\begin{pmatrix} \omega^0\\ \omega^1\\ \omega^2\\ \omega^3\\ \omega^4\\ \end{pmatrix} \end{align} where $s_0, s_1,s_2,s_3,s_4,s_5,s_7,s_8$ are smooth functions on $\mathcal{U}$ such that $s_0 s_5s_8\neq 0$.
Two marked contact twisted cubic structures represented by coframes $(\omega^0,\dots, \omega^4)$ on $\mathcal{U}$ and $(\bar{\omega}^0,\dots,\bar{\omega}^4)$ on $\mathcal{V}$ are (locally) equivalent if and only if there exists a (local) diffeomorphism $f:\mathcal{U}\to \mathcal{V}$ such that $(f^*(\bar{\omega}^0),\dots, f^*(\bar{\omega}^4))$ is related to $(\omega^0,\dots,\omega^4)$ by a transformation matrix of the form as in \eqref{equiv_punct}. \end{proposition}
Here, the bottom right $4\times 4$ block in the transformation matrix \eqref{equiv_punct} is the Borel subgroup $B\subset\mathrm{GL}(2,\mathbb{R})$, defined in \eqref{borel}, in the irreducible representation as in \eqref{irrepres}.
\begin{remark} Alternatively, we may describe a marked contact twisted cubic structure by considering the intersection of the null cones of only the two metrics $g_1$ and $g_3$ from \eqref{3metrics}.
\end{remark}
\subsection{Structure equations for marked contact Engel structures}
From now on we shall concentrate on marked contact Engel structures as defined in Definition \ref{contEngel}.
Consider the Maurer-Cartan equations of $\mathrm{G}_2$ as displayed in the Appendix in \eqref{MaurerCartan}, written with respect to the basis $(E_0,E_1,\dots,E_{13})$ as in \eqref{basis_g2} of $\mathfrak{g}$, which is adapted to the contact grading $$\mathfrak{g}=\mathfrak{g}_{-}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_{+}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2.$$ Then the kernel of the nine left-invariant forms $\theta^5, \theta^6,\dots,\theta^{13}$ from \eqref{MaurerCartan} defines an integrable distribution. The leaves of the corresponding foliation correspond to certain sections of $\mathrm{G}_2\to\mathrm{G}_2/P_{2}$. The pullbacks of the forms $\theta^5, \theta^6,\dots,\theta^{13}$ with respect to any of these sections vanish on $\mathrm{G}_2/\mathrm{P}_2$, and the pullbacks
of the remaining forms $\theta^0,\theta^1,\theta^2,\theta^3,\theta^4$ define an adapted coframe $(\alpha^0, \alpha^1, \alpha^2, \alpha^3, \alpha^4)$ for the contact Engel structure on $\mathrm{G}_2/\mathrm{P}_2$, which satisfies the system
\begin{equation}\label{MCg-}{\rm d}\alpha^0=\alpha^1\wedge\alpha^4-3\alpha^2\wedge\alpha^3, \quad {\rm d}\alpha^1=0, \quad{\rm d}\alpha^2=0, \quad {\rm d}\alpha^3=0, \quad {\rm d}\alpha^4=0.\end{equation}
Integrating this system yields local coordinates $(x^0, x^1, x^2, x^3, x^4)$ such that \begin{equation}\label{flatcoframe}\alpha^0=\mathrm{d}x^0+x^1 \mathrm{d}x^4- 3 x^2 \mathrm{d}x^3,\quad \alpha^1={\rm d} x^1,\quad \alpha^2={\rm d} x^2,\quad \alpha^3={\rm d} x^3,\quad \alpha^4={\rm d} x^4. \end{equation} Hence such a coframe $(\alpha^0, \alpha^1,\alpha^2,\alpha^3,\alpha^4)$ is an adapted coframe for the contact Engel structure.
\begin{remark}
Note that \eqref{MCg-} are the Maurer-Cartan equations of $G_{-}=\mathrm{exp}(\mathfrak{g}_{-})$ for the Maurer-Cartan form $\theta_{MC}$ of $G_{-}$.
Alternatively, the coordinate representation \eqref{flatcoframe} can be obtained from the parameterisation $\phi:\mathbb{R}^5\to G_{-} \, \hbox to 2.5pt{\hss$\ccdot$\hss} \, o \subset \mathrm{G}_2/P_{2}$ given by
$$\phi(x^0, x^1, x^2, x^3, x^4)=\mathrm{exp}(x^0E_0)\mathrm{exp}(x^1E_1)\mathrm{exp}(x^2 E_2)\mathrm{exp}(x^3 E_3)\mathrm{exp}(x^4 E_4) o,$$
with $E_0\in\mathfrak{g}_{-2}$ and $E_1, E_2, E_3, E_4\in\mathfrak{g}_{-1}$ and the well-known formula $\theta_{MC}=\phi^{-1}d\phi=\alpha^iE_i$.
\end{remark}
Now denote by $(X_0, X_1, X_2, X_3, X_4)$ the frame dual to the coframe $(\alpha^0, \alpha^1, \alpha^2, \alpha^3, \alpha^4)$ as in \eqref{flatcoframe}. We may assume that the section $\sigma:\mathcal{U}\to\gamma$ defining a general \emph{marked} contact Engel structure on $\mathrm{G}_2/\mathrm{P}_2$ is of the form \begin{equation}\label{section}\sigma=[-t^3 X_1+t^2 X_2-t X_3+ X_4],\end{equation} where $t=t(x^0, x^1, x^2, x^3, x^4)$ is a smooth function on $\mathcal{U}$. In this sense, the choice of a function $t$ determines a marked contact Engel structure, and up to local equivalence, all marked contact Engel structures can be obtained in this way. Note however, that different $t$'s can correspond to the same structure (up to local equivalence).
The osculating filtration from Proposition \ref{propfilt} of the marked Engel structure is of the form \begin{equation}\label{oscfilt} \ell^{\sigma}=\mathrm{Span}(\xi_4)\subset\mathcal{D}^{\sigma}=\mathrm{Span}(\xi_4,\xi_3)\subset\mathcal{H}^{\sigma}=\mathrm{Span}(\xi_4,\xi_3,\xi_2)\subset\mathcal{C}=\mathrm{Span}(\xi_4,\xi_3,\xi_2,\xi_1), \end{equation} where \begin{equation}\label{frame1} \begin{aligned} \xi_4&:=-t^3 X_1+t^2 X_2-t X_3+ X_4=\, -(x^1+3tx^2)\partial_{x^0}-t^3\partial_{x^1}+t^2\partial_{x^2}-t\partial_{x^3}+\partial_{x^4}\\ \xi_3&:= 3 t^2 X_1-2 t X_2+ X_3=\, 3x^2\partial_{x^0}+3t^2\partial_{x^1}-2t\partial_{x^2}+\partial_{x^3}\\ \xi_2&:=-3 t X_1+ X_2=\, -3 t \partial_{x^1}+\partial_{x^2}\\ \xi_1&:=X_1=\,\partial_{x^0}\\ \xi_0&:= X_0=\, \partial_{x^1}\\ \end{aligned} \end{equation}
Passing to the coframe $(\omega^0, \omega^1, \omega^2, \omega^3, \omega^4)$ dual to the frame $(\xi_0, \xi_1, \xi_2, \xi_3, \xi_4)$ yields the following.
\begin{lemma}\label{lemma1marked}
The most general marked contact Engel structure can be locally represented in terms of the following adapted coframe
\begin{align}\label{coframet} \begin{pmatrix} \omega^0\\ \omega^1\\ \omega^2\\ \omega^3\\ \omega^4\\ \end{pmatrix}=
\begin{pmatrix}\begin{aligned}
&{\rm d} x^0+x^1{\rm d} x^4- 3 x^2{\rm d} x^3\\
&{\rm d} x^1+3 t{\rm d} x^2+ 3 t^2 {\rm d} x^3+t^3{\rm d} x^4\\
&{\rm d} x^2 +2 t {\rm d} x^3 +t^2 {\rm d} x^4\\
&{\rm d} x^3+ t {\rm d} x^4\\
&{\rm d} x^4\\
\end{aligned}
\end{pmatrix}, \end{align} where $t=t(x^0, x^1, x^2, x^3, x^4)\in C^{\infty}(\mathcal{U})$. The filtration \eqref{oscfilt} associated to a marked contact Engel structure is given in terms of this coframe as \begin{equation}\label{eq.filtration.wo.w1.w2.w3} \ell^{\sigma}=\mathrm{ker}(\omega^0,\omega^1,\omega^2,\omega^3)\subset\mathcal{D}^{\sigma}=\mathrm{ker}(\omega^0,\omega^1,\omega^2)\subset\mathcal{H}^{\sigma}=\mathrm{ker}(\omega^0,\omega^1)\subset\mathcal{C}=\mathrm{ker}(\omega^0). \end{equation}
\end{lemma}
Our problem is to produce differential invariants that allow us to distinguish non-equivalent classes of marked contact Engel structures. In particular, all of these invariants should vanish for the simplest marked contact Engel structure, the one corresponding to $t=0$, which we call \emph{flat}.
\begin{definition}\label{flatEngel} A marked contact Engel structure is called \emph{flat} if it can be locally represented in terms of an adapted coframe $(\alpha^0,\alpha^1,\alpha^2,\alpha^3,\alpha^4)$ as in \eqref{flatcoframe}.
\end{definition}
Using Lemma \ref{lemma1marked}, we next observe the following.
\begin{lemma}\label{diffcof}
Any marked contact Engel structure admits an adapted coframe $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ satisfying
\begin{equation}\label{differentiatedcoframe2}
\begin{aligned}
&{\rm d}\omega^0=\omega^1\wedge\omega^4-3\omega^2\wedge\omega^3\\
&{\rm d}\omega^1=\tfrac{3}{4}(b^2-4ac+M-P)\omega^0\wedge\omega^2+3c\omega^1\wedge\omega^2-3 a \omega^2\wedge\omega^3 +3 J \omega^2\wedge\omega^4\\
&{\rm d}\omega^2= \tfrac{1}{2}(b^2-4ac+M-P) \omega^0\wedge\omega^3+2 c\omega^1\wedge\omega^3-2 b\omega^2\wedge\omega^3+2 J \omega^3\wedge\omega^4\\
& {\rm d}\omega^3=\tfrac{1}{4}(b^2-4ac+M-P)\omega^0\wedge\omega^4+c\omega^1\wedge\omega^4-b\omega^2\wedge\omega^4+a\omega^3\wedge\omega^4\\
&{\rm d}\omega^4=0\\
\end{aligned}
\end{equation}
for functions $a, b, c, J, M, P$.
\end{lemma}
\begin{proof}
We work in the representation from Lemma \ref{lemma1marked}.
Differentiating the coframe \eqref{coframet} gives
\begin{equation}\label{differentiatedcoframe1}
\begin{aligned}
{\rm d}\omega^0=&\ \omega^1\wedge\omega^4-3\omega^2\wedge\omega^3\\
{\rm d}\omega^1=&\ 3 {\rm d} t \wedge \omega^2\\
{\rm d}\omega^2= &\ 2 {\rm d} t \wedge \omega^3\\
{\rm d}\omega^3=&\ {\rm d} t\wedge\omega^4\\
{\rm d}\omega^4=&\ 0,\\
\end{aligned}
\end{equation}
Then one expands ${\rm d} t$ in terms of the coframe $(\omega^0, \omega^1, \omega^2, \omega^3, \omega^4)$ and then there is a unique solution for $a, b, c, J$ and $ M-P$
in terms of the function $t$ and its derivatives.
\end{proof}
\begin{remark}
Indeed, any marked contact twisted cubic structure admitting an adapted coframe as in Lemma \ref{diffcof} is flat as a contact twisted cubic structure, i.e., it is a marked contact Engel structure.
\end{remark}
Applying the exterior derivative on both sides of \eqref{differentiatedcoframe2} we get information about the exterior derivatives of the functions $a, b, c$ and $J$. Explicitly, we obtain the following lemmas. Recall that a subscript $\omega^i$ denotes the $i$th frame derivative as in Section \ref{notation}.
\begin{lemma}\label{lemma3marked}
The functions $a, b, c$ and $J$ from Lemma \ref{diffcof} satisfy
\begin{equation}\label{differentiatedJabc2}
\begin{aligned}
&{\rm d} J=J_{\omega^0}\omega^0+J_{\omega^1}\omega^1+J_{\omega^2}\omega^2+J_{\omega^3}\omega^3+J_{\omega^4}\omega^4\\
&{\rm d} a=a_{\omega^0}\omega^0+a_{\omega^1}\omega^1+\tfrac{1}{4}(-3b^2+M+3P)\omega^2+L\omega^3+(a^2-2bJ-J_{\omega^3})\omega^4\\
&{\rm d} b=\tfrac{1}{4}(-4 a_{\omega^1} b + 6 b^2 c - 8 a c^2 + 4 c M - M_{\omega^2} + P_{\omega^2} + 2 b Q - 4 a R)\omega^0+(2c^2+R)\omega^1+(2 a_{\omega^1}-3bc-Q)\omega^2\\&\quad+\tfrac{1}{2}(-b^2+M-3P)\omega^3+(ab-3cJ+J_{\omega^2})\omega^4\\
& {\rm d} c =c_{\omega^0}\omega^0+S\omega^1+(c^2-R)\omega^2+(a_{\omega^1}-2bc)\omega^3+\tfrac{1}{4}(b^2-4J_{\omega^1}+M-P)\omega^4 ,\\
\end{aligned}
\end{equation}
for functions $L, Q, R, S$ on ${\cal M}$.
\end{lemma}
\begin{lemma}
The functions $a, b, c, J, L, M, P, Q, R, S$ are uniquely determined by \eqref{differentiatedcoframe1} and \eqref{differentiatedJabc2}. Explicitly,
\begin{align*}
&a= t_{\omega^3}, \quad b=-t_{\omega^2},\quad c=t_{\omega^1},\quad J=-t_{\omega^4},\quad L=t_{\omega^3\omega^3},\quad M=6t_{\omega^0}-2(t_{\omega^2})^2+6 t_{\omega^3} t_{\omega^1}+ t_{\omega^2\omega^3},\\& P=2t_{\omega^0}-(t_{\omega^2})^2+2 t_{\omega^3} t_{\omega^1}+t_{\omega^2\omega^3},\quad
Q=2 t_{\omega^3\omega^1} +t_{\omega^2\omega^2}+3t_{\omega^2} t_{\omega^1},\quad R=-t_{\omega^2\omega^1}-2(t_{\omega^1})^2,\quad S=t_{\omega^1\omega^1}.
\end{align*}
\end{lemma}
\subsection{The main invariants and a characterization of the flat model}\label{themain} In this section we shall formulate our first main theorem, which in particular justifies the importance of the functions $J, L, M, P, Q, R, S$. Note that the flat marked contact Engel structure corresponding to $t=0$ in the parametrization from Lemma \ref{lemma1marked} satisfies $J=L=M=P=Q=R=S=0$.
Before stating the theorem, we introduce the following notation for the Maurer-Cartan equations, given in the Appendix by formula \eqref{MaurerCartan_Q}, of the $9$-dimensional parabolic subgroup $\mathrm{P}_1\subset\mathrm{G}_2$:
\begin{equation}\label{MaurerCartrew} \begin{aligned} &e^0= {\rm d}\theta^0 -(-6 \theta^0\wedge\theta^5 + \theta^1\wedge \theta^4 - 3\theta^2\wedge\theta^3)=0\\ &e^1= {\rm d}\theta^1 -(- 3\theta^1\wedge\theta^5 - 3\theta^1\wedge\theta^8)=0 \\ &e^2= {\rm d}\theta^2 -(\theta^1\wedge\theta^6 - 3\theta^2\wedge\theta^5 - \theta^2\wedge\theta^8)=0 \\ &e^3= {\rm d}\theta^3 -(2\theta^2\wedge\theta^6 -3\theta^3\wedge\theta^5 + \theta^3\wedge\theta^8)=0 \\ &e^4= {\rm d}\theta^4-(6 \theta^0\wedge\theta^{12}+3\theta^3\wedge\theta^6-3\theta^4\wedge\theta^5+3\theta^4\wedge\theta^8)=0\\ &e^5={\rm d}\theta^5-(-\theta^1\wedge\theta^{12})=0\\ &e^6={\rm d}\theta^6-(6\theta^2\wedge\theta^{12}+2\theta^6\wedge\theta^8)=0\\ &e^8={\rm d}\theta^8-(-3\theta^1\wedge\theta^{12})=0\\ &e^{12}={\rm d}\theta^{12}-( -3\theta^5\wedge\theta^{12}- 3\theta^8\wedge\theta^{12})=0\,.
\end{aligned}\end{equation}
\begin{remark} Anticipating the material that will be explained in Section \ref{sec_tanaka}, we advice a reader familiar with Tanaka theory to look at Proposition \ref{p_1} for the reason why we expect the parabolic subalgebra $\mathfrak{p}_1$ to be the infinitesimal symmetry algebra of the flat marked contact Engel structure.
\end{remark}
We call the group $\mathbf{S}\cong B\ltimes\mathbb{R}^5$,
\begin{equation} \label{matS} \begin{aligned}\mathbf{S}=\left\{(\mathbf{S}^{\mu}{}_{\nu})=\begin{pmatrix} s_0& 0&0&0&0\\ s_1&{s_5}^3& 0 & 0&0\\ s_2&{s_5}^2s_7& {s_5}^2 s_8& 0 &0\\ s_3& s_5 {s_7}^2& 2 s_7 s_5 s_8 & s_5{s_8}^2& 0\\ s_4& {s_7}^3& 3 {s_7}^2s_8&3s_7{s_8}^2&{s_8}^3
\end{pmatrix} :\, \mathrm{det}(\mathbf{S}^{\mu}{}_{\nu})=s_0 {s_5}^6{s_8}^6\neq 0\right\} \end{aligned} \end{equation}
the \emph{structure group} of the equivalence problem for marked contact twisted cubic structures.
\begin{theorem}\label{main} Given the most general marked contact Engel structure on $\mathcal{U}$, consider an adapted coframe $\omega=(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ that satisfies structure equations \eqref{differentiatedcoframe2}, and let $J, L, M, P, Q, R, S$ be the functions defined via \eqref{differentiatedcoframe2} and \eqref{differentiatedJabc2}.
\begin{enumerate}
\item Let $\hat{\omega}=(\hat{\omega}^0,\hat{\omega}^1,\hat{\omega}^2,\hat{\omega}^3,\hat{\omega}^4)$ be another coframe related to $\omega$ via $\hat{\omega}=A\hbox to 2.5pt{\hss$\ccdot$\hss}\phi^*(\omega)$, with $\phi:\mathcal{U}\to\mathcal{U}$ a diffeomorphism and $A:\mathcal{U}\to \mathbf{S}$ a function with values in the structure group \eqref{matS}. Further suppose that $\hat{\omega}$ satisfies the structure equations \eqref{differentiatedcoframe2} for some functions $\hat{a}, \hat{b}, \hat{c}, \hat{J}, \hat{M}, \hat{P}$, and let $\hat{Q}, \hat{R}, \hat{S}$ be the derived functions as in \eqref{differentiatedJabc2}. Then \begin{enumerate} \item $J=0$ iff $\hat{J}=0$ \item $J=L=0$ iff $\hat{J}=\hat{L}=0$ \item $J=L=M=0$ iff $\hat{J}=\hat{L}=\hat{M}=0$ \item $J=L=M=P=0$ iff $\hat{J}=\hat{L}=\hat{M}=\hat{P}=0$ \item $J=L=M=P=Q=0$ iff $\hat{J}=\hat{L}=\hat{M}=\hat{P}=\hat{Q}=0$ \item $J=L=M=P=Q=R=0$ iff $\hat{J}=\hat{L}=\hat{M}=\hat{P}=\hat{Q}=\hat{R}=0$ \item $J=L=M=P=Q=R=S=0$ iff $\hat{J}=\hat{L}=\hat{M}=\hat{P}=\hat{Q}=\hat{R}=\hat{S}=0$\label{flatness_co} \end{enumerate} \item A marked contact Engel structure is flat
if and only if
\begin{equation}\label{flatness_co} J=L=M=P=Q=R=S=0 \end{equation} holds. In this case the structure has a $9$-dimensional algebra of infinitesimal symmetries isomorphic to the parabolic subalgebra $\mathfrak{p}_1$. \end{enumerate} \end{theorem}
\begin{remark}\label{rem_cond}
Part 1. of the Theorem says that each of the below itemized differential conditions \begin{enumerate} \item $J=0$ \item $J=L=0$ \item $J=L=M=0$ \item $J=L=M=P=0$ \item $J=L=M=P=Q=0$ \item $J=L=M=P=Q=R=0$ \item $J=L=M=P=Q=R=S=0$ \end{enumerate} is an invariant condition on the marked contact Engel structure defined by the equivalence class $[\omega]$. Note however that e.g. $a=0$, or $L=0$ alone, is \emph{not} an invariant condition. \end{remark}
\begin{proof} of the Theorem \ref{main}.
We choose an adapted coframe $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ that satisfies \eqref{differentiatedcoframe2}. This determines a trivialization of the bundle of all adapted coframes, which may thus be identified with $\pi:\mathcal{U}\times \mathbf{S}\to \mathcal{U}$.
We can now lift $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ to the $5$ well-defined (tautological) $1$-forms
\begin{align}\label{thetas}
\theta^{\mu}=\mathbf{S}^{\mu}{}_{\nu}\omega^{\nu},\quad \mu=0,1,2,3,4,
\end{align} on $\mathcal{U}\times \mathbf{S}$.
Writing equations \eqref{differentiatedcoframe2} symbolically as \begin{equation}\label{diffcoframesymb}{\rm d}\omega^{\mu}=-\tfrac{1}{2}F^{\mu}{}_{\nu\rho}\omega^{\nu}\wedge\omega^{\rho},\end{equation} we express the differentials ${\rm d}\theta^0,...,{\rm d}\theta^4$ as $${\rm d}\theta^{\mu}={\rm d}(\mathbf{S}^{\mu}{}_{\nu}\omega^{\nu})={\rm d} \mathbf{S}^{\mu}{}_{\nu}\wedge\omega^{\nu}+ \mathbf{S}^{\mu}{}_{\nu}{\rm d}\omega^{\nu}={\rm d} \mathbf{S}^{\mu}{}_{\rho}(\mathbf{S}^{-1})^{\rho}{}_{\sigma}\wedge\theta^{\sigma}-\tfrac{1}{2}\mathbf{S}^{\mu}{}_{\nu} F^{\nu}{}_{\rho\sigma}(\mathbf{S}^{-1})^{\rho}{}_{\alpha}(\mathbf{S}^{-1})^{\sigma}{}_{\beta}\theta^{\alpha}\wedge\theta^{\beta}.$$ For computational reasons we set $$\delta=-s_5 s_8.$$
Now we will solve equations \eqref{MaurerCartrew}. The unknowns in these equations are the group parameters $s_0, s_1, s_2,$ $ s_3,s_4,s_5, s_7,$ $ \delta$ and the four $1$-forms $\theta^5$, $\theta^6$, $\theta^8$ and $\theta^{12}$. What is given is the coframe $\omega$ and the derived functions $a, b, c, J,$ etc, as defined in \eqref{differentiatedcoframe2} and \eqref{differentiatedJabc2}. Therefore, if we say that we solve equations $$e^0=0, \,e^1=0,\, ...,\,e^{12}=0,$$ we mean that we are searching for $s_0, s_1, s_2,$ $ s_3,s_4,s_5, s_7,$ $ \delta$ and $\theta^5$, $\theta^6$, $\theta^8$ and $\theta^{12}$ such that the equations are satisfied.
We start by solving equation $e^0=0$. Computing \begin{align*} {\rm d}\theta^0= \tfrac{1}{s_0} {\rm d} s_0\wedge \theta^0 - \tfrac{s_4}{\delta^3} \theta^0\wedge\theta^1 + \tfrac{3 s_3}{\delta^3} \theta^0\wedge\theta^2- \tfrac{3 s_2}{\delta^3}\theta^0\wedge\theta^3+\tfrac{s_1}{\delta^3}\theta^0\wedge\theta^4 - \tfrac{s_0}{\delta^3}\theta^1\wedge\theta^4 +\tfrac{3 s_0}{\delta^3}\theta^2\wedge\theta^3 \end{align*} and inserting it into $e^0\wedge\theta^0=0$ gives \begin{align*}
(-1- \tfrac{s_0}{\delta^3})\theta^1\wedge\theta^4\wedge\theta^0 +(3+\tfrac{3 s_0}{\delta^3})\theta^2\wedge\theta^3\wedge\theta^0=0, \end{align*} whose unique solution is \begin{equation}\label{s0} {s_0}=-{\delta^3}.\end{equation} Having established this, the most general solution of $e^0=0$ for $\theta^5$ is \begin{equation}\label{deftheta5}\theta^5:=\tfrac{1}{2\delta} {\rm d} \delta + \tfrac{s_4}{6\delta^3}\theta^1 - \tfrac{ s_3}{2\delta^3}\theta^2+ \tfrac{s_2}{2\delta^3}\theta^3-\tfrac{s_1}{6\delta^3}\theta^4 - \tfrac{1}{6}u_0\,\theta^0.\end{equation} Note that we had to introduce a new variable $u_0$, since adding to any particular solution for $\theta^5$ a functional multiple of $\theta^0$ is a solution as well. At this point the equation $e^0=0$ is satisfied.
We next consider the equation $e^1\wedge\theta^0\wedge\theta^1=0$, which reads \begin{equation}\label{theta1}\tfrac{3(s_1 \delta + a {s_5}^3\delta -3 J {s_5}^4 s_7)}{{\delta}^4}\theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^3+\tfrac{3 J {s_5}^5}{{\delta}^4}\theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^4=0.\end{equation} Since ${s_5}$ cannot be zero, the vanishing of the coefficient at the $\theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^4$-term in \eqref{theta1} is equivalent to $J=0$.
In other words, we have shown that under the most general transformation that maps one adapted coframe $\omega$ to another adapted coframe $\hat{\omega}$, the coefficient $F^1_{24}$ in the structure equations \eqref{diffcoframesymb} transforms as $$\hat{F}^1{}_{24}=\tfrac{3{s_5}^5}{{\delta}^4}F^1{}_{24}.$$
This shows that it defines a density invariant (or, \emph{relative invariant}) of the marked contact twisted cubic structure. In particular, its vanishing or not is an invariant property of the structure. For those coframes that satisfy the structure equations \eqref{differentiatedcoframe2}, the coefficient $\hat{F}^1{}_{24}$ is proportional to $ J$. Moreover, for the (particular) flat structure corresponding to $t=0$ we have $J=0$. This further shows that vanishing of this density invariant that we discovered is a necessary condition for flatness.
From now on we assume $$J=0$$
(which means that also the consequences $J_{\omega^0}=J_{\omega^1}=J_{\omega^2}=J_{\omega^3}=J_{\omega^4}=0$ hold). We return to equation \eqref{theta1}. We can now solve it by setting \begin{equation}\label{s1}s_1=- a {s_5}^3.\end{equation} Then we look at equation $e^1\wedge\theta^1=0$, which reads
$$ \tfrac{-{s_5}^2}{\delta^5}(M\delta +2 L s_5 s_7)\theta^0\wedge\theta^1\wedge\theta^2+\tfrac{{s_5}^4}{\delta^5}L\theta^0\wedge\theta^1\wedge\theta^3=0.
$$ The same argument as above applied to the second term in this equation shows that $L$ must be zero for $e^1\wedge\theta^1=0$ to admit a solution. We also infer from this that the simultaneous vanishing of $J$ and $L$ is an invariant condition on marked contact Engel structures, and that $J=L=0$ is another necessary condition for a structure to be flat. We now assume that
$$ J=L=0. $$
With this assumption $e^1\wedge\theta^1=0$ reads
$$ \tfrac{-{s_5}^2}{\delta^4}M\theta^0\wedge\theta^1\wedge\theta^2=0.
$$ As before, we may now conclude that the simultaneous vanishing of $J$, $L$ and $M$ is an invariant property and necessary for flatness. We will from now on assume that $$ J=L=M=0 $$ holds. Now the general solution for $e^1=0$ is \begin{equation}\begin{aligned}\label{deftheta8}\theta^8&=-\tfrac{1}{2\delta}{\rm d}\delta+\tfrac{1}{s_5}{\rm d}{s_5}-\tfrac{(-6 c s_2\delta^2 + 2 a_1 s_5 \delta^3 + 2 a s_4 {s_5}^4 - 6 a c {s_5}^2{s_7}\delta^2- 6 a {s_3}{s_5}^3 s_7 +6 a s_2 {s_5}^2{s_7}^2 +2 a^2 {s_5}^4 {s_7}^3 - s_5 u_0 {\delta}^6}{6 {s_5} {\delta}^6}\theta^0\\
&+\tfrac{2c\delta^2+s_3s_5-2a{s_5}^2{s_7}^2}{2{s_5}\delta^3}\theta^2-\tfrac{s_2-2a{s_5}^2s_7}{2\delta^3}\theta^3-\tfrac{a{s_5}^3}{2\delta^3}\theta^4 - \tfrac{1}{3}u_1\,\theta^1,\end{aligned}\end{equation} where we have introduced a new variable $u_1$. In this way, $e^1=0$ is solved.
We next attempt to solve $e^2=0$. We start with $e^2\wedge\theta^0\wedge\theta^1=0,$ which reads $$\tfrac{2(2 s_2-b s_5 \delta +2 a {s_5}^2 s_7)}{\delta^3}\theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^3=0.$$ Its unique solution is given by \begin{equation}\label{s2}s_2=\tfrac{1}{2}b s_5 \delta- a {s_5}^2s_7.\end{equation} Computing $e^2\wedge\theta^1=0$ and looking at the coefficient at the $\theta^0\wedge\theta^1\wedge\theta^3$ term, we see that in order to be able to solve the equation, $P$ has to be zero. We also conclude that $$ J=L=M=P=0 $$ is an invariant property, which we from now on assume to hold. Then the unique solution of $e^2\wedge\theta^1=0$ is \begin{equation}\label{defu0}u_0=\tfrac{(4a_1-6bc-3Q)\delta^3+3b^2s_5s_7\delta^2-3(s_3 +2a{s_7}^2s_5)b{s_5}\delta+2a{s_5}^2(-s_4s_5+3s_3s_7+2as_5{s_7}^3)}{2{\delta}^6}.\end{equation} Now the most general $1$-form $\theta^6$ such that $e^2=0$ holds is \begin{equation}\label{deftheta6}\begin{aligned} &\theta^6= \tfrac{s_7}{s_5\delta}{\rm d}\delta - \tfrac{s_7}{{s_5}^2}{\rm d} {s_5} -\tfrac{1}{s_5}{\rm d} {s_7}\\ &-\tfrac{2(2 c^2+R)\delta^4-2(4 b c +Q)s_5 s_7\delta^3 +(8cs_3 +5 b^2 {s_5} {s_7}^2+8 a c {s_5}{s_7}^2)s_5 \delta^2 +2(s_4{s_5}-4 s_3s_7-6a {s_5}{s_7}^3)b{s_5}^2\delta -4(s_4 {s_5}-3s_3 {s_7}-2 a{s_5}{s_7}^3)a{s_5}^3s_7}{4{s_5}^2{\delta}^6}\theta^0\\ &+\tfrac{2{s_5}^2u_1\delta^3+6cs_7\delta^2-12b s_5{s_7}^2\delta-3s_4{s_5}^2+18a {s_5}^2{s_7}^3}{6{s_5}^2\delta^3}\theta^2-\tfrac{2c \delta^2-2bs_5s_7\delta+3a{s_5}^2{s_7}^2}{s_5\delta^3}\theta^3 -\tfrac{s_5(b\delta - 2 a s_5s_7)}{2\delta^3}\theta^4 +u_2\theta^1.\end{aligned}\end{equation}
Next we compute $e^3\wedge\theta^0=0$, which can be solved
by \begin{equation}\label{s3}s_3=\tfrac{-c\delta^2+b s_5s_7\delta-a{s_5}^2{s_7}^2}{s_5},\end{equation} \begin{equation}\label{u1}u_1=\tfrac{-3(2 c s_7\delta^2-2b s_5{s_7}^2\delta+s_4{s_5}^2+2a{s_5}^2{s_7}^3)}{2{s_5}^2\delta^3},\end{equation} \begin{equation}\label{u2}u_2=\tfrac{-{s_7}^2(c\delta^2-bs_5s_7\delta+a{s_5}^2{s_7}^2)}{{s_5}^3\delta^3}.\end{equation} Equation $e^3=0$ now reads \begin{equation*}\begin{aligned} \tfrac{1}{{s_5}^4\delta^3}(-S\delta^2+2 Rs_5 s_7 \delta-Q{s_5}^2{s_7}^2)\theta^0\wedge\theta^1-\tfrac{2}{{s_5}^2\delta^3}(R\delta-Qs_5s_7)\theta^0\wedge\theta^2-\tfrac{1}{\delta^3}Q\theta^0\wedge\theta^3=0. \end{aligned}\end{equation*} From here we conclude that $$J=L=M=P=Q=0$$ is an invariant condition. Assuming that it be satisfied, we see that in order to be able to solve equation $e^3=0$, we also have to assume $R$ to be zero. We also see that $$J=L=M=P=Q=R=0$$ is an invariant condition. Assuming that it holds, we see that also $S$ has to be zero. Assuming that the invariant condition $$J=L=M=P=Q=R=S=0$$ holds, equation $e^3=0$ is now solved.
Now we consider $e^4=0$. The most general $1$-form $\theta^{12}$ solving this equation is \begin{equation}\label{deftheta12}\begin{aligned} \theta^{12}&=\tfrac{-(cs_7\delta^2-bs_5{s_7}^2\delta+s_4{s_5}^2+a{s_5}^2{s_7}^3)}{2{s_5}^2\delta^4}{\rm d}\delta+\tfrac{1}{6\delta^3}{\rm d}{s_4}+\tfrac{cs_7\delta^2-bs_5{s_7}^2\delta+s_4{s_5}^2+a{s_5}^2{s_7}^3}{2{s_5}^3\delta^3}{\rm d}{s_5} +\tfrac{c\delta^2-b s_5s_7\delta+a{s_5}^2{s_7}^2}{2{s_5}^2\delta^3}{\rm d}{s_7}\\ &+\tfrac{3c^2{s_7}^2\delta^4-6bcs_5{s_7}^3\delta^3+3(cs_4{s_5}^2s_7+b^2{s_5}^2{s_7}^4+2ac{s_5}^2{s_7}^4)\delta^2-3(bs_4{s_5}^3{s_7}^2+2 a b {s_5}^3{s_7}^5)\delta+{s_4}^2{s_5}^4+3a s_4{s_5}^4{s_7}^3+3a^2{s_5}^4{s_7}^6}{6{s_5}^4\delta^6}\theta^1\\ &+\tfrac{bc{s_7}^2\delta^3+(c s_4s_5-b^2s_5{s_7}^3-2acs_5{s_7}^3)\delta^2+3ab{s_5}^2{s_7}^4\delta-a s_4{s_5}^3{s_7}^2-2a^2{s_5}^3{s_7}^5}{2{s_5}^2\delta^6}\theta^2\\ &+\tfrac{c^2\delta^4-2bcs_5s_7{\delta}^3+(b^2{s_5}^2{s_7}^2+3ac{s_5}^2{s_7}^2)\delta^2-3ab{s_5}^3{s_7}^3\delta+as_4{s_5}^4{s_7}+2a^2{s_5}^4{s_7}^4}{2{s_5}^2\delta^6}\theta^3\\ &+\tfrac{4a_1\delta^3-(3b^2s_5s_7+12acs_5s_7)\delta^2+12ab{s_5}^2{s_7}^2\delta-4as_4{s_5}^3-8a^2{s_5}^3{s_7}^3}{24\delta^6}\theta^4+\tfrac{1}{6}u_3\theta^0, \end{aligned}\end{equation} where $u_3$ is a new variable. Next we consider equation $e^5=0$, which we solve for
\begin{equation}\label{u3}\begin{aligned} u_3&=\tfrac{-8c^3\delta^6+24bc^2s_5s_7\delta^5-(21b^2c{s_5}^2{s_7}^2+36ac^2{s_5}^2{s_7}^2)\delta^4+(6bcs_4{s_5}^3+5b^3{s_5}^3{s_7}^3+60abc{s_5}^3{s_7}^3)\delta^3}{4{s_5}^3\delta^9}\\ &-\tfrac{(3b^2s_4{s_5}^4{s_7}+24acs_4{s_5}^4{s_7}+21ab^2{s_5}^4{s_7}^4+36a^2c{s_5}^4{s_7}^4)\delta^2-(18abs_4{s_5}^5{s_7}^2+24a^2b{s_5}^5{s_7}^5)\delta+4{s_4}^2{s_5}^6+12a^2s_4{s_5}^6{s_7}^3+8a^3{s_5}^6{s_7}^6}{4{s_5}^3\delta^9}. \end{aligned}\end{equation} Computing shows that now $e^6=0,$ $e^8=0$ and $e^{12}=0$ are satisfied as well.
Concluding, we proved that the conditions displayed in Remark \ref{rem_cond}
are invariant conditions on marked contact Engel structures.
The flat marked contact Engel structure satisfies $J=L=M=P=Q=R=S=0$, so this is evidently a necessary condition for flatness.
Moreover, assuming $J=L=M=P=Q=R=S=0$, we uniquely determined
\begin{itemize}
\item a $9$-dimensional sub-bundle $\mathcal{P}$ of the $13$-dimensional bundle $\mathcal{U}\times \mathbf{S}\to \mathcal{U}$ we started out with (parametrized by the coordinates $x^0, x^1, x^2, x^3, x^4$ and the remaining fibre coordinates $s_4, s_5, \delta, s_7$)
\item and a well defined coframe $(\theta^0, \theta^1, \theta^2, \theta^3, \theta^4, \theta^5, \theta^6, \theta^8, \theta^{12})$ on $\mathcal{P}$ satisfying the Maurer-Cartan equations \eqref{MaurerCartan} whose first five forms $(\theta^0, \theta^1, \theta^2, \theta^3, \theta^4)$ when pulled back with respect to any section of $\mathcal{P}\to\mathcal{U}$ are contained in the equivalence class $[(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)]$.
\end{itemize} Hence a structure that satisfies these conditions has a $9$-dimensional algebra of infinitesimal symmetries isomorphic to the parabolic subalgebra $\mathfrak{p}_1$. Taking a section corresponding to a leaf of the integrable distribution given by the kernel of $\theta^5, \theta^6, \theta^8, \theta^{12}$, the pullbacks of $\theta^0, \theta^1, \theta^2, \theta^3, \theta^4$ to $\mathcal{U}$ satisfy
$${\rm d}\theta^0 = \theta^1\wedge \theta^4 - 3\theta^2\wedge\theta^3,\,{\rm d} \theta^1=0,\, {\rm d}\theta^2=0,\,{\rm d}\theta^3=0,\,{\rm d} \theta^4=0.$$
In particular, there exist local coordinates $(x^0, x^1, x^2, x^3, x^4)$ such that $\theta^0={\rm d} x^0+x^1{\rm d} x^4-3 x^2{\rm d} x^3,\, \theta^1={\rm d} x^1, \, \theta^2={\rm d} x^2,\, \theta^3={\rm d} x^3, \theta^4={\rm d} x^4$, which means that the marked Engel structure is flat.
\end{proof}
\subsection{A rigid coframe for marked contact Engel structures}
In the previous section we have explicitly constructed a rigid coframe on a $9$-dimensional bundle over the \emph{flat} marked contact Engel structure. In this section we apply Cartan's equivalence method to show how to associate a rigid coframe on a $9$-dimensional bundle to a \emph{general} marked contact Engel structure.
We start as in the proof of Theorem \ref{main}.
We choose an adapted coframe $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ that satisfies the structure equations \eqref{differentiatedcoframe2}
and as in the beginning of the proof of Theorem \ref{main} we lift it
to the $5$ well-defined (tautological) $1$-forms
\begin{align*}\label{thetas}
\theta^{\mu}=\mathbf{S}^{\mu}{}_{\nu}\omega^{\nu},\quad \mu=0,1,2,3,4,
\end{align*} on $\mathcal{U}\times \mathbf{S}$, where $\mathbf{S}$ is the structure group \eqref{matS}. We again reparametrize $\delta=-s_5 s_8$.
Since \begin{align*} {\rm d}\theta^0 \wedge\theta^0=- \tfrac{s_0}{\delta^3}\theta^1\wedge\theta^4\wedge\theta^0 +\tfrac{3 s_0}{\delta^3}\theta^2\wedge\theta^3\wedge\theta^0 \end{align*}
we normalize the coefficient of the $\theta^1\wedge\theta^4$--term in the expansion of ${\rm d}\theta^0$ to $1$ by setting
\begin{equation}\label{szero}s_0=-{\delta^3}.\end{equation} Then there exists a $1$-form $\theta^5$, which is uniquely defined up to addition of multiples of $\theta^0$,
satisfying the equation
$${\rm d}\theta^0= -6 \theta^0\wedge \theta^5 + \theta^1\wedge\theta^4 - 3 \theta^2\wedge\theta^3.$$
Computing \begin{align*} {\rm d}\theta^1 \wedge\theta^0\wedge\theta^1\wedge\theta^4=\tfrac{3(s_1\delta+a {s_5}^3\delta-3 J{s_5}^4{s_7})}{{\delta}^4}\theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^3\wedge\theta^4 \end{align*}
shows that we can further normalize the $\theta^2\wedge\theta^3$--coefficient in the expansion of ${\rm d}\theta^1$ to $0$ by setting
\begin{equation}\label{sone}s_1=\tfrac{-a \delta {s_5}^3+3 J {s_5}^4{s_7}}{\delta}.\end{equation}
Then there exists a $1$-form $\theta^8$, uniquely defined up to addition of multiples of $\theta^0$ and $\theta^1$, satisfying $${\rm d}\theta^1\wedge\theta^0=-3 \theta^0\wedge\theta^1\wedge\theta^5-3 \theta^0\wedge\theta^1\wedge\theta^8+\tfrac{3 J {s_5}^5}{\delta^4}\theta^0\wedge\theta^2\wedge\theta^4.$$
Now \begin{align*} {\rm d}\theta^2\wedge\theta^0\wedge\theta^1=&\tfrac{2(2 \delta s_2-b\delta^2 s_5+2 a \delta {s_5}^2s_7-3J {s_5}^3{s_7}^2)}{\delta^4}\theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^3-3 \theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^5\\&-\theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^8 +\tfrac{2 J{s_5}^5}{\delta^4}\theta^0\wedge\theta^1\wedge\theta^3\wedge\theta^4 \end{align*} shows that we can normalize the $\theta^2\wedge\theta^3$--term in the expansion of ${\rm d}\theta^2$ to $0$ by setting
\begin{equation}\label{stwo}s_2=\tfrac{s_5}{2\delta}(b\delta^2-2a\delta s_5 s_7+3J{s_5}^2{s_7}^2),\end{equation} and \begin{align*} {\rm d}\theta^3\wedge\theta^0&\wedge\theta^2\wedge\theta^3 = -\tfrac{c \delta^3+\delta s_3 s_5 - b \delta^2 s_5 s_7 + a\delta {s_5}^2 {s_7}^2-J{s_5}^3{s_7}^3}{\delta^4 s_5} \theta^0\wedge\theta^1\wedge\theta^2\wedge\theta^3\wedge\theta^4 \end{align*} shows that we can normalize the $\theta^1\wedge\theta^4$--term in the expansion of ${\rm d}\theta^3$ to $0$ by setting \begin{equation}\label{sthree}s_3=-\tfrac{1}{\delta s_5}(c\delta^3 - b \delta^2 s_5 {s_7}+a\delta {s_5}^2{s_7}^2-J {s_5}^3{s_7}^3).\end{equation}
Having performed these normalizations, on the subbundle $\mathcal{G}^9\subset(\mathcal{U}\times \mathbf{S})$ defined by \eqref{szero}, \eqref{sone}, \eqref{stwo}, \eqref{sthree}, we now have \begin{equation}\label{5forms} \begin{aligned} &\theta^0=-\delta^3\omega^0\\ &\theta^1=\tfrac{{s_5}^3(3J {s_5}{s_7}-a\delta)}{\delta}\omega^0+{s_5}^3\omega^1\\ &\theta^2=\tfrac{s_5(b\delta^2 - 2 a\delta s_5 s_7+3 J {s_5}^2{s_7}^2)}{2\delta}\omega^0+{s_5}^2 s_7\omega^1-\delta s_5\omega^2\\ &\theta^3=\tfrac{-c\delta^3+b\delta^2 s_5 s_7- a\delta {s_5}^2{s_7}^2+J{s_5}^3{s_7}^3}{s_5}\omega^0+s_5{s_7}^2\omega^1-2 \delta s_7 \omega^2 +\tfrac{\delta^2}{s_5}\omega^3\\ &\theta^4=s_4\omega^0+{s_7}^3\omega^1-\tfrac{3\delta {s_7}^2}{s_5}\omega^2+\tfrac{3\delta^2s_7}{{s_5}^2}\omega^3-\tfrac{\delta^3}{{s_5}^3}\omega^4. \end{aligned} \end{equation}
We have further introduced two additional forms $\theta^5$ and $\theta^8$, but on the $9$-dimensional bundle $\mathcal{G}^9$ given by \eqref{szero}, \eqref{sone}, \eqref{stwo}, \eqref{sthree} they are defined up to a certain freedom.
It turns out that imposing further normalizations determines forms $\theta^5$, $\theta^8$ uniquely and in addition picks up unique $1$-forms $\theta^6$ and $\theta^{12}$ that together with the five $1$-forms \eqref{5forms} constitute a coframe on $\mathcal{G}^9$. The normalizations needed are included in the following proposition:
\begin{proposition}\label{propo_coframe}
The five forms \eqref{5forms} on the $9$-dimensional subbundle $\mathcal{G}^9\subset \mathcal{U}\times \mathbf{S}$ given by \eqref{szero}, \eqref{sone}, \eqref{stwo}, \eqref{sthree} can be supplemented to a rigid coframe $(\theta^0, \theta^1, \theta^2, \theta^3, \theta^4, \theta^5, \theta^6, \theta^8, \theta^{12}) $ which is uniquely determined by the fact that it satisfies
\begin{equation}\label{5structureeq} \begin{aligned} {\rm d}\theta^0=&-6\theta^0\wedge\theta^5+\theta^1\wedge\theta^4-3\theta^2\wedge\theta^3\\ {\rm d}\theta^1=& - 3 \theta^1\wedge\theta^5 - 3 \theta^1\wedge\theta^8+
T^1{}_{02} \theta^0\wedge\theta^2 +
T^1{}_{03} \theta^0\wedge\theta^3 + T^1{}_{04} \theta^0\wedge\theta^4+ T^1{}_{06} \theta^0\wedge\theta^6 +
T^1{}_{24}\theta^2\wedge\theta^4\\
{\rm d}\theta^2=& \theta^1\wedge\theta^6- 3\theta^2\wedge\theta^5 -
\theta^2\wedge\theta^8 + T^2{}_{03} \theta^0\wedge\theta^3 - T^2{}_{04} \theta^0\wedge\theta^4 +T^2{}_{34}\theta^3\wedge\theta^4\\
{\rm d}\theta^3=&2 \theta^2\wedge\theta^6 - 3\theta^3\wedge\theta^5 + \theta^3\wedge\theta^8+T^3{}_{01}\theta^0\wedge\theta^1 + T^3{}_{02} \theta^0\wedge\theta^2 - T^3{}_{03} \theta^0\wedge\theta^3 + T^3{}_{04} \theta^0\wedge\theta^4\\ {\rm d}\theta^4=&6 \theta^0\wedge\theta^{12}+3\theta^3\wedge\theta^6-3\theta^4\wedge\theta^5+3\theta^4\wedge\theta^8,\\ \end{aligned} \end{equation} for some functions $T^i{}_{jk}$, and the additional normalization that ${\rm d}\theta^5$, when written with respect to the basis of forms $\theta^i\wedge\theta^j$, has zero coefficient at the $\theta^0\wedge\theta^1$ term.
\end{proposition} We remark that the normalizations given in Proposition \ref{propo_coframe} also uniquely determine the structure functions $T^k{}_{jl}$. In particular we have
\begin{equation}
T^1{}_{24}=-T^1{}_{06}=\tfrac{3}{2}T^2{}_{34}=\tfrac{3 J {s_5}^5}{\delta^4},\end{equation} and \begin{equation} T^1{}_{02}=\tfrac{{s_5}^2 (\delta^4 M - 6 \delta J s_4 {s_5}^3 - 9 c \delta^3 J s_5 s_7 -
3 \delta^3 J_{\omega^2} s_5 s_7 + 2 \delta^3 L s_5 s_7 - 9 b \delta^2 J {s_5}^2 {s_7}^2 - 9 \delta^2 J_{\omega^3} {s_5}^2 {s_7}^2 + 21 a \delta J {s_5}^3 {s_7}^3 - 9 \delta J_{\omega^4} {s_5}^3 {s_7}^3 - 27 J^2 {s_5}^4 {s_7}^4)}{\delta^8}. \end{equation}
\begin{remark} The bundle $\mathcal{G}^9\to \mathcal{U}$ has as structure group the subgroup of $\mathbf{S}$ of matrices of the form \begin{align*}\begin{pmatrix} -{\delta}^3& 0&0&0&0\\ 0&{s_5}^3& 0 & 0&0\\ 0&{s_5}^2 s_7& -\delta{s_5}& 0 &0\\ 0& s_5 {s_7}^2& -2 \delta s_7 & \tfrac{\delta^2}{s_5}& 0\\ s_4& {s_7}^3& -\tfrac{3 \delta{s_7}^2}{s_5}&\tfrac{3\delta^2 s_7}{{s_5}^2}&-\tfrac{\delta^3}{{s_5}^3}
\end{pmatrix}. \end{align*} \end{remark}
\begin{remark} The coframe constructed in Proposition \ref{propo_coframe} does not define a Cartan connection. In order to obtain a Cartan connection, more elaborate normalizations are necessary. \end{remark}
\subsection{Integrable structures and the submaximal models} Recall that any marked contact Engel structure is called \emph{integrable} if the rank $2$ distribution $\mathcal{D}^{\sigma}$, which in terms of an adapted coframe is given by $$\mathcal{D}^\sigma=\mathrm{ker}(\omega^0,\omega^1,\omega^2),$$ is integrable.
The following proposition shows that integrability of a marked contact Engel structure precisely corresponds to the vanishing of the first (relative) invariant from Theorem \ref{main}. \begin{proposition}\label{prop.integrability} A marked contact Engel structure is integrable if and only if $J=0$.
\end{proposition}
\begin{proof} Let $(\omega^0,\omega^1,\omega^2,\omega^3,\omega^4)$ be any adapted coframe that satisfies the structure equations \eqref{differentiatedcoframe2} with associated function $J$. A direct computation shows that \begin{align*}&{\rm d}\omega^0\wedge\omega^0\wedge\omega^1\wedge\omega^2=0\\ &{\rm d}\omega^1\wedge\omega^0\wedge\omega^1\wedge\omega^2=0\\ &{\rm d}\omega^2\wedge\omega^0\wedge\omega^1\wedge\omega^2= 2 \ J\; \omega^0\wedge\omega^1\wedge\omega^2\wedge\omega^3\wedge\omega^4. \end{align*} \end{proof}
For integrable marked contact Engel structures the structure equations simplify as follows. \begin{proposition}\label{propo_coframe_int} Consider an integrable marked contact Engel structure. Then the five forms \eqref{5forms} on the $9$-dimensional subbundle $\mathcal{G}^9$ of $\mathcal{U}\times \mathbf{S}$ given by \eqref{szero}, \eqref{sone}, \eqref{stwo}, \eqref{sthree} can be supplemented to a rigid coframe $(\theta^0, \theta^1, \theta^2, \theta^3, \theta^4, \theta^5, \theta^6, \theta^8, \theta^{12}) $ which is uniquely determined by the structure equations
\begin{equation}\label{5structureeq} \begin{aligned} {\rm d}\theta^0=&-6\theta^0\wedge\theta^5+\theta^1\wedge\theta^4-3\theta^2\wedge\theta^3\\ {\rm d}\theta^1=&-3 \theta^1\wedge\theta^5-3\theta^1\wedge\theta^8+\tfrac{{s_5}^2(\delta M +2 L {s_5}{s_7})}{{\delta}^5}\theta^0\wedge\theta^2-\tfrac{{s_5}^4 L}{{\delta}^5} \theta^0\wedge\theta^3\\ {\rm d}\theta^2 = & \theta^1\wedge\theta^6 - 3\theta^2\wedge\theta^5 - \theta^2\wedge\theta^8 -\tfrac{{s_5}^2(5\delta P-3 \delta M +4 L {s_5}{s_7})}{4{\delta}^5}\theta^0\wedge\theta^3\\ {\rm d}\theta^3 = & 2\theta^2\wedge\theta^6 -3\theta^3\wedge\theta^5 + \theta^3\wedge\theta^8-\tfrac{{\delta}^4 U -2 {\delta}^3 R {s_5}{s_7}+{\delta}^2 Q {s_5}^2{s_7}^2+2\delta P {s_5}^3{s_7}^3 + L {s_5}^4{s_7}^4}{{\delta}^5{s_5}^5}\theta^0\wedge\theta^1 \\
&-\tfrac{2(\delta^3 R-{\delta}^2 Q {s_5}{s_7} - 3 {\delta} P {s_5}^2{s_7}^2 -2 L {s_5}^3{s_7}^3)}{\delta^5{s_5}^2}\theta^0\wedge\theta^2-\tfrac{\delta^2 Q+6\delta P s_5 s_7 +6 L {s_5}^2{s_7}^2}{\delta^3}\theta^0\wedge\theta^3+\tfrac{(M-P){s_5}^2}{2{\delta}^4}\theta^0\wedge\theta^4\\ {\rm d}\theta^4=&6 \theta^0\wedge\theta^{12}+3\theta^3\wedge\theta^6-3\theta^4\wedge\theta^5+3\theta^4\wedge\theta^8\\ \end{aligned} \end{equation} and the additional normalization that ${\rm d}\theta^5$, when expressed with respect to the basis of forms $\theta^i\wedge\theta^j$, has zero coefficient at the $\theta^0\wedge\theta^1$ term.
\end{proposition}
In particular, the structure equations for integrable structures exhibit a new relative invariant for these structures that is independent of the filtration of invariant conditions from Section \ref{themain}.
\begin{proposition}\label{prop_seconddi}
Consider an integrable marked contact Engel structure on $\mathcal{U}$. Let $(\omega^0,\omega^1,\omega^2, \omega^3,\omega^4)$ be an adapted coframe satisfying the structure equations \eqref{differentiatedcoframe2} with $J=0$, and let $\phi$ be the $3$-form defined as \begin{equation}\phi=\omega^1\wedge\omega^2\wedge\omega^3- a \omega^0\wedge\omega^2\wedge\omega^3+\tfrac{1}{2}b \omega^0\wedge\omega^1\wedge\omega^3- c \omega^0\wedge\omega^1\wedge\omega^2.\end{equation} \begin{enumerate} \item Then the rank $2$-distribution \begin{align*}\mathcal{R}^{\sigma}=\mathrm{ker}(\phi)
\end{align*} on $\mathcal{U}$ is invariantly associated to the marked contact twisted cubic structure. \item This distribution is integrable if and only if $M-P$ vanishes. \end{enumerate}
\end{proposition}
\begin{proof} Let $\theta^0,\theta^1,\theta^2,\theta^3,\theta^4$ be the invariant forms \eqref{5forms} on $\mathcal{G}^9$ with $J=0$. A direct calculation gives \begin{align*} \theta^1\wedge\theta^2\wedge\theta^3=\delta^3 {s_5}^3(\omega^1\wedge\omega^2\wedge\omega^3- a \omega^0\wedge\omega^2\wedge\omega^3+\tfrac{1}{2}b \omega^0\wedge\omega^1\wedge\omega^3- c \omega^0\wedge\omega^1\wedge\omega^2). \end{align*} This shows that the kernel of $\theta^1\wedge\theta^2\wedge\theta^3$ descends to a distribution $\mathcal{R}^{\sigma}=\mathrm{ker}(\phi)$ on $\mathcal{U}$, which is independent of the choice of adapted coframe, and thus invariantly associated to the marked contact twisted cubic structure.
Since $$\phi=(\omega^1-a \omega^0)\wedge ( \omega^2-\tfrac{1}{2}b\omega^0)\wedge ( \omega^3- c \omega^0)$$
and
\begin{equation*} \begin{aligned} &{\rm d}(\omega^1- a\omega^0)\wedge(\omega^1-a\omega^0)\wedge(\omega^2-\tfrac{b}{2}\omega^0)\wedge(\omega^3-c\omega^0)=0,\\ &{\rm d}(\omega^2- \tfrac{b}{2}\omega^0)\wedge(\omega^1-a\omega^0)\wedge(\omega^2-\tfrac{b}{2}\omega^0)\wedge(\omega^3-c\omega^0)=0,\\ &{\rm d}(\omega^3- c\omega^0)\wedge(\omega^1-a\omega^0)\wedge(\omega^2-\tfrac{b}{2}\omega^0)\wedge(\omega^3-c\omega^0)=\tfrac{1}{2} (M-P) {s_5}^2\omega^0\wedge\omega^1\wedge\omega^2\wedge\omega^3\wedge\omega^4 , \end{aligned} \end{equation*}
integrability of $\mathcal{R}^{\sigma}$ is equivalent to the vanishing of $M-P$.
\end{proof}
\begin{remark}{\textbf{(Submaximal branch)}}
The structure equations for integrable marked contact Engel structures displayed in Proposition \ref{propo_coframe_int} show that for the subclass of structures with nowhere vanishing relative invariant $M-P$, we can further normalize the coefficient $T^3{}_{04}=\tfrac{(M-P){s_5}^2}{2\delta^4}$. It is also visible that the sign of $M-P$ is an invariant of integrable marked contact Engel structures.
We could now proceed as follows. We could normalize the coefficient $T^3{}_{04}$ to $\tfrac{\epsilon}{2}$ with $\epsilon=\mathrm{sign}(M-P)$ (or any non-zero multiple of $\epsilon$).
This means that we restrict to the $8$-dimensional subset $\mathcal{G}^8\subset \mathcal{G}^9$ defined by $$ s_5 =\tfrac{\delta^2}{\sqrt{\epsilon (M-P)}}.$$ On this subset, the pullbacks of the $1$-form $\theta^8$ is linearly dependent on the pullbacks of the remaining forms $\theta^0,\theta^1,\theta^2,\theta^3,\theta^4,\theta^5,\theta^6,\theta^{12},$ which define a coframe on $\mathcal{G}^8$. If we now compute the structure equations with respect to the coframe on $\mathcal{G}^8$ and assume that all of the structure functions are constants, we arrive at the structure equations \eqref{submaxsyst}. These are Maurer-Cartan equations for $\mathfrak{sl}(3,\mathbb{R})$ if $\epsilon> 0$ and Maurer-Cartan equations for $\mathfrak{su}(2,1)$ if $\epsilon< 0$. The analysis in Section \ref{sec_tree} (where we will start by normalizing $T^1{}_{03}=-\tfrac{{s_5}^4 L}{{\delta}^5}$ rather than $T^3{}_{04}$) will show that these are, up to local equivalence, the only marked contact Engel structures with $8$-dimensional transitive symmetry algebra, and we will refer to these structures as the \emph{submaximal} marked contact Engel structures. \end{remark}
\subsection{A tree of homogeneous models}\label{sec_tree}
The goal of this section is to find all locally non-equivalent homogeneous marked contact Engel structures with symmetry group of dimension $\geq 6$. To this end, we return to the conditions from Theorem \ref{main}, which divide marked contact Engel structures into classes of mutually non-equivalent structures.
We apply Cartan's reduction procedure to determine the maximally symmetric homogeneous structures in each of the branches determined by the conditions from Theorem \ref{main}.
We will, in the following, often abuse notation. In particular, we will denote various different subbundles $\mathcal{G}^i\subset\mathcal{G}^9$ of dimension $i$ by the same symbol. Moreover, we will frequently pullback the forms $\theta^0,\theta^1,\theta^2, \theta^3,\theta^4,\theta^5,\theta^6,\theta^8,\theta^{12}$ to these various subbundles and always reuse the same names for the pulled back forms. For different $\mathcal{G}^i$, we will be choosing subsets of these forms that constitute coframes on the subbundles $\mathcal{G}^i$. We will express the exterior derivatives ${\rm d}\theta^i$ of these coframe forms in terms of the bases of $2$-forms given by the wedge products $\theta^i\wedge\theta^j$ of the coframe forms, and refer to the equations $${\rm d}\theta^k=T^k{}_{ij}\theta^i\wedge\theta^j,$$ as the \emph{structure equations} and to the functions $T^k{}_{ij}$ as the \emph{structure functions} (with respect to the coframe).
\subsubsection{The branch $J\neq 0$}
Here we shall assume that $J\neq0$. This assumption allows us to perform a number of normalizations. We proceed as follows. First, looking at ${\rm d}\theta^1$ in Proposition \ref{propo_coframe}, we see that we can normalize the coefficient $T^1_{24}=\tfrac{3 {s_5}^5}{\delta^4} J$ to any non-zero value, and we shall normalize it to $3$. We also see that we can normalize the coefficient $T^1_{02}$ to zero. This means that we restrict to a subbundle $\mathcal{G}^7\subset\mathcal{G}^9$ given by $$s_5=\left(\tfrac{\delta^4}{J}\right)^{\frac{1}{5}},\, s_4=\tfrac{\delta^4 M - 9 c \delta^3 J s_5 s_7 - 3 \delta^3 J_{\omega^2} s_5 s_7 + 2 \delta^3 L s_5 s_7 - 9 b \delta^2 J {s_5}^2 {s_7}^2 - 9 \delta^2 J_{\omega^3} {s_5}^2 {s_7}^2 +
21 a \delta J {s_5}^3 {s_7}^3 - 9 \delta J_{\omega^4} {s_5}^3 {s_7}^3 - 27 J^2 {s_5}^4 {s_7}^4}{6 \delta J {s_5}^3}.$$
We pullback the forms $\theta^0, \theta^1, \theta^2, \theta^3, \theta^4, \theta^5, \theta^6, \theta^8, \theta^{12}$ to $\mathcal{G}^7$, where they are no longer independent, and express $\theta^8$ and $\theta^{12}$ as linear combinations with functional coefficients of the remaining forms. Now we compute the structure equations with respect to the coframe on $\mathcal{G}^7$ given by $\theta^0,\dots,\theta^6$. Looking at these structure equations shows that we can now normalize the coefficient of ${\rm d}\theta^1$ at the $\theta^1\wedge\theta^4$ term to zero, which determines a $6$-dimensional subbundle $\mathcal{G}^6\subset\mathcal{G}^7$ given by
$$s_7=\tfrac{\delta^{\frac{1}{5}}(3 a J -J_{\omega^4})}{14 J^{\frac{9}{5}}}.$$
On this subbundle, which is parametrized by the coordinates on $\mathcal{U}$ and the fibre coordinate $\delta$, the forms $\theta^0,\dots,\theta^5$ define a coframe
that satisfies structure equations of the form \begin{equation}\label{Jnot0coframe} \begin{aligned} {\rm d}\theta^0=&-6 \theta^0\wedge\theta^5+\theta^1\wedge\theta^4-3 \theta^2\wedge\theta^3\\ {\rm d}\theta^1=&\tfrac{\alpha_1}{\delta^3} \theta^0 \wedge\theta^1 +\tfrac{\alpha_2}{\delta^{\frac{12}{5}}}\theta^0\wedge\theta^2+ \tfrac{\alpha_3}{\delta^{\frac{9}{5}}}\theta^0\wedge\theta^3 +
\tfrac{\alpha_4}{\delta^{\frac{6}{5}}} \theta^0\wedge\theta^4 + \tfrac{\alpha_5}{\delta^{\frac{9}{5}}} \theta^1\wedge\theta^2 +
\tfrac{\alpha_6}{\delta^{\frac{6}{5}}} \theta^1\wedge\theta^3 \\&
- \tfrac{24}{5} \theta^1\wedge\theta^5 + 3 \theta^2\wedge\theta^4\\ {\rm d}\theta^2=&\tfrac{\alpha_7}{\delta^{\frac{18}{5}}} \theta^0 \wedge\theta^1 + \tfrac{\alpha_8}{\delta^3} \theta^0 \wedge\theta^2 +
\tfrac{\alpha_9}{\delta^{\frac{12}{5}}} \theta^0 \wedge\theta^3 + \tfrac{5\alpha_5}{6\delta^{\frac{9}{5}}} \theta^0 \wedge\theta^4 +
\tfrac{\alpha_{10}}{\delta^{\frac{12}{5}}} \theta^1 \wedge\theta^2 + \tfrac{\alpha_{11}}{\delta^{\frac{9}{5}}} \theta^1 \wedge\theta^3\\ &
-\tfrac{3\alpha_4 + 5 \alpha_6}{9\delta^{\frac{6}{5}}} \theta^1 \wedge\theta^4 + \tfrac{\alpha_6}{3\delta^{\frac{6}{5}}} \theta^2 \wedge\theta^3 - \tfrac{18}{5} \theta^2 \wedge\theta^5 + 2 \theta^3 \wedge\theta^4\\ {\rm d}\theta^3=&\tfrac{\alpha_{12}}{\delta^{\frac{21}{5}}} \theta^0 \wedge\theta^1 + \tfrac{\alpha_{13}}{\delta^{\frac{18}{5}}}\theta^0\wedge\theta^2+
\tfrac{\alpha_{14}}{\delta^3} \theta^0 \wedge\theta^3
+ \tfrac{6 \alpha_9 +75\alpha_{10}+25\alpha_2}{15 \delta^{\frac{12}{5}}} \theta^0 \wedge\theta^4 +\tfrac{2(\alpha_1-3\alpha_8)}{3\delta^3}\theta^1\wedge\theta^2 \\& -\tfrac{3 \alpha_{10}+\alpha_2}{3\delta^{\frac{12}{5}}} \theta^1 \wedge\theta^3 + \tfrac{\alpha_5 +6 \alpha_{11}}{3\delta^{\frac{9}{5}}} \theta^2 \wedge\theta^3 -
\tfrac{6 \alpha_4+10\alpha_6}{9\delta^{\frac{6}{5}}} \theta^2 \wedge\theta^4
- \tfrac{12}{5} \theta^3 \wedge\theta^5\\ {\rm d}\theta^4=&\tfrac{\alpha_{15}}{\delta^{\frac{24}{5}}} \theta^0 \wedge\theta^1 + \tfrac{\alpha_{16}}{\delta^{\frac{21}{5}}} \theta^0 \wedge\theta^2 + \tfrac{\alpha_{17}}{\delta^{\frac{18}{5}}} \theta^0 \wedge\theta^3 + \tfrac{\alpha_{18}}{\delta^3} \theta^0 \wedge\theta^4+\tfrac{\alpha_1-3\alpha_8}{\delta^3}\theta^1\wedge\theta^3 -\tfrac{3\alpha_{10}+\alpha_2}{\delta^{\frac{12}{5}}} \theta^1 \wedge\theta^4 \\& +\tfrac{\alpha_2}{\delta^{\frac{12}{5}}} \theta^2 \wedge\theta^3 +
\tfrac{\alpha_5}{\delta^{\frac{9}{5}}} \theta^2 \wedge\theta^4 - \tfrac{3 \alpha_4+2\alpha_6}{3\delta^{\frac{6}{5}}} \theta^3 \wedge\theta^4 -
\tfrac{6}{5} \theta^4 \wedge\theta^5\\
{\rm d}\theta^5=&\tfrac{\alpha_{19}}{\delta^{\frac{27}{5}}} \theta^0 \wedge\theta^1 + \tfrac{\alpha_{20}}{\delta^{\frac{24}{5}}} \theta^0 \wedge\theta^2 + \tfrac{\alpha_{21}}{\delta^{\frac{21}{5}}} \theta^0 \wedge\theta^3 + \tfrac{\alpha_{22}}{\delta^{\frac{18}{5}}} \theta^0 \wedge\theta^4 -\tfrac{3 \alpha_{12}+\alpha_{16}}{6\delta^{\frac{21}{5}}} \theta^1 \wedge\theta^2 - \tfrac{\alpha_{17}-3\alpha_7}{6 \delta^{\frac{18}{5}}} \theta^1 \wedge\theta^3\\ & -\tfrac{\alpha1+\alpha_{18}}{6\delta^3} \theta^1 \wedge\theta^4 + \tfrac{\alpha_8+\alpha_{14}}{2\delta^3} \theta^2 \wedge\theta^3 +
\tfrac{6\alpha_9+75\alpha_{10}+20\alpha_2}{30\delta^{\frac{12}{5}}} \theta^2 \wedge\theta^4 -
\tfrac{2\alpha_3 - 5 \alpha_5}{12\delta^{\frac{9}{5}}} \theta^3 \wedge\theta^4,
\end{aligned}
\end{equation}
where $\alpha_1,\dots,\alpha_{21}$ are the pullbacks of functions on $\mathcal{U}$, that is, as functions on $\mathcal{G}^6$ they do not depend on $\delta$.
Now we are looking for homogeneous structures with six dimensional symmetry algebra in this branch. For such structures all of the structure functions are constants. In particular, all of those that depend on $\delta$ have to be identically zero. On the other hand, one easily checks that this constant coefficient system
\begin{equation}\label{Jnotzerosyst} \begin{aligned} {\rm d}\theta^0=&-6 \theta^0\wedge\theta^5+\theta^1\wedge\theta^4-3 \theta^2\wedge\theta^3\\ {\rm d}\theta^1=& - \tfrac{24}{5} \theta^1\wedge\theta^5 + 3 \theta^2\wedge\theta^4\\ {\rm d}\theta^2=&- \tfrac{18}{5} \theta^2 \wedge\theta^5 + 2 \theta^3 \wedge\theta^4\\ {\rm d}\theta^3=& - \tfrac{12}{5} \theta^3 \wedge\theta^5\\ {\rm d}\theta^4=&- \tfrac{6}{5} \theta^4 \wedge\theta^5\\ {\rm d}\theta^5=&0.
\end{aligned} \end{equation}
is closed, that is, ${\rm d}^2\theta^i=0$, for all $i=0,1,2,3,4,5$. This means that there is a unique local model with $6$-dimensional symmetry algebra in this branch whose symmetry algebra has Maurer-Cartan equations \eqref{Jnotzerosyst}.
There may be homogeneous models with $5$-dimensional symmetry algebra in this branch as well.
\subsubsection{The branch $J=0$, $L\neq 0$}
For integrable structures, we have seen that $L$ defines a relative invariant. We shall assume here that it be nowhere vanishing. Similar as before, this assumption allows us to perform normalisations. We normalize the coefficient $T^1_{02}$
to zero, and the coefficient $T^1_{03}$
to $1$. On the subbundle determined by these normalizations, $\theta^6$ and $\theta^8$ are expressible in terms of the remaining forms, which constitute a coframe. Looking at ${\rm d}\theta^3$ (with the expressions for $\theta^6$ and $\theta^8$ inserted) we now see that the coefficient at the $\theta^1\wedge\theta^3$ term can be normalized to zero. Together, these normalizations determine a $6$-dimensional subbundle $\mathcal{G}^6\subset\mathcal{G}^9$ defined by
$$s_7=\tfrac{M}{2 L^{\frac{4}{5}} {s_5}^{\frac{1}{5}}},\quad \delta=-L^{\frac{1}{5}} {s_5}^{\frac{4}{5}},\quad s_4=-\tfrac{8 L^2 L_{\omega^1} +16 c L^2 M -4 L L_{\omega^2} M +8 bL M^2 +2 L_{\omega^3} M^2 +a M^3}{8 L^{\frac{12}{5}}{s_5}^{\frac{3}{5}} },$$ on which (the pullbacks of) the forms $\theta^0, \theta^1,\theta^2, \theta^3, \theta^4, \theta^5$ define a coframe.
Now, if there were homogeneous structures with $6$-dimensional symmetry algebra in this branch, then for these structures all of the structure functions of the structure equations with respect to the coframe $(\theta^0, \theta^1,\theta^2, \theta^3, \theta^4, \theta^5)$ on $\mathcal{G}^6$ must be constant. However, this assumption leads to a contradiction, and we conclude that there are no homogeneous models with $6$-dimensional symmetry algebra in this branch.
It turns out that there are structures with $5$-dimensional transitive symmetry algebra in this branch, and below we describe how to find them.
The structure equations lead us to distinguish two subclasses of structures, those for which the relative invariant $M-P$ vanishes and those for which it does not vanish.
We first consider the class of structures for which $M-P\neq 0$, which allows us to normalize
the coefficient at the $\theta^0\wedge\theta^3$ term of ${\rm d}\theta^2$. This determines a $5$-dimensional subbundle of $\mathcal{G}^6\to \mathcal{U}$, and thus a rigid coframe $\theta^0,\theta^1,\theta^2,\theta^3,\theta^4$ on $\mathcal{U}$. However, assuming that the structure equations with respect to this coframe have only constant structure functions quickly leads to a contradiction, and we conclude that there are no homogeneous structures in this branch.
We shall henceforth assume that $M-P=0$. In this case, the structure equations exhibit a new relative invariant, namely $5bL+2L_{\omega^3}$. This leads us to branch further into the subclass of structures for which $5bL+2L_{\omega^3}$ is vanishing and the subclass for which is non-vanishing.
Assuming that $5bL+2L_{\omega^3}\neq 0$ allows us to normalize, namely we normalize the coefficient of ${\rm d}\theta^1$ at the $\theta^1\wedge\theta^3$ term to $\frac{3}{8}$. This determines a $5$-dimensional subbundle $\mathcal{G}^5\subset\mathcal{G}^6,$ given by $$s_5=\left(\tfrac{(5b L+2 L_{\omega^3})^5}{L^7}\right)^{\frac{1}{3}},$$ and a rigid coframe $\theta^0,\theta^1,\theta^2,\theta^3,\theta^4$ on $\mathcal{U}$. Assuming that all of the structure functions with respect to this coframe are constant and using that ${\rm d}^2=0$, we find that there is a locally unique homogeneous model with $5$-dimensional symmetry algebra in this branch. It has Maurer-Cartan equations \begin{equation}\begin{aligned}\label{J=0Lneq0} &{\rm d}\theta^0 = -\tfrac{5}{6} \theta^0\wedge\theta^3 -24 \theta^0\wedge \theta^4 + \theta^1\wedge\theta^4 -3 \theta^2\wedge\theta^3\\ &{\rm d}\theta^1 = \theta^0\wedge\theta^3 -\tfrac{2}{3}\theta^1\wedge\theta^3 -30 \theta^1\wedge\theta^4\\ &{\rm d}\theta^2 = -\tfrac{1}{2} \theta^2\wedge\theta^3 -18 \theta^2\wedge\theta^4\\ &{\rm d}\theta^3 = -6\theta^3\wedge\theta^4 \\ &{\rm d}\theta^4=\tfrac{1}{6} \theta^3\wedge\theta^4.\\
\end{aligned}\end{equation}
Further analysis shows that there are no homogeneous models in the branch $5bL+2L_{\omega^3}=0$.
\subsubsection{The branch $J=L=0$, $M\neq0$}
Looking at \eqref{5structureeq}, we see that under the assumption $J=L=0$, the coefficient $T^1_{02}$ reads $\tfrac{{s_5}^2 M}{\delta^4}$. This shows that the sign of $M$ is an invariant, and we normalize this coefficient to $\epsilon=\mathrm{sign}(M)$.
More precisely, we restrict to a hypersurface $\mathcal{G}^8$ in $\mathcal{G}^9$ defined by $$s_5=\tfrac{\delta^2}{\sqrt{\epsilon M}}.$$ We pullback the $1$-forms $\theta^0, \theta^1, \theta^2, \theta^3,\theta^4, \theta^5, \theta^6, \theta^8, \theta^{12}$ to $\mathcal{G}^8$, and find that on this hypersurface $\theta^8$ is linearly dependent on the other $1$-forms.
Having done that, we compute ${\rm d} \theta^i$, for all $i=0,1,2,3,4,5,6,12,$ on $\mathcal{G}^8$ in terms of the basis of $2$-forms $\theta^i\wedge\theta^j$.
Inspecting the system shows that the coefficient of ${\rm d}\theta^5$ at the $\theta^2\wedge\theta^3$ term reads
$$\tfrac{\sqrt{\epsilon M} Q+6 \delta P s_7}{2\delta^3\sqrt{\epsilon M}}.$$
We now branch according to whether $P$ vanishes or not.
Assuming that $P\neq 0$, allows to normalize the above coefficient to zero. This determines a $7$-dimensional subbundle $\mathcal{G}^7$ of $\mathcal{G}^8$, given by $s_7= -\tfrac{\sqrt{\epsilon M} Q}{6 \delta P}.$ We pullback the forms $\theta^i$, $i=0,1,2,3,4,5,6,12,$ and express $\theta^6$ as a combination of the remaining forms. We compute the structure equations with respect to the coframe on $\mathcal{G}^7$, and note that we can now normalize the coefficient
of ${\rm d}\theta^4$ at the $\theta^2\wedge\theta^3$ term to zero. This determines a $6$-dimensional subbundle $\mathcal{G}^6$ of $\mathcal{G}^7$, given by
$$s_4 =\tfrac{(\epsilon M)^{\frac{3}{2}} (60 c M P Q - 30 M_{\omega^2} P Q + 180 c P^2 Q + 126 P P_{\omega^2} Q + 84 a_{\omega^0} Q^2 + 84 a a_{\omega^1} Q^2 - 21 b^3 Q^2 - 21 b M Q^2 + 81 b P Q^2 + 2 a Q^3 - 72 P^2 Q_{\omega^2})}{432 (\delta P)^3},$$
on which we express $\theta^{12}$ in terms of the remaining forms. Moreover, as a consequence of the assumption that $P\neq 0,$ we have $2 c M-M_{\omega^2}\neq 0.$ This allows to further normalize the coefficient of ${\rm d}\theta^1$ at the $\theta^1\wedge\theta^2$ term (with respect to the coframe on $\mathcal{G}^6$) to any non-zero value, and we shall normalize it to $12$. This determines a $5$-dimensional subbundle $\mathcal{G}^5\subset\mathcal{G}^6$, given by $\delta=\left(\tfrac{2 c M-M_{\omega^2}}{8\sqrt{\epsilon M}}\right)^{\frac{1}{3}}$. We have now obtained
a unique coframe on the $5$-manifold ${\cal M}$. Inspecting the structure equations of this coframe shows that there are two locally non-equivalent homogeneous models with $5$-dimensional symmetry algebras in this branch, whose Maurer-Cartan equations read
\begin{equation}\begin{aligned}\label{J=0L=0Mneq0Pneq0} &{\rm d}\theta^0 = -\tfrac{15}{2} \theta^0\wedge\theta^2 -\tfrac{1}{6}\epsilon \theta^0\wedge \theta^4 + \theta^1\wedge\theta^4 -3 \theta^2\wedge\theta^3\\ &{\rm d}\theta^1 = \epsilon \theta^0\wedge\theta^2 -3 \theta^1\wedge\theta^2 -\tfrac{1}{3}\epsilon \theta^1\wedge\theta^4\\ &{\rm d}\theta^2 = \tfrac{1}{4} \theta^0\wedge\theta^1 -\tfrac{1}{12}\epsilon \theta^0\wedge\theta^3-\tfrac{1}{2}\theta^1\wedge\theta^3-\tfrac{1}{6}\epsilon\theta^2\wedge\theta^4 \\ &{\rm d}\theta^3 = \tfrac{9}{2}\theta^0\wedge\theta^2+\tfrac{1}{6}\epsilon\theta^0\wedge\theta^4+9\epsilon\theta^1\wedge\theta^2+3\theta^2\wedge\theta^3 \\ &{\rm d}\theta^4=-\tfrac{27}{4}\epsilon \theta^0\wedge\theta^1+\tfrac{9}{4}\theta^0\wedge\theta^3+\tfrac{27}{2}\epsilon\theta^1\wedge\theta^3+\tfrac{9}{2}\theta^2\wedge\theta^4\\
\end{aligned}\end{equation}
where $\epsilon=\pm 1$. For these structures $3P-2M=0$.
Next we assumes that $P=0$. Analysing the differential consequences of this assumption, we obtain that for such structures
$${\rm d}\theta^1=-\tfrac{9 Q}{2\delta^3}\theta^0\wedge\theta^1+\epsilon\theta^0\wedge\theta^2+\tfrac{3(2 c M-M_{\omega^2})}{2\delta^3\sqrt{\epsilon M}}\theta^1\wedge\theta^2-12\theta^1\wedge\theta^5.$$
In particular, we can further branch into those structures for which $Q$ vanishes and those for which it does not vanish. The assumption $Q\neq 0$ allows to perform further normalizations, which determine a unique coframe on the $5$-dimensional manifold. Further analysis shows that there are no homogeneous models in this branch.
On the other hand, assuming that $Q=0$ and analyzing the differential consequences one obtains also that $R=S=0$. The only structures satisfying these assumptions are the submaximally symmetric structures, with structure equations
\begin{equation}\label{submaxsyst} \begin{aligned} &{\rm d}\theta^0 =-6\theta^0\wedge\theta^5+\theta^1\wedge\theta^4-3\theta^2\wedge\theta^3\\ &{\rm d}\theta^1 = \epsilon \theta^0\wedge\theta^2-12\theta^1\wedge\theta^5 \\ &{\rm d}\theta^2 =\tfrac{3}{4} \epsilon \theta^0\wedge\theta^3+\theta^1\wedge\theta^6-6\theta^2\wedge\theta^5\\ &{\rm d}\theta^3 =\tfrac{1}{2}\epsilon\theta^0\wedge\theta^4+2\theta^2\wedge\theta^6 \\ &{\rm d}\theta^4=6\theta^0\wedge\theta^{12}+3\theta^3\wedge\theta^6+6\theta^4\wedge\theta^5\\ &{\rm d}\theta^5=-\tfrac{1}{12}\epsilon\theta^0\wedge\theta^6-\theta^1\wedge\theta^{12}+\tfrac{1}{12}\epsilon\theta^2\wedge\theta^4\\ &{\rm d}\theta^6=6\theta^2\wedge\theta^{12}-\tfrac{3}{4}\epsilon\theta^3\wedge\theta^4-6\theta^5\wedge\theta^6\\ &{\rm d}\theta^{12}=\tfrac{1}{6}\epsilon\theta^4\wedge\theta^6-12\theta^5\wedge\theta^{12}.\\ \end{aligned} \end{equation} These are Maurer-Cartan equations for $\mathfrak{sl}(3,\mathbb{R})$ if $\epsilon<0$ and Maurer-Cartan equations for $\mathfrak{su}(2,1)$ if $\epsilon> 0$.
\subsubsection{The branch $J=L=M=0$, $P\neq0$}
Looking at the structure equations \eqref{5structureeq}, we see that under the assumptions $J=L=M=0$ and $P\neq0$ we can normalize the coefficient $T^3_{03}$
to zero, and then, on the subbundle determined by this reduction, express $\theta^6$ in terms of the other forms. Having done that, we compute ${\rm d}\theta^4$ and normalize the coefficient at the $\theta^2\wedge\theta^3$ term to zero, and then we normalize the coefficient at the $\theta^0\wedge\theta^3$ term in ${\rm d}\theta^2$ to $-\tfrac{5 \epsilon}{4}$, where $\epsilon=\mathrm{sign}(P)$. These normalizations determine a $6$-dimensional subbundle $\mathcal{G}^6\subset\mathcal{G}^9$
on which $\theta^6, \theta^8, \theta^{12}$ are expressible in terms of the remaining forms $\theta^0,\dots,\theta^5$, which form a coframe. Assuming that the structure equations have only constant coefficients yields a contradiction, and we conclude that there are no homogeneous models with $6$-dimensional symmetry algebra in this branch. There may be models with $5$-dimensional transitive symmetry algebra in this branch.
\subsubsection{The branch $J=0$, $L=0$, $M=0$, $P=0$, $Q\neq 0$}
Here the assumptions allow to normalize the coefficient at $\theta^0\wedge\theta^2$ of ${\rm d}\theta^3$ to zero, the coefficient at $\theta^0\wedge\theta^3$ of ${\rm d}\theta^3$ to one, and the coefficient at $\theta^0\wedge\theta^2$ of ${\rm d}\theta^6$ to zero. This determines a $6$-dimensional subbundle $\mathcal{G}^6\subset\mathcal{G}^9$ given by $$s_7=-\tfrac{R}{Q^{\frac{2}{3}}s_5},\quad s_5=-Q^{\frac{1}{3}},\quad s_4=\tfrac{2a_{\omega^1\omega^1}Q^2-4c^3Q^2+8cQ^2R+2QQ_{\omega^2}R-3bQR^2+2aR^3-2Q^2R_{\omega^2}-3bQ^2S}{2Q^2{s_5}^3}.$$ We express the pullbacks of the forms $\theta^5, \theta^6$ and $\theta^{12}$ to $\mathcal{G}^6$ in terms of $\theta^0, \theta^1, \theta^2, \theta^3, \theta^4, \theta^8$. Assuming that the structure equations have only constant coefficients then quickly implies that they are of the form
\begin{equation}\begin{aligned}\label{J=0L=0M=0P=0Qneq0} &{\rm d}\theta^0 = \theta^1\wedge\theta^4 -3 \theta^2\wedge\theta^3\\ &{\rm d}\theta^1 = \tfrac{1}{2}\theta^0\wedge\theta^1 -3 \theta^1\wedge\theta^8\\ &{\rm d}\theta^2 = \tfrac{1}{2}\theta^0\wedge\theta^2 - \theta^2\wedge\theta^8\\ &{\rm d}\theta^3 = -\tfrac{1}{2}\theta^0\wedge\theta^3 + \theta^3\wedge\theta^8\\ &{\rm d}\theta^4= -\tfrac{1}{2}\theta^0\wedge\theta^4 +3 \theta^4\wedge\theta^8\\ &{\rm d}\theta^8= -\tfrac{1}{2}\theta^1\wedge\theta^4 +\tfrac{1}{2} \theta^2\wedge\theta^3\,. \end{aligned}\end{equation}
This system is closed, and can be viewed as the Maurer-Cartan equations of $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$ with respect to a basis of left-invariant forms. In particular, there is a locally unique maximally symmetric homogeneous model in this branch with $6$-dimensional symmetry algebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$.
There may be homogeneous models with $5$-dimensional symmetry algebras in this branch as well.
\subsection{Summary}
\tikzset{
round/.style = {circle,draw},
notround/.style = {draw, rounded corners=6pt, thin,align=center,
text width=6em},
longer/.style = {draw, rounded corners=6pt, thin,align=center,
text width=8em},
shorter/.style = {draw, rounded corners=6pt, thin,align=center,
text width=3em}
}
\begin{table}[h] \label{table} \caption{The following graph shows the maximal symmetry dimension for homogeneous models in various branches of marked contact Engel structures. }
\centering \begin{tikzpicture}[
level 1/.style={sibling distance=30mm},
>=latex]
\node [circle,draw] (j){$J$}
child {node [round] (l) {$L$}
child {node [round] (m) {$M$}
child {node [round] (p) {$P$}
child {node [round] (q) {$Q$}
child {node [round] (r) {$R$}
child {node[round] (u) {$S$}
child {node [notround] (max) {maximal symmetry (Theorem \ref{main})}
}
child {node [notround] (g) {no homog. model} };
} child {node [notround] (f) {no homog. model} }; } child {node [notround] (e) {
6-dim. symmetry (\ref{J=0L=0M=0P=0Qneq0})} };
} child {node [shorter] (d) {dim. < 6} };
} child {node [round] (c) {$P$}
child[grow = down,level distance=2.3cm]{node[round] (q2){$Q$} child{node[notround](submax){submaximal symmetry (\ref{submaxsyst})} } child{node[notround](k){no homog. model} }; } child [grow = right,level distance=3.5cm] {node [longer] (h){
5-dim. symmetry (\ref{J=0L=0Mneq0Pneq0})} }
};
} child {node [notround] (b) {
5-dim. symmetry (\ref{J=0Lneq0}) } }; } child {node [longer] (a) {
6-dim. symmetry (\ref{Jnotzerosyst})} };
\path (j) edge node[anchor=south, above] {$= 0\quad\quad$} (l);
\path (j) edge node[anchor=north, above] {$\quad\quad\neq 0$} (a);
\path (l) edge node[anchor=south, above] {$= 0\quad\quad$} (m);
\path (l) edge node[anchor=north, above] {$\quad\quad\neq 0$} (b);
\path (m) edge node[anchor=south, above] {$= 0\quad\quad$} (p);
\path (m) edge node[anchor=north, above] {$\quad\quad\neq 0$} (c);
\path (p) edge node[anchor=south, above] {$= 0\quad\quad$} (q);
\path (p) edge node[anchor=north, above] {$\quad\quad\neq 0$} (d); \path (q) edge node[anchor=south, above] {$= 0\quad\quad$} (r);
\path (q) edge node[anchor=north, above] {$\quad\quad\neq 0$} (e);
\path (r) edge node[anchor=south, above] {$= 0\quad\quad$} (u);
\path (r) edge node[anchor=north, above] {$\quad\quad\neq 0$} (f);
\path (u) edge node[anchor=south, above] {$= 0\quad\quad$} (max);
\path (u) edge node[anchor=north, above] {$\quad\quad\neq 0$} (g);
\path (c) edge node[anchor=south, above] {$= 0\quad\quad$} (q2);
\path (c) edge node[anchor=north, above] {$\neq 0$} (h);
\path (q2) edge node[anchor=south, above] {$= 0\quad\quad$} (submax);
\path (q2) edge node[anchor=north, above] {$\quad\quad\neq 0$} (k); \end{tikzpicture}
\end{table}
We summarize the main results of this section in the following theorem:
\begin{theorem}\label{thmsummary}
\begin{itemize}
\
\item Up to local equivalence, there is a unique homogeneous marked contact Engel structure with $9$-dimensional infinitesimal symmetry algebra. The infinitesimal symmetry algebra is isomorphic to $\mathfrak{p}_1$. The structure is characterized by $$J=L=M=P=Q=R=S=0.$$
\item Up to local equivalence, there are precisely two homogeneous marked contact Engel structures with $8$-dimensional infinitesimal symmetry algebra. The infinitesimal symmetry algebras are isomorphic to $\mathfrak{sl}(3,\mathbb{R})$ and $\mathfrak{su}(1,2)$, respectively. The structures are characterized by $$J=L=P=Q=0\quad \mbox{and}\quad M\neq0.$$
\item There are no homogeneous marked contact Engel structure with $7$-dimensional infinitesimal symmetry algebra.
\item Up to local equivalence, there are precisely two homogeneous marked contact Engel structures with $6$-dimensional infinitesimal symmetry algebras. The respective Maurer-Cartan equations are given in \eqref{Jnotzerosyst} and \eqref{J=0L=0M=0P=0Qneq0}; the second symmetry algebra is isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$.
\item There are examples of homogeneous marked contact Engel structures with $5$-dimensional infinitesimal symmetry algebra, whose Maurer-Cartan equations are given in \eqref{J=0Lneq0} and \eqref{J=0L=0Mneq0Pneq0}. \end{itemize}
\end{theorem} There may be other, locally non-equivalent homogeneous marked contact Engel structures with $5$-dimensional symmetry algebra as well.
\section{Geometric characterizations of certain branches of marked contact Engel structures}\label{sec_123}
In this section we geometrically interpret some of the invariant conditions on marked contact Engel structures from Theorem \ref{main}.
Namely, we shall see how the first three of these conditions can be understood as properties of the filtration $$\ell^{\sigma}\subset\mathcal{D}^{\sigma}\subset \mathcal{H}^{\sigma}\subset\mathcal{C}\subset T{\cal M}$$ from Proposition \ref{propfilt} associated with a marked contact Engel structure. We have already shown that the first condition for Theorem \ref{main}, $J=0$, is equivalent to the integrability of the rank two distribution $\mathcal{D}^{\sigma}$. In Section \ref{equivalent} we show that locally only two cases can occur: either $\mathcal{D}^{\sigma}$ is indeed integrable, or $\mathcal{D}^{\sigma}$ is $(2,3,5)$ (see Definition \ref{235} below).
We further characterize the integrability of $\mathcal{D}^{\sigma}$ in terms of special properties, introduced in Section \ref{type}, of the line field $\ell^{\sigma}$.
Moreover, in Section \ref{morecond}, we show how to characterize, in the integrable case, further geometric conditions starting from $\ell^{\sigma}$.
\subsection{Various types of vector fields inside a contact distribution}\label{type}
Let ${\cal M}$ be a manifold with a distribution $\mathcal{D}\subset T{\cal M}$. Taking Lie brackets of sections of $\mathcal{D}$ defines a filtration of the tangent bundle of ${\cal M}$, called the (weak) derived flag $\mathcal{D}\subset\mathcal{D}'\subset\mathcal{D}''\subset\dots$ of $\mathcal{D}$, where $$ \mathcal{D}'_x:=[\mathcal{D},\mathcal{D}]_x=\mathrm{Span}\{\xi_x, [\xi,\eta]_x \,:\, \xi, \eta \in\Gamma(\mathcal{D}) \},\,\,\mathcal{D}''_x:=[\mathcal{D},\mathcal{D}']_x=\mathrm{Span}\{\xi_x, [\xi,\eta]_x \,:\, \xi\in\Gamma(\mathcal{D}'),\,\eta\in\Gamma(\mathcal{D}) \} $$ and so on. The sequence $(\mathrm{dim}(\mathcal{D}_x), \mathrm{dim}(\mathcal{D}'_x),\dots)$ is called the growth (vector) at a point $x\in {\cal M}$.
\begin{definition}\label{235} We say that a distribution $\mathcal{D}$ on a $5$-dimensional manifold ${\cal M}$ is $(2,3,5)$ if its growth vector is $(2,3,5)$, i.e., $\mathrm{dim}(\mathcal{D}_x)=2$, $\mathrm{dim}(\mathcal{D}'_x)=3$ and $\mathrm{dim}(\mathcal{D}''_x)=5$ at all points $x\in{\cal M}$.
\end{definition}
Let us focus on a $5$-dimensional contact manifold $({\cal M},\mathcal{C})$, where, locally, $\mathcal{C}=\ker\theta$ for a contact form $\theta$. Let $\mathcal{D}\subset\mathcal{C}$ be a $2$-dimensional Legendrian distribution. The growth of $\mathcal{D}$ is strictly related to the notion of \emph{type} (see \cite{GianniDGA,GianniAnnales} for more details) of a vector field inside $\mathcal{C}$ defined below. \begin{definition} The \emph{type} of a vector field $Y\in\Gamma(\mathcal{C})$ is the rank of the following system:
$$ \theta\,,\,\, \mathcal{L}_Y(\theta)\,,\,\, \mathcal{L}_Y^2(\theta)\,,\,\,\mathcal{L}_Y^3(\theta)\,. $$ \end{definition} Note that, due to the complete non-integrability of the contact distribution, one cannot have vector field of type $1$. Note also that the type depends neither on the choice of $\theta$ nor on the length of $Y$, i.e. it is well defined the type of a line distribution contained in the contact distribution $\mathcal{C}$. By choosing a contact form $\theta$, any $1$-form $\alpha$ on ${\cal M}$ determines a vector field $Y_\alpha$ lying in the contact distribution by the relations $$ \mathcal{L}_{Y_\alpha}(\theta) = d\theta(Y_\alpha,\,\hbox to 2.5pt{\hss$\ccdot$\hss}\,) = \alpha - \alpha(Z)\theta\,,\quad \theta(Y_\alpha) = 0\,, $$ where $Z$ is the Reeb vector field associated with $\theta$. Although $Y_\alpha$ depends on the choice of $\theta$, its direction does not. In the case $\alpha=df$ where $f\in C^\infty({\cal M})$, we simply write $Y_f$ instead of $Y_{df}$ and it will be called the \emph{Hamiltonian vector field associated with $f$}. Hamiltonian vector fields are a special kind of vector field of type 2.
We quote the following propositions, whose proofs are contained in \cite{GianniDGA}. We shall use them in Sections \ref{equivalent} and \ref{morecond} for a geometrical interpretation of some invariants of a marked contact Engel structure. \begin{proposition}[\cite{GianniDGA}]\label{prop.equiv.type} The following statements are equivalent. \begin{enumerate} \item The vector field $Y\in\mathcal{C}$ is of type $2$. \item $Y$ is a characteristic symmetry of the distribution $Y^\perp$. \item the derived distribution $(Y{^\perp})'$ has dimension 4. \end{enumerate} \end{proposition}
\begin{proposition}[\cite{GianniDGA}]\label{prop.Hamiltionian.4.dim} $Y$ is a multiple of a hamiltonian field $Y_f$ if and only if $(Y^\perp)'$ is $4$--dimensional and integrable. \end{proposition}
\subsection{Equivalent descriptions of Integrability}\label{equivalent} In this section we shall use the notions introduced in Section \ref{type} to provide equivalent descriptions of integrable marked contact Engel structures.
First, using the coordinate description \eqref{oscfilt} of the osculating filtration $\ell^{\sigma}\subset\mathcal{D}^{\sigma}\subset \mathcal{H}^{\sigma}\subset\mathcal{C}\subset T{\cal M}$ it is straightforward to verify the following Proposition. \begin{proposition}\label{prop.integrability} \ \begin{enumerate} \item We always have an inclusion $(\mathcal{D}^{\sigma})'\subset\mathcal{H}^{\sigma}.$ \item There exists a well-defined invariant map $$ \Phi_{J}:\Lambda^2\mathcal{D}^{\sigma}\to\mathcal{H}^{\sigma}/\mathcal{D}^{\sigma},\quad \xi_x\wedge\eta_x\mapsto [\xi,\eta]_x\quad\mathrm{mod}\,\mathcal{D}^{\sigma}. $$ whose vanishing is equivalent to integrability of the distribution $\mathcal{D}^{\sigma}$. \item In the parametrization \eqref{section}, integrability of $\mathcal{D}^{\sigma}$ is equivalent to \begin{equation}\label{J=0eq} J=-\xi_4(t)=(x^1+3tx^2)t_{x^0}+t^3 t_{x^1} - t^2 t_{x^2} + t t_{x^3} - t_{x^4}=0. \end{equation} \end{enumerate} \end{proposition}
\begin{proposition}\label{prop.integr.2.3.5} The distribution $\mathcal{D}^\sigma$ is either integrable or of $(2,3,5)$--type. \end{proposition} \begin{proof} Let us assume $\mathcal{D}^\sigma$ non-integrable. Then \begin{equation}\label{eq.derivato.secondo} {\mathcal{D}^\sigma}''=\langle \xi_4\,, \xi_3\,, [\xi_4,\xi_3]\,, [\xi_4,[\xi_4,\xi_3]] \,, [\xi_3,[\xi_4,\xi_3]] \rangle\,. \end{equation} The dimension of ${\mathcal{D}^\sigma}''$ is less than $5$ if and only if the determinant of the $5\times 5$ matrix formed by the components of vector fields of \eqref{eq.derivato.secondo} is zero. Such condition is $\xi_4(t)=0$, that in view of Proposition \ref{prop.integrability} implies the integrability of $\mathcal{D}^\sigma$, that contradicts our initial hypothesis. \end{proof}
Recall that we denote by $\mathcal{H}^{\sigma}=(\ell^{\sigma})^{\perp}$ the symplectic orthogonal to $\ell^{\sigma}\subset\mathcal{C}$.
\begin{proposition}\label{prop.J} The following statements are equivalent: \begin{enumerate} \item $J=0$. \item $\mathcal{D}^\sigma$ is integrable. \item $\dim\left({\mathcal{H}^{\sigma}}'\right)=4$. \item Any vector field in $\ell^{\sigma}$ is of type $2$. \item Any vector field in $\ell^{\sigma}$ is a characteristic symmetry of the distribution $\mathcal{H}^{\sigma}$. \end{enumerate} \end{proposition} \begin{proof} The equivalence between point $1$ and $2$ has already been proven in Proposition \ref{prop.integrability}.
\noindent $2$ implies $3$. In general, since $\mathcal{H}^{\sigma}=\langle \xi_4,\xi_3,\xi_2\rangle$, we have ${\mathcal{H}^{\sigma}}'=\langle \xi_4,\xi_3,\xi_2,[\xi_4,\xi_3],[\xi_4,\xi_2],[\xi_3,\xi_2] \rangle$. If $\mathcal{D}^\sigma=\langle \xi_4,\xi_3 \rangle$ is integrable, then ${\mathcal{H}^{\sigma}}'$ is spanned by $\xi_4,\xi_3,\xi_2,[\xi_4,\xi_2],[\xi_3,\xi_2]$, and a direct calculation shows that the condition that it has rank equal to $4$ is precisely $J=0$.
\noindent $3$ implies $2$. By contradiction, let us suppose that $\mathcal{D}^\sigma$ is not integrable. Then ${{\mathcal{D}^\sigma}'}^\perp=\ell^\sigma$ that implies ${{\mathcal{D}^\sigma}''}={\mathcal{H}^{\sigma}}'$, that in view of Proposition \ref{prop.integr.2.3.5} is $5$--dimensional, a contradiction.
\noindent $3$, $4$ and $5$ are equivalent because of Proposition \ref{prop.equiv.type}. \end{proof}
\begin{remark} Proposition \ref{prop.J} shows that for integrable marked contact Engel structures, the filtration $$\mathcal{D}^{\sigma}\subset\mathcal{H}^{\sigma}\subset{\mathcal{H}^{\sigma}}'\subset T{\cal M}$$ is preserved under the Lie derivative of any vector field contained in $\ell^{\sigma}$. In particular, it descends to a filtration on the local leaf space of the foliation determined by $\ell^{\sigma}$. \end{remark}
\subsection{Two more conditions on integrable marked contact Engel structures}\label{morecond}
Suppose that $J=0$. Then, by Proposition \ref{prop.J}, any vector field in $\ell^{\sigma}$ is a characteristic symmetry of the distribution $\mathcal{H}^{\sigma}$ and consequently also of ${\mathcal{H}^{\sigma}}'$. It follows that, if $J$ vanishes, the Lie bracket of vector fields induces a well defined map $$\Phi_L:\mathcal{D}^{\sigma}/\ell^{\sigma}\otimes( {\mathcal{H}^{\sigma}}'/\mathcal{H}^{\sigma})\to T{\cal M}/{\mathcal{H}^{\sigma}}'.$$ With respect to the frame \eqref{frame1}, the map is determined by a single function. Vanishing of $\Phi_L$ is equivalent to $L=0$.
\begin{proposition}\label{prop.J.L} Suppose that $J=0$. The following statements are equivalent: \begin{enumerate} \item $L=0$. \item Any vector field contained in the distribution $\mathcal{D}^\sigma$ is an internal symmetry of ${\mathcal{H}^{\sigma}}'$. \end{enumerate} \end{proposition}
\begin{proof} Since $\mathcal{D}^\sigma$ is integrable, in view of Proposition \ref{prop.J} the distribution ${\mathcal{H}^{\sigma}}'$ is $4$--dimensional. It is spanned by vectors $\xi_4$, $\xi_3$, $\xi_2$ (that are inside $\mathcal{H}^{\sigma}$) and by an extra vector $$ [\xi_3,\xi_2]=-3\partial_{x^0} -3(3x^2t_{x^0} -3t^2t_{x^1} +t_{x^3})\partial_{x^1} -2(3tt_{x^1} - t_{x^2})\partial_{x^2} $$ In view of the integrability of $\mathcal{D}^\sigma$ and in view of the fact that $\xi_4$ is a characteristic symmetry of ${\mathcal{H}^{\sigma}}$ (and then also of ${\mathcal{H}^{\sigma}}'$), see Proposition \ref{prop.equiv.type}, we have that the vector fields in $\mathcal{D}^\sigma$ are symmetries of ${\mathcal{H}^{\sigma}}'$ if and only if $[\xi_3,[\xi_3,\xi_2]]\in {\mathcal{H}^{\sigma}}'$. This is equivalent to $L=0$.
\end{proof}
By Proposition \ref{prop.J.L}, if $J=L=0$, then the Lie bracket of vector fields induces a well-defined map $$\Phi_M: \Lambda^2{\mathcal{H}^{\sigma}}'/\mathcal{D}^{\sigma}\to T{\cal M}/{\mathcal{H}^{\sigma}}'.$$ With respect to the frame \eqref{frame1}, it corresponds to a single function. Vanishing of $\Phi_M$ means precisely that $M=0$.
\begin{proposition}\label{prop.J.L.M} Suppose that $J=L=0$. The following statements are equivalent: \begin{enumerate} \item $M=0$. \item The distribution ${\mathcal{H}^{\sigma}}'$ is $4$-dimensional and integrable.
\item The direction $\ell^\sigma$ is a Hamiltonian direction. \end{enumerate} \end{proposition} \begin{proof} $1$ is equivalent to $2$. In fact, in view of the reasonings contained in the proof of Proposition \ref{prop.J.L}, under our assumption ${\mathcal{H}^{\sigma}}'$ is integrable if and only if $[\xi_2,[\xi_3,\xi_2]]\in {\mathcal{H}^{\sigma}}'$. Recalling that ${\mathcal{H}^{\sigma}}'=\langle \xi_4,\xi_3,\xi_2,[\xi_3,\xi_2] \rangle$ (see Proposition \ref{prop.J.L}), it is straightforward to realize that $[\xi_2,[\xi_3,\xi_2]]\in \langle \xi_4,\xi_3,\xi_2,[\xi_3,\xi_2] \rangle$ if and only if $M=0$.
\noindent $2$ is equivalent to $3$. It follows from Proposition \ref{prop.Hamiltionian.4.dim}. \end{proof}
\section{A Kerr theorem for contact Engel structures} \label{SecKerr}
In Section \ref{secKerr1} we show how to construct a general integrable marked contact Engel structure.
We state this result in Theorem \ref{Kerr1} in analogy to Penrose's formulation of Kerr's theorem from relativity. In Section \ref{secKerr2} we give a twistorial interpretation of the result. We show that integrable marked contact Engel structures are in local 1-1 correspondence with generic hypersurfaces in the twistor space $\mathrm{G}_2/\mathrm{P}_1$, see Corollary \ref{Kerr2}. Via this correspondence, highly symmetric integrable marked contact Engel structures correspond to highly symmetric hypersurfaces of $\mathrm{G}_2/\mathrm{P}_1$. We use this correspondence to give a description of the maximal and submaximal models, having symmetry algebras $\mathfrak{p}_1$, $\mathfrak{sl}(3,\mathbb{R})$ and $\mathfrak{su}(1,2)$, respectively, in Section \ref{maxsubmax}. Moreover, we investigate the geometric structures hypersurfaces in $\mathrm{G}_2/\mathrm{P}_1$ inherit from the geometry of the ambient space.
\subsection{Local description of integrable marked contact Engel structures: the Kerr theorem} \label{secKerr1} In this section we show how to find the general solution to the
non-linear PDE \begin{equation}\label{eq_Jvanishes} J=(x^1+3tx^2)t_{x^0}+t^3 t_{x^1} - t^2 t_{x^2} + t t_{x^3} -t_{x^4}=0. \end{equation}
This is analogous to a result from relativity attributed to Kerr, see e.g.
\cite{penroserindler2, Tafel}. We thus refer to it as a Kerr theorem for Engel structures\footnote{We state our theorem in parallel to Penrose's formulation of the original Kerr theorem, as in \cite[Theorem 7.4.8]{penroserindler2}.}.
\begin{theorem}[Kerr theorem for contact Engel structures]\label{Kerr1} The general smooth solution to the equation \eqref{eq_Jvanishes} is obtainable locally by choosing an arbitrary smooth function $F$ of five variables and solving the equation $$F(x^0+x^1x^4+3tx^2x^4-t^3(x^4)^2, x^1+t^3x^4, x^2-t^2x^4, x^3+t x^4, t)=0$$ for $t$ in terms of $x^0, x^1, x^2, x^3, x^4$. \end{theorem}
\begin{proof} We introduce the following variables \begin{equation}\label{new_var}\begin{aligned} y^0=x^0+x^1x^4+3t x^2x^4-t^3(x^4)^2, \quad y^1=x^1+t^3x^4,\quad y^2=x^2-t^2x^4,\quad y^3=x^3+tx^4. \end{aligned} \end{equation} As in the proof of Proposition \ref{prop.integrability} one sees that ${\rm d}\omega^0\wedge\omega^0\wedge\omega^1\wedge\omega^2=0,$ $ {\rm d}\omega^1\wedge\omega^0\wedge\omega^1\wedge\omega^2=0$ and in the new variables we have \begin{equation*} {\rm d}\omega^2\wedge\omega^0\wedge\omega^1\wedge\omega^2= -2 \; {\rm d} t\wedge{\rm d} y^0\wedge{\rm d} y^1\wedge{\rm d} y^2\wedge{\rm d} y^3. \end{equation*} The latter expression vanishes if and only if there exists a smooth function $F$ of five variables such that $F(t, y^0,y^1,y^2,y^3)=0$.
On the other hand, the proof of Proposition \ref{prop.integrability} shows that vanishing of ${\rm d}\omega^2\wedge\omega^0\wedge\omega^1\wedge\omega^2$
is equivalent to $J=0$.
\end{proof}
\begin{example} To give an example how Theorem \ref{Kerr1} works, we consider $F(t,y^0,y^1,y^2,y^3)=t-\frac{s y^3 -y^1}{y^2}$, where $s$ is an arbitrary constant. Then we find $t$ as a function of $x^0,x^1,x^2,x^3,x^4$ from $$t=\frac{s y^3 -y^1}{y^2}=\frac{sx^3+tsx^4-x^1-t^3x^4}{x^2-t^2x^4}.$$ This gives $$t=\frac{x^1-sx^3}{-x^2+sx^4},$$ and one can check by a direct calculation that it satisfies \eqref{eq_Jvanishes}. \end{example}
\begin{remark}
An operational answer to how the variables \eqref{new_var} were obtained is that we were rewriting the co-frame forms from \eqref{coframet} as \begin{align*} \omega^4&={\rm d} x^4\\ \omega^3&
={\rm d} (x^3+t x^4)-x^4{\rm d} t\\
\omega^2&
={\rm d} (x^2- t^2 x^4)+2 t \omega^3+2 t x^4 {\rm d} t\\
\omega^1&
={\rm d} (x^1+t^3 x^4)-3t^2x^4{\rm d} t+3 t \omega^2-3 t^2 \omega^3\\
\omega^0&
={\rm d} (x^0+3t x^2x^4+x^1x^4-t^3{x^4}^2)-x^4\omega^1 -3 (x^2-t^2x^4)\omega^3-3x^4(x^2-t^2x^4){\rm d} t.
\end{align*}
\end{remark}
\subsection{Local coordinates adapted to the $\mathrm{G}_2$ double fibration}\label{double_coord} In analogy with the classical Kerr Theorem, we also have a geometrical interpretation of Theorem \ref{Kerr1} in terms of a twistorial correspondence, which is given in Corollary \ref{Kerr2} in the next section.
Our proof of this correspondence uses local coordinates adapted to the double filtration for $\mathrm{G}_2$ depicted below.
\begin{center}
\begin{tikzpicture}[sharp corners=2pt,inner sep=7pt,node distance=.7cm,every text node part/.style={align=center}]
\node[draw, minimum height = 2.5cm, minimum width = 3cm] (state0){$\mathrm{G}_2/\mathrm{P}_{1,2}$\\\\$(x^0,x^1,x^2,x^3,x^4, x^5)$ \\\\$(y^0,y^1,y^2,y^3, y^4, y^5)$}; \node[below=2cm of state0, minimum height = 2.5cm, minimum width = 3cm](state2){}; \node[draw,left=2cm of state2, minimum height = 2cm, minimum width = 3cm](state1){$\mathrm{G}_2/\mathrm{P}_2$\\\\$(x^0,x^1,x^2,x^3,x^4, x^5)$}; \node[draw,right=2cm of state2, minimum height = 2cm, minimum width = 3cm](state3){$\mathrm{G}_2/\mathrm{P}_1$\\\\$(y^0,y^1,y^2,y^3, y^4, y^5)$};
\draw[-triangle 60] (state0) -- (state1) node [midway, above, rotate = 45]{$\pi_2$}; \draw[-triangle 60] (state0) -- (state1) node [midway, below, rotate = 45]{$\partial_{x^5}$};
\draw[-triangle 60] (state0) -- (state3) node [midway, above, rotate = -45]{$\pi_1$}; \draw[-triangle 60] (state0) -- (state3) node [midway, below, rotate = -45]{$\partial_{y^5}$}; \end{tikzpicture} \end{center}
Let $(\theta^0,\theta^1,\dots, \theta^{13})$ be the coframe of left-invariant forms on $\mathrm{G}_2$ corresponding to a basis of $\mathfrak{g}$ as in \eqref{basis_g2}. This coframe is adapted to the grading of the Lie algebra $\mathfrak{g}$
in such a way that each leaf of the integrable distribution of the kernel of the eight left-invariant forms $\theta^5, \theta^6, \theta^8, \theta^9, \theta^{10}, \theta^{11}, \theta^{12}, \theta^{13}$ on $\mathrm{G}_2$ from \eqref{MaurerCartan}
corresponds to a section of $\mathrm{G}_2\to\mathrm{G}_2/\mathrm{P}_{1,2}$. The pullbacks $\omega^0,\omega^1,\omega^2,\omega^3,\omega^4,\omega^7$ of the forms $\theta^0,\theta^1,\theta^2,\theta^3,\theta^4, \theta^7$ to a leaf satisfy
$${\rm d}\omega^0=\omega^1\wedge\omega^4-3\omega^2\wedge\omega^3, \quad {\rm d}\omega^1=3\omega^2\wedge\omega^7, \quad{\rm d}\omega^2=2\omega^3\wedge\omega^7, \quad {\rm d}\omega^3=\omega^4\wedge\omega^7, \quad {\rm d}\omega^4=0,\quad {\rm d}\omega^7=0.$$
We integrate this system in two ways. One yields local coordinates $(x^0,x^1,x^2,x^3,x^4, x^5)$ on $\mathrm{G}_2/\mathrm{P}_{1,2}$ such that
\begin{equation}\label{omegas} \begin{aligned} & \omega^0={\rm d} x^0+x^1{\rm d} x^4- 3 x^2{\rm d} x^3\\
&\omega^1={\rm d} x^1+3 x^5{\rm d} x^2+ 3 (x^5)^2 {\rm d} x^3+(x^5)^3{\rm d} x^4\\
& \omega^2={\rm d} x^2 +2 {x^5} {\rm d} x^3 +(x^5)^2 {\rm d} x^4\\
& \omega^3={\rm d} x^3+ x^5 {\rm d} x^4\\
& \omega^4={\rm d} x^4,\\
&\omega^7=-{\rm d} x^5,
\end{aligned}
\end{equation}
Denoting by $\xi_0,\xi_1,\xi_2,\xi_3,\xi_4,\xi_7$ the dual frame, the vertical bundle for $\pi_1$ is spanned by $$\xi_4=-(x^1+3x^5x^2)\partial_{x^0}-(x^5)^3\partial_{x^1}+(x^5)^2\partial_{x^2}-(x^5)\partial_{x^3}+\partial_{x^4},$$ the vertical bundle for $\pi_2$ is spanned by $$\xi_7=-\partial_{x^5}.$$ We can view $(x^0,x^1,x^2,x^3,x^4)$ as local coordinates on $\mathrm{G}_2/\mathrm{P}_2$, then $$\pi_2:(x^0,x^1,x^2,x^3,x^4,x^5)\mapsto (x^0,x^1,x^2,x^3,x^4),$$ i.e., $x^5$ is the fibre coordinate for $\pi_2$.
The other way of integrating yields local coordinates $(y^0,y^1,y^2,y^3,y^4,y^5)$ on $\mathrm{G}_2/\mathrm{P}_{1,2}$ such that \begin{equation}\label{newomegas} \begin{aligned} & \omega^0={\rm d} y^0-y^5{\rm d} y^1- 3 y^4 y^5{\rm d} y^2-3(y^2+y^5(y^4)^2){\rm d} y^3\\
&\omega^1={\rm d} y^1+3 y^4{\rm d} y^2+ 3 (y^4)^2 {\rm d} y^3\\
& \omega^2={\rm d} y^2 +2 {y^4} {\rm d} y^3 \\
& \omega^3={\rm d} y^3- y^5 {\rm d} y^4\\
& \omega^4={\rm d} y^5,\\
&\omega^7=-{\rm d} y^4.
\end{aligned}
\end{equation}
In these coordinates the field $\xi_4$ spanning the vertical bundle for $\pi_1$, is rectified, i.e., we have $$\xi_4=\partial_{y^5},$$ and $$\xi_7=-3 y^5 y^2 \partial_{y^0}-3 (y^4)^2 y^5 \partial_{y^1}+2 y^4 y^5 \partial_{y^2}-y^5 \partial_{y^3}-\partial_{y^4}\, .$$
We can view $(y^0,y^1,y^2,y^3,y^4)$ as coordinates on $\mathrm{G}_2/\mathrm{P}_1$. Then
$$\pi_1: (y^0,y^1,y^2,y^3,y^4,y^5)\mapsto (y^0,y^1,y^2,y^3,y^4),$$
i.e., $y^5$ is the fibre coordinate for $\pi_1$.
A change of coordinates from $(x^0,x^1,x^2,x^3,x^4, x^5)$ to $(y^0,y^1,y^2,y^3, y^4, y^5)$ is given by
\begin{equation}\label{coordchange} \begin{aligned} &y^0=x^0+x^1x^4+3x^5 x^2x^4-(x^5)^3(x^4)^2, \quad y^1=x^1+(x^5)^3x^4,\\ & y^2=x^2-(x^5)^2x^4, \quad y^3=x^3+x^5x^4,\quad y^4=x^5, \quad y^5=x^4. \end{aligned} \end{equation} Similar coordinate transformations can be found e.g. in \cite{machida, ishi}.
\subsection{Geometrical interpretation of the Kerr theorem for contact Engel structures}\label{secKerr2}
Having set up the coordinate systems, the geometrical interpretation of Theorem \ref{Kerr1}, given in Corollary \ref{Kerr2}, is now almost immediate.
\begin{corollary}\label{Kerr2} Consider the double fibration \begin{align}\label{doubfibcor}
\xymatrix{
&\mathrm{G}_2/\mathrm{P}_{1,2} \ar[dl]_{\pi_2}^{\xi_4} \ar[dr]^{\pi_1}_{\xi_7} & \\
\mathrm{G}_2/\mathrm{P}_2 & & \mathrm{G}_2/\mathrm{P}_1 .} \end{align} There is a local bijective correspondence between integrable sections of $\pi_2$ and hypersurfaces $\Sigma\subset\mathrm{G}_2/\mathrm{P}_1$ which are generic in the sense that their preimages ${\pi_1}^{-1}(\Sigma)$ intersect the fibres ${\pi_2}^{-1}(x)$
transversally.
\end{corollary}
\begin{proof}
Any local section $\sigma:\mathcal{U}\to\mathrm{G}_2/\mathrm{P}_{1,2}$, with\, $\mathcal{U}\subset \mathrm{G}_2/\mathrm{P}_2,$
defines a hypersurface in $\mathrm{G}_2/\mathrm{P}_{1,2}$ locally given in terms of coordinates $(x^0,x^1,x^2,x^3,x^4,x^5)$ by its graph $$x^5=t(x^0,x^1,x^2,x^3,x^4).$$ By Proposition \ref{prop.integrability}, the integrability condition reads
$$0=-(x^1+3tx^2)t_{x^0}-t^3 t_{x^1}+t^2 t_{x^2}-t t_{x^3}+ t_{x^4}=\xi_4(t)\vert_{\sigma (\mathcal{U})}.$$
Since $\xi_4$ spans the vertical bundle of $\pi_1$, this means that $\sigma(\mathcal{U})$ is tangential to the fibres of $\pi_1$, which implies that $\sigma$ defines a hypersurface in $\mathrm{G}_2/\mathrm{P}_1$.
Conversely, let $\Sigma$ be a hypersurface in $\mathrm{G}_2/\mathrm{P}_1$ such that $\pi_1^{-1}(\Sigma)$ is transversal to the fibres of $\pi_2$.
Because of this genericity assumption on $\Sigma$, we may apply the implicit function theorem and write $\pi_1^{-1}(\Sigma)$, locally, as the graph of a section $x^5=t(x^0,x^1,x^2,x^3,x^4)$. By construction $\xi_4\hbox to 2.5pt{\hss$\ccdot$\hss} t\vert_{\sigma (\mathcal{U})}=0$, that is, the section is integrable. \end{proof}
We conclude this section with a number of remarks, each of which deserves further investigations. Recall that a marked contact Engel structure can be viewed as a (local) foliation of $\mathrm{G}_2/\mathrm{P}_2$ by unparametrized curves whose tangent directions are contained in $\gamma\subset\mathbb{P}(\mathcal{C})$. We called such a foliation a $\gamma$-congruence in Proposition \ref{prop_punctured}. Note that $\Sigma$ appearing in the Corollary \ref{Kerr2} can be locally identified with its leaf space.
\begin{remark}{(\textbf{On geodesics for Weyl connections})} \
For contact twisted cubic structures, there exists a class of distinguished connections on the tangent bundle preserving the geometric structure, which are known as \emph{Weyl connections}.
A choice of contact form uniquely determines a connection from the class of Weyl connections. It is an algebraic computation to determine how a Weyl connection transforms under a change of contact form, see \cite[Proposition 5.1.6]{book}. In particular, using the transformation formula, it is straightforward to verify that if an unparametrised curve whose tangent directions are contained in $\gamma\subset\mathbb{P}(\mathcal{C})$ is a geodesic for one Weyl connection, i.e., $\nabla_{c'}c'\propto c',$ then it is a geodesic for any other Weyl connection as well. We shall call these curves $\gamma$-geodesics.
In the case of the flat model, i.e., the contact Engel structure, the $\gamma$-geodesics are then just curves of the form $g\,\mathrm{exp}(tX)\hbox to 2.5pt{\hss$\ccdot$\hss} o\subset\mathrm{G}_2/\mathrm{P}_2$ with $X$ an element in the highest weight orbit of $G_0$ on $\mathfrak{g}_{-1}$.
Returning to the coordinate representation \eqref{section} of marked contact Engel structures, here the Weyl connection $\nabla$ determined by the contact form $\alpha^0$ is such that it preserves the coframe $(\alpha^0,\alpha^1,\alpha^2,\alpha^3,\alpha^4)$ in all horizontal directions, i.e., $\nabla_{X}\alpha^i=0$ for all $X\in\Gamma(\mathcal{C})$.
In terms of this Weyl connection, $\nabla_{\xi_4}\xi_4=-t_{\omega^4}\xi_3=J\xi_3,$ where $\ell^{\sigma}=\mathrm{Span}(\xi_4)$. Hence, the condition that a $\gamma$-congruence consists entirely of $\gamma$-geodesics is precisely the integrability condition $J=0$. (Note that this means that the relative invariant $J$ is an obstruction against the existence of a Weyl connection that preserves the marked contact Engel structure.)
\end{remark} There are further viewpoints on the $\mathrm{G}_2$-correspondence discussed here and results that should be useful in this context, we refer e.g. to \cite{BryantCartan, BryantHsu, DoubZel, machida, ishi, Ishiusw2}.
Our next remarks concern the geometric structures that a hypersurface $\Sigma\subset\mathrm{G}_2/\mathrm{P}_1$ inherits from the ambient geometry on $\mathrm{G}_2/\mathrm{P}_1$. The $\mathrm{G}_2$-homogeneous space $\mathrm{G}_2/\mathrm{P}_1$ is equipped with a $\mathrm{G}_2$-invariant $(2,3,5)$ distribution $\mathcal{D}^{(2,3,5)}$ (see Definition \ref{235}), first discovered by Cartan and Engel \cite{cartan, engel}. Taking the pullback of the $1$-forms $\omega^0, \omega^1, \omega^2, \omega^3, \omega^7$ on $\mathrm{G}_2/\mathrm{P}_{1,2}$ as in \eqref{newomegas} by any section of $\pi_1:\mathrm{G}_2/\mathrm{P}_{1,2}\to\mathrm{G}_2/\mathrm{P}_1$ defines a co-frame on $\mathrm{G}_2/\mathrm{P}_1$. This coframe is adapted to the $\mathrm{G}_2$-invariant $(2,3,5)$-distribution $\mathcal{D}^{(2,3,5)}$ in the sense that
$$\mathcal{D}^{(2,3,5)}=\mathrm{ker}(\omega^0,\omega^1,\omega^2),$$ with derived rank $3$ distribution $$(\mathcal{D}^{(2,3,5)})'=[\mathcal{D}^{(2,3,5)},\mathcal{D}^{(2,3,5)}]=\mathrm{ker}(\omega^0,\omega^1).$$
\begin{remark}{\textbf{(On 3rd order ODEs 1)}}\ Consider the section of $\,\pi_2:\mathrm{G}_2/\mathrm{P}_{1,2}\to\mathrm{G}_2/\mathrm{P}_1$ corresponding to $y^5=0$, rename the coordinates as usual jet coordinates as follows $$y^0=y,\, y^1=z,\, y^2=y',\, 3y^3=x,\, -\tfrac{2}{3}y^4=y'',$$ and change the co-frame by an admissible transformation
(in other words, we are putting it into Goursat normal form): \begin{equation*} \begin{aligned} &\hat{ \omega}^0=\omega^0={\rm d} y-y'{\rm d} x\\
&\hat{\omega}^1=\omega^1-3y^4\omega^2={\rm d} z -\tfrac{9}{4} (y'')^2{\rm d} x \\
& \hat{\omega}^2 ={\rm d} y'-y'' {\rm d} x\\
& \hat{ \omega}^3= 3\omega^3={\rm d} x\\
&\hat{\omega}^7=\tfrac{3}{2}\omega^7={\rm d} y''.
\end{aligned}
\end{equation*} This shows that integral curves $c(x)=(x,y(x),y'(x),y''(x),z(x))$ of the distribution $\mathrm{ker}(\omega^0,\omega^1,\omega^2)$ are solutions to the Hilbert-Cartan equations $z'=\frac{9}{4}{y''}^2$.
Now consider a hypersurface $\Sigma\subset \mathrm{G}_2/\mathrm{P}_1$ given as as $H(x,y,y',y'',z)=0$. Differentiating and inserting the Hilbert-Cartan equation, we get an explicit third order ODE on $y=y(x)$, $$y'''=-\frac{1}{H_{y''}}(\tfrac{9}{4}{y''}^2+H_x-H_yy'-H_{y'}y'').$$ \end{remark}
\begin{remark}{\textbf{(On 3rd order ODEs 2)}} \ Here we take another viewpoint.
Recall that a distribution with growth vector $(2,3,4)$ is called an \emph{Engel distribution} (see e.g. \cite{BryantHsu, BryGovEasNeu}). It is well known that the derived rank $3$ distribution of an Engel distribution admits a unique line field spanned by a characteristic symmetry contained in the Engel distribution. We refer to it as the \emph{characteristic line field}. More precisely, there exist local coordinates $(x,y,y',y'')$ such that the Engel distribution is generated by $$\tfrac{\mathrm{d}\,}{\mathrm{d} x}=\partial_x+y'\partial_y+y''\partial_{y'}, \quad \partial_{y''},$$ where $\partial_{y''}$ spans the characteristic line field. Any line field transversal to $\partial_{y''}$ is generated by $\mathrm{D}=\tfrac{\mathrm{d}\,}{\mathrm{d} x}+F\,\partial_{y''}$, for some smooth function $F$, to which is associated the third order ODE $$y'''=F(x,y,y',y'').$$ The geometry consisting of an Engel disribution together with a transversal line field is itself a parabolic geometry, modeled on $\mathrm{Sp}(4,\mathbb{R})/P,$ where $P$ is the Borel subgroup \cite{thirdorder, BryGovEasNeu}.
Now let $\Sigma$ be a hypersurface in $\mathrm{G}_2/\mathrm{P}_1$. One verifies that in terms of the geometry on $\mathrm{G}_2/\mathrm{P}_1$, the genericity condition of Corollary \ref{Kerr2}, namely, that $\pi_1^{-1}(\Sigma)$ be transversal to fibres of $\pi_2$,
can be rephrased as the condition that at each point $p\in\Sigma$ the tangent space of $\Sigma$ and the $(2,3,5)$-distribution $\mathcal{D}^{(2,3,5)}$ intersect in a line. In particular, this yields a line distribution $\mathcal{L}^{\Sigma}\subset T\Sigma$ on $\Sigma$ (and $\Sigma$ is thus foliated by integral curves). Likewise, the rank three distribution $(\mathcal{D}^{(2,3,5)})'$ on $\mathrm{G}_1/\mathrm{P}_1$ gives rise to a rank two distribution $\mathcal{H}^{\Sigma}\subset T \Sigma$.
It turns out that distribution $\mathcal{H}^{\Sigma}$ is maximally non-integrable, i.e., it is an Engel distribution, if and only if an additional genericity condition on the hypersurface $\Sigma$ is satisfied. Computing shows that this condition is equivalent to $L\neq 0$ as in Theorem \ref{main}. Suppose that $L\neq 0$ and
let $\mathcal{K}^{\Sigma}\subset\mathcal{H}^{\Sigma}$ be the characteristic line field of the Engel distribution $\mathcal{H}^{\Sigma}$.
Then one further verifies that the fields $\mathcal{K}^{\Sigma}$ and $\mathcal{L}^{\Sigma}$ are linearly independent, and thus one has a direct sum decomposition $\mathcal{H}^{\Sigma}=\mathcal{K}^{\Sigma}\oplus\mathcal{L}^{\Sigma}$. By the above discussion, this equips $\Sigma$ with the structure of a third order ODE (considered modulo contact transformations), or equivalently, a parabolic geometry modeled on $\mathrm{Sp}(4,\mathbb{R})/P$.
\end{remark}
\begin{remark}{\textbf{(On the induced conformal structures)}}\ For our final remark, we recall that $\mathrm{G}_2/\mathrm{P}_1$ carries a $\mathrm{G}_2$-invariant conformal class of metrics $[g]$ of signature $(2,3)$, with respect to which $\mathcal{D}^{(2,3,5)}$ is totally null, see \cite{nurowskiconf}. When $\mathrm{G}_2/\mathrm{P}_1$ is identified with the projectivized null cone $\mathbb{P}(\mathcal{N})=\{[X]\in\mathbb{R}^{3,4}: h(X,X)=0\}$, then this conformal structure is induced from the $\mathrm{G}_2$-invariant metric $h$ on $\mathbb{R}^{3,4}$.
One can pullback the $\mathrm{G}_2$-invariant conformal class $[g]$ to the hypersurface $\Sigma\subset \mathrm{G}_2/\mathrm{P}_1$,
which yields an induced \emph{non-degenerate} conformal structure on $\Sigma$ if and only if the relative invariant $M-P$ as in Proposition \ref{prop_seconddi} is non-vanishing.
\end{remark}
\subsection{Maximal and submaximal models for marked contact Engel structures revisited}\label{maxsubmax}
We shall use the correspondence between integrable marked contact Engel structures and hypersurfaces in the twistor space to describe the maximal and submaximal models derived in Section \ref{CartanEquiv}.
Let $\Phi\in\Lambda^3(\mathbb{R}^{3,4})^*$ be the defining three form of the group $\mathrm{G}_2$ and let $h\in\bigodot^2(\mathbb{R}^{3,4})^*$ be the $\mathrm{G}_2$-invariant bilinear form of signature $(3,4)$. Then homogeneous spaces occurring in the double fibration \eqref{doublefib} admit the following descriptions (see e.g. \cite{BryantCartan, machida, SplitOct}):
\begin{itemize}
\item $\mathrm{G}_2/\mathrm{P}_1$ can be identified with the projectivized null cone $\mathbb{P}(\mathcal{N})$ of all $1$-dimensional subspaces $\mathbb{L}\subset\mathbb{R}^{3,4}$ that are null with respect to $h$,
\item $\mathrm{G}_2/\mathrm{P}_2$ can be identified with the set of $2$-dimensional totally null subspaces $\Pi\subset\mathbb{R}^{3,4}$ that insert trivially into the defining $3$-form $\Phi$,
\item $\mathrm{G}_2/\mathrm{P}_{1,2}$ can be identified with the correspondence space of all pairs $(\mathbb{L}, \Pi)\in\mathrm{G}_2/\mathrm{P}_1\times \mathrm{G}_2/\mathrm{P}_2$, where $\mathbb{L}\subset \Pi$.
\end{itemize}
A fibre ${\pi_2}^{-1}(\Pi)$ can be identified with the set of all $1$-dimensional subspaces contained in $\Pi$ and is thus isomorphic to $\mathbb{R}\mathbb{P}^1$. A fibre ${\pi_1}^{-1}(\mathbb{L})$ can be identified with the set of all totally null $2$-dimensional subspaces $\Pi$ that insert trivially into $\Phi$ and contain $\mathbb{L}$; this is the set of $2$-dimensional subspaces of the $3$-dimensional null subspace \begin{equation*} \mathrm{Ann}_{\Phi}(\mathbb{L})=\{ X\in \mathbb{R}^{3,4}\mid \Phi (\mathbb{L}, X, \hbox to 2.5pt{\hss$\ccdot$\hss})=0\}\subset \mathbb{R}^{3,4} , \end{equation*} and hence also isomorphic to $\mathbb{R}\mathbb{P}^1$.
Viewing $\mathrm{G}_2/\mathrm{P}_1=\mathbb{P}(\mathcal{N})$ as a projectivized null cone, the simplest kinds of hypersurfaces in $\mathrm{G}_2/\mathrm{P}_1$ are obtained by intersecting the null cone with a $6$-dimensional subspace $\mathbb{W}\subset\mathbb{R}^{3,4}$ and projectivizing. Such hyperplanes $\mathbb{W}=\mathbb{L}^{\perp}$ split into three classes according to whether its annihilator $\mathbb{L}$ is a lightlike, timelike or spacelike line.
It is further known that the group $\mathrm{G}_2$ acts transitively on the set of, respectively, lightlike, timelike, spacelike lines $\mathbb{L}\subset \mathbb{R}^{3,4}$ and that \begin{itemize} \item $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})=\mathrm{P}_1$ iff $\left\langle\mathbb{L},\mathbb{L}\right\rangle=0$, \item $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})=\mathrm{SU}(1,2)$ iff $\left\langle\mathbb{L},\mathbb{L}\right\rangle >0$, \item $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})=\mathrm{SL}(3,\mathbb{R})$ iff $\left\langle\mathbb{L},\mathbb{L}\right\rangle<0$. \end{itemize} Each of these groups has a unique open orbit in $\mathbb{P}(\mathcal{N}),$ which is contained in the space $\mathbb{P}(\mathcal{N}\cap\mathbb{L}^\perp)$,
see e.g. \cite{meTravis}.
According to Theorem \ref{Kerr2}, there are corresponding marked contact Engel structures, which we can easily describe explicitly:
\begin{proposition} \label{submaxmod} The subset \begin{equation}\label{eqSubMaxMod} {\cal M}_{\mathbb{L}}:=\{ \Pi\in \mathrm{G}_2/\mathrm{P}_2\mid \dim(\Pi\cap\mathbb{L}^\perp)=1\}\subset \mathrm{G}_2/\mathrm{P}_2 \end{equation} is equipped with a canonical $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})$-invariant marked contact Engel structure
\begin{equation}\label{def_sig} \sigma (\Pi):=( \Pi,\Pi\cap\mathbb{L}^\perp)\in \mathrm{G}_2/\mathrm{P}_{1,2}\, . \end{equation}
\end{proposition} Clearly, if we
fit \eqref{eqSubMaxMod} into the double fibration \eqref{doublefib},
then for $\sigma:{\cal M}_{\mathbb{L}}\to \pi_2^{-1}({\cal M}_{\mathbb{L}})\subset\mathrm{G}_2/\mathrm{P}_{1,2}$ defined as in \eqref{def_sig}, the corresponding hypersurface $\Sigma_{\mathbb{L}}:=\pi_1(\sigma({\cal M}_{\mathbb{L}}))$ is contained in $\mathbb{P}(\mathcal{N}\cap \mathbb{L}^\perp)$.
\begin{remark} By looking at the three cases individually we can see that $\Sigma_{\mathbb{L}}$ indeed coincides with the open $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})$-orbit in $\mathbb{P}(\mathcal{N}\cap \mathbb{L}^\perp)$.
If $\left\langle \mathbb{L},\mathbb{L}\right\rangle=0$, the $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})\cong \mathrm{P}_1$ preserves a filtration $$\mathbb{L}\subset\mathbb{D}\subset\mathbb{D}^{\perp}\subset\mathbb{L}^{\perp}\subset\mathbb{V},$$ where
$\mathbb{D}:=\mathrm{Ann}_{\Phi}(\mathbb{L})=\{ X\in \mathbb{R}^{3,4}\mid \Phi (\mathbb{L}, X, \hbox to 2.5pt{\hss$\ccdot$\hss})=0\}\subset \mathbb{R}^{3,4}.$
The open $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})$-orbit consists of all null lines contained in $\mathbb{L}^\perp$ but transversal to $\mathbb{D}^\perp$. Now suppose that a $2$-plane $\Pi\in \mathrm{G}_2/\mathrm{P}_2$ has non-trivial intersection with $\mathbb{D}^{\perp}$. Then, since $\mathbb{D}$ is maximally isotropic, a null line contained in the intersection has to be already contained in $\mathbb{D}$. Using the terminology from \cite{BaezHuerta}, this implies that any element $X\in\mathbb{L}$ and any element $Y\in\Pi$ are two rolls away from each other and then Theorem 10 in \cite{BaezHuerta} shows that $\left\langle X,Y\right\rangle=0$, hence $\Pi\subset\mathbb{L}^{\perp}$. This shows that $\Sigma_{\mathbb{L}}$ is contained in the open $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})$-orbit and equality follows from the fact that $\Sigma_{\mathbb{L}}$ is also invariant under the $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})$-action.
If $\left\langle \mathbb{L},\mathbb{L}\right\rangle<0$, we have $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})\cong\mathrm{SL}(3,\mathbb{R})$ which preserves the following decomposition $$\mathbb{R}^7=\mathbb{L}\oplus\mathbb{L}^{\perp}=\mathbb{L}\oplus\mathbb{U}\oplus\mathbb{U}^*.$$ The group $\mathrm{SL}(3,\mathbb{R})$ acts transitively on $\mathbb{P}\mathbb{U}$, $\mathbb{P}\mathbb{U}^*$ and the open orbit of all null lines in $\mathbb{L}^{\perp}$ that are neither contained in $\mathbb{U}$ nor $\mathbb{U}^*$,
respectively, see \cite{meTravis}. The open orbit is $\Sigma_{\mathbb{L}}$; this follows from the fact that if a null line $\mathbb{L'}$ is contained in one of the spaces $\mathbb{U}$ or $\mathbb{U}^*$, then its $\Phi$-annihilator $\mathrm{Ann}_{\Phi}(\mathbb{L}')$ is contained in $\mathbb{L}^{\perp}$.
If $\left\langle \mathbb{L},\mathbb{L}\right\rangle>0,$ the group $\mathrm{Stab}_{\mathrm{G}_2}(\mathbb{L})\cong\mathrm{SU}(1,2)$ acts transitively on $\mathbb{P}(\mathcal{N}\cap \mathbb{L}^\perp)$. \end{remark}
\begin{proposition}\label{submaxprop} The structures from Proposition \ref{submaxmod} realize maximally symmetric and submaximally symmetric models of marked contact twisted cubic structures.
Their infinitesimal symmetry algebras are $\mathfrak{p}_1$, $\mathfrak{sl}(3,\mathbb{R})$, and $\mathfrak{su}(1,2)$, respectively. \end{proposition}
\begin{proof} It is known that the infinitesimal symmetry algebra of a contact twisted cubic structure is either of dimension $14$, in which case it is the Lie algebra $\mathfrak{g}$ of $\mathrm{G}_2$, or else the dimension is $\leq 7$, see \cite{gap}. This implies that if the infinitesimal symmetry algebra of a marked contact twisted cubic structure has dimension $8$ or $9$, then it is a subalgebra of the Lie algebra $\mathfrak{g}$ of $\mathrm{G}_2$ and the underlying contact twisted cubic structure is a contact Engel structure.
By construction, the marked contact Engel structures from Proposition \ref{submaxmod} are invariant under $\mathfrak{p}_1$, $\mathfrak{sl}(3,\mathbb{R})$ and $\mathfrak{su}(1,2)$, respectively. It remains to show that the infinitesimal symmetry algebras of these structures are not bigger, but this follows from the fact that $\mathfrak{p}_1$, $\mathfrak{sl}(3,\mathbb{R})$ and $\mathfrak{su}(1,2)$ are maximal subalgebras of $\mathfrak{g}$ \cite{subalg_g2}. \end{proof}
\begin{remark} Of course, it follows from the analysis in Section \ref{CartanEquiv} that, up to local equivalence, the structures from Proposition \ref{submaxmod} are the unique homogeneous marked contact Engel structures having infinitesimal symmetry algebras of dimension eight or nine. Alternatively, with a little more work, we could recover this fact from purely algebraic considerations at this point using that we know the subalgebras of $\mathfrak{g}$. \end{remark}
\section{Considerations about general marked contact twisted cubic structures}\label{sec_general} The discussion of this section applies to general marked contact twisted cubic structures, i.e., here we shall \emph{not} restrict our considerations to marked contact Engel structures. We will regard marked contact twisted cubic structures as particular types of filtered $G_0$-structures in this section. For references on the general material used in this section see \cite{tanaka, Morimoto, yamaguchi, zelenko, book, CapCartan}.
In Section \ref{sec_tanaka} we review the (algebraic) Tanaka prolongation and some of its implications. The computation of the Tanaka prolongation implies the existence of a canonical coframe on a $9$-dimensional bundle associated with any marked contact twisted cubic structure in a natural manner.
In Section \ref{Cartan}, we briefly address the existence question of a canonical Cartan connection for marked contact twisted cubic structures, that is, of a canonical coframe with particularly nice properties. We show that, for algebraic reasons, the constructions of canonical Cartan connections from \cite{Morimoto} or \cite{CapCartan} are not applicable to our case.
In particular, for the filtered $G_0$-structures we are considering, a normalization condition in the sense of \cite{CapCartan} does not exist.
\subsection{Tanaka prolongation and applications}
\label{sec_tanaka} Recall, see Proposition \ref{prop_twisted}, that a contact twisted cubic structure can be equivalently regarded as a contact structure $\mathcal{C}\subset T{\cal M}$ together with a reduction of the graded frame bundle $\mathcal{F}\to {\cal M}$ with respect to an irreducible representation $\rho:\mathrm{GL}(2,\mathbb{R})\to\mathrm{CSp}(2,\mathbb{R})$. A marked contact twisted cubic structure, see Proposition \ref{prop_punctured}, can be seen as a further reduction of $\mathcal{F}\to {\cal M}$ with respect to the restriction $\rho:B\to\mathrm{CSp}(2,\mathbb{R})$ of $\rho$ to the Borel subgroup $B\subset\mathrm{GL}(2,\mathbb{R})$. In the terminology of \cite{Morimoto, CapCartan}, this means that \begin{itemize} \item a contact twisted cubic structure is a filtered $G_0$-structures of type $\mathfrak{m}$, where $G_0$ is the irreducible $\mathrm{GL}(2,\mathbb{R})$, and \item a marked contact twisted cubic structure is a filtered $Q_0$-structures of type $\mathfrak{m}$, where $Q_0$ is the Borel subgroup $B\subset \mathrm{GL}(2,\mathbb{R}).$ \end{itemize} In both cases $\mathfrak{m}=\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-1}$ is the $5$-dimensional Heisenberg Lie algebra.
Now suppose $\mathfrak{m}=\mathfrak{m}_{-k}\oplus\cdots\oplus\mathfrak{m}_{-1}$ is any fundamental graded Lie algebra, where fundamental means that it is generated as a Lie algebra by $\mathfrak{m}_{-1}$. Let $\mathfrak{g}_0\subset\mathrm{Der}_{gr}(\mathfrak{m})$ be a subalgebra of the Lie algebra $\mathrm{Der}_{gr}(\mathfrak{m})$ of $\mathrm{Aut}_{gr}(\mathfrak{m})$. Tanaka introduced the following algebraic object, which plays a fundamental role in his approach to the equivalence problem of filtered $G_0$-structures.
\begin{proposition}\label{tanakaprop} (\cite{tanaka}) There exists a unique, up to isomorphism, graded Lie algebra $\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0)$, called the (algebraic) \emph{Tanaka prolongation} of the pair $(\mathfrak{m},\mathfrak{g}_0)$, satisfying the following conditions: \begin{enumerate} \item The non-positive part is $\mathfrak{m}\oplus\mathfrak{g}_0$, i.e., $\mathfrak{g}(\mathfrak{m},\mathfrak{s})_i=\mathfrak{m}_i$ for $i<0$ and $\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0)_0=\mathfrak{g}_0$. \item If $X\in\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0)_i$ for some $i>0$ satisfies $[X,\mathfrak{m}_{-1}]=\{0\}$, then $X=0$. \item $\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0)$ is maximal among the graded Lie algebras satisfying (1) and (2). \end{enumerate}
\end{proposition} Let $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}}\mathfrak{g}_i$ be a graded Lie algebra
satisfying (1) and (2) from Proposition \ref{tanakaprop}. The condition that $\mathfrak{g}$ be
the Tanaka prolongation of $(\mathfrak{m},\mathfrak{g}_0)$ can be expressed in terms of the Lie algebra cohomology $H^*(\mathfrak{m},\mathfrak{g})$ with respect to the representation $\mathrm{ad}:\mathfrak{m}\to\mathfrak{gl}(\mathfrak{g})$; this is the cohomology of the cochain complex $(C(\mathfrak{m},\mathfrak{g}),\partial)$ where $C^q(\mathfrak{m},\mathfrak{g}):=\Lambda^q\mathfrak{m}^*\otimes\mathfrak{g}$ and $\partial:C^q(\mathfrak{m},\mathfrak{g})\to C^{q+1}(\mathfrak{m},\mathfrak{g})$ is the standard differential. Note that since $\mathfrak{m}$ and $\mathfrak{g}$ are graded Lie algebras, also the cochain spaces are naturally graded, and since $\partial$ preserves the homogeneous degree of maps, we have an induced grading on the cohomology spaces. We shall denote the $l$th grading component by a subscript $l$. Then (see e.g. \cite{yamaguchi}) the graded Lie algebra $\mathfrak{g}$ is the prolongation of $(\mathfrak{m},\mathfrak{g}_0)$ if and only if $H^1(\mathfrak{m},\mathfrak{g})_l=0$ for all $l>0$. If $\mathfrak{g}$ is simple, the Lie algebra cohomologies can be computed using Kostant's theorem (see e.g. \cite{book} for an account of Kostant's theorem).
\begin{example} Let $\mathfrak{g}$ be the Lie algebra of $\mathrm{G}_2$ equipped with its contact grading $$\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_{1}\oplus\mathfrak{g}_2$$ as discussed in Section \ref{LieTheory}. Then $\mathfrak{m}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}$ is the $5$-dimensional Lie Heisenberg algebra and, via the restriction of the adjoint representation, $\mathfrak{g}_0$ is a subalgebra of $\mathrm{Der}_{gr}(\mathfrak{m})$.
Utilizing Kostant's theorem, one shows that $H^1(\mathfrak{m},\mathfrak{g})_l=0$ for all $l>0$, see \cite{yamaguchi}, and therefore $\mathfrak{g}$ is the Tanaka prolongation of $(\mathfrak{m},\mathfrak{g}_0)$. \end{example}
Let $\mathfrak{q}_0\subset\mathfrak{g}_0\subset \mathrm{Der}_{gr}(\mathfrak{m})$ be a subalgebra, then the Tanaka prolongation $\mathfrak{q}=\mathfrak{g}(\mathfrak{m},\mathfrak{q}_0)$ of the pair $(\mathfrak{m},\mathfrak{q}_0)$ is a graded subalgebra of $\mathfrak{g}=\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0)$, where, for positive $i$, $$\mathfrak{q}_{i}=\{X\in\mathfrak{g}_{i}:\, [X,\mathfrak{g}_{-1}]\subset\mathfrak{q}_{i-1} \}.$$
This immediately leads to the following: \begin{proposition}
\label{p_1} Let $\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_{1}\oplus\mathfrak{g}_2$ be the Lie algebra of $\mathrm{G}_2$ equipped with its contact grading, $\mathfrak{m}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}$ the $5$-dimensional Heisenberg Lie algebra, and let $\mathfrak{q}_0\subset\mathfrak{g}_0\cong\mathfrak{gl}(2,\mathbb{R})$ be the Borel subalgebra. Then the Tanaka prolongation $\mathfrak{q}$ of $(\mathfrak{m},\mathfrak{q}_0)$ is a $9$-dimensional Lie algebra isomorphic to the parabolic subalgebra $\mathfrak{p}_1\subset\mathfrak{g}$.
\end{proposition}
\begin{proof} Let $\mathfrak{q}=\mathfrak{q}_{-2}\oplus\mathfrak{q}_{-1}\oplus\mathfrak{q}_0\oplus\mathfrak{q}_1$ be the subalgebra of $\mathfrak{g}$
spanned by the Cartan subalgebra and all root spaces corresponding to black nodes in the following root diagram of $\mathrm{G}_2$ : \begin{center} \begin{tikzpicture}[scale=1]
\draw (-1.85, -0.60) rectangle (1.85,-2);
\draw (-0.4, -0.2) rectangle (1.4,0.2); \draw (0.85, 0.65) rectangle (1.85,1.1);
\draw[thick] (0,-1.732) -- (0,1.732);
\draw[thick](0,1.732)circle(0.08);
\filldraw[white](0,1.732)circle(0.06);
\filldraw[black](0,-1.732)circle(0.08);
\draw[thick](-1.5,-0.866)--(1.5,0.866); \draw[thick](-1.5,-0.866)circle(0.08); \filldraw[black](-1.5,-0.866)circle(0.06);
\draw[thick](1.5,0.866)circle(0.08);
\filldraw[black](1.5,0.866)circle(0.06);
\draw[thick](1.5,-0.866)--(-1.5,0.866); \filldraw[white](-1.5,0.866)circle(0.06); \draw[thick](-1.5,0.866)circle(0.08);
\filldraw(1.5,-0.866)circle(0.08);
\draw[thick](-1,0)--(1,0); \draw[thick](-1,0)circle(0.08); \filldraw[white](-1,0)circle(0.06);
\draw[thick](1,0)circle(0.08);
\filldraw[black](1,0)circle(0.06); \draw[thick](1,0)circle(0.08);
\filldraw[thick](0.5,-0.866)--(-0.5,0.866);
\filldraw[black](0.5,-0.866)circle(0.08);
\draw[thick](-0.5,0.866)circle(0.08);
\filldraw[white](-0.5,0.866)circle(0.06);
\draw[thick](-0.5,-0.866)--(0.5,0.866);
\filldraw[black](-0.5,-0.866)circle(0.08);
\draw[thick](0.5,0.866)circle(0.08);
\filldraw[white](0.5,0.866)circle(0.06);
\draw[dashed](- 2, 1.2)--(2, 1.2);
\draw[dashed](- 2, 0.4)--(2, 0.4);
\draw[dashed](- 2, -1.2)--(2, -1.2);
\draw[dashed](- 2, -0.4)--(2, -0.4);
\draw (-2.7,1.7) node {$\mathfrak{g}_{2}$}; \draw (-2.5,-1.7) node {$\mathfrak{g}_{-2\,}$}; \draw (-2.7,0.9) node {$\mathfrak{g}_{1}$}; \draw (-2.5,-0.9) node {$\mathfrak{g}_{-1\,}$}; \draw (-2.6,0) node {$\mathfrak{g}_{0}\ $};
\draw (3,-1.7) node {$\mathfrak{q}_{-2\,}$}; \draw (2.8,0.9) node {$\mathfrak{q}_{1}$}; \draw (3,-0.9) node {$\mathfrak{q}_{-1\,}$}; \draw (2.9,0) node{$\mathfrak{q}_{0}\ $};
\draw (2.5,1.2) node {};
\draw (-2.5,-1.2) node {};
\draw (0.9,1.2) node {};
\draw (-0.8,-1.2) node {};
\draw (1.55,0) node {};
\draw (-1.55,0) node {}; \draw(2.1,-1.2) node {} ; \draw(-2.1,1.2) node {} ;
\draw(0.8,-1.2) node {};
\draw(-0.8,1.2) node {};
\draw (0,-2.1) node {};
\draw (0,2.1) node {};
\end{tikzpicture} \end{center} Then $\mathfrak{q}$ is a graded Lie algebra satisfying properties (1) and (2) from Proposition \ref{tanakaprop}. Moreover, there is no proper subalgebra $\mathfrak{q}'\subset\mathfrak{g}$ containing $\mathfrak{q}$. This can be either deduced from the above root diagram, by observing that any subalgebra $\mathfrak{q}'$ containing $\mathfrak{q}$ and in addition a root space corresponding to a white root has to be all of $\mathfrak{g}$. Alternatively, it immediately follows from the fact that a Lie algebra of root type $\mathrm{G}_2$ has no subalgebra of dimension bigger than $9$. Hence property (3) of Proposition \ref{tanakaprop} is satisfied as well.
\end{proof} \begin{remark} Identifying $\mathfrak{g}_{-1}\cong S^3\mathbb{R}^2$, the Borel subalgebra $\mathfrak{q}_0\subset\mathfrak{g}_0$ is the stabilizer of a line $\mathrm{Span}(l)\subset\mathbb{R}^2$, equivalently, of a line $\mathrm{Span}(l\odot l\odot l)\subset \smash{\bigodot^3\mathbb{R}^2}$. Recall that $\mathfrak{g}_1=(\mathfrak{g}_{-1})^*$ via the Killing form, and then $\mathfrak{q}_1$ can be viewed as the annihilator of the $3$-dimensional subspace $\mathrm{Span}(\{X\odot Y\odot l: X,Y\in\mathbb{R}^2\})$ of $\smash{\bigodot^3\mathbb{R}^2}=\mathfrak{g}_{-1}.$ \end{remark}
Given a filtered $G_0$-structure of type $\mathfrak{m}$ such that the Tanaka prolongation of the pair $(\mathfrak{m},\mathfrak{g}_0)$ is finite-dimensional, Tanaka theory \begin{itemize} \item
provides a procedure to construct, in a natural manner,
a bundle $\mathcal{G}\to {\cal M}$ of dimension $\mathrm{dim}(\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0))$
together with a coframe $\omega$ (an \emph{absolute parallelism}) on $\mathcal{G}$ (and it predicts the number of prolongation steps to be done to arrive there), \item and it establishes $\mathrm{dim}(\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0))$ as a sharp upper bound for the dimension of the infinitesimal symmetry algebra of the filtered $G_0$-structure. \end{itemize}
Applied to marked contact twisted cubic structures, as a Corollary to Proposition \ref{p_1}, this yields the following: \begin{corollary}\label{maximal_sym}
\
\begin{itemize} \item To any marked contact twisted cubic structure there is a naturally associated $9$-dimensional bundle equipped with a canonical coframe. \item The dimension of the Lie algebra of infinitesimal symmetries of a marked contact twisted cubic structure is
$\leq 9$. \end{itemize} \end{corollary}
\subsection{Canonical Cartan connections and the problem of finding a normalization condition} \label{Cartan} Given a filtered $G_0$-structure of type $\mathfrak{m}$ with algebraic Tanaka prolongation $\mathfrak{g}=\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0)$, it is a natural question to ask whether there exists a canonical Cartan connection associated with the structure. This question has been studied in \cite{Morimoto}, where a general criterion (the ``condition (C)'') ensuring the existence of a canonical Cartan connection is given, and more recently in \cite{CapCartan}, where the essential step to obtaining a canonical Cartan connection is to find a normalization condition with certain algebraic properties.
\subsubsection{Cartan geometries}
For a comprehensive introduction to Cartan geometries and applications of the concept see \cite{book}.
Let $G/P$ be a homogeneous space, let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mathfrak{p}$ the Lie algebra of $P$.
A Cartan geometry of type $(\mathfrak{g},P)$ on a manifold ${\cal M}$ is a pair $(\mathcal{G} \to {\cal M}, \omega)$, where $\mathcal{G} \to {\cal M}$ is a $P$-principal bundle and $\omega\in\Omega^1(\mathcal{G},\mathfrak{g})$ a Cartan connection, i.e., a Lie algebra valued $1$-form satisfying \begin{enumerate} \item $\omega_u : T_u \mathcal{G} \to \mathfrak{g}$ is an isomorphism for all $u \in \mathcal{G}$,
\item $\omega(\zeta_X) = X$ for all $X \in \mathfrak{p}$,
\item $(r^p)^*\omega = \mathrm{Ad}(p^{-1})\omega$,
\end{enumerate} where $r^p$ denotes the right action of $P$ on $\mathcal{G}$ and $\zeta_{X}$ the fundamental vector field generated by $X\in\mathfrak{p}$. The homogeneous (flat) model of a Cartan geometry of type $(\mathfrak{g},P)$ is the principal bundle $G\to G/P$ together with the Maurer-Cartan form $\omega_{MC}$ on $G$. The curvature of a Cartan geometry is the $2$-form $K = d\omega + \frac{1}{2} [\omega, \omega] \in \Omega^2(\mathcal{G},\mathfrak{g}).$ It is equivariant for the principal $P$-action and horizontal, i.e. $K(\zeta_{X},\hbox to 2.5pt{\hss$\ccdot$\hss})=0$ for any $X\in\mathfrak{p}$, which implies that it can be equivalently viewed as an equivariant function $K:\mathcal{G}\to\Lambda^2(\mathfrak{g}/\mathfrak{p})^*\otimes\mathfrak{g}$. The curvature vanishes if and only if the Cartan geometry is locally isomorphic to the homogeneous model; in this case the Cartan geometry is called flat.
\subsubsection{Normalization conditions} Given a filtered $G_0$-structure of type $\mathfrak{m}$, let $\mathfrak{g}=\mathfrak{g}(\mathfrak{m},\mathfrak{g}_0)$ be the algebraic Tanaka prolongation. Let $P$ be a Lie group with Lie algebra the non-negative part $\mathfrak{g}^0$ of $\mathfrak{g}$. Then the curvature function of any Cartan connection of type $(\mathfrak{g}, P)$ takes values in $\Lambda^2(\mathfrak{g}/\mathfrak{g}^0)^*\otimes\mathfrak{g}$, which is naturally filtered, and the associated graded space $\mathrm{gr}(\Lambda^2(\mathfrak{g}/\mathfrak{g}^0)^*\otimes\mathfrak{g})$ can be identified with $\Lambda^2\mathfrak{m}^*\otimes\mathfrak{g}$. The latter space is the space of $2$-cochains in the standard complex computing the Lie algebra cohomology $H^*(\mathfrak{m},\mathfrak{g})$. As before we denote by $\partial:\Lambda^k\mathfrak{m}^*\otimes\mathfrak{g}\to\Lambda^{k+1}\mathfrak{m}^*\otimes\mathfrak{g}$ the coboundary operators in that complex and we denote the $i$th grading component by a subscript $i$.
\begin{definition}\label{normcond}\cite[Definition 3.3]{CapCartan} A normalization condition for Cartan geometries of type $(\mathfrak{g},P)$ is a $P$-invariant linear subspace $\mathcal{N}\subset\Lambda^2(\mathfrak{g}/\mathfrak{g}^0)^*\otimes\mathfrak{g}$ such that for each $i>0$ the subspace $\mathrm{gr}(\mathcal{N})_i\subset (\Lambda^2\mathfrak{m}^*\otimes\mathfrak{g})_i$ is complementary to the image of
$\partial:(\mathfrak{m}^*\otimes\mathfrak{g})_i\to (\Lambda^2\mathfrak{m}^*\otimes\mathfrak{g})_i$. \end{definition}
\subsubsection{Analysis for marked contact twisted cubic structures}
Recall the algebraic setup: Let $\mathfrak{g}$ be the Lie algebra of $\mathrm{G}_2$ endowed with its contact grading $\mathfrak{g}=\bigoplus_{i=-2}^{2}\mathfrak{g}_i$ and $\mathfrak{q}=\bigoplus_{i=-2}^{1}\mathfrak{q}_i$
the graded subalgebra from Proposition \ref{p_1}.
In particular, $\mathfrak{m}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}=\mathfrak{q}_{-2}\oplus\mathfrak{q}_{-1}$ is the $5$-dimensional Heisenberg algebra.
We ask whether we can find a normalization condition for Cartan geometries of type $(\mathfrak{q},Q^0),$ where $Q^0$ is a Lie group with Lie algebra $\mathfrak{q}^0=\mathfrak{q}_0\oplus\mathfrak{q}_1$.
The inclusion $\mathfrak{q}\hookrightarrow\mathfrak{g}$ induces inclusions of the corresponding cochain spaces and we obtain the following commuting diagram $$\begin{array}{cccccccccccc} 0&\longrightarrow& \mathfrak{g}& \stackrel{\tilde{\partial}}{\longrightarrow}&\mathfrak{m}^*\otimes\mathfrak{g}&\stackrel{\tilde{\partial}}{\longrightarrow}&\Lambda^2\mathfrak{m}^*\otimes\mathfrak{g}&\stackrel{\tilde{\partial}}{\longrightarrow}
&\Lambda^3\mathfrak{m}^*\otimes\mathfrak{g}&\stackrel{\tilde{\partial}}{\longrightarrow}&\dots\\ & &\uparrow& &\uparrow & &\uparrow& &\uparrow& \\ 0&\longrightarrow&\mathfrak{q}&\stackrel{\partial}{\longrightarrow}&\mathfrak{m}^*\otimes\mathfrak{q}&\stackrel{\partial}{\longrightarrow}&\Lambda^2\mathfrak{m}^*\otimes\mathfrak{q}&\stackrel{\partial}{\longrightarrow}&\Lambda^3\mathfrak{m}^*\otimes\mathfrak{g}&\stackrel{\partial}{\longrightarrow}&\dots \end{array}$$
We know that $H^1(\mathfrak{m},\mathfrak{g})_l=0$ and $H^1(\mathfrak{m},\mathfrak{q})_l=0$ for all $l>0$, since this is implied by the fact that $\mathfrak{g}$ and $\mathfrak{q}$ are the Tanaka prolongations of $(\mathfrak{m},\mathfrak{g}_0)$ and $(\mathfrak{m},\mathfrak{q}_0)$, respectively.
The space of $2$-cochains of homogeneity one
\begin{equation}\label{2cochain}(\Lambda^2\mathfrak{m}^*\otimes\mathfrak{g})_1=\Lambda^2\mathfrak{g}_{-1}^*\otimes\mathfrak{g}_{-1}\oplus\mathfrak{g}_{-2}^*\otimes\mathfrak{g}_{-1}\otimes\mathfrak{g}_{-2}\end{equation}
is a completely reducible $\mathfrak{g}_0\cong\mathfrak{gl}(2,\mathbb{R})$ representation
isomorphic, as a representation of the semisimple part ${\mathfrak{g}_0}^{ss}$, to
\begin{align}\label{decomp} \overbrace{\underbrace{\smash{\bigodot}^5\mathbb{R}^2\oplus\smash{\bigodot}^3\mathbb{R}^2\oplus\smash{\bigodot}^3\mathbb{R}^2\oplus\mathbb{R}^2}_{\mathrm{Im}(\tilde{\partial})}\oplus\smash{\bigodot}^7\mathbb{R}^2}^{\mathrm{ker}(\tilde{\partial})}\oplus\smash{\bigodot}^3\mathbb{R}^2\,. \end{align} Hence $$H^2(\mathfrak{m},\mathfrak{g})_1\cong \smash{\bigodot}^7\mathbb{R}^2.$$ This fact can also be derived using Kostant's theorem (see \cite{yamaguchi, book}).
Next, it is visible from the decomposition \eqref{2cochain} that the inclusion $\mathfrak{q}\hookrightarrow\mathfrak{g}$ induces an identification $(\Lambda^2\mathfrak{m}^*\otimes\mathfrak{q})_1 =(\Lambda^2\mathfrak{m}^*\otimes\mathfrak{g})_1$. Likewise $(\Lambda^3\mathfrak{m}^*\otimes\mathfrak{q})_1= (\Lambda^3\mathfrak{m}^*\otimes\mathfrak{g})_1,$ and thus $\mathrm{ker}(\partial)=\mathrm{ker}(\tilde{\partial})\subset(\Lambda^2\mathfrak{m}^*\otimes\mathfrak{g})_1$. We can see from \eqref{decomp} that $\mathrm{Im}(\tilde{\partial})$ has dimension $16$, and that it has an invariant complement isomorphic to $\smash{\bigodot}^7\mathbb{R}^2$ in $\mathrm{ker}(\tilde{\partial})$. The image of $\partial:(\mathfrak{m}^*\otimes\mathfrak{q})_1\to(\Lambda^2\mathfrak{m}^*\otimes\mathfrak{q})_1$ is a $\mathfrak{q}_0$-submodule $\mathrm{Im}(\partial)\subset\mathrm{Im}(\tilde{\partial})$ of dimension $\mathrm{dim}((\mathfrak{m}^*\otimes\mathfrak{q})_1)-\mathrm{dim}(\mathfrak{q}_1)=15$, where $(\mathfrak{m}^*\otimes\mathfrak{q})_1=\mathfrak{q}_{-1}^*\otimes\mathfrak{q}_0\oplus\mathfrak{q}_{-2}^*\otimes\mathfrak{q}_{-1}$, and hence of codimension $1$ in $\mathrm{Im}(\tilde{\partial})$. In particular, $$H^2(\mathfrak{m},\mathfrak{q})_1\cong H^2(\mathfrak{m},\mathfrak{g})_1\oplus\mathbb{R}\cong \smash{\bigodot}^7\mathbb{R}^2\oplus\mathbb{R}.$$
On the other hand, we have the following:
\begin{proposition}\label{no_norm} There is no $\mathfrak{q}_0$-invariant subspace complementary to the image of $$\partial: (\mathfrak{m}^*\otimes\mathfrak{q})_1\to (\Lambda^2\mathfrak{m}^*\otimes\mathfrak{q})_1\,.$$ In particular, there exists no normalization condition in the sense of Definition \ref{normcond} for Cartan geometries of type $(\mathfrak{q}, Q^0)$. \end{proposition} \begin{proof} Suppose such a $\mathfrak{q}_0$-invariant complement $\mathbb{W}$ exists, i.e., we have a $\mathfrak{q}_0$-invariant decomposition $$\mathbb{W}\oplus\mathrm{Im}(\partial)= (\Lambda^2\mathfrak{m}^*\otimes\mathfrak{q})_1.$$ To simplify the discussion, recall that $\mathrm{Im}(\partial)$ is a codimension one subspace of $\mathrm{Im}(\tilde{\partial})$, and consider
$\mathbb{U}:=\mathbb{W}\cap\mathrm{Im}(\tilde{\partial})$; this is now a $1$-dimensional $\mathfrak{q}_0$-subrepresentation of the $\mathfrak{g}_0$-representation $\mathrm{Im}(\tilde{\partial})$ such that
$$\mathbb{U}\oplus\mathrm{Im}(\partial)=\mathrm{Im}(\tilde{\partial}).$$
Now let $\tilde{\mathbb{U}}$ be the irreducible $\mathfrak{g}_0$-subrepresentation of $\mathrm{Im}(\tilde{\partial})$ generated by $\mathbb{U}$. The dimension of $\tilde{\mathbb{U}}$ is $>1$, since \eqref{decomp} shows that there is no $1$-dimensional $\mathfrak{g}_0$-subrepresentation in $\mathrm{Im}(\tilde{\partial})$. In particular, $\tilde{\mathbb{U}}$ has non-zero intersection with $\mathrm{Im}(\partial)$. So we now have a non-trivial $\mathfrak{q}_0$-invariant decomposition
$$\mathbb{U}\oplus(\tilde{\mathbb{U}}\cap\mathrm{Im}(\tilde{\partial}))=\tilde{\mathbb{U}},$$
where $\tilde{\mathbb{U}}$ is now a finite-dimensional \emph{irreducible} $\mathfrak{g}_0$-representation. But this is impossible.
\end{proof}
Proposition \ref{no_norm} also shows that there exists no Lie group $Q_0$ with Lie algebra $\mathfrak{q}_0$ such that Morimoto's ``Condition C'' (see \cite[Definition 3.10.1]{Morimoto}) is satisfied.
\section{Appendix}
\label{appendixg2}
For explicit computations we use the following basis of the Lie algebra $\mathfrak{g}$ of $\mathrm{G}_2$. Consider the $7\times 7$ matrices \begin{equation*} \begin{aligned} &A=\tiny{ \begin{pmatrix} 0&\tfrac43\alpha_2&\tfrac43\alpha_0&\tfrac49\alpha_1-\alpha_3&-\tfrac49\alpha_1-\alpha_3&-\tfrac43\alpha_0&0\\ -\tfrac43\alpha_2&0&2\alpha_3&0&0&-2\alpha_3&-\tfrac43\alpha_2\\ -\tfrac43\alpha_0&-2\alpha_3&0&-\tfrac23\alpha_2+\tfrac32\alpha_4&\tfrac23\alpha_2+\tfrac32\alpha_4&0&-\tfrac43\alpha_0\\ -\tfrac49\alpha_1+\alpha_3&0&\tfrac23\alpha_2-\tfrac32\alpha_4&0&0&-\tfrac23\alpha_2+\tfrac32\alpha_4&-\tfrac49\alpha_1+\alpha_3\\ -\tfrac49\alpha_1-\alpha_3&0&\tfrac23\alpha_2+\tfrac32\alpha_4&0&0&-\tfrac23\alpha_2-\tfrac32\alpha_4&-\tfrac49\alpha_1-\alpha_3\\ -\tfrac43\alpha_0&-2\alpha_3&0&-\tfrac23\alpha_2+\tfrac32\alpha_4&\tfrac23\alpha_2+\tfrac32\alpha_4&0&-\tfrac43\alpha_0\\ 0&-\tfrac43\alpha_2&-\tfrac43\alpha_0&-\tfrac49\alpha_1+\alpha_3&\tfrac49\alpha_1+\alpha_3&\tfrac43\alpha_0&0 \end{pmatrix}},\\ &B=\tiny{ \begin{pmatrix} 0&0&\tfrac34\beta_2-\tfrac13\beta_3&0&0&-\tfrac34\beta_2-\tfrac13\beta_3&3\beta_1+\beta_4\\ 0&0&0&\tfrac32\beta_2-\tfrac23\beta_3&\tfrac32\beta_2+\tfrac23\beta_3&0&0\\ -\tfrac34\beta_2+\tfrac13\beta_3&0&0&0&0&-3\beta_1+\beta_4&\tfrac34\beta_2+\tfrac13\beta_3\\ 0&-\tfrac32\beta_2+\tfrac23\beta_3&0&0&-2\beta_4&0&0\\ 0&\tfrac32\beta_2+\tfrac23\beta_3&0&-2\beta_4&0&0&0\\ -\tfrac34\beta_2-\tfrac13\beta_3&0&-3\beta_1+\beta_4&0&0&0&\tfrac34\beta_2-\tfrac13\beta_3\\ 3\beta_1+\beta_4&0&\tfrac34\beta_2+\tfrac13\beta_3&0&0&-\tfrac34\beta_2+\tfrac13\beta_3&0 \end{pmatrix}},\\ &C=\tiny{ \begin{pmatrix} 0&-\tfrac32\gamma_3&-\tfrac98\gamma_0&-\tfrac12\gamma_2+\tfrac{27}{8}\gamma_4&\tfrac12\gamma_2+\tfrac{27}{8}\gamma_4&-\tfrac98\gamma_0&0\\ \tfrac32\gamma_3&0&\gamma_2&0&0&\gamma_2&-\tfrac32\gamma_3\\ \tfrac98\gamma_0&-\gamma_2&0&-\gamma_1+\tfrac34\gamma_3&\gamma_1+\tfrac34\gamma_3&0&-\tfrac98\gamma_0\\ \tfrac12\gamma_2-\tfrac{27}{8}\gamma_4&0&\gamma_1-\tfrac34\gamma_3&0&0&\gamma_1-\tfrac34\gamma_3&-\tfrac12\gamma_2+\tfrac{27}{8}\gamma_4\\ \tfrac12\gamma_2+\tfrac{27}{8}\gamma_4&0&\gamma_1+\tfrac34\gamma_3&0&0&\gamma_1+\tfrac34\gamma_3&-\tfrac12\gamma_2-\tfrac{27}{8}\gamma_4\\ -\tfrac98\gamma_0&\gamma_2&0&\gamma_1-\tfrac34\gamma_3&-\gamma_1-\tfrac34\gamma_3&0&\tfrac98\gamma_0\\ 0&-\tfrac32\gamma_3&-\tfrac98\gamma_0&-\tfrac12\gamma_2+\tfrac{27}{8}\gamma_4&\tfrac12\gamma_2+\tfrac{27}{8}\gamma_4&-\tfrac98\gamma_0&0 \end{pmatrix}}, \end{aligned} \end{equation*} where $\alpha_0,\alpha_1,\alpha_2,\alpha_3,\alpha_4,\beta_1,\beta_2,\beta_3,\beta_4,\gamma_0,\gamma_1,\gamma_2,\gamma_3,\gamma_4$ are real constants. Then \begin{equation} \label{basis_g2} E_0=\tfrac{{\rm d} A}{{\rm d}\alpha_0},\quad E_i=\tfrac{{\rm d} A}{{\rm d}\alpha_i},\quad E_{4+I}=\tfrac{{\rm d} B}{{\rm d}\beta_I},\quad E_{8+i}=\tfrac{{\rm d} C}{{\rm d}\gamma_i},\quad E_{13}=\tfrac{{\rm d} C}{{\rm d}\gamma_0},\end{equation} where $i=1,2,3,4$, and $I=1,2,3,4,$ define (as one can verify) a basis for $\mathfrak{g}_2$.
The basis is adapted to the contact grading of $\mathfrak{g}=\mathfrak{g}_2$ in the sense that $E_0$ is contained in the grading component $\mathfrak{g}_{-2}$, $E_1, E_2, E_3, E_4$ are contained in $\mathfrak{g}_{-1}$, $E_5, E_6, E_7, E_8$ are contained in $\mathfrak{g}_0$, $E_9, E_{10}, E_{11},E_{12}$ in $\mathfrak{g}_1$, and $E_{13}$ in $\mathfrak{g}_{2}$.
Writing the Maurer-Cartan form $\Omega_{G_2}$ as $\Omega_{G_2}=\theta^i E_i$, where the $\theta^i$ are now left-invariant $\mathbb{R}$-valued $1$-forms,
the Maurer-Cartan equations for $\mathrm{G}_2$ are of the following form:
\begin{equation}\begin{aligned}\label{MaurerCartan} &{\rm d}\theta^0=-6\theta^0\wedge\theta^5+\theta^1\wedge\theta^4-3\theta^2\wedge\theta^3\\ &\\ &{\rm d}\theta^1=6\theta^0\wedge\theta^9-3\theta^1\wedge\theta^5-3\theta^1\wedge\theta^8+3\theta^2\wedge\theta^7\\ &{\rm d}\theta^2=2\theta^0\wedge\theta^{10}+\theta^1\wedge\theta^6-3\theta^2\wedge\theta^5-\theta^2\wedge\theta^8+2\theta^3\wedge\theta^7\\ &{\rm d}\theta^3=2\theta^0\wedge\theta^{11}+2\theta^2\wedge\theta^6-3\theta^3\wedge\theta^5+\theta^3\wedge\theta^8+\theta^4\wedge\theta^7\\ &{\rm d}\theta^4=6\theta^0\wedge\theta^{12}+3\theta^3\wedge\theta^6-3\theta^4\wedge\theta^5+3\theta^4\wedge\theta^8\\ &\\ &{\rm d}\theta^5=2\theta^0\wedge\theta^{13}-\theta^1\wedge\theta^{12}+\theta^2\wedge\theta^{11}-\theta^3\wedge\theta^{10}+\theta^4\wedge\theta^9\\ &{\rm d}\theta^6=6\theta^2\wedge\theta^{12}-4\theta^3\wedge\theta^{11}+2\theta^4\wedge\theta^{10}+2\theta^6\wedge\theta^8\\ &{\rm d}\theta^7=-2\theta^1\wedge\theta^{11}+4\theta^2\wedge\theta^{10}-6\theta^3\wedge\theta^9-2\theta^7\wedge\theta^8\\ &{\rm d}\theta^8=-3\theta^1\wedge\theta^{12}+\theta^2\wedge\theta^{11}+\theta^3\wedge\theta^{10}-3\theta^4\wedge\theta^9-\theta^6\wedge\theta^7\\ &\\ &{\rm d}\theta^9=-\theta^1\wedge\theta^{13}-3\theta^5\wedge\theta^9-\theta^7\wedge\theta^{10}+3\theta^8\wedge\theta^9\\ &{\rm d}\theta^{10}=-3\theta^2\wedge\theta^{13}-3\theta^5\wedge\theta^{10}-3\theta^6\wedge\theta^9-2\theta^7\wedge\theta^{11}+\theta^8\wedge\theta^{10}\\ &{\rm d}\theta^{11}=-3\theta^3\wedge\theta^{13}-3\theta^5\wedge\theta^{11}-2\theta^6\wedge\theta^{10}-3\theta^7\wedge\theta^{12}-\theta^8\wedge\theta^{11}\\ &{\rm d}\theta^{12}=-\theta^4\wedge\theta^{13}-3\theta^5\wedge\theta^{12}-\theta^6\wedge\theta^{11}-3\theta^8\wedge\theta^{12}\\ &\\ &{\rm d}\theta^{13}=-6\theta^5\wedge\theta^{13}-6\theta^9\wedge\theta^{12}+2\theta^{10}\wedge\theta^{11}.
\end{aligned}\end{equation}
The nine generators $(E_0, E_1, E_2, E_3, E_4, E_5, E_6, E_8, E_{12})$ marked by black dots below are a basis for a subalgebra $\mathfrak{q}\cong{\mathfrak{p}_1}$ having minimal intersection with $\mathfrak{p}_2=\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2$.
\begin{center} \begin{tikzpicture}[scale=1]
\draw[dashed] (0,-1.732) -- (0,1.732);
\draw[thick](0,1.732)circle(0.08);
\filldraw[white](0,1.732)circle(0.06);
\filldraw[black](0,-1.732)circle(0.08);
\draw[dashed](-1.5,-0.866)--(1.5,0.866); \draw[thick](-1.5,-0.866)circle(0.08); \filldraw[black](-1.5,-0.866)circle(0.06);
\draw[thick](1.5,0.866)circle(0.08);
\filldraw[black](1.5,0.866)circle(0.06);
\draw[dashed](1.5,-0.866)--(-1.5,0.866); \filldraw[white](-1.5,0.866)circle(0.06); \draw[thick](-1.5,0.866)circle(0.08);
\filldraw(1.5,-0.866)circle(0.08);
\draw[dashed](-1,0)--(1,0); \draw[thick](-1,0)circle(0.08); \filldraw[white](-1,0)circle(0.06);
\draw[thick](1,0)circle(0.08);
\filldraw[black](1,0)circle(0.06); \draw[thick](1,0)circle(0.08);
\filldraw[dashed](0.5,-0.866)--(-0.5,0.866);
\filldraw[black](0.5,-0.866)circle(0.08);
\draw[thick](-0.5,0.866)circle(0.08);
\filldraw[white](-0.5,0.866)circle(0.06);
\draw[dashed](-0.5,-0.866)--(0.5,0.866);
\filldraw[black](-0.5,-0.866)circle(0.08);
\draw[thick](0.5,0.866)circle(0.08);
\filldraw[white](0.5,0.866)circle(0.06);
\draw(0.1,2.1) node {$E_{13}$};
\draw(0.9,1.2) node {$E_{11}$};
\draw (-0.7,1.2) node {$E_{10}$};
\draw (-1.8,1.2) node {$E_{9}$}; \draw (-1.6,0) node {$E_7$};
\draw (2,1.2) node {$E_{12}$};
\draw (1.6,0) node {$E_8$};
\draw (2,-1.2) node {$E_{4}$};
\draw (0.9,-1.2) node {$E_{3}$};
\draw (-0.7,-1.2) node {$E_{2}$};
\draw (-1.8,-1.2) node {$E_{1}$};
\draw (-0.3,-0.2) node {$E_5$};
\draw (0.3,-0.2) node {$E_6$};
\draw (0.1,-2.1) node {$E_0$}; \end{tikzpicture} \end{center} The kernel of the forms $\theta^7, \theta^9, \theta^{10}, \theta^{11}, \theta^{13}$ is an integrable distribution. On each leaf of the foliation defined by this distribution the forms $\theta^7, \theta^9, \theta^{10}, \theta^{11}, \theta^{13}$ vanish and the system \eqref{MaurerCartan} reduces to the Maurer-Cartan equations for $Q\cong \mathrm{P}_1$:
\begin{equation}\begin{aligned}\label{MaurerCartan_Q} &{\rm d}\theta^0 = -6 \theta^0\wedge\theta^5 + \theta^1\wedge \theta^4 - 3\theta^2\wedge\theta^3\\ &{\rm d}\theta^1 = - 3\theta^1\wedge\theta^5 - 3\theta^1\wedge\theta^8 \\ &{\rm d}\theta^2 = \theta^1\wedge\theta^6 - 3\theta^2\wedge\theta^5 - \theta^2\wedge\theta^8 \\ &{\rm d}\theta^3 = 2\theta^2\wedge\theta^6 -3\theta^3\wedge\theta^5 + \theta^3\wedge\theta^8 \\ &{\rm d}\theta^4=6 \theta^0\wedge\theta^{12}+3\theta^3\wedge\theta^6-3\theta^4\wedge\theta^5+3\theta^4\wedge\theta^8\\ &{\rm d}\theta^5=-\theta^1\wedge\theta^{12}\\ &{\rm d}\theta^6=6\theta^2\wedge\theta^{12}+2\theta^6\wedge\theta^8\\ &{\rm d}\theta^8=-3\theta^1\wedge\theta^{12}\\ &{\rm d}\theta^{12}= -3\theta^5\wedge\theta^{12}- 3\theta^8\wedge\theta^{12}
\end{aligned}\end{equation}
\end{document}
\end{document} | arXiv |
Evaluate $\left\lceil\sqrt{\frac{9}{4}}\right\rceil+\left\lceil\frac{9}{4}\right\rceil+\left\lceil\left(\frac{9}{4}\right)^2\right\rceil$.
The equation can be rewritten as $\left\lceil\frac{3}{2}\right\rceil+\left\lceil\frac{9}{4}\right\rceil+\left\lceil\frac{81}{16}\right\rceil$. The smallest integer greater than $\frac{3}{2}$ is $2$. The smallest integer greater than $\frac{9}{4}$ is $3$. The smallest integer greater than $\frac{81}{16}$ is $6$. Therefore $2+3+6=\boxed{11}$. | Math Dataset |
\begin{document}
\date{} \baselineskip 14pt \newcommand{\displaystyle}{\displaystyle} \thispagestyle{empty} \title{Differential Harnack estimates for conjugate heat equation under the Ricci flow} \author{Abimbola Abolarinwa\thanks{Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom.} \thanks{E-mail: [email protected]}} \maketitle
\begin{abstract} We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$ \Delta - R(x,t)$", where $R$ is the scalar curvature of the Ricci flow, which is well generalised to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by product, we obtain some Li-Yau type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncampact case. \paragraph{Keywords:}Ricci Flow, Conjugate Heat Equation, Harnack inequality, Gradient Estimate, Laplace-Beltrami operator, Laplacian Comparison Theorem. \paragraph {2010 Mathematics Subject Classification:} 35K08, 53C44, 58J35 58J60 \end{abstract}
\section{Introduction}
Let $M$ be an $n$-dimensional compact (or noncompact without boundary) manifold on which a one parameter family of Riemannian metrics $g(t), t \in [0, T)$ is defined. We say $(M, g(t))$ is a solution to the Ricci flow if it is evolving by the following nonlinear weakly parabolic partial differential equation \begin{equation}\label{eq11} \frac{\partial }{ \partial t} g (x, t) = - 2 Ric(x,t), \hspace{2cm} (x, t) \in M \times [0, T) \end{equation} with $g(x, 0) = g(0)$, where $Ric$ is the Ricci curvature and $T \leq \infty$. By the positive solution to the heat equation on the manifold, we mean a smooth function atleast $C^2$ in $x$ and $C^1$ in $t$, $ u \in C^{2, 1}(M \times [0,T])$ which satisfies the following equation \begin{equation}\label{eq12} \Big( \Delta - \frac{\partial }{ \partial t} \Big) u( x, t) = 0, \hspace{2cm} (x, t) \in M \times [0, T], \end{equation} where the symbol $ \Delta $ is the Laplace-Beltrami operator acting on function in space with respect to metric $g(t)$ in time. We can couple the Ricci flow (\ref{eq11}) to equation of the form (\ref{eq12}), either forward, backward, or perturbed with some potential function, see the author's papers \cite{[Ab1],[Ab2]} for more details. In the present we consider a more generalized situation where the heat equation is perturbed, in this case, the Laplacian is replaced with $ \Delta - R(x,t)$, where $R$ is the scalar curvature of the Ricci flow $g(t)$ and we obtain some Harnack and gradient estimates on the logarithm of the positive solutions. The author also obtained various estimates on positive solutions and fundamental solution in \cite{[Ab3]}. Throughout, we assume, that the manifold is endowed with bounded curvature, we remark that boundedness and nonnegativity of the curvature is preserved as long as Ricci flow exists \cite{[CLN06]}.
Heat equation coupled to the Ricci flow can be associated with some physical interpretation in terms of heat conduction process. Precisely, the manifold $M$ with initial metric $g(x, 0)$ can be thought of as having the temperature distribution $u(x, 0)$ at $ t = 0.$ If we now allow the manifold to evolve under the Ricci flow and simultaneously allow the heat to diffuse on $M$, then, the solution $u(x, t)$ will represent the space-time temperature on $M$. Moreover, if $u(x, t)$ approaches $\delta$-function at the initial time, we know that $u(x,t) >0$, this gives another physical interpretation that temperature is always positive, whence we can consider the potential $ f = \log u$ as an entropy or unit mass of heat supplied and the local production entropy is given by $ | \nabla f|^2 = \frac{ |\nabla u|^2}{u^2} $.
Harnack inequalities are indeed very powerful tools in geometric analysis. The paper of Li and Yau \cite{[LY86]} paved way for the rigorous studies and many interesting applications of Harnack inequalities. They derived gradient estimates for positive solutions to the heat operator defined on closed manifold with bounded Ricci curvature and from where their Harnack inequalities follows. These inequalities were in turn used to establish various lower and upper bounds on the heat kernel. They also studied manifold satisfying Dirichlet and Neumann conditions. On the other hand, Perelman in \cite{[Pe02]} obtained differential Harnack estimate for the fundamental solution to the conjugate heat equation on compact manifold evolving by the Ricci flow. Perelman's results are unprecedented as they play a key factor in the proof of Poincar\'e conjecture. Meanwhile, shortly before Perelman's paper appeared online, C. Guenther \cite{[Gu02]} had found gradient estimates for positive solutions to the heat equation under the Ricci flow by adapting the methodology of Bakry and Qian \cite{[BQ99]} to time dependent metric case. As an application of her results, she got a Harnack-type inequality and obtain a lower bound for fundamental solutions. These techniques were first brought into the study of Ricci flow by R. Hamilton, see \cite{[Ha93a]} for instance. As useful as Harnack inequalities are, they have also been discovered in other geometric flows; See the following- H-D. Cao \cite{[CaNi]} for heat equation on K\"ahler manifolds, B. Chow \cite{[Ch91b]} for Gaussian curvature flow and \cite{[Ch92]} for Yamabe flow, also B. Chow and R. Hamilton \cite{[CH97]}, and R. Hamilton \cite{[Ha95b]} on mean curvature flow. The following references among many others are found relevant \cite{[Ab4],[Cao08],[Gu02],[Zh06]}. See also the following monographs \cite{[SY94]} on Gradient estimates and \cite{[CCG$^+$07],[CK04],[CLN06],[Mu06]} for theory and application of Ricci flow.
Recently, \cite{[BCP]} and \cite{[KZh08]} have extended results in \cite{[Zh06]} to heat equation and its conjugate respectively. We remark that our results are similar to those of \cite{[KZh08]} but with different approach, the application to heat conduction that we have in mind has greatly motivated our approach. The detail descriptions of our results are presented in Sections 2 and 3 (localized version), while we collect some elements of the Ricci flow used in our calculation as an appendix in the last section.
\section{Estimates on Positive Solutions to the Conjugate Heat Equation}\label{sec2} Let $\square := \partial_t - \Delta $ be the heat operator acting on functions $u: M \times [0, T] \rightarrow \mathbb{R}$, where $M \times [0, T]$ is endowed with the volume form $ d \mu(x) dt $. The conjugate to the heat operator $ \Gamma$ is defined by \begin{equation}\label{Ceq1} \square^* = - \partial_t - \Delta_x + R, \end{equation} where $R$ is the scalar curvature. We remark that for any solution $g(t), t \in [0, T]$ to the Ricci flow and smooth functions $u, v : M \times [0, T] \rightarrow \mathbb{R}$, the following identity holds \begin{equation} \int_0^T \int_M ( \square u) v d \mu(x) dt = \int_0^T \int_M u( \square^* v) d \mu(x) dt. \end{equation} By direct application of integration by parts with the fact that the functions $u$ and $v$ are $C^2$ with compact support (or if $M$ is compact) and using evolution of $d \mu$ under the Ricci flow the last identity can be shown easily.
In a special case $u \equiv 1$, we have
$$ \frac{d}{dt} \int_M v d\mu = - \int_M \square^* d\mu. $$
\begin{proposition}\label{prop21}
Let $u = (4 \pi \tau )^{-\frac{n}{2}} e^{-f}$ be a positive solution to the conjugate heat equation. The evolution equation
\begin{equation}
\frac{\partial f}{ \partial t} = - \Delta f + | \nabla f|^2 - R + \frac{n}{ 2 \tau}
\end{equation}
is equivalent to the following evolution
\begin{equation}
\square^* u = 0.
\end{equation}
\end{proposition}
\begin{proof}
$$ \square^* u = (- \partial_t - \Delta_x + R)(4 \pi \tau )^{-\frac{n}{2}} e^{-f}.$$
By direct calculation, it follows that
$$ \partial_t [(4 \pi \tau )^{-\frac{n}{2}} e^{-f}] = ( \frac{n}{ 2\tau}- \partial_t f ) (4 \pi \tau )^{-\frac{n}{2}} e^{-f} $$
$$ \Delta [(4 \pi \tau )^{-\frac{n}{2}} e^{-f}] =( - \Delta f + | \nabla f|^2 )(4 \pi \tau )^{-\frac{n}{2}} e^{-f}.$$
Then
$$\square^* u = \Big( -\frac{n}{ 2\tau}+ \partial_t f + \Delta f - | \nabla f|^2 + R \Big)u = 0, $$
where we have made use of $ \partial_t \tau = -1$ and since $ u >0$, the claimed is then proved.
\end{proof}
Let $(M, g(t)), t \in [0,T]$ be a solution of the Ricci flow on a closed manifold. Here $T > 0$ is taken to be the maximum time of existence for the flow. Let $u$ be a positive solution to the conjugate heat equation, then we have the following coupled system.
\begin{equation}
\left \{ \begin{array}{l}\label{Heqn1}
\displaystyle \frac{\partial g_{ij}}{\partial t} = - 2 R_{ij} \\
\ \\
\displaystyle - \frac{\partial u}{\partial t} - \Delta_{g(t)} u + R_{g(t)} u = 0,
\end{array} \right.
\end{equation} which we refer to as Perelman's conjugate heat equation coupled to the Ricci flow. We will prove differential Harnack and gradient estimates for all positive solutions of the conjugate heat equation in the above system. A differential Harnack estimate of Li-Yau type yields a space-time gradient estimate for a positive solution to a heat-type equation, which when integrated compares the solution at different points in space and time. We will later apply the maximum principle to obtain a localized version of the estimates.
\subsection{Main Result I. (Differential Harnack Inequality and Gradient Estimates)} The main result of this subsection is contained in Theorem (\ref{thm Heqn1}) and as an application we arrived at Theorem (\ref{cor35}) which gives the corresponding Li-Yau type gradient estimate for all positive solution to the conjugate heat equation in the system (\ref{Heqn1}).
\begin{theorem} \label{thm Heqn1} Let $ u \in C^{2, 1}(M \times [0,T])$ be a positive solution to the conjugate heat equation $ \square^* u = ( - \partial_t - \Delta + R )u = 0$ and the metric $g(t)$ evolve by the Ricci flow in the interval $[0, T)$ on a closed manifold $M$ with nonnegative scalar curvature. Suppose further that $u = ( 4 \pi \tau )^{-\frac{n}{2}} e^{-f}$, where $ \tau = T-t$, then for all points $(x,t) \in (M \times [0,T])$, we have the Harnack quantity
\begin{equation}\label{Heqn2}
P = 2 \Delta f -| \nabla f|^2 + R - \frac{ 2n}{\tau} \leq 0.
\end{equation} Then $P$ evolves as
\begin{equation}
\frac{\partial }{\partial t} P = - \Delta P + 2 \langle \nabla f, \nabla P \rangle + 2 \Big| R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij} \Big|^2 + \frac{2}{\tau} P + \frac{2}{\tau} | \nabla f|^2 + \frac{4n}{\tau^2} + \frac{2}{\tau} R.
\end{equation}
for all $ t > 0$. Moreover $P \leq 0$ for all $t \in [0, T].$ \end{theorem}
Note that $u = ( 4 \pi \tau )^{-\frac{n}{2}} e^{-f}$ implies $ \ln u = - f - \frac{n}{2} \ln( 4 \pi \tau)$ and we can write (\ref{Heqn2}) as
\begin{equation}
\frac{| \nabla u|^2}{ u^2} - 2 \frac{u_t}{u}- R - \frac{ 2n}{\tau} \leq 0,
\end{equation}
which is similar to the celebrated Li-Yau \cite{[LY86]} gradient estimate for the heat equation on manifold with nonnegative Ricci curvature.
We need the usual routine computations as in the following;
\begin{lemma}\label{lem332}
Let $(g, f)$ solve the system (\ref{Heqn1}) above. Suppose further that $u = ( 4 \pi \tau )^{-\frac{n}{2}} e^{-f}$ with $ \tau = T -t$. Then we have
$$ ( \frac{\partial}{\partial t } + \Delta ) \Delta f = 2 R^{ij} \nabla_i \nabla_j f + \Delta | \nabla f|^2 - \Delta R$$
and
$$( \frac{\partial}{\partial t } + \Delta )| \nabla f|^2 = 4 R_{ij} \nabla_i f \nabla_j f + 2 \langle \nabla f, \nabla | \nabla f|^2 \rangle + 2 | \nabla \nabla f |^2 - 2 \langle \nabla f, |\nabla R|^2 \rangle. $$
\end{lemma}
\begin{proof}
By direct calculation and Proposition \ref{prop21}
\begin{align*}
\frac{\partial}{\partial t } ( \Delta f ) = \frac{\partial}{\partial t } ( g^{ij} \partial_i \partial_j f ) &
= \frac{\partial}{\partial t } (g^{ij} ) \partial_i \partial_j f + g^{ij} \partial_i \partial_j \frac{\partial}{\partial t }f \\
\displaystyle &= 2R^{ij} \partial_i \partial_j f + \Delta ( - \Delta f + | \nabla f|^2 - R + \frac{n}{ 2 \tau } ) \\
\displaystyle &= 2R^{ij} \nabla_i \nabla_j f - \Delta( \Delta f) + \Delta | \nabla f|^2 - \Delta R
\end{align*}
then,
\begin{align*}
\Big( \frac{\partial}{\partial t } +\Delta \Big) \Delta f &= 2R^{ij} \nabla_i \nabla_j f - \Delta( \Delta f) + \Delta | \nabla f|^2 - \Delta R + \Delta( \Delta f) \\
\displaystyle &= 2R^{ij} \nabla_i \nabla_j f + \Delta | \nabla f|^2 - \Delta R
\end{align*}
Part 1 is proved.
\begin{align*}
\frac{\partial}{\partial t } | \nabla f|^2 & = 2 R^{ij} \partial_i f \partial_j f + 2 g^{ij} \partial_i f \partial_j \frac{\partial}{ \partial t} f \\
\displaystyle & = 2 R^{ij} \partial_i f \partial_j f + 2 \langle \nabla f, \nabla( - \Delta f + | \nabla f|^2 - R + \frac{n}{ 2 \tau } ) \rangle \\
\displaystyle & = 2 R^{ij} \nabla_i f \nabla_j f + 2 \langle \nabla f, \nabla | \nabla f|^2 \rangle - 2 \langle \nabla f, \nabla \Delta f \rangle - 2 \langle \nabla f, \nabla R \rangle
\end{align*}
then,
\begin{align*}
\Big( \frac{\partial}{\partial t } +\Delta \Big) | \nabla f|^2 = 2 R^{ij} \partial_i f \partial_j f + 2 \langle \nabla f, \nabla | \nabla f|^2 \rangle - 2 \langle \nabla f, \nabla \Delta f \rangle - 2 \langle \nabla f, \nabla R \rangle + \Delta | \nabla f|^2.
\end{align*}
Using the Bochner identity
$$ \Delta | \nabla f|^2 = 2 | \nabla \nabla f|^2 + 2 \langle \nabla f, \nabla \Delta f \rangle + 2 Rc ( \nabla f, \nabla f) $$
we obtain the identity in part (2).
\end{proof}
\begin{proof} Proof of Theorem \ref{thm Heqn1}.
Since $ P = 2 \Delta f - | \nabla f |^2 + R - \frac{2n}{\tau}$ and by direct computation and using Lemma \ref{lem332}, we have
\begin{align*}
\Big( \frac{\partial}{\partial t } +\Delta \Big) P &= 2 \Big( \frac{\partial}{\partial t } +\Delta \Big) \Delta f - \Big( \frac{\partial}{\partial t } +\Delta \Big) |\nabla f |^2 + \Big( \frac{\partial}{\partial t } +\Delta \Big) R - \frac{\partial}{\partial t } \Big(\frac{2n}{\tau} \Big) \\
\displaystyle & = 4 R^{ij} \nabla_i \nabla_j f + 2 \Delta | \nabla f|^2 - 2\Delta R - 4 Rc( \nabla f, \nabla f) - 2 \langle \nabla f, \nabla | \nabla f|^2 \rangle \\
\displaystyle & \hspace{1.5cm} - 2 | \nabla \nabla f|^2 + 2 \langle \nabla f, \nabla R \rangle + 2 \Delta R + 2 | Rc|^2 + \frac{2n}{\tau^2} \\
\displaystyle & = 4 R^{ij} \nabla_i \nabla_j f + 2 | Rc|^2 + \frac{2n}{\tau^2} - 2 \langle \nabla f, \nabla | \nabla f|^2 \rangle + 2 \langle \nabla f, \nabla R \rangle \\
\displaystyle & \hspace{1.5cm} + 2 \Delta | \nabla f|^2 - 4 Rc ( \nabla f, \nabla f) - 2| \nabla \nabla f|^2 \\
\displaystyle & = 4 R^{ij} \nabla_i \nabla_j f + 2 | Rc|^2 + \frac{2n}{\tau^2} - 2 \langle \nabla f, \nabla | \nabla f|^2 \rangle + 2 \langle \nabla f, \nabla R \rangle \\
\displaystyle & \hspace{1.5cm} + \Delta | \nabla f|^2 - 2 Rc ( \nabla f, \nabla f) + 2 \langle \nabla f, \nabla \Delta f \rangle \\
\displaystyle & = 4 R^{ij} \nabla_i \nabla_j f + 2 | Rc|^2 + \frac{2n}{\tau^2} + 2| \nabla \nabla f|^2 - 2 \langle \nabla f, \nabla | \nabla f|^2 \rangle \\ \displaystyle & \hspace{1.5cm}+ 2 \langle \nabla f, \nabla R \rangle + 4\langle \nabla f, \nabla \Delta f \rangle \\
\displaystyle & = 4 R^{ij} \nabla_i \nabla_j f + 2 | Rc|^2 + \frac{2n}{\tau^2} + 2| \nabla \nabla f|^2 + 2 \langle \nabla f, \nabla P \rangle\\
\displaystyle & = 2 | R_{ij} + \nabla_i \nabla_j f |^2 +\frac{2n}{\tau^2} + 2 \langle \nabla f, \nabla P \rangle. \end{align*} By direct computation we notice that \begin{align*}
\Big| R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij} \Big|^2 = | R_{ij} + \nabla_i \nabla_j f |^2 - \frac{2}{\tau}( R + \Delta f ) + \frac{n}{\tau^2}, \end{align*} which implies \begin{align*}
2 | R_{ij} + \nabla_i \nabla_j f |^2 + \frac{2n}{\tau^2} = 2 \Big| R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij} \Big|^2 + \frac{4}{\tau}( R + \Delta f ). \end{align*} Also \begin{align*}
\displaystyle \frac{4}{\tau}( R + \Delta f ) &= \frac{2}{\tau}( R + 2\Delta f ) + \frac{2}{\tau} R \\
\displaystyle & = \frac{2}{\tau} P + \frac{2}{\tau} | \nabla f|^2 + \frac{4n}{\tau^2} + \frac{2}{\tau} R. \end{align*} Therefore, by putting these together we have \begin{align*}
\Big( \frac{\partial}{\partial t } +\Delta \Big) P = 2 \langle \nabla f, \nabla P \rangle + 2 \Big| R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij} \Big|^2 +\frac{2}{\tau} P + \frac{2}{\tau} | \nabla f|^2 + \frac{4n}{\tau^2} + \frac{2}{\tau} R, \end{align*} which proves the evolution equation for $P$.
To prove that $ P \leq 0$ for all time $t \in [ 0, T]$, we know that for small $ \tau$, $ P(\tau) <0$. We can use the Maximum principle to conclude this. Notice that by the Perelman's $\mathcal{W}$-entropy monotonicity
$$ R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij} \geq 0$$
and strictly positive except when $g(t)$ is a shrinking gradient soliton. So our conclusion will follow from a theorem in \cite[Theorem 4]{[CxZz]}.
For completeness we show this;
by Cauchy-Schwarz inequality and the fact that $ R = g^{ij} R_{ij}$ and $ \sum_{i,j} g_{ij} = n$, we have
$$ | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij} |^2 \geq \frac{1}{n} ( R + \Delta f - \frac{n}{\tau} )^2 $$
and by definition of $P$
$$ P + R + | \nabla f |^2 = 2 ( R + \Delta f - \frac{n}{\tau} ). $$
Hence
$$2 \Big| R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} \Big|^2 \geq \frac{1}{2n} (P + R + | \nabla f |^2)^2.$$ Putting the last identity into the evolution equation for $P$ yields
\begin{align*}
\frac{\partial P}{\partial t } & \geq - \Delta P + 2 \langle \nabla P, \nabla f \rangle + \frac{1}{2n} (P + R + | \nabla f |^2)^2 + \frac{2}{\tau } (P + R + | \nabla f |^2) + \frac{4 n}{\tau^2}\\
\displaystyle & = - \Delta P+ 2 \langle \nabla P, \nabla f \rangle + \frac{1}{2n} (P + R + | \nabla f |^2 + \frac{2n }{\tau^2})^2 + \frac{2n }{\tau^2 }.
\end{align*} This implies that
\begin{align*}
\frac{\partial P}{\partial \tau } \leq \Delta P- 2 \langle \nabla P, \nabla f \rangle - \frac{1}{2n} (P + R + | \nabla f |^2 + \frac{2n }{\tau^2})^2 - \frac{2n }{\tau^2 }.
\end{align*} Then
\begin{align}\label{evq}
\frac{\partial P}{\partial \tau } \leq \Delta P- 2 \langle \nabla P, \nabla f \rangle.
\end{align} Applying the maximum principle to the evolution equation (\ref{evq}) yields clearly that $P \leq 0$ for all $\tau$, hence, for all $t \in [0, T).$
\end{proof}
The result here is an improvement on Kuang and Zhang's \cite{[KZh08]} since it holds with no assumption on the curvature.
This result can also be compared with those of \cite{[CaH09],[Cao08]} where they define a general Harnack quantity for conjugate heat equation and derive its evolution under the Ricci flow.
\subsection{Main Result II. (Pointwise Harnack Estimates)} The aim of this subsection is to state and proof the Li-Yau type pointwise Harnack estimate corresponding to the Harnack inequality proved in the last subsection. We introduce some notations. Given $x_1 , x_2 \in M$ and $t_1, t_2 \in [0,T]$ satisfying $ t_1 < t_2$
$$ \Theta( x_1, t_1; x_2, t_2) = \inf_{\gamma} \int_{t_1}^{t_2} \Big| \frac{d}{dt} \gamma(t) \Big|^2 dt,$$
where the infimum is taken over all the smooth path $ \gamma :[t_1, x_2] \rightarrow M$ connecting $x_1$ and $x_2$. The norm $|.|$ depends on $t$. We now present a lemma which is crucial to the proof of our main result in this subsection.
\begin{lemma} Let $( M, g(t))$ be a complete solution to the Ricci flow. Let $u: M \times [0,T] \rightarrow \mathbb{R}$ be a smooth positive solution to the heat equation (\ref{eq11}). Define $ f = \log u$ and assumed that
$$ - \frac{ \partial f}{\partial t} \leq \frac{1}{ \alpha} \Big( \frac{\beta}{ t} - | \nabla f|^2 \Big), \ \ \ ( x,t) \in M \times [0, T] $$ for some $ \alpha, \beta > 0$. Then, the inequality \begin{equation}\label{CLem} u(x_2, t_2 ) \leq u( x_1, t_1) \Big ( \frac{t_2}{t_1}\Big)^{\frac{\alpha}{\beta}} \exp \Big( \frac{\alpha}{4} \Theta( x_1, t_1; x_2, t_2) \Big) \end{equation} holds for all $( x_1, t_1)$ and $(x_2, t_2)$ such that $ t_1 < t_2.$ \end{lemma} \begin{proof} Obtain the time differential of a function $f$ depending on the path $\gamma$ as follows \begin{align*}
\frac{d}{d t} f( \gamma(t), t) &= \nabla f( \gamma(t), t) \frac{d}{d t} \gamma(t) - \frac{\partial }{\partial s}( \gamma(t), s) \Big|_{ s= t} \\
& \leq \Big| \nabla f \Big| \Big| \frac{d}{d t} \gamma(t) \Big| + \frac{1}{ \alpha} \Big( \frac{\beta}{ t} - | \nabla f |^2 \Big) \\
& \leq \frac{\alpha}{4} \Big| \frac{d}{d t} \gamma(t)\Big|^2 + \frac{\beta}{ \alpha t}. \end{align*} The last inequality was obtained by the application of completing the square method in form of a quadratic inequality satisfying $ax^2 - b x \geq - \frac{b^2}{ 4a} , \ \ ( a , b > 0 )$. Then integrating over the path from $ t_1$ to $ t_2$, we have \begin{align*}
f ( x_2, t_2) - f( x_1, t_1) &= \int_{t_1}^{t_2} \frac{d}{d t} f( \gamma(t), t) dt \\
& \leq \frac{\alpha}{4} \int_{t_1}^{t_2} \Big| \frac{d}{d t} \gamma(t)\Big|^2 dt + \frac{\beta}{ \alpha } \log t \Big|_{t_1}^{t^2}. \end{align*} The required estimate ( \ref{CLem}) follows immediately after exponentiation. \end{proof}
We have the following as an immediate consequence of the above theorem
\begin{corollary}\label{cor35} (Harnack Estimates).
Let $ u \in C^{2,1} ( M \times[0,T))$ be a positive solution to the conjugate heat equation $ \Gamma^* u = 0 $ and $g(t), t \in [0,T)$ evolve by the Ricci flow on a closed manifold $M$ with nonnegative scalar curvature $R$. Then for any points $ (x_1, t_1)$ and $ (x_2, t_2)$ in $ M \times (0,T)$ such that $ 0 < t_1 \leq t_2 < T$, the following estimate holds
\begin{equation}
\frac{u(x_2, t_2)}{u(x_1, t_1)} \leq \Big( \frac{\tau_1}{\tau_2}\Big)^n \exp \Big[ \int_0^1 \frac{ | \gamma'(s) |^2 }{2 (\tau_1 - \tau_2)} ds + \frac{ (\tau_1 - \tau_2)}{2} R \Big],
\end{equation}
where $\tau_i =T - t_i, i =1,2$ and $\gamma : [0,1]$ is a geodesic curve connecting points $x_1$ and $x_2$ in $M.$
\end{corollary}
\begin{proof}
Let $\gamma : [0,1]$ be a minimizing geodesic connecting points $x_1$ and $x_2$ in $M$ such that $ \gamma(0) = x_1$ and $ \gamma(1) = x_2$ with $ | \gamma'(s)|$ being the length of the vector $\gamma'(s)$ at time $ \tau(s) = (1-s) \tau_1 + s \tau_2, \ \ \ 0 \leq \tau_2 \leq \tau_1 \leq T.$
Define $\eta(s) = \ln u( \gamma(s), (1-s) \tau_1 + s \tau_2)$. Clearly, $\eta(0) = \ln u(x_1, t_1)$ and $\eta(1) = \ln u(x_2, t_2).$
Integrating along $ \eta(s)$, we obtain
$$ \ln u(x_2, t_2) - \ln u(x_1, t_1) = \int_0^1 \Big( \frac{ \partial }{\partial s} \ln u(\gamma(s), (1-s) \tau_1 + s \tau_2) \Big) ds $$
i.e.,
$$ \ln \Big( \frac{u(x_2, t_2)}{u(x_1, t_1)} \Big) = \ \ln u( \gamma(t), t) \Big|_0^1 .$$
By direct computation, we have on the path $ \gamma(s)$ that
\begin{align*}
\frac{ \partial }{\partial s} \eta(s) = \frac{ d}{d s} \ln u &= \nabla \ln u \cdot \gamma'(s) +\frac{\partial }{\partial t} \ln u \\
\displaystyle & = \frac{\nabla u}{u} \cdot \gamma'(s) - \frac{u_t (\tau_1 - \tau_2)}{u} \\
\displaystyle & = (\tau_1 - \tau_2) \Big( \frac{\nabla u}{u} \cdot \frac{\gamma'(s)}{\tau_1 - \tau_2} - \frac{u_t }{u} \Big).
\end{align*}
From Theorem \ref{thm Heqn1}, we have
$$ \frac{| \nabla u |^2}{ u^2} - 2 \frac{u_t}{u} \leq R + \frac{2n}{\tau}, $$
which implies
$$ - \frac{u_t}{u} \leq \frac{1}{2} ( R + \frac{2n}{ \tau} ) - \frac{| \nabla u |^2}{ 2 u^2}.$$
By this, we have
\begin{align*}
\frac{d}{d s} \ln u &\leq (\tau_1 - \tau_2) \Big( \frac{\nabla u}{u} \cdot \frac{\gamma'(s)}{(\tau_1 -\tau_2)} - \frac{|\nabla u |^2}{2 u^2} + \frac{1}{2}( R + \frac{2 n}{ \tau}) \Big) \\
\displaystyle & = - \frac{(\tau_1 -\tau_2)}{2}\Big( \frac{\nabla u}{u} - \frac{\gamma'(s)}{(\tau_1 -\tau_2)} \Big)^2 \\
& \hspace{1.5cm} + \frac{(\tau_1 -\tau_2)}{2} \frac{|\gamma'(s)|^2}{(\tau_1 -\tau_2)^2} + \frac{(\tau_1 -\tau_2)}{2}\Big( R + \frac{2 n}{ \tau}\Big)\\
\displaystyle & \leq \frac{|\gamma'(s)|^2}{2 (\tau_1 -\tau_2)}
+ \frac{(\tau_1 -\tau_2)}{2} \Big( R + \frac{2 n}{ \tau}\Big).
\end{align*}
Now integrating with respect to $s$, from $0$ to $1$, we have
\begin{equation}
\ln u \Big|_0^1 \leq \int_0^1 \frac{|\gamma'(s)|^2}{2 (\tau_1 -\tau_2)} + \frac{(\tau_1 -\tau_2)}{2} \int_0^1 R ds + \ln \Big( \frac{\tau_1}{\tau_2} \Big)^n,
\end{equation} exponentiating both sides, we get
\begin{equation*}
\frac{u(x_2, t_2)}{u(x_1, t_1)} \leq \Big( \frac{\tau_1}{\tau_2}\Big)^n \exp \Big[ \int_0^1 \frac{ | \gamma'(s) |^2 }{2 (\tau_1 - \tau_2)} ds + \frac{ (\tau_1 - \tau_2)}{2} R \Big].
\end{equation*}
\end{proof}
\section{Main Result III. (Localising the Harnack and Gradient Estimates)}\label{sec3}
We establish a localised form of the Harnack and gradient estimates obtained in the last subsection. The main idea is the application of the Maximum principle on some smooth cut-off function. It was also the basic idea used by Li and Yau in \cite{[LY86]}, this type of approach has since become tradition. It has been systematically developed over the years since the paper of Cheng and Yau \cite{[CY75]}, see also \cite{[SY94],[Yau75]}, however our computation is more involved as the metric is also evolving.
A natural function that will be defined on $M$ is the distance function from a given point, namely, let $ p \in M$ and define $d(x, p)$ for all $ x\in M,$ where $dist(\cdot, \cdot )$ is the geodesic distance. Note that $d(x, p)$ is only Lipschitz continuous, i.e., everywhere continuously differentiable except on the cut locus of $p$ and on the point where $x$ and $p$ coincide. It is then easy to see that
$$ | \nabla d | = g^{ij} \partial_i d \ \partial_j d = 1 \ \ on \ \ \ M \setminus \{ \{ p \} \cup cut(p) \} .$$ Let $d(x, y, t)$ be the geodesic distance between $x$ and $y$ with respect to the metric $g(t)$, we define a smooth cut-off function $ \varphi(x, t)$ with support in the geodesic cube $$ \mathcal{Q}_{ 2\rho, T} := \{ ( x, t) \in M \times (0, T] : d(x, p, t) \leq 2\rho \} ,$$ for any $C^2$-function $ \psi( s)$ on $[0, + \infty )$ with \begin{equation*} \psi(s) = \left \{ \begin{array}{l}
\displaystyle 1 , \hspace{1.5cm} s \in [0,1], \\
\displaystyle 0, \hspace{1.5cm} s \in [2,+ \infty) \end{array} \right. \end{equation*} and
$$ \psi'(s) \leq 0, \ \ \ \frac{| \psi'|^2}{\psi} \leq C_1 \ \ \ and \ \ | \psi''(s)| \leq C_2,$$ where $C_1, C_2$ are absolute constants depending only on the dimension of the manifold, such that
$$ \varphi(x, t) = \psi \Big( \frac{d( x, p, t)}{\rho } \Big) \ \ \ \ and \ \ \ \varphi \Big|_{ \mathcal{Q}_{ 2\rho, T} } =1 .$$ We will apply the maximum principle and invoke Calabi's trick \cite{[Ca58]} to assume everywhere smoothness of $ \varphi(x, t)$ since $ \psi(s)$ is in general Lipschitz. We need Laplacian comparison theorem to do some calculation on $\varphi(x, t)$. Here is the statement of the theorem; Let $M$ be a complete $n$-dimensional Riemannian manifold whose Ricci curvature is bounded from below by $ Rc \geq (n-1)k $ for some constant $ k \in \mathbb{R}$. Then the Laplacian of the distance function satisfies \begin{equation}\label{eq338} \Delta d(x, p) \leq \left \{ \begin{array}{l}
\displaystyle (n-1) \sqrt{k} \cot ( \sqrt{k} \rho) , \hspace{1.8cm} \ k >0 \\ \ \\
\displaystyle (n-1) \rho^{-1}, \hspace{3.4cm} k =0 \\ \ \\
\displaystyle (n-1) \sqrt{|k|} \coth ( \sqrt{|k|} \rho) , \hspace{1.1cm} \ k < 0. \end{array} \right. \end{equation} For detail of the Laplacian comparison theorem see \cite[Theorem 1.128]{[CLN06]} or the book \cite{[SY94]}. We are now set to prove the localized version of the gradient estimate for the system (\ref{Heqn1}).
\begin{theorem} \label{thmL} Let $ u \in C^{2, 1}(M \times [0,T])$ be a positive solution to the conjugate heat equation $ \Gamma^* u = ( - \partial_t - \Delta + R )u = 0$ defined in geodesic cube $ \mathcal{Q}_{ 2\rho, T}$ and the metric $g(t)$ evolves by the Ricci flow in the interval $[0, T]$ on a closed manifold $M$ with bounded Ricci curvature, say $ Rc \geq -Kg$, for some constant $K > 0$. Suppose further that $u = ( 4 \pi \tau )^{-\frac{n}{2}} e^{-f}$, where $ \tau = T-t$, then for all points in $ \mathcal{Q}_{ 2\rho, T}$ we have the following estimate
\begin{equation}\label{eq410}
\frac{| \nabla u|^2}{ u^2} - 2 \frac{u_t}{u}- R \leq \frac{4n}{ 1 - 4 \delta n} \Bigg \{ \frac{1}{\tau} + C \Bigg( \frac{1}{\rho^2} + \frac{\sqrt{K}}{\rho} + \frac{K}{\rho} + \frac{1}{T} \Bigg) \Bigg \},
\end{equation}
where $C$ is an absolute constant depending only on the dimension of the manifold and $\delta$ such that $\delta < \frac{1}{4n}.$ \end{theorem} \begin{proof}
Recall the evolution equation for the differential Harnack quantity $$ P = 2 \Delta f -| \nabla f|^2 + R - \frac{ 2n}{\tau}, $$
\begin{equation*}
\frac{\partial }{\partial t} P \geq - \Delta P + 2 \langle \nabla f, \nabla P \rangle + 2 | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2 + \frac{2}{\tau} P + \frac{ 4n}{\tau^2} + \frac{ 2}{\tau} |\nabla f|^2,
\end{equation*}
using the non negativity of the scalar curvature
Multiplying the quantity $P$ by $t \varphi$, since $ \varphi$ is time-dependent we have at any
point where $ \varphi \neq 0$ that \begin{align*}
\frac{1}{\tau } \frac{\partial }{\partial t}( \tau \varphi P)& = \varphi \frac{\partial P}{\partial t} + \frac{\partial \varphi }{\partial t} P -\frac{ \varphi P}{\tau}\\
& \geq \varphi \Big( - \Delta P + 2 \langle \nabla f, \nabla P \rangle + \frac{2}{\tau} P + \frac{ 4n}{\tau^2} + \frac{ 2}{\tau} |\nabla f|^2 \Big) \\
& + 2 \varphi | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2 + \frac{\partial \varphi }{\partial t} P - \frac{ \varphi P}{\tau}\\ & = - \Delta ( \varphi P) + 2 \nabla \varphi \nabla P + 2 \langle \nabla f, \nabla P \rangle \varphi + P( \Delta + \partial_t ) \varphi \\
& + \frac{ 4n}{\tau^2} \varphi + \frac{ \varphi P}{\tau} + \frac{ 2}{\tau} \varphi |\nabla f|^2 + 2 \varphi | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2. \end{align*} The last equality is due to derivative test on $(\varphi P)$ at the minimum point as obtained in the condition (\ref{mincond}) below. The approach is to estimate $ \frac{\partial}{\partial t}( \tau \varphi P)$ at the point where minimum (or maximum) value for $(\tau \varphi P)$ is attained and do some analysis at the minimum (or maximum) point. We know that the support of $(\tau \varphi P)(x, t)$ is contained in $ \mathcal{Q}_{2\rho} \times [0, T]$ since $$ Supp ( \varphi) \subset \mathcal{Q}_{ 2\rho, T} := \{ ( x, t) \in M \times (0, T] : d(x, p, t) \leq 2\rho \} .$$ Now let $ (x_0, t_0)$ be a point in $\mathcal{Q}_{ 2\rho, T}$ at which $( \tau \varphi P)$ attains its minimum value. At this point, we have to assume that $P$ is positive since if $P \leq 0$, we have the same estimate and $( \tau \varphi P)(x_0, t_0) \leq 0$ implies $( \tau \varphi P)(x, t) \leq 0$ for all $ x \in M$ such that the distance $d( x, x_0, t) \leq 2\rho $ and the theorem will follow trivially.
Note that at the minimum point $( x_0, t_0)$ we have by the derivative test that $( 0 \leq \varphi \leq 1)$
\begin{align}\label{eq215}
\nabla ( \tau \varphi P) (x_0, t_0) = 0 , \ \ \ \frac{\partial }{\partial t} ( \tau \varphi P) (x_0, t_0) \leq 0 \ \ \ and \ \ \ \Delta ( \tau \varphi P)(x_0, t_0) \geq 0.
\end{align}
We shall obtain a lower bound for $ \tau \varphi P$ at this minimum point. Therefore
\begin{align}\label{eq400}
\left. \begin{array}{l}
\displaystyle 0 \geq - \Delta ( \varphi P) + 2 \nabla \varphi \nabla P + 2 \langle \nabla f, \nabla P \rangle \varphi + P( \Delta + \partial_t ) \varphi + \frac{ \varphi P}{\tau} \\
\displaystyle \hspace{1cm} + \frac{ 4n}{\tau^2} \varphi + \frac{ 2}{\tau} \varphi |\nabla f|^2 + 2 \varphi | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2. \end{array} \right.
\end{align} By the argument in (\ref{eq215}) and product rule we have
$$ \nabla( \varphi P) (x_0, t_0) - P \nabla \varphi (x_0, t_0) = \varphi \nabla P (x_0, t_0) $$
which means $ \varphi \nabla P $ can always be replaced by $ - P \nabla \varphi$. Similarly,
\begin{align}\label{mincond}
- \varphi \Delta P = - \Delta ( \varphi P) + P \Delta \varphi + 2 \nabla \varphi \nabla P,
\end{align}
which we have already used before the last inequality.
Notice that by direct calculation using product rule
$$ \nabla \varphi \nabla P = \frac{ \nabla \varphi }{\varphi} \cdot \nabla( \varphi P ) - \frac{| \nabla \varphi|^2}{\varphi} P$$ and $$ 2 \langle \nabla f, \nabla P \rangle \varphi = \langle \nabla f, \nabla( \varphi P) \rangle - \langle \nabla f, \nabla \varphi \rangle P. $$ Putting the last two equations into (\ref{eq400}) we have \begin{align*}
0&\geq - \Delta ( \varphi P) + 2 \frac{ \nabla \varphi }{\varphi} \cdot \nabla( \varphi P ) - 2 \frac{| \nabla \varphi|^2}{\varphi} P + 2 \langle \nabla f, \nabla( \varphi P) \rangle -2 \langle \nabla f, \nabla \varphi \rangle P \\
& + P( \Delta + \partial_t ) \varphi + \frac{ \varphi P}{\tau} + \frac{ 4n}{\tau^2} \varphi + \frac{ 2}{\tau} \varphi |\nabla f|^2 + 2 \varphi | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2 . \end{align*} By using the argument in (\ref{eq215})
\begin{align}\label{eq413}
\left. \begin{array}{l}
\displaystyle 0 \geq - 2 \frac{| \nabla \varphi|^2}{\varphi} P -2 \langle \nabla f, \nabla \varphi \rangle P + P( \Delta + \partial_t ) \varphi + \frac{\varphi P}{\tau}\\
\displaystyle \hspace{1cm} + \frac{ 4n}{\tau^2} \varphi + \frac{ 2}{\tau} \varphi |\nabla f|^2 + 2 \varphi | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2 \end{array} \right \}.
\end{align}
Observe that for any $ \delta >0$,
$$ 2 | \nabla f| | \nabla \varphi| P = 2 \varphi| \nabla f| \frac{| \nabla \varphi |}{\varphi} P \leq \delta \varphi | \nabla f|^2 P + \delta^{-1} \frac{| \nabla \varphi|^2}{\varphi} P $$
\begin{equation}\label{eq414}
2 | \nabla f| | \nabla \varphi| P \leq \delta \varphi | \nabla f|^4 P + \delta \varphi P^2 + \delta^{-1} \frac{| \nabla \varphi|^2}{\varphi} P
\end{equation}
and also that
$$ | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2 \geq \frac{1}{n} \Big( R + \Delta f - \frac{n}{\tau} \Big)^2. $$
It is equally clear that
$$ P = 2 \Delta f - | \nabla f |^2 + R - \frac{ 2n}{\tau} = 2 \Big( R + \Delta f - \frac{n}{\tau} \Big)^2 - | \nabla f |^2 - R ,$$
which implies
$$( P + | \nabla f |^2 + R ) = 2 \Big( \Delta f + R - \frac{n}{\tau} \Big).$$
Therefore
$$ 2 \varphi | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2 \geq \frac{ \varphi}{2 n} \Big( P + | \nabla f |^2 + R \Big)^2. $$ Notice also that \begin{align*}
( P + | \nabla f |^2 + R )^2 (y, s) & = (P + | \nabla f |^2 + R^+ - R^- )^2(y, s) \\
& \geq \frac{1}{2}( P + | \nabla f |^2 + R^+)^2(y,s) - (R^-)^2(y,s) \\
& \geq \frac{1}{2}( P + | \nabla f |^2 )^2(y,s) - (R^-)^2(y,s) \\
& \geq \frac{1}{2}( P^2 + | \nabla f |^4 )(y,s) - ( \sup_{\mathcal{Q}_{2 \rho, T}} R^-)^2 \\
& \geq \frac{1}{2}( P^2 + | \nabla f |^4 )(y,s) - n^2 K^2, \end{align*} where we have applied some inequalities, namely; $2(a -b)^2 \geq a^2 - 2 b^2$ and $( a +b)^2 \geq a^2 + b^2 $ with $ a , b \geq 0$ and a lower bound assumption on Ricci curvature, $ R_{ij} \geq - K,$ which implies $R = -nK\ \implies R^- \leq nK$ and $R = -R^-$.
Hence
\begin{equation}\label{eq415}
2 \varphi | R_{ij} + \nabla_i \nabla_j f - \frac{1}{\tau} g_{ij}|^2 \geq \frac{\varphi}{4 n} P^2 + \frac{\varphi}{4 n} | \nabla f|^4 .
\end{equation}
Whereever $ P < 0$, we then obtain from (\ref{eq413}) - (\ref{eq415}) that
\begin{align*}
0& \geq \Big( \frac{1 }{4 n} - \delta \Big) \varphi P^2 + \Bigg \{ ( \delta^{-1} - 2) \frac{| \nabla \varphi|^2}{\varphi} + ( \Delta + \partial_t ) \varphi + \frac{\varphi}{\tau} \Bigg \} P \\
& \hspace{3cm} - \Big( \delta - \frac{1 }{4 n} \Big) \varphi | \nabla f|^4 + \frac{2 }{\tau} \varphi | \nabla f|^2 + \frac{ 4 n}{\tau^2} \varphi, \end{align*}
using the inequality of the form $ m | \nabla f|^4 - n | \nabla f|^2 \geq - \frac{n^2}{4 m}$ and multiplying by $ \varphi$ again ($ \varphi \neq 0$), we have a quadratic polynomial in $ ( \varphi P )$ which we use to bound $( \varphi P)$ in the following
\begin{align}\label{eq416}
\left. \begin{array}{l}
\displaystyle \Big( \frac{1 }{4 n} - \delta \Big) ( \varphi P )^2 + \Bigg \{ ( \delta^{-1} - 2) \frac{| \nabla \varphi|^2}{\varphi} + ( \Delta + \partial_t ) \varphi + \frac{\varphi}{\tau} \Bigg \} ( \varphi P) \\ \ \\
\displaystyle \hspace{3cm} - \frac{ 4 n}{\tau^2} \Big( \frac{1}{ 1 - 4 n \delta } - 1 \Big) \varphi^2 \leq 0. \end{array} \right \}. \end{align} Note that if there is a number $x \in \mathbb{R}$ satisfying inequality $ px^2 + q x + r \leq 0, $ when $ p > 0, q > 0$ and $r <0$, then $ q^2 - 4pr > 0$ and we then have the bounds $$ \frac{-q - \sqrt{q^2 - 4pr}}{ 2p} \leq x \leq \frac{-q + \sqrt{q^2 - 4pr}}{ 2p}, $$ which clearly implies $$ \frac{-q - \sqrt{ - 4pr}}{ p} \leq x \leq \frac{q + \sqrt{- 4pr}}{ p}. $$ Now, choosing $\delta$ such that $ \delta < \frac{1}{4n}$ and denoting
$$ Z = ( \delta^{-1} - 2) \frac{| \nabla \varphi|^2}{\varphi} + ( \Delta + \partial_t ) \varphi, $$ we obtain $$ \tau_0 \varphi P \geq - \frac{4n}{ 1 - 4 \delta n} \Bigg \{ \tau_0 Z + \varphi + 4 \varphi\sqrt{ { \delta n}} \Bigg \}.$$ Moreover, since $\tau_0 \leq \tau \leq T $ and $ 0 \leq \varphi \leq 1$, we have $$ \tau P \geq - \frac{4n}{ 1 - 4 \delta n} \Big \{ \tau Z + 1 + C_3 \Big \},$$ where $C_3$ depends on $n$ and $ \delta$. It remains to estimate $Z$ via appropriate choice of a cut function $ \varphi : M \times [0, T] \rightarrow [0, 1]$
such that $ \frac{\partial }{\partial t} \varphi, \Delta \varphi$ and
$ \frac{| \nabla \varphi|^2}{\varphi} $ have appropriate upper bounds.
The main difficulty with this kind of approach lies in the fact that for any
cut-off function, one gets different kind of estimates and therefore the
cut-off function in use must be chosen so related to the result one is looking for.
Define a $ C^2$-function $ 0 \leq \psi \leq 1$, on $[0, \infty )$ satisfying
$$ \psi'(s) \leq 0, \ \ \ \frac{| \psi'|^2}{\psi} \leq C_1 \ \ \ and \ \ | \psi''(s)| \leq C_2$$ and define $\varphi$ by $$ \varphi(x, t) = \psi \Big( \frac{d( x, x_0, t)}{\rho } \Big) $$ and we have the following after some computations \begin{align*}
\frac{| \nabla \varphi |^2}{ \varphi} = \frac{| \psi' |^2 \cdot | \nabla d|^2 }{\rho^2 \varphi} \leq \frac{C_2}{\rho^2},
\end{align*} and by the Laplacian comparison Theorem (\ref{eq338}) we have \begin{align*}
\Delta \varphi &= \frac{ \psi' \Delta d}{\rho} + \frac{ \psi'' | \nabla d |^2}{\rho^2} \leq \frac{ C_1}{\rho} \sqrt{K} + \frac{C_2}{\rho^2}
\end{align*}
Next is to estimate time derivative of $\varphi$: consider a fixed smooth path $\gamma :[a, b] \to M$ whose length at time $t$ is given by $d(\gamma) = \int_a^b |\gamma'(t)|_{g(t)} dr$, where $r$ is the arc length. Differentiating we get
$$ \frac{\partial}{\partial t} (d(\gamma)) = \frac{1}{2} \int_a^b \Big|\gamma'(t) \Big|^{-1}_{g(t)} \frac{\partial g}{\partial t} \Big(\gamma'(t), \gamma'(t)\Big) dr = \int_\gamma Rc(\xi, \xi) dr,$$
where $\xi$ is the unit tangent vector to the path $\gamma$. For detail see \cite[Lemma 3.11]{[CK04]}. Now \begin{align*} \frac{\partial}{\partial t} \varphi = \psi' \Big(\frac{d}{\rho} \Big) \frac{1}{\rho} \frac{d}{ dt} (d(x, p, t)) & = \psi' \Big(\frac{d}{\rho} \Big) \frac{1}{\rho} \int_{\gamma} Rc ( \xi(s), \xi(s) ) ds \\ & \leq \frac{\sqrt{C_1}}{\rho} \psi^{\frac{1}{2}} K. \end{align*} Therefore $$ Z \leq \frac{C_2'}{\rho^2} + \frac{ C_1}{\rho} \sqrt{K} + \frac{\sqrt{C_1}}{\rho} K + \frac{C_2}{\rho^2},$$ where $C_2'$ depends on $n$ and $ \delta.$ Hence \begin{align*} \varphi P \geq - \frac{4n}{ 1 - 4 \delta n} \Bigg \{ \frac{1}{\tau} + C \Bigg( \frac{1}{\rho^2} + \frac{\sqrt{K}}{\rho} + \frac{K}{\rho} + \frac{1}{\tau} \Bigg) \Bigg \}, \end{align*} where $ C = \max \{C_1, C_2, C_3 \}.$ The required estimate follows since both minimum and maximum points for $( \varphi P)$ are contained in the cube $\mathcal{Q}_{2\rho, T}.$ \end{proof}
\section{Concluding Remarks} Our main results in Section \ref{sec2} hold for all positive solutions and calculations are done without recourse to reduced length, therefore, they can be seen as improvement on Perelman's which only works for the fundamental solution via his reduced distance function. After a simple modification and $\epsilon $
regularisation method we can get a corresponding result for heat kernel-type function, namely,
if the function $ u(x,t)$ is defined on $ M \times (0, T]$ instead of $ M \times [0, T]$,
it suffices to replace $u(x, t)$ and $g(x,t)$ with $u(x, t+ \epsilon )$ and $g(x,t + \epsilon )$
for a sufficiently small $\epsilon >0$, do similar analysis and later send $\epsilon $ to $0$. The local estimate in Section \ref{sec3} is desirable to extend our result to the case the manifold is noncompact, for example, in the local monotonicity formula and mean value theorem considered in \cite{[EKNT]} a local version is needed. The estimates obtained here are used to prove on diagonal and gaussian-type upper bound for heat kernel under a mild assumption on curvature and a technical lemma involving the best constant in the Sobolev embedding. This will be announced in a forthcoming paper.
\section*{Appendix} \subsection*{Elements of the Ricci Flow} Given an $n$-dimensional Riemannaian manifold $M$ endowed with metric $g$. In local coordinate $ \{ x^i \}, 1 \leq i \leq n $, we can write the metric in component form $$ ds^2 = g = g_{ij} dx^i d x^j. $$ Consider a smooth function $f$ defined on $M$, then, the Laplace-Beltrami operator acting on $f$ is defined by
$$ \Delta f = \frac{1}{ \sqrt{ |g|} }\sum_{i,j}^n \frac{\partial}{\partial x^i} \Big( \sqrt{|g|} g^{ij} \frac{\partial}{\partial x^j} f \Big) = g^{ij} \Big( \partial_i \partial_j f - \Gamma_{ij}^k \partial_k f \Big),$$
where $( g^{ij} ) = ( g_{ij})^{-1} $ is the metric inverse, $|g|$ is the matrix determinant of $( g^{ij} )$ and $ \Gamma_{ij}^k$ are the Christoffel's symbols. The degenerate parabolic partial differential equation $$ \frac{\partial }{ \partial t} g_{ij} = - 2 R_{ij} $$ is the Ricci flow on $(M, g(t))$, where $R_{ij}$ is the component of the Ricci curvature tensor and $g(t)$ is a one-parameter family of Riemannian metrics. The degeneracy of the pde is due to the group of differomorphism invariance, but we are sure of the existence of solution at least for short a time (Hamilton \cite{[Ha82]}).
Interestingly all geometric quantities associated with $M$ also evolve along the Ricci flow, in the present study we have made use of the following evolutions \begin{align*}
\displaystyle & metric \ inverse : \ \ \ \ \ \ \ \ \ \ \ \frac{\partial}{\partial t } g^{ij} = 2 R^{ij}\\
\displaystyle & volume \ element : \ \ \ \ \ \ \ \ \ \ \frac{\partial}{\partial t } d \mu \ = - R d \mu \\
\displaystyle & Scalar \ curvature : \ \ \ \ \ \ \ \ \frac{\partial}{\partial t} R = \Delta R + 2 | R_{ij}|^2 \\
\displaystyle & Laplacian: \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\partial}{\partial t} \Delta_{g(t)} = 2 R_{ij} \cdot \nabla^i \nabla^j \\
\displaystyle & Christoffel's \ symbols: \ \frac{\partial}{\partial t } \Gamma^k_{ij} = - g^{kl} \Big( \partial_i R_{jl} + \partial_j R_{il} - \partial_l R_{ij} \Big), \end{align*} where $R_{ij}$ is the Ricci curvature ($R = g^{ij} R_{ij}$) and $ \sum_{ij} g^{ij} = n$. The metric is bounded under the Ricci flow (Cf. \cite{[CCG$^+$07],[CK04],[CLN06]}, for more on this and detail of geometric and analytical aspect of the Ricci flow).
\section*{Acknowledgment} The author wishes to acknowledge his PhD thesis advisor Prof. Ali Taheri for constant encouragement. He also thanks the anonymous reviewers for their valuable comments. His research is supported by TETFund of Federal Government of Nigeria and University of Sussex, United Kingdom.
\end{document} | arXiv |
\begin{document}
\title{Binary and ternary quasi-perfect codes with small dimensions}
\begin{center} {\large Tsonka Baicheva, Iliya Bouyukliev, Stefan Dodunekov} \\ {\it Institute of Mathematics and Informatics\\ Bulgarian Academy of Sciences, Bulgaria}\\ {\large Veerle Fack}\\ {\it Vakgroep Toegepaste Wiskunde en Informatica, Universiteit Gent, Belgium}\\ \end{center}
\section{Introduction}
Let $F_q^n$ be the $n$-dimensional vector space over the finite field with $q$ elements $GF(q)$. A {\it linear code} $C$ is a $k$-dimensional subspace of $F_q^n$. For $x,y\in F_q^n$ let $d(x,y)$ denote the Hamming distance between $x$ and $y$, which is equal to the number of positions where $x$ and $y$ differ. The minimum Hamming distance for a code $C$ is defined by $$d(C)=\min_{c_1,c_2\in C,c_1\ne c_2} d(c_1,c_2)$$ and the Hamming weight $w(x)$ of a vector $x\in F_q^n$ is defined by $$w(x)=d(x,{\bf 0})$$ where ${\bf 0}$ is the all zero vector. The packing radius $e(C)$ of the code is $$e(C)=\left\lfloor{{d(C)-1}\over 2}\right\rfloor$$ and this is the maximum weight of successfully correctable errors. The ball of radius $t$ around a word $y\in F_q^n$ is defined by $$\{x\vert x\in F_q^n, d(x,y)\leq t\}.$$ Then $e(C)$ is the largest possible integer number such that the balls of radius $e(C)$ around the codewords are disjoint. The covering radius $R(C)$ of a code $C$ is defined as the least possible integer number such that the balls of radius $R(C)$ around the codewords cover the whole $F_q^n$, i.e. $$R(C)=\max_{x\in F_q^n} \min_{c\in C} d(x,c).$$ With these notations a $q$-ary linear code of length $n$, dimension $k$, minimum distance $d$ and covering radius $R$ is denoted by $[n,k,d]_qR$.
A coset of the code $C$ defined by the vector $x \in F_q^n$ is the set $x+C=\{x+c \ \vert \ c \in C\}$. A coset leader of $x+C$ is a vector in $x+C$ of smallest weight. When the code is linear its covering radius is equal to the weight of the heaviest coset leader. The covering radius of a linear code can also be defined in terms of the parity check matrix.
\begin{theorem} \cite{CHLL} Let $C$ be a $[n,k]$ code with parity check matrix $H$. The covering radius of $C$ is the smallest integer $R$ such that every $q$-ary $(n-k)$-tuple can be written as a linear combination of at most $R$ columns of $H$. \end{theorem}
The special case are codes for which $R(C)=e(C)$ and such codes are called {\it perfect} codes. The problem of finding all perfect codes was begun by Golay in 1949 and completed in 1973 by Tiet\"av\"ainen \cite{T} and independently by Zinov'ev and Leont'ev \cite{ZL}. The only perfect codes are: $[n,n,1]_q0$ codes for each $n\geq 1$; $[2s+1,1,2s+1]_2s$ repetition codes for each $s\geq 1$; code of length $n$ containing only one codeword; $q$-ary codes with the parameters of Hamming codes; the $[23,12,7]_23$ binary Golay code; the $[11,6,5]_32$ ternary Golay code.
The next step in this direction is to consider codes for which packing and covering radii differ by 1, i.e.\ {\it quasi-perfect} codes. A code is called quasi-perfect (QP) if its packing radius is $e$ and its covering radius is $e+1$, for some nonnegative integer $e$. Clearly, the minimum distance of such a code is $2e+1$ or $2e+2$. Then a natural question is which codes are quasi-perfect? It is clear that any code with covering radius 1 and minimum distance 1 or 2 is quasi-perfect. Therefore, quasi-perfect codes with covering radius 1 are not interesting and we will focus on the investigation of quasi-perfect codes with covering radius greater than 1.
\section {Known results about quasi-perfect codes with covering radius greater than 1}
Quasi-perfect codes with covering radius 2 and 3 were extensively studied and many infinite families of binary, ternary and quaternary QP codes are known. In particular, codes with parameters $[n,k,d]_q2$ $d=3,4$ are QP. These codes are connected with 1-saturating sets in projective spaces $PG(n-k-1,q)$ and a lot of infinite families of such codes are described in the literature (see \cite{GS} - \cite{GP}). The following theorem leads to a chain of QP codes.
\begin{theorem} Assume that an $[n,k,d]_q2$ QP code with ${\displaystyle n\leq {{q^{n-k}-1}\over {q-1}} - 2}$ and $3\leq d\leq 4$ exists. Then an$[n+1,k+1,3]_q2$ QP code exists. \end{theorem}
{\it Proof.} Let we add a column to a parity check matrix of a $[n,k,d]_q2$ code to obtain a new $[n+1,k+1,d_{new}]_qR_{new}$ code. According to the Theorem 1 $R_{new}\leq 2$. As the new length ${\displaystyle n+1\leq {{q^{n-k}-1}\over {q-1}} - 1}$, it is impossible that $R_{new}=1$. Also for any new column it holds that $d_{new}\leq 3$. If the new column is not obtained by multiplication of an old column by an element of the field $GF(q)$ then $d_{new}=3$. The right choice of the new column is possible as ${\displaystyle n\leq {{q^{n-k}-1}\over {q-1}} - 2}$.
$[n,n-4,5]_q3$ QP codes correspond to complete arcs in the projective space $PG(3,q)$ and are investigated in \cite{DMP1}, \cite{DFMP}. Survey and new results for the binary QP codes can be found in \cite{EM}. It appears that there is a great variety of QP codes of minimum distance up to 5.
Considerably less is known for $q$-ary QP codes with $q>2$. One infinite family of ternary codes is known due to Gashkov and Sidel'nikov \cite{GS}. The family members are $[(3^s+1)/2,(3^s+1)/2-2s,5]_33$ codes. Quasi-perfectness of two families of quaternary codes, namely $[(4^s-1)/3,(4^s-1)/3-2s,5]_43$ and $[(2^{2s+1}+1)/3,(2^{2s+1}+1)/3-2s-1,5]_43$, presented in \cite{G} and \cite{DZ} was shown by Dodunekov \cite{Dod,Dod1}.
The first computer searches to find new quasi-perfect codes were by Wagner in 1966~\cite{W}. He proposed a tree-search program which uses the properties of parity-check matrices of binary linear quasi-perfect codes to find such codes. Fixing the number of check digits and the number of errors to be corrected, the program finds one quasi-perfect code for each block length if such a code exists. Using this program 27 new binary linear QP codes were found \cite{W,W1}. The codes have lengths between 19 and 55 and all have covering radius 3. Later Simonis \cite{Sim} proved that one of Wagner's codes, namely $[23,14,5]$, is unique.
Baicheva, Dodunekov and K\"otter \cite{BDK} investigated the weight structure and error-correcting performance of the ternary $[13,7,5]$ quadratic-residue code and showed that the covering radius of the code is equal to three, i.e.\ it is a quasi-perfect code. Recently Danev and Dodunekov \cite{DD} proved that this code is the first member of a family of ternary QP codes with parameters $[(3^s-1)/2,(3^s-1)/2-2s,5]_3$ for all odd $s\geq 3$.
All these results lead to the following question: How restrictive is quasi-perfectness, i.e.\ are there inequivalent quasi-perfect codes? In our work we classify all binary of dimension up to 9 and ternary of dimension up to 6 linear QP codes as well as give some partial classifications for dimensions up to 14 and 13 respectively. It turned out that there are many cases where more than one QP code for fixed length and dimension exists. In this way we answer the above question.
\section{Classification of binary and ternary quasi-perfect linear codes}
The approach used in this work is based on the classification of codes with given parameters. First we fix the dimension of the code and determine the possible lengths and minimum distances of the codes which could be quasi-perfect. Then we classify all such codes and finally compute their covering radii. In this way we determine all quasi-perfect codes with the fixed parameters. In order to determine the parameters of possible candidates for quasi-perfect codes with covering radius $e+1$ we take into account that the minimum distance of these codes can only be $2e+1$ or $2e+2$. Brouwer's tables of bounds on the size of linear codes \cite{Bro} and the tables for the least covering radius of binary \cite{CHLL} and ternary linear codes \cite{BV} are used to find the possible lengths of QP codes when the dimension is fixed. Once the parameters (length, dimension and minimum distance) are determined all codes with these parameters are classified up to equivalence using the approach of \cite{Bou1}.
In the classification of the codes two main approaches were used. The first one is based on puncturing, the second one on shortening. While in general the dimension of the code is unchanged by puncturing, this is not true if all non-zero positions of a codeword are deleted. Let $G$ be a generator matrix of a linear $[n,k,d]_q$ code $C$. Then the residual code $\mbox{Res}(C,{\bf c})$ of $C$ with respect to a codeword $c$ is the code generated by the restriction of $G$ to the columns where $c$ has a zero entry. A lower bound on the minimum distance of the residual code is given by
\begin{lemma} \cite{D} Suppose $C$ is an $[n,k,d]_q$ code and suppose ${\bf c} \in C$ has weight w, where $d > w(q-1)/q$. Then $\mbox{Res}(C,{\bf c})$ is an $[n-w,k-1,d']_q$ code with $d' \geq d - w + \lceil{w/q}\rceil$. \end{lemma}
Inverting this operation, we search for an $[n,k,d]_q$ code on the basis of an $[n-w,k-1,d']_q$ code (its residual code with respect to a codeword of weight $w$) or an $[n-i,k,d']_q$ code (punctured on $i$ coordinates code). We can apply the same operation for the residual or punctured code respectively. This procedure is repeated until we obtain as a start code a code with small parameters such that all codes having these parameters can easily be classified. For example, starting from the $[3,2,2]_2$ code, we obtain all $[8,3,5]_2$ codes, take only the nonequivalent of them and then obtain all nonequivalent $[28,4,20]_2$ codes.
The second approach increases both the length and the dimension of the code, i.e.\ we construct $[n,k,d]_q$ codes extending \mbox{$[n{-}i,k{-}i,d]_q$} or $[n-i-1,k-i,d]_q$ codes. The following result shows when the latter type of code can be used \cite[p.\ 592]{MS}.
\begin{lemma} Let $C$ be an $[n,k,d]_q$ code. If there exists a codeword ${\bf c} \in C^{\perp}$ with $wt({\bf c})=i$, then there is an $[n-i,k-i+1,d]_q$ code. \end{lemma}
If $G$ is a generator matrix for an $[n-i,k-i,d]_q$ or an $[n-i-1,k-i,d]_q$ code, we extend it (in all possible ways) to \begin{equation} \left(
\begin{array}{c|c} * & {\bf I}_{i} \\ \hline {\bf G} & {\bf 0} \end{array} \right) \ \ \rm{or} \ \ \left(
\begin{array}{c|c} * & {\bf 1} \ {\bf I}_{i} \\ \hline {\bf G} & {\bf 0} \end{array} \right), \end{equation} respectively, where ${\bf I}_{i}$ is the $i\times i$ identity matrix, {\bf 1} is an all-1 column vector, and the starred submatrix is to be determined. If we let the matrix $G$ be in systematic form, we can fix $k$ more columns to get \begin{equation} \left(
\begin{array}{c|c|c} * & {\bf 0} & {\bf I}_{i} \\ \hline {\bf G}_{1} & {\bf I}_{k} & {\bf 0} \end{array} \right) \ \ \rm{or} \ \ \left(
\begin{array}{c|c|c} * & {\bf 0} & {\bf 1} \ {\bf I}_{i} \\ \hline {\bf G}_{1} & {\bf I}_{k} & {\bf 0} \end{array} \right). \end{equation}
Again we can apply recursively the same approach to obtain $G_1$ while on the bottom of this hierarchy of extensions is the trivial $[k,k,1]_q$ code.
In our investigation we are interested in codes with covering radius greater than 1, therefore they have minimum distance at least 3. Thus their dual codes are projective codes. To classify the binary codes with codimension $n-k$ up to 6 we use the results from \cite{Bou} where all binary projective codes with dimensions up to 6 are classified. Then among the codes from \cite{Bou} we consider only those having the necessary minimum distance of the dual code. For example, to classify all $[8,2,5]$ codes we consider 14 $[8,6]$ projective codes. The dual code of only one of them has minimum distance 5 and thus we have only one $[8,2,5]$ code. In the same way using results from \cite{BB} where all ternary projective codes of dimension 4 are classified we classify the ternary linear codes of codimension 4 which could be quasi-perfect.
After the classification was completed we proceed with the determination of the covering radii of the codes. We recall that in the case of linear codes this is equivalent to the determination of the heaviest coset leader. To do this, we use the fact that if the code is in a systematic form, a representative of each coset can be found by generating all words of the form $(\underbrace {0,\dots ,0}_{k},a)$, $a\in F_q^{n-k}$. Taking into account that each vector of weight less than or equal to $e$ is a unique coset leader, we test only words of the above form and weight greater than $e$. Therefore we have to test at most $\sum_{i=e}^{n-k}{{n-k}\choose{i+1}}(q-1)^{i+1}$ words because if we obtain a coset leader of weight greater than $e+1$ we stop the check.
{\it Remark.} Let us denote by $\alpha_i$ for $i=0,1,\dots ,n$ the number of coset leaders of weight $i$. The set of the coset leaders of each QP code is known. All vectors of weights less than or equal to $e$ are coset leaders and thus $\alpha_i={n\choose i}(q-1)^i$ for $i=0,\dots ,e$. Then for $\alpha_{e+1}$ we get $\alpha_{e+1}=q^k-\sum_{i=0}^e\alpha_i$.
\section{Results} By the approach described in the previous section all binary and ternary quasi-perfect codes of dimensions up to 9 and 6 correspondingly are determined, as well as some partial results for binary codes of dimensions up to 14 and ternary codes of dimensions up to 13 are obtained. The results are summarized in Table I.
\begin{table} \caption{Binary and Ternary Quasi-perfect Codes} \centering
\begin{tabular}{|l|l|l||l|l|l|} \hline
\multicolumn{6}{|c|}{Binary quasi-perfect codes}\\ \hline Code&All&QP&Code&All&QP\\ \hline $[5,2,3]$ & 1 & 1&$[14,9,3]$ & 126 &113 \\ $[6,3,3]$ & 1 & 1&$[15,9,3]$& 11464& 380\\ $[8,2,5]$* & 1 & 1&$[17,9,5]$& 1 & 1\\ $[7,3,3]$ & 3 &2&$[14,10,3]$& 1& 1\\ $[8,4,4]$*& 1 & 1&$[15,10,3]$& 142&131 \\ $[8,4,3]$ & 4 & 4&$[16,10,3]$& 28900& 2296\\ $[9,4,4]$* & 4 & 1&$[19,10,5]$& 31237 & 13\\ $[9,4,3]$ & 19 & 1&$[16,11,4]$*& 1 & 1\\ $[11,4,5]$& 1 & 1&$[16,11,3]$& 143 & 143\\ $[9,5,3]$ & 5 & 5 &$[17,11,4]$*& 39 & 5\\ $[10,5,4]$* & 4 & 1&$[17,11,3]$&70416& 12221\\ $[10,5,3]$& 37 & 12&$[20,11,5]$&13924 & 565\\ $[10,6,3]$& 4 & 4&$[17,12,3]$& 129 & 129\\ $[11,6,3]$& 58& 25&$[18,12,4]$*& 33 & 1\\ $[14,6,5]$& 11 & 1&$[21,12,5]$&2373 & 666\\ $[11,7,3]$& 3 & 3&$[22,12,6]$& 128 & 1\\ $[12,7,3]$& 84 & 55&$[24,12,8]$& 1 & 1\\ $[13,7,4]$*& 45 & 1&$[24,12,7]$& 11& 11\\ $[13,7,3]$& 1660 & 7&$[25,12,8]$& 7 & 2\\ $[15,7,5]$& 6 & 4&$[18,13,3]$& 113& 113 \\ $[12,8,3]$& 2& 2&$[19,13,3]$& 366064& 185208\\ $[13,8,3]$& 109 & 88&$[22,13,5]$&128 & 120\\ $[14,8,3]$& 4419 & 65&$[19,14,3]$& 91 & 91\\ $[13,9,3]$& 1 & 1&$[20,14,4]$*& 24 & 1\\
\hline \hline
\multicolumn{6}{|c|}{Ternary quasi-perfect codes}
\\ \hline Code&All&QP&Code&All&QP\\ \hline $[5,2,3]$* & 2 &2&$[11,7,3]$& 339 & 319\\ $[6,3,3]$ &1 & 1&$[12,7,3]$& 60910 & 1\\ $[7,4,3]$ & 4 &4& $[13,7,5]$& 6 & 5\\ $[8,4,4]$* & 3 &2&$[11,8,3]$& 1 & 1\\ $[8,4,3]$ & 37 &5&$[12,8,3]$& 805 & 753\\ $[8,5,3]$ & 3 &3&$[14,8,5]$ &1 &1\\ $[9,5,3]$ & 87 &23&$[12,9,3]$ &1 &1\\ $[9,6,3]$ & 3 &3&$[13,9,3]$ &1504 &1479\\ $[10,6,4]$*& 1 &1&$[14,10,3]$ &2695 &2659\\ $[10,6,3]$& 195&102&$[15,11,3]$ &4304 &4304\\ $[12,6,6]$& 1 &1&$[16,12,3]$ &6472 &6472\\ $[12,6,5]$& 36 &18&$[17,13,3]$ &8846 &8846\\ $[10,7,3]$& 2 & 2 & & &\\ \hline \end{tabular} \end{table}
Some of the codes from the table are not new and have already been constructed in previous works. We will note that QP codes with minimum distances 3 or 4 and covering radius 2 are connected with 1-saturating sets in projective spaces $PG(n-k-1,q)$ in the following way: the points of a 1-saturating $n$-set can be considered as $n-k$-dimensional columns of a parity-check matrix of an $[n,k]_q2$ code. Also QP codes with minimum distance 4 are complete caps in $PG(n-k-1,q)$. Constructions of minimal 1-saturating sets and complete caps in binary projective spaces $PG(k-1,2)$ are described in \cite{FMMP}, \cite{Dav}, \cite{DMP} - \cite{DMP2}, \cite{KL}. Codes obtained in these works are marked with a *. Some of the marked codes are also obtained in \cite{KL} where recursive constructions of complete caps in $PG(n-k-1,2)$ are given. Existing of codes with parameters $[10,5,3]_22$, $[14,6,5]_23$ and $[13,7,3]_22$ is shown in \cite{GS}. The $[17,9,5]_23$ code is the first representative from the infinite family of $[2^{2s}+1,2^{2s}+1-4,5]_2$, $s\geq 2$ Zetterberg's codes which are proved to be quasi-perfect by Dodunekov \cite{Dod}. $[19,10,5]_23$, $[20,11,5]_23$, $[23,14,5]_23$ and $[24,14,6]_23$ are among the QP codes obtained by a computer search by Wagner. He obtained only one representative for each of the parameters. Our classification shows that QP codes with the first two parameters are not unique. There are additionally 12 $[19,10,5]_2$ and 564 $[20,11,5]_2$ quasi-perfect codes. $[24,12,8]_2$ is the well known extended Golay code which is also known to be a quasi-perfect one.
For a completeness of the classification results about QP codes, we will note some not classified in this work such codes. In \cite{DFMP} the unique $[6,1,5]_23$ and in \cite{DMP1} the unique $[5,1,5]_33$ codes are presented. As complete caps in $PG(4,3)$ the $[16,11,4]_32$, $[17,12,4]_32$ and $[18,13,4]_32$ codes in \cite{FMMP} and in $PG(6,2)$ the $[21,14,4]_22$ code in \cite{GDT} are obtained. In \cite{KL} it is shown that there are 5 nonequivalent $[21,14,4]_22$ codes. Also applying Theorem 2 to codes from the Table the following chains of QP codes' parameters can be obtained.\\ $[5,2,3]_22\rightarrow [6,3,3]_22;$\\ $[8,4,4]_22\rightarrow \dots \rightarrow [14,10,3]_22;$\\ $[9,4,4]_22\rightarrow \dots \rightarrow [30,25,3]_22;$\\ $[13,7,4]_22\rightarrow \dots \rightarrow {\bf [18,12,3]_22}\rightarrow [19,13,3]_22\rightarrow {\bf [20,14,3]_22}\rightarrow \dots \rightarrow {\bf[62,56,3]_22};$\\ $[5,2,3]_32\rightarrow [12,3,3]_32;$\\ $[8,4,4]_32\rightarrow \dots \rightarrow [17,13,3]_32\rightarrow {\bf [18,14,3]_32}\rightarrow \dots \rightarrow {\bf [40,36,3]_32}$;\\ $[12,7,3]_32\rightarrow {\bf [13,8,3]_32}\rightarrow \dots \rightarrow {\bf [121,116,3]_32}$.\\
Codes not classified in this work are boldfaced.
Until this work the only known examples of QP codes with minimum distance greater than 5 were binary repetition codes, the $[24,12,8]_24$ extended Golay code, the $[22,12,6]_23$ punctured Golay code, $[7,1,7]_34$ and $[8,1,7]_24$ codes classified in \cite{DFMP}. We provide examples of more such codes and in this way answer the first open question from the recent paper of Etzion and Mounits \cite{EM} where to find new or to prove the nonexistence of QP codes with $d>5$ is suggested. The most interesting are $[24,12,7]_24$ and $[25,12,8]_24$ codes which are the first examples of quasi-perfect codes with $R=4$ except the $[24,12,8]_24$ extended Golay and the $[8,1,8]_24$ repetition codes. The generator matrices of these codes are given in the Appendix. The codes are in a systematic form with generator matrix $G=[I_k\vert A]$ and the identity matrix $I_k$ is omitted in order to save space.
\section{Conclusions}
In this work classification results about binary and ternary linear quasi-perfect codes of small dimensions are obtained. More precisely, all binary QP codes of dimensions up to 9 and ternary QP codes of dimensions up to 6 are classified as well as some partial classifications about QP codes of dimensions up to 14 are got. The results show that for each dimension there are only few possible lengths for which quasi-perfect codes exist. For some parameters hundreds and thousands of nonequivalent QP codes are found which means that quasi-perfectness is not so restrictive characteristic of the code. QP codes of minimum distance greater than 5 are obtained and therefore it could be expected that at greater dimensions QP codes with bigger covering radii exist. Thus it will be an interesting research problem to answer the following questions:
$\bullet$ Are there quasi-perfect codes with minimum distance greater than 8 except the binary repetition code?
$\bullet$ Is there an upper bound about minimum distance of a QP code?
At the end we will conclude with the observation that the classification of all parameters of QP codes would be much more difficult than the similar one for perfect codes.
\section {Acknowledgement} The authors wish to express her appreciation to the anonymous reviewers whose comments and suggestions greatly improved the paper. Theorem 2 from section II is due to one of the reviewers.
\section {Appendix. Generator matrices of binary quasi-perfect codes with minimum distance 7 and 8}
\centerline{A. $[24,12,7]_24$ QP codes} {\small $A_1=\left( \begin{array}{c} 010010110101\\ 101000111001\\ 111010010010\\ 000110011011\\ 010101010110\\ 101100001110\\ 111111011101\\ 111100110100\\ 101101010011\\ 010101111001\\ 011000011111\\ 000011111110 \end{array} \right )$ $A_2=\left( \begin{array}{c} 110001001011\\ 001011000111\\ 111010010010\\ 100101100101\\ 010101010110\\ 101100001110\\ 011100100011\\ 111100110100\\ 001110101101\\ 110110000111\\ 111011100001\\ 000011111110\end{array} \right )$
$A_3=\left( \begin{array}{c} 010101001011\\ 001100111001\\ 011110010010\\ 100010011011\\ 110110101000\\ 101100001110\\ 111111011101\\ 111011001010\\ 101010101101\\ 110110000111\\ 011000011111\\ 000011111110\end{array} \right )$ $A_4=\left(
\begin{array}{c} 100010111101\\ 001011010011\\ 101110001110\\ 110011101010\\ 110111010100\\ 111101111111\\ 111110100001\\ 011011001101\\ 111010110110\\ 110110011011\\ 001111111000\\ 000111100111\end{array} \right ) $
$A_5=\left( \begin{array}{c} 000010111101\\ 011100110100\\ 011001101001\\ 110100001101\\ 110000110011\\ 101101111111\\ 101110100001\\ 111100101010\\ 111101010001\\ 100110011011\\ 001111111000\\ 010111100111\end{array} \right )$ $A_6=\left(
\begin{array}{c} 000011011110\\ 101010101011\\ 101110010101\\ 010011110001\\ 010110101100\\ 111101111111\\ 111111000010\\ 111011001101\\ 111010110110\\ 010110011011\\ 001111111000\\ 000111100111\end{array} \right )$
$A_7=\left( \begin{array}{c} 110001010101\\ 001011011001\\ 111010010010\\ 000110010111\\ 110110100100\\ 001111100010\\ 111111001111\\ 111100101010\\ 001110101101\\ 110110011001\\ 111011100001\\ 000011111110\end{array} \right ) $ $A_8=\left(
\begin{array}{c} 000011011011\\ 001010101101\\ 001110010110\\ 100011100110\\ 100110110001\\ 101101111111\\ 101111001000\\ 101011010101\\ 101010111010\\ 100110001111\\ 001111100011\\ 010111111100\end{array} \right ) $
$A_9=\left( \begin{array}{c} 010101011101\\ 011011101100\\ 001110001111\\ 100001101011\\ 110111000110\\ 111100100001\\ 101010110010\\ 101101010111\\ 111110011100\\ 110011110101\\ 011000111011\\ 000111111010\end{array} \right )A_{10}=\left(
\begin{array}{c} 110001101110\\ 111111011111\\ 101010111100\\ 010110101101\\ 100100110011\\ 001011100111\\ 011101110100\\ 111110100010\\ 101101101001\\ 010111000110\\ 011000111011\\ 000111111010\end{array} \right )$
$A_{11}=\left( \begin{array}{c} 000111100110\\ 101101010101\\ 101100101011\\ 110000110111\\ 110011011100\\ 111011100001\\ 111110010010\\ 011010101110\\ 011001011011\\ 010101101101\\ 001111111000\\ 000110011111\end{array} \right )$
\centerline{ B. $[25,12,8]_24$ QP codes}
$A_1=\left( \begin{array}{c}1101101100100\\ 1101000111001\\ 1110100001101\\ 1110110110000\\ 1011001110010\\ 1011010101100\\ 1000100111110\\ 0111100101010\\ 0111101010001\\ 0100110011011\\ 0001111111000\\ 0010111100111\end{array} \right ) $ $A_2=\left( \begin{array}{c}1101101110000\\ 1101000100111\\ 1100011101001\\ 1100001011110\\ 1001110010110\\ 1001101001101\\ 1000100111011\\ 0101011010101\\ 0101010111010\\ 0100110001111\\ 0001111100011\\ 0010111111100\end{array} \right)$ }
\end{document} | arXiv |
\begin{document}
\title{\bf Cohomology of compatible BiHom-Lie algebras}
\author{\bf Asif Sania, Basdouri Imed, Bouzid Mosbahi, Nacib Saber }
\author{{ Asif Sania \footnote{Corresponding author, E-mail: [email protected]}, Basdouri Imed $^{1}$
\footnote { Corresponding author, E-mail: [email protected]}
,\ Bouzid Mosbahi$^{2}$
\footnote { Corresponding author, E-mail: [email protected]}
,\ Nacib Saber $^{3}$
\footnote { Corresponding author, E-mail: [email protected]}
}\\
{\small 1. Nanjing University of Information Science and Technology, Nanjing, PR China.}\\
{\small 2. University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, Tunisia } \\
{\small 3. University of Sfax, Faculty of Sciences Sfax, BP
1171, 3038 Sfax, Tunisia}\\
{\small 4. University of Sfax, Faculty of Sciences Sfax, BP
1171, 3038 Sfax, Tunisia}}
\date{}
\maketitle
\begin{abstract} This paper defines compatible BiHom-Lie algebras by twisting the compatible Lie algebras by two linear commuting maps. We show the characterization of compatible BiHom-Lie algebra as a Maurer-Cartan element in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible BiHom-Lie algebras.
\end{abstract}
\textbf{Key words}:\, BiHom-Lie algebra, Compatible BiHom-Lie algebra, Cohomology, graded Lie algebra.
\textbf{Mathematics Subject Classification}:\,16R60, 17B05, 17B40, 17B37
\numberwithin{equation}{section} \tableofcontents
\section{Introduction} The motivation to study BiHom-Lie algebra structure lies in its importance in the physics and deformations theory of Lie algebras, in particular Lie algebras of vector fields. BiHom-Lie algebra was first introduced by Cheng and Qi in \cite{2}, is a $4$-tuple $(g, [\cdot,\cdot], \alpha, \beta)$, where $g$ is a $\mathbb{K}$-linear space, $\alpha, \beta: g \rightarrow g,$ are linear maps and $[\cdot,\cdot] : g \otimes g \longrightarrow g$ is Lie bracket operation on $g$, with notation $[\cdot,\cdot](p \otimes q) = [p,q]$, satisfying the following conditions, for all $p, q, r \in g:$ \begin{center}
$\alpha \circ \beta = \beta \circ \alpha,$\\
$[\beta(p), \alpha(q)] = -[\beta(q),\alpha(p)]$ (skew-symmetry),\\
$[\beta^{2}(p),[\beta(q),\alpha(r)]] + [\beta^{2}(q),[\beta(r),\alpha(p)]] + [\beta^{2}(r),[\beta(p),\alpha(q)]]=0$ (BiHom-Jacobi condition). \end{center} The primary characteristic of BiHom-Lie algebras is that they are an extension of Hom-Lie algebra with one twist map $\alpha$, which is defined in \cite{4, 10}, where the identities characterizing BiHom-Lie algebras are twisted by two twist maps $\alpha, \beta$. If we consider $\alpha=\beta$ then the theory of BiHom-Lie algebras deform to Hom-Lie algebras and by putting $\alpha=\beta=id$, then we get a Lie algebra structure.
It's important to note that (linearly) compatible products have gained a significant amount of attention. In this respect, two products of a particular kind defined on the same vector space are said to be compatible if their sum also defines the same type of algebraic structure. They appeared in a variety of mathematical physics and mathematics problems. For instance, the idea of two Hom-Lie structures being compatible first arose in \cite{3}, where the author developed a generalized algebraic structure with a single commuting multiplicative linear map. This structure was referred to as compatible Hom-Lie algebra. Das recently developed a cohomology theory for the compatible Hom-Lie algebras. This cohomology is based on the characterization of a compatible Hom-Lie algebra as a Maurer-Cartan element in a bidifferential graded Lie algebra. By realizing the importance of both the BiHom Lie algebras and compatible Lie algebras, we study the compatible BiHom-Lie algebras and provide its cohomology theory. Same as Bihom Lie algebras, compatible BiHom-Lie algebras also return to compatible Hom-Lie algebras by choosing the same twist map.
The paper is organized as follows. In Section 2 (preliminary section), we recall some important definitions and notions of BiHom-Lie algebras and bidifferential graded Lie algebras. Section 3 introduces compatible BiHom-Lie algebras and their Maurer-Cartan characterizations in a suitably constructed bidifferential graded Lie algebra. We also define the notion of representation of a compatible BiHom-Lie algebra and construct the semidirect product. The cohomology of a compatible BiHom-Lie algebra with coefficients in a representation is given in Section 4.
\section{BiHom-Lie algebras and bidifferential graded Lie algebras } \subsection{BiHom-Lie algebras} We start by recalling some important remarks about BiHom-Lie algebras defined in the introductory Section.
\begin{re} A BiHom-Lie algebra is called a multiplicative BiHom-Lie algebra if $\alpha$ and $\beta$ are algebra morphisms, i.e. we have $\alpha([p,q]) = [\alpha(p), \alpha(q)], \beta([p,q]) = [\beta(p), \beta(q)]$, for any $p,q \in g$. \end{re} \begin{re} A multiplicative BiHom-Lie algebra is called a regular BiHom-Lie algebra if $\alpha, \beta$ are bijective maps. \end{re} \begin{re} Obviously, a BiHom-Lie algebra $(g, [\cdot,\cdot], \alpha,\beta)$ for which $\alpha = \beta$ is just a Hom-Lie algebra $(g, [\cdot,\cdot], \alpha)$. \end{re} \begin{ex} Let $(g, \mu, \alpha, \beta)$ be a BiHom-associative algebra with a bilinear map $\mu: g\otimes g\to g$ defined by $\mu(p,q)=p\cdot q$ and bijective linear maps $\alpha, \beta:g\to g$. Define a new multiplication $[\cdot,\cdot]$ by $[p,q]= p.q -(\alpha^{-1} \beta(q)).(\alpha\beta^{-1}(p))$, for every $p, q \in g$. We say $(g, [\cdot,\cdot], \alpha, \beta)$ is a BiHom Lie algebra. \end{ex} \begin{ex}
Let $(g,[\cdot,\cdot])$ be an ordinary Lie algebra over a field $\mathbb{K}$ and let $\alpha,\beta:g \rightarrow g$ two commuting linear maps such that $\alpha([p,q])=[\alpha(p),\alpha(q)]$ and $\beta([p,q])=[\beta(p),\beta(q)]$ for all $a, a^{\prime}\in g$. Define the linear maps $\{\cdot,\cdot\}:g\otimes g \longrightarrow g$ by
$\{p,q\}= [\alpha(p),\beta(q)]$ for all $p,q \in g$. Then $g_{(\alpha,\beta)}=( g,\{\cdot,\cdot\},\alpha,\beta)$ is a BiHom-Lie algebra called the twist of Lie algebra $(g, [\cdot,\cdot])$. \end{ex} \begin{defn} Let $(g, [\cdot,\cdot], \alpha,\beta)$ and $(g^{\prime}, [\cdot,\cdot]^{\prime}, \alpha^{\prime},\beta^{\prime})$ be two BiHom-Lie algebras. A linear map $\phi : g \rightarrow g^{\prime}$ is said to be a BiHom-Lie algebra morphism if $\alpha^{\prime} \circ \phi = \phi \circ \alpha,\beta^{\prime} \circ \phi = \phi \circ \beta$ and $\phi[p, q] = [\phi(p), \phi(q)]^{\prime},$ for $p, q \in g$. \end{defn} \begin{ex} Given two BiHom-Lie algebras $(g, [\cdot,\cdot], \alpha,\beta)$ and $(g^{\prime}, [\cdot,\cdot]^{\prime}, \alpha^{\prime},\beta^{\prime})$, there is a BiHom-Lie algebra $(g\oplus g^{\prime}, [\cdot,\cdot ]_{g\oplus g^{\prime}}, \alpha+ \alpha^{\prime}, \beta+ \beta^{\prime})$, where the skew-symmetric bilinear map $[\cdot,\cdot]_{g\oplus g^{\prime}}: \wedge^{2}(g\oplus g^{\prime})\longrightarrow g\oplus g^{\prime}$ is given by $[(p_{1},q_{1}),(p_{2},q_{2})]_{g\oplus g^{\prime}}= ([p_{1},p_{2}],[q_{1},q_{2}]^{\prime})$ for all $p_{1}, p_{2} \in g, q_{1}, q_{2} \in g^{\prime}$ and the linear maps $(\alpha+ \alpha^{\prime}),(\beta+ \beta^{\prime}): g\oplus g^{\prime}\longrightarrow g\oplus g^{\prime}$ are given by
$$(\alpha+\alpha^{\prime})(p, q)= (\alpha(p),\alpha^{\prime}(q)), ~~(\beta+ \beta^{\prime})(p, q)= (\beta(p),\beta^{\prime}(q)),~~ \forall p\in g, q\in g^{\prime}$$ \end{ex}
Next, we recall the graded Lie bracket (called the Nijenhuis-Richardson bracket) whose Maurer-Cartan elements are given by the BiHom-Lie algebra structure. This generalizes the classical Nijenhuis-Richardson bracket in the context of Lie algebra. Let $g$ be a vector space and $\alpha, \beta: g \rightarrow g$ be two linear maps. For each $n \geq 0$, consider the spaces $$C^{0}_{BiHom}(g, g)= \{p \in g ~|~ \alpha(p)= p\textit{ and }\beta(p)= p\}$$ and $$C^{n}_{BiHom}(g, g) = \{f : \otimes^{n}g \rightarrow g~|~\alpha \circ f = f \circ \alpha^{\otimes^{n}}, \beta \circ f = f \circ \beta^{\otimes ^{n}}\},~~for ~~n \geq 1. $$ Then the shifted graded vector space $C^{\ast+ 1}_{BiHom}(g, g)= \oplus_{n\geq0}C^{n+ 1}_{BiHom}(g, g)$ carries a graded Lie bracket defined as follows:\\ For $P \in C^{m+1}_{BiHom} (g, g)$ and $Q \in C^{n+1} _{BiHom}(g,g)$, the Nijenhuis-Richardson bracket $[P,Q]_{NR} \in C^{m+n+1}_{BiHom} (g, g)$ given by $$[P,Q]_{NR} = P \diamond Q - (-1)^{mn} Q \diamond P,$$ where $$(P \diamond Q)(p_{1},..., p_{m+n+1}) = \sum_{\sigma \in Sh(n+1,m)} (-1)^{\sigma} P(Q(p_{\sigma(1)},..., p_{\sigma(n+1)}), \alpha\beta^{n}(p_{\sigma(n+2)}),..., \alpha\beta^{n}(p_{\sigma(n+m+1)})).$$ With this notation, we have the following. \begin{prop}
Let $g$ be a vector space with $\alpha,\beta: g \rightarrow g $ being two linear maps.
Then the BiHom-Lie bracket on $g$ is precisely the Maurer-Cartan element in the graded Lie algebra $(C^{\ast+1}_{BiHom(g, g)}, [\cdot,\cdot]_{NR})$. \end{prop}In the following, we recall the Chevalley-Eilenberg cohomology of a BiHom-Lie algebra $(g, [. , .], \alpha, \beta)$ with coefficients in a representation. \begin{defn}
Let $(g, [\cdot, \cdot], \alpha, \beta)$ be a BiHom-Lie algebra. A representation of $g$ is a
$4$-tuple $(V, \bullet, \alpha_{V}, \beta_{V})$, where $V$ is a linear space, $\alpha_{V}, \beta_{V} : V \rightarrow V$ are two commuting
linear maps and representation $\bullet: g \otimes V \longrightarrow V$ defined by $(p, v)\longmapsto p \bullet v$ is a bilinear operation (called the action) satisfying: \begin{enumerate}
\item $\alpha(p) \bullet \alpha_{V}(v)=\alpha_{V}(p \bullet v)$,
\item$\beta(p) \bullet \beta_{V}(v)=\beta_{V}(p \bullet v)$,
\item$[\beta(p),q]\bullet \beta_{V}(v)= \alpha\beta(p)\bullet(q\bullet v)- \beta(y)\bullet(\alpha(p)\bullet v)$, \end{enumerate} for all $p, q \in g$ and $v\in V$. \end{defn} \begin{ex}
Let $(g, [\cdot, \cdot], \alpha, \beta)$ be a BiHom-Lie algebra and $(V, \bullet, \alpha_{V}, \beta_{V})$ be a representation of $g$. Assume that the maps $\alpha, \beta$ and $\alpha_{V}, \beta_{V}$ are bijective. Then the semi direct products $g\ltimes V= (g\oplus V, [.,.], \alpha\oplus\alpha_{V}, \beta\oplus\beta_{V})$ is a BiHom-Lie algebra. Where $\alpha\oplus\alpha_{V},\beta\oplus\beta_{V}:g\oplus V \longrightarrow g\oplus V$ are defined by
$$(\alpha\oplus\alpha_{V})(p,a)=(\alpha(p),\alpha_{V}(a)), (\beta\oplus\beta_{V})(p,a)=(\beta(x),\beta_{V}(a))$$ and the bracket $[\cdot, \cdot]$ is defined by $$[(p,a),(q,b)]=([p,q], p\bullet b- \alpha^{-1}\beta(y) \bullet \alpha_{V}\beta^{-1}_{V}(a))$$
for all $p, q \in g$ and $a,b \in V$.
\end{ex}It follows that any BiHom-Lie algebra $(g, [\cdot,\cdot], \alpha,\beta)$ is a representation of itself with the action given by the bracket $[. , .]$. This is called the adjoint representation. Let $(g, [. , .], \alpha,\beta)$ be a BiHom-Lie algebra and $(V,\bullet,\alpha_{V},\beta_{V})$ be a representation on it. For each $n \geq 0$, we define the $n$-th cochain on $g$ with the coefficients in the representation $(V, \bullet, \alpha,\beta)$ which is the set of skew symmetric $n$-linear maps from $g\otimes g \otimes \cdots\otimes g$ ($n$-times) to $V,$ denote by $C^{n}(g, V)$. More specifically $$C^{n}(g, V ) = \{f ~|~f : \otimes^{n}g \longrightarrow V ~\textit{ is a multilinear map}\}.$$ A $n$-BiHom-cochain on $g$ with the coefficients in $V$ is defined to be a $n$-cochain $f \in C^{n}(g,V)$ such that is compatible with $\alpha,\beta$ and $\alpha_{V}, \beta_{V}$ in the sense that $\alpha_{V}\circ f= f\circ\alpha$ and $\beta_{V}\circ f=f\circ\beta$ i.e$$\alpha_{V}(f(p_{1},...,p_{n}))= f(\alpha(p_{1}),...,\alpha(p_{n})),~~\beta_{V}(f(p_{1},...,p_{n}))= f(\beta(p_{1}),...,\beta(p_{n}))$$ denoted by
$$C^{0}_{BiHom}(g,V)=\{v \in V | \alpha_{V}(v)=v , \beta_{V}(v)=v\}$$and
$$C^{n} _{BiHom}(g,V)=\{f \in C^{n}(g,V) | \alpha_{V}\circ f=f\circ\alpha \textit{ and } \beta_{V}\circ f=f\circ\beta\}$$ The coboundary operator $\delta_{BiHom}: C^{n}_{BiHom}(g,V )\longrightarrow C^{n+1}_{ BiHom}(g,V )$, for $n \geq 0$ is given by \begin{enumerate}\item \begin{equation*} \delta_{BiHom} (p)(v)= \alpha\beta^{-1}(p) \bullet v,~~ for ~~v \in C^{0}_{BiHom}(g,V )~~ and~~ p \in g. \end{equation*} \item\begin{eqnarray*}&\delta_{BiHom}f (p_{1},\cdots, p_{n+1}) \\=&\sum_{i=1}^{n+1}(-1)^{i} \alpha\beta^{ n-1}(p_{i})\bullet f(p_{1},\cdots, \hat{p_{i}},\cdots,p_{n+1}) \\&+\sum_{1\leq i<j\leq n+1}(-1)^{i+j+1}f([\alpha^{-1}\beta(p_{i}),p_{j}],\beta(p_{1}),\cdots,\hat{\beta(p_{i})},\cdots,\hat{\beta(p_{j})},\cdots,\beta(p_{n+1}))\end{eqnarray*} \end{enumerate}for $f \in C^{n}_{BiHom}(g, V )$ and $p_{1},\cdots, p_{n+1} \in g$. The cohomology groups of the cochain complex $$\{C^{\ast}_{BiHom}(g,V ),\delta_{BiHom}\}$$ are called the Chevalley-Eilenberg cohomology groups, denoted by $H^{\ast}_{BiHom}(g, V)$. It is important to note that the coboundary operator for the Chevalley-Eilenberg cohomology of the BiHom-Lie algebra $(g, [\cdot,\cdot], \alpha, \beta)$ with coefficients in itself is simply given by $$\delta_{BiHom}f = (-1)^{n-1}[\mu, f]_{NR} , ~~\forall ~f \in C^{n}_{BiHom}(g, g),$$ where $\mu \in C^{2}_{BiHom}(g,g)$ corresponds to the BiHom-Lie bracket $[\cdot,\cdot]$.
\subsection{Bidifferential graded Lie algebras} Before presenting the notion of bidifferential graded Lie algebras, let us first give the definition of a differential graded Lie algebra. \begin{defn}A differential graded Lie algebra is a triple $(L = \oplus L^{i}, [\cdot , \cdot], d)$ consisting of a graded Lie algebra together with a differential $d : L \longrightarrow L$ of degree $+1$ which is a derivation for the bracket $[\cdot ,\cdot ]$. An element $\theta \in L^{1}$ is said to be a Maurer-Cartan element in the differential graded Lie algebra $(L, [\cdot,\cdot ], d)$, if $\theta$ satisfies Maurer-Cartan equation: $$d\theta + \frac{1}{2}[\theta, \theta] = 0.$$ \end{defn}\begin{defn} A bidifferential graded Lie algebra is a quadruple $(L = \oplus L^{i}, [\cdot , \cdot], d_{1}, d_{2})$ in which the triples $(L, [\cdot , \cdot], d_{1})$ and $(L, [\cdot , \cdot], d_{2})$ are differential graded Lie algebras additionally satisfying $d_{1} \circ d_{2} + d_{2} \circ d_{1} = 0$. \end{defn}\begin{re} Any graded Lie algebra can be considered as a bidifferential graded Lie algebra with both the differentials $d_{1}$ and $d_{2}$ to be trivial. \end{re}\begin{defn} Let $(L, [\cdot , \cdot], d_{1}, d_{2})$ be a bidifferential graded Lie algebra. A pair of elements $(\theta_{1}, \theta_{2}) \in L^{1} \oplus L^{1}$ is said to be a Maurer-Cartan element if \begin{enumerate}
\item $ \theta_{1}$ is a Maurer-Cartan element in the differential graded Lie algebra $(L, [ , ], d_{1});$ \item $\theta_{2}$ is a Maurer-Cartan element in the differential graded Lie algebra $(L, [ , ], d_{2});$
\item the following compatibility condition holds $$d_{1}\theta_{2} + d_{2}\theta_{1} + [\theta_{1}, \theta_{2}] = 0.$$ \end{enumerate}Like a differential graded Lie algebra can be twisted by a Maurer-Cartan element, the same result holds for bidifferential graded Lie algebras. \end{defn}\begin{prop} Let $(L, [\cdot , \cdot], d_{1}, d_{2})$ be a bidifferential graded Lie algebra and let $(\theta_{1}, \theta_{2})$ be a Maurer-Cartan element. Then the quadruple $(L, [\cdot , \cdot], d_{1}^{\theta_{1}}, d_{2}^{\theta_{2}})$ is a bidifferential graded Lie algebra, where $$d_{1}^{\theta_{1}} = d_{1} + [\theta_{1},-]~~and ~~d_{2}^{\theta_{2}} = d_{2} + [\theta_{2},-].$$ For any $v_{1}, v_{2} \in L^{1}$, the pair $(\theta_{1} + v_{1}, \theta_{2} + v_{2})$ is a Maurer-Cartan element in the bidifferential graded Lie algebra $(L, [\cdot , \cdot], d_{1}, d_{2})$ if and only if $(v_{1}, v_{2})$ is a Maurer-Cartan element in the bidifferential graded Lie algebra $(L, [\cdot , \cdot], d_{1}^{\theta_{1}} , d_{2}^{\theta_{2}} )$. \end{prop} \section{Compatible BiHom-Lie algebras} In this section, we introduce compatible Bihom-Lie algebras and give a Maurer-Cartan characterization. We end this section by defining representations of compatible Bihom-Lie algebras. Let $g$ be a vector space and $\alpha , \beta: g \rightarrow g$ be two linear commuting maps. \begin{defn} Two BiHom-Lie algebras $(g, [\cdot ,\cdot ]_{1},\alpha,\beta)$ and $(g, [\cdot , \cdot]_{2},\alpha,\beta)$ are said to be compatible if for all $\lambda,\eta \in \mathbb{K}$, the $4$-tuple $(g,\lambda[\cdot,\cdot]_{1} + \eta[\cdot,\cdot]_{2},\alpha,\beta)$ is a BiHom-Lie algebra. The compatibility condition in this definition is equivalent to the following: \begin{eqnarray*} &[\beta^{2}(p), [\beta(q), \alpha(r)]_{1}]_{2} + [\beta^{2}(q), [\beta(r), \alpha(p)]_{1}]_{2} + [\beta^{2}(r), [\beta(p), \alpha(q)]_{1}]_{2}\\& + [\beta^{2}(p), [\beta(q), \alpha(r)]_{2}]_{1} + [\beta^{2}(q), [\beta(r), \alpha(p)]_{2}]_{1} + [\beta^{2}(r), [\beta(p), \alpha(q)]_{2}]_{1} = 0 \end{eqnarray*} for all $p, q, r \in g$. \end{defn}\begin{defn} A compatible BiHom-Lie algebra is a $5$-tuple $(g, [\cdot ,\cdot ]_{1}, [\cdot ,\cdot ]_{2}, \alpha,\beta)$ in which $(g, [\cdot ,\cdot ]_{1},\alpha,\beta)$ and $(g, [\cdot , \cdot]_{2},\alpha,\beta)$ are both BiHom-Lie algebras and are compatible. In this case, we say that the pair $([\cdot ,\cdot ]_{1}, [\cdot , \cdot]_{2})$ is a compatible BiHom-Lie algebra structure on $g$ when the twisting maps $\alpha$ and $\beta$ are clear from the context.\\ Compatible BiHom-Lie algebras are twisted version of compatible Lie algebras. Recall that a compatible Lie algebra is a triple $(g, [\cdot ,\cdot ]_{1}, [\cdot ,\cdot ]_{2})$ in which $(g, [\cdot ,\cdot ]_{1})$ and $(g, [\cdot,\cdot]_{2})$ are Lie algebras and are compatible in the sense that $\lambda[\cdot ,\cdot ]_{1} +\eta[\cdot , \cdot]_{2}$ is a Lie bracket on $g$, for all $\lambda, \eta \in \mathbb{K}$. Thus, a compatible BiHom-Lie algebra $(g, [\cdot ,\cdot ]_{1}, [\cdot ,\cdot ]_{2},\alpha,\beta)$ with $\alpha = \beta = id$ is nothing but a compatible Lie algebra. \end{defn} \begin{defn} Let $(g, [\cdot ,\cdot ]_{1}, [\cdot ,\cdot ]_{2}, \alpha,\beta)$ and $(g^{\prime}, [\cdot, \cdot ]_{1}^{\prime} ,[\cdot ,\cdot ]_{2}^{\prime}, \alpha^{\prime},\beta^{\prime})$ be two compatible BiHom-Lie algebras. A morphism between them is a linear map $\phi : g\rightarrow g^{\prime}$ which is a BiHom-Lie algebra morphism from $(g, [\cdot,\cdot]_{1}, \alpha,\beta)$ to $(g^{\prime}, [\cdot ,\cdot ]_{1}^{\prime},\alpha^{\prime},\beta^{\prime})$, and a BiHom-Lie algebra morphism from $(g, [\cdot ,\cdot ]_{2}, \alpha,\beta)$ to $(g^{\prime}, {[\cdot ,\cdot ]}_{2}^{\prime}, \alpha^{\prime}, \beta^{\prime}).$ \end{defn} \begin{ex} Let $(g, [\cdot , \cdot],\alpha,\beta)$ be a BiHom-Lie algebra. A linear operator $N: g\longrightarrow g$ is called a BiHom-Nijenhuis operator if $[N(p), N(q)]= N([ N(p), q]- [ N(q), p]- N([p, q]))$, for all $p, q \in g$. Then there is a deformed BiHom-Lie Bracket on $g$ given by $$[p, q]_{N}= [N(p), q]- [ N(q), p]- N([p, q])$$ In the other words $(g,[\cdot , \cdot]_{N},\alpha,\beta)$ is a BiHom-Lie algebra. It is easy to see that the 5-tuple $(g,[\cdot , \cdot],[\cdot , \cdot]_{N},\alpha,\beta)$ is a compatible BiHom-Lie algebra. \end{ex} \begin{defn} Let $(g, [\cdot , \cdot], \alpha,\beta)$ be a BiHom-Lie algebra and $s,l$ be a non-negative integer, $\lambda \in \mathbb{K}$. If A linear operator $R : g \longrightarrow g$ satisfying\begin{align*}
\alpha \circ R &= R \circ \alpha\\
\beta \circ R &= R \circ \beta\\
[R(p),R(q)] &= R([\alpha^{s}\beta^{l}R(p),q] + [p,\alpha^{s}\beta^{l}R(q)] +\lambda[p,q])
\end{align*} for all $p, q \in g$, then $R$ is called an $sl$-Rota-Baxter operator of weight $\lambda$ on $(g, [\cdot , \cdot], \alpha, \beta)$ \end{defn} A $sl$-Rota-Baxter operator $R$ induces a new BiHom-Lie algebra structure on $g$ with the BiHom-Lie Bracket \begin{center} $[p,q]_{R} = [\alpha^{s}\beta^{l}R(p),q] + [p,\alpha^{s}\beta^{l}R(q)] +\lambda[p,q]$ \end{center}\begin{defn} Two $sl$-Rota-Baxter operators $R$ and $S$ of same $\lambda \in \mathbb{K}$ on a BiHom-Lie algebra $(g, [\cdot , \cdot],\alpha,\beta)$ are said to be compatible if $$[R(p),S(q)]+[S(p),R(q)]=R([\alpha^{s}\beta^{l}S(p),q]+[p,\alpha^{s}\beta^{l}S(q)])+ S([\alpha^{s}\beta^{l}R(p),q]+[p,\alpha^{s}\beta^{l}R(q)]).$$ \end{defn} \begin{prop} Let $R$ and $S$ be two compatible $sl$-Rota-Baxter operators of weight $\lambda \in \mathbb{K}$ on a BiHom-Lie algebra $(g,[\cdot , \cdot],\alpha,\beta)$, Then $(g,[\cdot , \cdot]_{R},[\cdot , \cdot]_{S},\alpha,\beta)$ is a compatible BiHom-Lie algebra. \end{prop} Let $g$ be a vector space and $\alpha, \beta : g \longrightarrow g$ be a linear map. Consider the graded Lie algebra $(C^{\ast+ 1}_{BiHom}(g, g), [\cdot , \cdot]_{NR})$ the quadruple $$(C^{\ast+1} _{BiHom}(g, g), [\cdot , \cdot]_{NR}, d_{1} = 0, d_{2} = 0)$$ is a bidifferential graded Lie algebra. Then we have the following Maurer-Cartan characterization of compatible BiHom-Lie algebras. \begin{thm}
There is a one-to-one correspondence between compatible BiHom-Lie algebra structures on $g$
and Maurer-Cartan elements in the bidifferential graded Lie algebra
$(C^{\ast+1} _{BiHom}(g,g), [\cdot , \cdot]_{NR}, d_{1} = 0, d_{2} = 0)$. \end{thm} \begin{proof}
Let $[\cdot , \cdot]_{1}$ and $[\cdot , \cdot]_{2}$ be two multiplicative skew-symmetric bilinear brackets on $g$. Then the brackets $[\cdot , \cdot]_{1}$ and $[\cdot , \cdot]_{2}$ correspond to elements (say, $\mu_{1}$ and $\mu_{2}$, respectively) in $C^{2}_{BiHom}(g,g)$. Then
\begin{enumerate}
\item $[\cdot , \cdot]_{1}$ is a BiHom-Lie bracket $\Longleftrightarrow[\mu_{1}, \mu_{1}]_{NR} = 0;$ \\
\item $[\cdot , \cdot]_{2}$ is a BiHom-Lie bracket $\Longleftrightarrow [\mu_{2}, \mu_{2}]_{NR} = 0;$\\
\item compatibility condition (3)$\Longleftrightarrow [\mu_{1}, \mu_{2}]_{NR} = 0.$
\end{enumerate}Hence $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2}, \alpha,\beta)$ is a compatible BiHom-Lie algebra if and only if $(\mu_{1}, \mu_{2})$ is a Maurer-Cartan element in the bidifferential graded Lie algebra $(C^{\ast+1}_{BiHom}(g, g), [\cdot , \cdot]_{NR}, d_{1} = 0, d_{2} = 0)$. \end{proof}\begin{prop} Let $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2},\alpha,\beta)$ be a compatible BiHom-Lie algebra. Then for any multiplicative skew-symmetric bilinear operations $[\cdot , \cdot]^{\prime}_{1}$ and $[\cdot , \cdot]^{\prime}_{2}$ on $g$, the 5-tuple $$(g, [\cdot , \cdot]_{1} + [\cdot , \cdot]^{\prime}_{1}, [\cdot , \cdot]_{2} + [\cdot , \cdot]^{\prime}_{2},\alpha,\beta)$$ is a compatible BiHom-Lie algebra if and only if $(\mu^{\prime}_{1}, \mu^{\prime}_{2})$ is a Maurer-Cartan element in the bidifferential graded Lie algebra $(C^{\ast+1}_{BiHom}(g, g), [\cdot , \cdot]_{NR}, d_{1} = [\mu_{1},-], d_{2} = [\mu_{2},-])$. Here $\mu^{\prime}_{1}, \mu^{\prime}_{2} \in C^{2}_{BiHom}(g, g)$ denote the elements corresponding to the brackets $[\cdot , \cdot]^{\prime}_{1}$ and $[\cdot , \cdot]^{\prime}_{2}$, respectively. \end{prop}In the following, we define representations of a compatible BiHom-Lie algebra.\begin{defn} A representation of the compatible BiHom-Lie algebra $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2}, \alpha,\beta)$ consists of a 5-tuple $(V, \bullet_{1}, \bullet_{2}, \alpha_{V},\beta_{V})$ such that\\ (i)$ (V, \bullet_{1},\alpha_{V},\beta_{V})$ is a representation of the BiHom-Lie algebra $(g,[. , .]_{1},\alpha,\beta)$\\ (ii) $(V, \bullet_{2},\alpha_{V}, \beta_{V})$ is a representation of the BiHom-Lie algebra $(g,[. , .]_{2},\alpha,\beta)$\\ (iii) the following compatibility condition holds\\ $[\beta(p),q]_{1} \bullet_{2} \beta_{V}(v) + [\beta(p), q]_{2} \bullet_{1} \beta_{V}(v) = \alpha\beta(p) \bullet_{1} (q \bullet_{2} v) - \beta(q) \bullet_{2} (\alpha(p) \bullet_{1} v) + \alpha\beta(p) \bullet_{2} (q \bullet_{1} v) - \beta(q) \bullet_{1} (\alpha(p) \bullet_{2} v)$, for all $p, q \in g$ and $v \in V $. It follows that any compatible BiHom-Lie algebra $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2},\alpha,\beta)$ is a representation of itself, where $\bullet_{1} = [\cdot , \cdot]_{1}$ and $\bullet_{2} = [\cdot , \cdot]_{2}$. This is called the adjoint representation. \end{defn} \begin{re}Let $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2}, \alpha,\beta)$ be a compatible BiHom-Lie algebra and $(V, \bullet_{1}, \bullet_{2},\alpha_{V},\beta_{V})$ be a representation of it. Then for any $\lambda, \eta \in \mathbb{K}$, the 4-tuple $(g, \lambda[\cdot , \cdot]_{1} + \eta[\cdot , \cdot]_{2}, \alpha,\beta)$ is a BiHom-Lie algebra and $(V, \lambda \bullet_{1} +\eta \bullet_{2},\alpha_{V},\beta_{V})$ is a representation of it. The proof of the following proposition is similar to the standard case. \end{re}\begin{prop} Let $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2},\alpha,\beta)$ be a compatible BiHom-Lie algebra and $(V, \bullet_{1}, \bullet_{2},\alpha_{V},\beta_{V})$ be a representation of it. Then the direct sum $g \oplus V$ carries a compatible BiHom-Lie algebra structure with the linear homomorphism $\alpha \oplus \alpha_{V}$ and $\beta\oplus\beta_{V}$, and BiHom-Lie brackets $$[(p, a), (q, b)]_{i}^{\ltimes} = ([p, q]_{i}, x \bullet_{i} b - (\alpha^{-1}\beta(y)) \bullet_{i}( \alpha_{V}\beta_{V}^{-1}(a)),~~for~~~ i = 1, 2 ~~and~~ (p, a), (q, b) \in g \oplus V.$$ This is called the semidirect product. \end{prop}\section{Cohomology of compatible BiHom-Lie algebras} In this section, we introduce the cohomology of a compatible BiHom-Lie algebra with coefficients in a representation. Let $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2}, \alpha,\beta)$ be a compatible BiHom-Lie algebra and $(V, \bullet_{1}, \bullet_{2}, \alpha_{V}, \beta_{V})$ be a representation of it. Let $^{1}\delta_{BiHom} : C^{n} _{BiHom}(g, V )\longrightarrow C^{n+1} _{BiHom}(g, V )$ (resp.$^{2}\delta_{BiHom}: C^{n} _{BiHom}(g, V )\longrightarrow C^{n+1} _{BiHom}(g, V )$, for $n \geq 0$, be the coboundary operator for the Chevalley-Eilenberg cohomology of the BiHom-Lie algebra $(g, [\cdot, \cdot]_{1},\alpha,\beta)$ with coefficients in the representation $(V, \bullet_{1},\alpha_{V},\beta_{V})$ (resp. of the BiHom-Lie algebra $(g, [\cdot , \cdot]_{2},\alpha,\beta)$ with coefficients in the representation $(V, \bullet_{2},\alpha_{V},\beta_{V})$. Then we have\\ \begin{center} $(^{1}\delta_{BiHom})^{2} = 0$ and $(^{2}\delta_{BiHom})^{2} = 0$. \end{center} Moreover, we have the following.\begin{prop}\label{prop4.1} \begin{center} The coboundary operators $^{1}\delta_{BiHom}$ and $^{2} \delta_{BiHom}$ satisfy the following compatibility\\ $^{1}\delta_{BiHom} \circ ^{2}\delta_{BiHom} + ^{2}\delta_{BiHom} \circ ^{1}\delta_{BiHom} = 0.$ \end{center} \end{prop} Before we prove the above proposition, we first observe the followings. For a compatible BiHom-Lie algebra $(g, [\cdot , \cdot]_{1}, [\cdot , \cdot]_{2},\alpha,\beta)$ and a representation $(V,\bullet_{1}, \bullet_{2},\alpha_{V},\beta_{V})$, we consider the semidirect product compatible BiHom-Lie algebra structure on $g\oplus V$ given in Proposition 3.6. We denote by\\ $\pi_{1}, \pi_{2} \in C^{2}_{BiHom}(g\oplus V, g\oplus V )$ the elements corresponding to the BiHom-Lie brackets $[\cdot , \cdot]^{\ltimes}_{1}$ and $[\cdot , \cdot]^{\ltimes}_{2}$ on $g \oplus V$ , respectively. Let \begin{center}
$^{1}\delta_{BiHom} : C^{n} _{BiHom}(g \oplus V, g \oplus V ) \longrightarrow C^{n+1} _{BiHom}(g \oplus V, g \oplus V )$, for $n \geq0$,\\ $^{2}\delta_{BiHom} : C^{n} _{BiHom}(g \oplus V, g \oplus V ) \longrightarrow C^{n+1} _{BiHom}(g \oplus V, g \oplus V )$, for $n \geq 0$, \end{center}denote respectively the coboundary operator for the Chevalley-Eilenberg cohomology of the BiHom-Lie algebra $(g \oplus V, [\cdot , \cdot]^{\ltimes}_{1} , \alpha\oplus \alpha_{V}, \beta\oplus \beta_{V})$ (resp. of the BiHom-Lie algebra $(g \oplus V, [\cdot , \cdot]^{\ltimes}_{2} , \alpha\oplus \alpha_{V}, \beta\oplus \beta_{V})$ with coefficients in itself. Note that any map $f \in C^{n} _{BiHom}(g \oplus V)$ can be lifted to a map $\tilde{f} \in C_{BiHom}^{n}(g\oplus V,g \oplus V)$ by \begin{center} $\tilde{f}((p_{1},v_{1}),...,(p_{n},v_{n}))=(0, f(p_{1},...,p_{n})).$ \end{center} Then $f = 0$ if and only if $\tilde{f}$ = 0.\\ With these notations, for any $f \in C^{n} _{BiHom}(g,V)$, we have\\ $(\widetilde{^{1}\delta_{BiHom}f})=\delta^{1}_{BiHom}(\tilde{f})=(-1)^{n-1}[\pi_{1},\tilde{f}]_{NR}$ and $(\widetilde{^{2}\delta_{BiHom}f})=\delta^{2}_{BiHom}(\tilde{f})=(-1)^{n-1}[\pi_{2},\tilde{f}]_{NR}$.\begin{proof}(Proposition \ref{prop4.1}) For any $f \in C^{n}_{BiHom}(g,V)$, we have\begin{align*}
\widetilde{(^{1}\delta_{BiHom}\circ^{2}\delta_{BiHom}+{^{2}\delta_{BiHom}}\circ^{1}\delta_{BiHom})(f)}&=\widetilde{^{1}\delta_{BiHom}(^{2}\delta_{BiHom}f)}+\widetilde{^{2}\delta_{BiHom}(^{1}\delta_{BiHom}f)}\\
&=(-1)^{n}[\pi_{1},\widetilde{^{2}\delta_{BiHom}f}]_{NR}+(-1)^{n}[\pi_{2},\widetilde{^{1}\delta_{BiHom}f}]_{NR}\\
&=-[\pi_{1}, [\pi_{2},\tilde{f}]_{NR}]_{NR} - [\pi_{2}, [\pi_{1},\tilde{f}]_{NR}]_{NR}\\
&= [\pi_{2}, [\pi_{1},\tilde{f}]_{NR}]_{NR}-[[\pi_{1}, \pi_{2}]_{NR}, \tilde{f}]_{NR} - [\pi_{2}, [\pi_{1},\tilde{f}]_{NR}]_{NR}\\
&=0\end{align*} \begin{center}
where $([\pi_{1},\pi_{2}]_{NR} = 0)$.\end{center}Therefore, it follows that $(^{1}\delta_{BiHom}\circ ^{2}\delta_{BiHom}+^{2}\delta_{BiHom}\circ ^{1}\delta_{BiHom})(f)= 0$. Hence the result follows.\end{proof}We are now in a position to define the cohomology of a compatible BiHom-Lie algebra\\$(g, [\cdot, \cdot]_{1}, [\cdot, \cdot]_{2},\alpha,\beta)$ with coefficients in a representation $(V, \bullet_{1}, \bullet_{2},\alpha_{V},\beta_{V})$. For each $n \geq 0$, we define an abelian group $C^{n}_{cBiHom}(g, V )$ as follows: \begin{center} \begin{align*}
C^{0}_{cBiHom}(g, V) &= C^{0}_{BiHom}(g,V) \cap \{v \in V | \alpha\beta^{-1}(p) \bullet_{1} v = \alpha\beta^{-1}(p)\bullet_{2} v, \forall p \in g\}\\
&= \{v \in V | \alpha_{V}(v) = v\textit{ , } \beta_{V}(v) = v \textit{ and } \alpha\beta^{-1}(p) \bullet_{1} v = \alpha\beta^{-1} (p) \bullet_{2} v, \forall p \in g\}, \end{align*} $C^{n}_{cBiHom}(g, V ) = \underbrace{C^{n}_{BiHom}(g, V )\oplus...\oplus C^{n}_{BiHom}(g, V )}_{\textit{n- copies}}$, for $n \geq 1$. \end{center}Define a map $\delta_{cBiHom }: C^{n}_{cBiHom}(g, V ) \longrightarrow C^{n+1}_{cBiHom}(g,V ), for n \geq 0$ by\begin{center} $\delta_{cBiHom}(v)(p) = \alpha\beta^{-1}(p) \bullet_{1} v = \alpha\beta^{-1}(p) \bullet_{2} v$, for $v \in C^{0}_{ cBiHom}(g, V )$ and $p \in g,$\\ $\delta_{cBiHom}(f_{1},...,f_{n}) =(^{1}\delta_{BiHom}f_{1},...,\underbrace{^{1}\delta_{BiHom}f_{i} + ^{2}\delta_{BiHom}f_{i-1}}_{\textit{i-th position}},...,^{2}\delta_{BiHom}f_{n})$, \end{center} for $(f_{1},...,f_{n}) \in C^{n}_{cBiHom}(g, V )$.Then we have the following. \begin{prop}
The map $\delta_{cBiHom}$ is a coboundary map, i.e., $(\delta_{cBiHom})^{2} = 0$. \end{prop}\begin{proof} For any $v \in C^{0}_{cBiHom}(g,V)$, we have \begin{center}
$(\delta_{cBiHom})^{2}(v) = \delta_{cBiHom}(\delta_{cBiHom}v) = (^{1}\delta_{BiHom}\delta_{cBiHom}v , ^{2}\delta_{BiHom}\delta_{cBiHom}v)$\\
$=(^{1}\delta_{BiHom}$ $^{1}\delta_{BiHom}v $,$^{ 2}\delta_{BiHom}$ $^{2}\delta_{BiHom}v) = 0.$
\end{center}Moreover, for any $(f_{1},...,f_{n})\in C^{n}_{cBiHom}(g,V), n \geq 1$, we have \begin{center}\begin{align*}&(\delta_{cBiHom})^{2}(f_{1},...,f_{n})\\&= \delta_{cBiHom}( ^{1} \delta_{BiHom}f_{1},...,^{1}\delta_{BiHom}f_{i} + ^{2}\delta_{BiHom}f_{i-1},...,^{2}\delta_{BiHom}f_{n})\\ &=( {^{1}\delta_{BiHom}} {^{1}\delta_{BiHom}}f_{1} , {^{2}\delta_{BiHom}}{^{1}\delta_{BiHom}}f_{1} + {^{1}\delta_{BiHom}} {^{2}\delta_{BiHom}}f_{1} + {^{1}\delta_{BiHom}} {^{1}\delta_{BiHom}f_{2}} ,...,\\&\underbrace{{^{2}\delta_{BiHom}} {^{2}\delta_{BiHom}}f_{i-2} + {^{2}\delta_{BiHom}} {^{1}\delta_{BiHom}}f_{i-1} + {^{1}\delta_{BiHom}} {^{2}\delta_{BiHom}}f_{i-1} + {^{1}\delta_{BiHom}} {^{1}\delta_{BiHom}f_{i}} ,...,}_{3\leq i \leq n-1}\\&{^{2}\delta_{BiHom}} {^{2}\delta_{BiHom}}f_{n-1} + {^{2}\delta_{BiHom}} {^{1}\delta_{BiHom}}f_{n} + {^{1}\delta_{BiHom}} {^{2}\delta_{BiHom}}f_{n}, { ^{2}\delta_{BiHom}} {^{2}\delta_{BiHom}}f_{n} \\&= 0\end{align*}\end{center} This proves that $(\delta_{cBiHom})^{2} = 0$.
\end{proof}It follows from the above proposition that $\{C^{\ast}_{cBiHom}(g,V), \delta_{cBiHom}\}$ is a cochain complex. The corresponding cohomology groups\begin{center}
$H^{n}_{cBiHom}(g, V ) =\frac{ Z^{n}_{cBiHom}(g,V)}{B^{n}_{cBiHom}(g,V)}= \frac{ Ker \delta_{cBiHom} : C^{n}_{cBiHom}(g,V) \longrightarrow C^{n+1}_{cBiHom} (g,V)}{ Im \delta_{cBiHom} : C^{n-1}_{cBiHom}(g,V) \longrightarrow C^{n}_{cBiHom}(g,V)}$ , for $n \geq 0$ \end{center}are called the cohomology of the compatible BiHom-Lie algebra $(g,[. , .]_{1}, [. , .]_{2},\alpha,\beta)$ with coefficients in the representation $(V,\bullet_{1},\bullet_{2},\alpha_{V},\beta_{V})$.
\par Let $g = (g, [\cdot , \cdot]_1, [\cdot , \cdot]_2, \alpha,\beta)$ be a compatible BiHom-Lie algebra and $V = (V, \bullet_{1}, \bullet_{2}, \alpha_V,\beta_V)$ be a representation of it. Then we know from Remark 3.15 that $g_+ = (g, [\cdot , \cdot]_1 + [\cdot , \cdot]_2, \alpha+\beta)$ is a BiHom-Lie algebra and $V_+ = (V, \bullet_{1}+ \bullet_{2} ,\alpha_V+\beta_V)$ is a representation of it. Consider the cochain complex $\{C^*_{cBiHom}(g, V), \delta_{BiHom}\}$ of the compatible BiHom-Lie algebra $g$ with coefficients in the representation $V$ , and the cochain complex $\{C^*_{BiHom}(g_+, V_+),\delta_{BiHom}\}$ of the BiHom-Lie algebra $g_+$ with coefficients in $V$. Like wise in \cite{1}, we can establish a following theorem: \begin{thm} The collection $\{\varphi\}_{n\geq0}$ defines a morphism of cochain complexes from to $\{C^*_{BiHom}(g_+, V_+), \delta_{BiHom}\}$. Hence, it induces a morphism $H^*_{cBiHom}(g, V ) \to H^{*}_{BiHom} (g_{+}, V_+)$ between corresponding cohomologies. where $\varphi_n$ is a map $\varphi_{n\geq0} : C^n_{cBiHom}(g,V) \to C^n_{BiHom}(g_+, V_+)$ defined by\begin{eqnarray}\varphi_{n\geq0}=\biggl\{\varphi_{0}(v)= &\frac12(v),~~ v \in C^0_{cBiHom}(g,V) \\\varphi_{n\geq1}((f_1, \cdots, f_n))=& f_1+ \cdots+ f_n, ~~ (f_1, \cdots, f_n)\in C^{n\geq1}_{cBiHom}(g,V)\biggr\}
\end{eqnarray} \end{thm} \noindent {\bf Acknowledgment:} The authors would like to thank the referee for valuable comments and suggestions on this article.
\end{document} | arXiv |
\begin{document}
\begin{abstract}
\noindent We give an explicit version of Shimura's
reciprocity law for singular values
of Siegel modular functions.
We use this to construct the first examples of
class invariants of quartic CM fields
that are smaller than Igusa invariants.
Our statement also enables a new proof of Shimura's
reciprocity law by Tonghai Yang.
\end{abstract}
\title{An explicit version of Shimura's reciprocity law for Siegel modular functions}
\section{Introduction}
The values of the
modular function $j$ in imaginary quadratic numbers~$\tau$
generate abelian extensions of imaginary quadratic fields~$K=\mathbf{Q}(\tau)$. These values enable explicit computation of the Hilbert class field of~$K$ and of elliptic curves over finite fields with a prescribed number of points (the ``CM method'') for primality testing and cryptography.
These algebraic numbers $j(\tau)$ have very large height, which limits their usefulness in such explicit applications. So we consider arbitrary modular functions~$f$ instead, whose values are again abelian over~$K$, hoping to find numbers of smaller height. If these values $f(\tau)$ lie in the same field as $j(\tau)$, then we call them \emph{class invariants}, and they can take the place of $j(\tau)$ in applications, which leads to great speed-ups~\cite{enge-cm}.
The values $f(\tau)$ are acted upon by ideals (and id\`eles) of~$K$ via the Artin isomorphism. Shimura's reciprocity law expresses this action in terms of an action on the modular functions~$f$ themselves, and an explicit version of this reciprocity law~\cite{MR563924,gee-stevenhagen} allows one to search for class invariants in a systematic way.
There exists a higher-dimensional CM-method, with applications to hyperelliptic curve cryptography and a more general analytic construction of class fields~\cite{spallek,HEHCC}.
A significant speedup will be obtained by replacing the \emph{Igusa invariants} in this construction by \emph{smaller} class invariants.
Shimura gave various higher-dimensional analogues of his reciprocity law~\cite{shimura-models-I, shimura-models-II, shimura-arithmetic, shimura-fourier, shimura-theta-cm, shimura-reciprocity-theta}. Our main result (Theorems \ref{thm:general}, \ref{thm:idealgroup}, and~\ref{thm:special} below) is a new and explicit version, suitable for finding class invariants of higher-degree CM-fields, generated by square roots of
totally negative algebraic numbers.
We use our explicit formulation of Shimura's reciprocity law
to find the first examples of small class invariants of quartic CM-fields. Our formulation of Shimura's reciprocity law also enabled a new proof of Shimura's reciprocity law by Tonghai Yang~\cite{yang-shimura}, which postdates our work. As a third application, Andreas Enge and the author~\cite{enge-streng} use the explicit reciprocity law for generalizing Schertz' work on class invariants~\cite{schertz} to higher dimension.
\subsection{Summary of results}
Let $\mathcal{F}_N$ be the field of Siegel modular functions of level $N$ with $q$-expansion coefficients in $\mathbf{Q}(\zeta_N)$ (c.f.~\eqref{eq:FN}). Let $f\in\mathcal{F}_N$ be such a function. Let $\tau$ be a symmetric $g\times g$ matrix with positive definite imaginary part (that is, a point in the Siegel upper half space $\mathbf{H}_g$). If $\tau$ is a primitive CM point (Section~\ref{ssec:cm}), then $f(\tau)$ is an algebraic number and is in fact abelian over a field known as the \emph{reflex field} $K^{\mathrm{r}}$ of $\tau$ (Section~\ref{sec:reflex}).
Now given $\sigma\in\mathrm{Gal}(\overline{\mathbf{Q}}/K^{\mathrm{r}})$, there are various reasons why we would like to be able to compute $f(\tau)^\sigma$. For example, it allows us to decide whether $f(\tau)$ is in certain subfields of $\overline{\mathbf{Q}}$ and to find its minimal polynomial over~$K^{\mathrm{r}}$. This minimal polynomial can be used to speed up explicit class field theory and explicit CM constructions of curves and Jacobians \cite{enge-morain, sutherlandclassinv}.
Shimura's reciprocity law~\cite{shimura-models-I, shimura-models-II, shimura-arithmetic, shimura-fourier, shimura-theta-cm, shimura-reciprocity-theta} expresses $f(\tau)^\sigma$ in the form $F(\tau)$ where $F$ is obtained from $f$ and $\sigma$.
The function $F$ is obtained in loc.~cit.~from
a group action of uncountable ad{\`e}lic groups, where only the actions of certain subgroups are explicit. So in order to use such actions, one needs to approximate the ad{\`e}lic group elements by products of elements in particular subgroups. We did this, and the result is an explicit reciprocity law in terms of ideals and ray class groups, rather than id{\`e}le class groups.
Let $\mathfrak{a}$ be an ideal.
Then Theorem~\ref{thm:general} gives (in terms of $\mathfrak{a})$ efficiently computable $U\in \GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ and $\tau'\in\mathbf{H}_g$ with \begin{equation}\label{eq:action} f(\tau)^{[\mathfrak{a}]} = f^U(\tau'), \end{equation} where the action of $U$
on $\mathcal{F}_N$ can be computed in one of the various practical ways explained in Section~\ref{sec:actioncompute}.
Moreover, the point $\tau'$ can be constructed in such a way that it is in a fundamental region, allowing for efficient numerical evaluation of $f^U(\tau')$.
We use this reciprocity law to prove (Theorem~\ref{thm:idealgroup}) a formula for the ideal group corresponding to the abelian extension \[\cmext{N} = K^{\mathrm{r}}(f(\tau) : f\in\mathcal{F}_N)\qquad\mbox{of}\qquad K^{\mathrm{r}}.\]
Computations with $f(\tau)$ become even more efficient when it is real instead of complex.
Proposition~\ref{prop:complexconjugation} gives a sufficient condition for this to happen.
The author has implemented the actions
in SageMath~\cite{sage} (which uses PARI~\cite{pari}) and made the program available online at~\cite{cmcode}.
\subsection{Overview of content}
Section~\ref{sec:statement} states the results and Sections \ref{sec:ad}--\ref{sec:conjugationproof} contain a proof. The action of~$U$ in~\eqref{eq:action} becomes most explicit when expressing the function~$f$ in terms of theta constants, which happens in Section~\ref{sec:theta}.
Section~\ref{sec:examples} gives a detailed example of how to obtain useful class invariants.
Finally, Section~\ref{sec:applications} gives applications to computational class field theory and to the construction of curves over finite fields. The final three sections (\ref{sec:theta}--\ref{sec:applications}) can be read independently of Sections \ref{sec:ad}--\ref{sec:conjugationproof}.
\section{Definitions and statement of the main results}\label{sec:statement}
\subsection{The upper half space}
Fix a positive integer~$g$.
The \emph{Siegel upper half space}
$\mathbf{H}=\mathbf{H}_g$ is the set of~$g\times g$ symmetric
complex
matrices with positive definite imaginary part.
It parametrizes $g$-dimensional principally
polarized abelian varieties $A$ over $\mathbf{C}$
together with a \emph{symplectic} basis $b_1,\ldots,b_{2g}$
of their first homology.
In more detail, every abelian variety over $\mathbf{C}$ is of the form $A = \mathbf{C}^g / \Lambda$ for a lattice $\Lambda$ of rank~$2g$. A polarization is given by a Riemann form, i.e., an $\mathbf{R}$-bilinear form $E$ on $\mathbf{C}^g$ that restricts to an alternating bilinear form $\Lambda\times \Lambda\rightarrow \mathbf{Z}$
such that~$(u,v)\mapsto E(iu,v)$ is symmetric and positive definite. Given a $\mathbf{Z}$-basis of~$\Lambda$, there is a matrix, which by abuse of notation we also denote by~$E$, such that~$E(u,v) = u^{\mathrm{t}} E v$. We say that~$E$ is \emph{principal} if $E$ has determinant~$1$. In that case, there exists a \emph{symplectic basis}, i.e., a basis such that $E$ is given in terms of~$(g\times g)$-blocks as \[E = \Omega := \left(\begin{array}{rr} 0 & 1\\ -1 & 0\end{array}\right).\]
To a point $\tau\in\mathbf{H}_g$, we associate the principally polarized abelian variety with $\Lambda = \tau\mathbf{Z}^g + \mathbf{Z}^g$ and symplectic basis $\tau e_1,\ldots,\tau e_g, e_1,\ldots, e_g$, where $e_i$ is the $i$-th basis element of~$\mathbf{Z}^g$. Conversely, given a principally polarized abelian variety and a symplectic basis, we can apply a $\mathbf{C}$-linear transformation of~$\mathbf{C}^g$ to write it in this form~\cite[Chapter~8]{birkenhake-lange}.
\subsection{The algebraic groups}
Given a commutative ring~$R$, let
$$\GSpspecific{2g}(R) = \{ A\in\matrixring{2g}{R} : A^{\mathrm{t}} \Omega A = \nu\Omega\text{ with $\nu\in R^\times$}\}.$$ Note that~$\nu$ defines a homomorphism of algebraic groups $\GSpspecific{2g}\rightarrow \mathbf{G}_{\mathrm{m}}$, and denote its kernel by~$\Spspecific{2g}$. For~$g=1$, we have simply $\GSpspecific{2}=\mathrm{GL}_2$, $\nu=\det$, $\Spspecific{2}=\mathrm{SL}_2$.
The homomorphism $\nu$ has a section~$i$, satisfying $\nu\circ i =\mathrm{id}_{\mathbf{G}_{\mathrm{m}}}$, given by\footnote{Warning: our $i$ differs from Shimura's $\iota$ in the sense that $i(t)=\iota(t)^{-1}$} $$i(t) = \tbt{1 & 0 \\ 0 & t}.$$ For any ring $R$ for which this makes sense, we also define $$\GSpspecific{2g}(R)^+ = \{A\in\GSpspecific{2g}(R) : \nu(A)>0\}.$$ The group $\GSpspecific{2g}(\mathbf{R})^+$ acts on $\mathbf{H}_g$ by $$\tbt{a & b \\ c & d}\tau = (a \tau + b) (c \tau + d)^{-1},$$ where $a$, $b$, $c$, $d$ are $(g\times g)$-blocks. Changes of symplectic bases correspond to the action of $\Spspecific{2g}(\mathbf{Z})\subset \GSpspecific{2g}(\mathbf{R})^+$ on $\mathbf{H}_g$ (see Lemma~\ref{lem:transpose} below), leading to the well-known fact that
$\Spspecific{2g}(\mathbf{Z})\backslash \mathbf{H}$ parametrizes the set of isomorphism classes of principally polarized abelian varieties of dimension~$g$.
The natural map $\Spspecific{2g}(\mathbf{Z})\rightarrow \Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ is surjective~\cite[Thm.~VII.21]{newman}. Its kernel $\Gamma_N$ is called the principal congruence subgroup of level~$N$.
\subsection{Modular forms and group actions}\label{ssec:statementfirst}
A \emph{Siegel modular form} of weight $k$ and level~$N$ is a holomorphic function $f:\mathbf{H}_g\rightarrow \mathbf{C}$ such that for all $A=\tbtinl{a}{b}{c}{d}\in \Gamma_N$, we have $f(A \tau) =\det (c\tau+d)^k f(\tau)$, and which is ``holomorphic at the cusps''. We will not define holomorphicity at the cusps, as it is automatically satisfied for~$g>1$ by the Koecher principle~\cite{koecher}, and is a textbook condition for~$g=1$.
Every Siegel modular form $f$ has a \emph{Fourier expansion} or \emph{$q$-expansion} \begin{equation}\label{eq:fourier} f(\tau) = \sum_{\xi} a_{\xi} q^{\xi}, \quad a_{\xi}\in\mathbf{C},\quad q^{\xi}:=\exp (2\pi i \mathrm{Tr}(\xi \tau) / N), \end{equation} where $\xi$ runs over the symmetric matrices in $\matrixring{g}{\frac{1}{2}\mathbf{Z}}$ with integral diagonal entries. The numbers $a_{\xi}$ are the \emph{coefficients} of the $q$-expansion.
Let $\mathcal{F}_N$
be the field
\begin{equation}\label{eq:FN}
\mathcal{F}_N=\left\{\frac{g_1}{g_2} : \begin{tabular}{c} $g_i$ are Siegel modular forms of equal weight and level N,\\ with $q$-expansion coefficients in $\mathbf{Q}(\zeta_N)$, and $g_2\not=0$ \end{tabular}\right\}. \end{equation}
\begin{proposition}\label{prop:groupaction}
There is a right action of $\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ on $\mathcal{F}_N$
given as follows.
For $A\in \GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$, let $t = \nu(A)$ and $B = i(t)^{-1} A$. Then \[f^A = (f^{i(t)})^B,\] where we have:
\begin{enumerate}\item
For $B\in\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$, let $\widetilde{B} \in \Spspecific{2g}(\mathbf{Z})$
be such that $B = (\widetilde{B}\ \mathrm{mod}\ N)$.
Then $f^B(\tau) = f(\widetilde{B}\tau)$ for all $f\in \mathcal{F}_N$.
\item
For $t \in (\mathbf{Z}/N\mathbf{Z})^{\times}$, the matrix $i(t)$ acts by the natual Galois
action of $(\mathbf{Z}/N\mathbf{Z})^{\times}$ on $q$-expansion coefficients,
that is, if $f = \sum_{\xi, k} a(\xi, k)\zeta_N^k q^\xi\in\mathcal{F}_N$
with $a(\xi,k)\in\mathbf{Q}$, then
$f^{i(t)} = \sum_{\xi,k} a(\xi, k)\zeta_{N}^{k t} q^\xi$.
\end{enumerate} \end{proposition} We give detailed references in Section~\ref{sec:ad}.
\begin{remark}
As it is a group action, the action also satisfies $f^A = (f^{B'})^{i(t)}$
for $B' = A i(t)^{-1} = i(t) B i(t)^{-1}$. \end{remark}
\begin{remark}\label{rem:moduliaction}
The action may look a bit ad hoc here, but it has a natural interpretation
in terms of moduli spaces of abelian varieties. We will not need
this interpretation, but give it now for context
in the case $N\geq 3$
(see also~\cite{yang-shimura}).
There is a smooth variety $Y(N)$ (irreducible over $\mathbf{Q}$, reducible over $\mathbf{Q}(\zeta_N)$)
for which all of the following hold.
For every field $L\supset \mathbf{Q}$, the set
$X(N)(L)$ is the set of isomorphism classes of pairs $(A, \phi)$ where
$A$ is a principally polarized
abelian variety of dimension $g$ and $\phi : (\mathbf{Z}/N\mathbf{Z})^{2g}\rightarrow A[N]$
is an isomorphism such that the $N$-Weil pairing $e_N$ satisfies
\begin{equation}\label{eq:elementz}e_N(\phi(u), \phi(v)) = z^{ u \Omega v}\end{equation}
for some primitive $N$th root of unity $z\in L^\times$.
Here we call two such pairs $(A_1,\phi_1)$ and $(A_2,\phi_2)$ isomorphic
if there exists an isomorphism $\psi :A_1\rightarrow A_2$ of principally polarized
abelian varieties such that
$\psi\circ\phi_1 = \phi_2$.
The function field $\mathbf{Q}(Y(N))$ has a primitive $N$-th root of unity
$z$ sending $(A,\phi)\in Y(N)(L)$ to the element $z$ of~\eqref{eq:elementz}.
Let $Y'(N)$ be the subvariety over $\mathbf{Q}(\zeta_N)$ given by $z=\zeta_N$.
Then there is an isomorphism $\mathbf{Q}(Y(N))\rightarrow \mathbf{Q}(\zeta_N)(Y'(N))$
sending $z$ to $\zeta_N$ and we identify these function fields via this isomorphism.
We have an isomorphism of manifolds
$m : \Gamma_N \backslash \mathbf{H}_g\rightarrow Y'(N)(\mathbf{C})$
sending the class of $\tau\in\mathbf{H}_g$ to the pair $(A,\phi)$ with
$A = \mathbf{C}^g/(\tau\mathbf{Z}^g+\mathbf{Z}^g)$
and $\phi(v) = \frac{1}{N}(\tau, 1_g) v \in A[N]$ for all $v\in(\mathbf{Z}/N\mathbf{Z})^g$.
This embedding induces a further isomorphism $m^{\times} : \mathbf{Q}(\zeta_N)(Y'(N))\rightarrow \mathcal{F}_N$
sending $f$ to $f\circ m$.
The group $\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ acts on $Y(N)$ via
\begin{equation}\label{eq:actionmoduli}M\cdot (A,\phi) = (A, \phi\circ M^{\mathrm{t}}).\end{equation}
This in turn induces a right action on $\mathbf{Q}(Y(N))$ and hence via $m^{\times}$ on $\mathcal{F}_N$.
We now check in two cases that this action agrees with the action of
Proposition~\ref{prop:groupaction}.
For $M\in\Spspecific{2g}(\mathbf{Z})$, we have $m(M\tau) = M\cdot m(\tau)$,
which yields \ref{prop:groupaction}(1).
We compute $\zeta_N^{i(t)}$ by computing
$z^{i(t)}(A,\phi) =
z((A,\phi\circ i(t))) =
z((A,\phi))^t$
(because $i(t)\Omega i(t)=t\Omega$).
In particular, this induces the action $\zeta_N^{i(t)} = \zeta_N^t$,
which is consistent with \ref{prop:groupaction}(2). \end{remark}
\subsection{Computing the group action}\label{sec:actioncompute}
There are roughly four ways in which, given $A\in\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ and $f\in\mathcal{F}_N$, we could compute $f^A$ and $f^A(\tau)$.
\textbf{1. From the definition.} The most obvious is to first take $t = \nu(A)$, and write $A = i(t) B$ with $B=i(t)^{-1} A \in\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$.
Next, compute a lift $\widetilde{B}\in \Spspecific{2g}(\mathbf{Z})$ of $B$. This can be done by following the steps of the proof of \cite[Thm.~VII.21]{newman}. Alternatively, one could express an element of $\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ as a product of standard generators of $\Spspecific{2g}(\mathbf{Z})$ (and in the case $g=1$, this results in explicit formulas as in \cite[Lemma~6]{MR1730432}).
Then we compute $f^{i(t)}$ and evaluate it in $\widetilde{B}\tau$. The disadvantage of using this method in practice is that while $\tau$ can often be engineered to be in a fundamental region where modular functions converge quickly, we have no control over $\widetilde{B}\tau$.
For this reason, we will not take this approach, and we promote the methods 2--4 instead.
\textbf{2. Using theta functions.} In many practical situations, the function $f\in\mathcal{F}_N$ is expressed in terms of \emph{theta constants}. In that case, there is a more direct formula for the action of~$\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ on $f$,
which does not require finding a lift to $\Spspecific{2g}(\mathbf{Z})$. We give the action in Section~\ref{sec:theta} and use this in our examples in Section~\ref{sec:examples}.
\textbf{3. Using the moduli interpretation.} In some cases, one could use the action as in Remark~\ref{rem:moduliaction}. We do not follow this approach in the present article, but we do illustrate it with the following example.
For $g=1$ and $N=2$, we have $\Fcal{2}=\mathbf{Q}(\lambda)$, where $\lambda(\tau)$ is the Legendre invariant given as follows. To $\lambda\in\mathbf{C}\setminus\{0,1\}$, associate the elliptic curve $E : y^2 = x(x-1)(x-\lambda)$ and the isomorphism $\phi : (\mathbf{Z}/2\mathbf{Z})^2\rightarrow E[2]$ given by $\phi(e_1)=(0,0)$, $\phi(e_2) = (1,0)$. This gives a complex analytic isomorphism $\mathbf{C}\setminus\{0,1\}\rightarrow Y(2)(\mathbf{C})$. In particular, we have that $\phi(e_1+e_2)$ is the remaining point of order two, which is $(\lambda, 0)$.
The action of $\mathrm{GSp}_{2}(\mathbf{Z}/2\mathbf{Z})$
on $(\mathbf{Z}/2\mathbf{Z})^2$ is by permutation of $e_1$, $e_2$ and $e_3:=e_1+e_2$. That is, we have the isomorphism $\sigma: \mathrm{GSp}_{2}(\mathbf{Z}/2\mathbf{Z})\rightarrow S_3$
given by $M e_i = e_{\sigma(M)(i)}$ for $i=1,2,3$.
Writing $\lambda_1:=0$, $\lambda_2:=1$, $\lambda_3:=\lambda\in\Fcal{2}$, the action of \eqref{eq:actionmoduli} becomes
\[\lambda^{M} = \frac{\lambda_{\tau(3)} - \lambda_{\tau(1)}} {\lambda_{\tau(2)}-\lambda_{\tau(1)}},\quad\mbox{where}\quad\tau = \sigma(M^{\mathrm{t}}).\]
With Rosenhain invariants and the appropriate isomorphism $\mathrm{GSp}_{4}(\mathbf{Z}/2\mathbf{Z})\cong S_6$, one gets a similar formula for $g=N=2$.
Similar things can be done for other small values of $g$ and $N$ on a case-by-case basis.
\textbf{4. By selecting $f$ in such a way that the action is easy}
Work in progress of the author with Andreas Enge~\cite{enge-streng} uses our main results to pick $f$ and $\tau$ in such a way that many of the relevant $f^A$ are simply equal to~$f$ and others are easy to find as well.
\subsection{Complex multiplication}\label{ssec:cm}
Suppose $A$ is a $g$-dimensional polarized abelian variety over~$\mathbf{C}$ with \emph{complex multiplication}, i.e., such that $\mathrm{End}(A)\otimes\mathbf{Q}$ contains a CM-field $K$ of degree~$2g$.
It is known that every such~$A$ can be obtained as follows.
Let~$\Phi=\{\phi_1,\ldots,\phi_g\}$ be a \emph{CM-type}, i.e., a set of~$g$ embeddings $K\rightarrow \mathbf{C}$ such that no two are complex conjugate. By abuse of notation, write $\Phi(x)=(\phi_1(x),\ldots,\phi_g(x))\in\mathbf{C}^g$ for~$x\in K$. Let $\mathfrak{b}$ be a lattice in~$K$, i.e., a non-zero fractional ideal of an order of~$K$. Let $\xi\in K$ be such that for all $\phi\in\Phi$, the complex number $\phi(\xi)$ lies on the positive imaginary axis, and such that the bilinear form $E_{\xi} : K\times K\rightarrow \mathbf{Q}: (x,y)\mapsto \mathrm{Tr}(\overline{x}y\xi)$ maps $\mathfrak{b}\times\mathfrak{b}$ to~$\mathbf{Z}$. Take $A = \mathbf{C}^g/\Phi(\mathfrak{b})$ and let a polarization on $A$ be given by $E_{\xi}$ extended $\mathbf{R}$-linearly from $\mathfrak{b}$ to~$\mathbf{C}^g$. Finally, let $\mathcal{O}=\{x\in K : x\mathfrak{b}\subset\mathfrak{b}\}$ be the multiplier ring of~$\mathfrak{b}$, and embed it into $\mathrm{End}(A)$ by taking $x\Phi(u) = \Phi(xu)$ and extending this linearly.
Any CM-point $\tau$ can thus be obtained from a quadruple $(\Phi, \mathfrak{b}, \xi, B)$ with $\Phi$, $\mathfrak{b}$ and $\xi$ as above and $B=(b_1,\ldots,b_{2g})$ a symplectic basis of~$\mathfrak{b}$ for the pairing~$E_{\xi}$. We will make the reciprocity law explicit in terms of such quadruples, and note that one obtains $\tau$ with the formula
$$\tau = (\Phi(b_{g+1})|\cdots|\Phi(b_{2g}))^{-1}(\Phi(b_1)|\cdots|\Phi(b_g)).$$ We denote this $\tau$ by $\tau(\Phi, \mathfrak{b}, \xi, B)$ or simply by $\tau(\Phi, B)$.
We will assume that~$\Phi$ is a \emph{primitive} CM-type, or, equivalently, that we have $K=\mathrm{End}(A)\otimes\mathbf{Q}$ (\cite[{Thms.~1.3.3 and~1.3.5}]{lang-cm}). We then have $\mathcal{O}=\mathrm{End}(A)$. In that case, we call $\tau$ a \emph{primitive CM point}.
\subsubsection{Computing the period matrices} \label{sec:algperiodmatrices}
Given $K$, finding representatives $(\Phi, \mathfrak{b},\xi)$ for all isomorphism
classes of principally polarized abelian varieties with CM by $\mathcal{O}_K$
can be done with van Wamelen's algorithm~\cite[Algorithm~1]{vanwamelen}.
For a version of this algorithm without duplicates, see
\cite[Algorithm~4.12]{runtime}.
\label{sec:basis}
It is well-known how to then compute the symplectic bases~$B$.
This is available as the \verb!symplectic_form! method
of integer matrices in SageMath~\cite{sage} or the
\verb!FrobeniusFormAlternating! function in Magma~\cite{magma}, but see
for example \cite[Algorithm~5.2]{runtime} for details.
Together, this gives a method for finding all period matrices,
and we implemented this as \verb!CM_Field(...).period_matrices()!
in \cite{cmcode},
which returns period matrices in the form of SageMath objects \verb!tau!
that include the data of $\Phi$, $\mathfrak{b}$,
$\xi$, and the basis~$B$ and can produce arbitrary-precision approximations
of~$\tau$.
In practical computations, one wants to take $B$ such that numerical
formulas for modular forms converge quickly when evaluated
in~$\tau$.
This can be done by first taking $B$ arbitrary and then applying
an $\Spspecific{2g}(\mathbf{Z})$-reduction algorithm to~$\tau$ to move it to
a nice region such as a fundamental domain,
and adjusting $B$
accordingly.
We implemented this for $g\leq 2$ as \verb!tau.reduce()! in~\cite{cmcode}.
For more information about reduction for arbitrary~$g$, see
\cite{DHBHS} and \cite[Section~1.3]{KLLRSS}.
The specific case $g=1$ comes down to Gauss reduction of quadratic forms,
and details for $g=2$ are given in Dupont's thesis~\cite{dupont}.
For an implementation for $g=3$, see Kılıçer~\cite{Genusthreereduction}.
\subsection{The type norm}\label{sec:reflex}\label{sec:typenorm}
The type norm of $\Phi$ is the map $N_{\Phi}: K\rightarrow \mathbf{C}: x\mapsto \prod_{\phi\in\Phi} \phi(x)$. Its image generates the \emph{reflex field} $K^{\mathrm{r}}$ of~$\Phi$, and there is a reflex type norm map \[ N_{\Phi^{\mathrm{r}}} : K^{\mathrm{r}}\rightarrow K : x \mapsto \prod_{\psi\in\Phi^{\mathrm{r}}} \psi(x),\] where the product is taken over the \emph{reflex type} $\Phi^{\mathrm{r}}$, i.e., the set of embeddings $\psi : K^{\mathrm{r}} \rightarrow \overline{K}$ such that there is a map $\phi:\overline{K}\rightarrow\mathbf{C}$ with $\phi\circ\psi =\mathrm{id}_{K^{\mathrm{r}}}$
and $\phi_{|K}\in\Phi$.
The reflex type norm extends to ideals via (\cite[Proposition~29 in \S8.3]{shimura-taniyama}) \[ N_{\Phi^{\mathrm{r}}}(\mathfrak{a})\mathcal{O}_L = \prod_{\psi\in\Phi^{\mathrm{r}}} (\psi(\mathfrak{a})\mathcal{O}_L)\] for any number field $L\subset \overline{K}$ containing the images $\psi(K^{\mathrm{r}})$ for all $\psi\in\Phi^{\mathrm{r}}$. We implemented this as \verb!Phir.type_norm()! in \cite{cmcode}.
\subsection{The main theorem}
\newcommand{\mult}[3]{[#1]^{#2}_{#3}} \newcommand{\change}[2]{\mult{1}{#1}{#2}}
Given a number field $K$, two bases $B = (b_1,\ldots,b_n)$ and $C=(c_1,\ldots, c_n)$ of $K$ over $\mathbf{Q}$ and an element $x\in K$, we denote by $\mult{x}{C}{B}$ the $n\times n$ matrix over $\mathbf{Q}$ such that for each $j$ the $j$th column is $xc_j$ expressed in terms of~$B$. In other words, it is the matrix of multiplication by $x$ as a map from $K$ with basis $C$ to $K$ with basis $B$. If we interpret $B$ and $C$ as row vectors in $K^n$, then we have \begin{equation}\label{eq:matrixofmult} x\cdot C = B\cdot \mult{x}{C}{B}. \end{equation}
We say that a matrix $M\in\matrixring{d}{\mathbf{Q}}$ is \emph{invertible mod $N$} if the numerator of the determinant and the denominators of all coefficients are coprime to~$N$. In that case, reduction modulo $N$ defines a matrix $(M\ \mathrm{mod}\ N)\in\mathrm{GL}_d(\mathbf{Z}/N\mathbf{Z})$. \begin{theorem}\label{thm:general}
Let $\tau=\tau(\Phi, \mathfrak{b}, \xi, B)\in\mathbf{H}_g$ be a primitive CM point with CM field~$K$,
let $N$ be a positive integer and let $f\in\mathcal{F}_N$
be a function that does not have a pole at $\tau$.
Let $F$ be the smallest positive integer such that $F\mathcal{O}_K$ is contained in the multiplier ring
$\mathcal{O}$ of~$\mathfrak{b}$ ($F=1$ if $\mathcal{O}=\mathcal{O}_K$).
Then $f(\tau)$ lies in the ray class field of $K^{\mathrm{r}}$ for the modulus $NF$.
For any fractional ideal
$\mathfrak{a}\in \IKr{NF}$, if $[\mathfrak{a}]$
is the class of $\mathfrak{a}$ in the ray class group mod~$NF$,
then $f^{[\mathfrak{a}]}$ is given as follows.
Choose a symplectic basis
$C$
of~$N_{\Phi^{\mathrm{r}},\mathcal{O}}(\mathfrak{a})^{-1}\mathfrak{b}$
with respect to $E_{
N(\mathfrak{a})\xi}$
and let \[\tau' = \tau(\Phi, N_{\Phi^{\mathrm{r}},\mathcal{O}}(\mathfrak{a})^{-1}\mathfrak{b},
N(\mathfrak{a})\xi, C).\]
Then $M:=(\change{C}{B})^{\mathrm{t}}$
is in
$\GSpspecific{2g}(\mathbf{Q})^{+}$,
with $\nu(M) = N(\mathfrak{a})^{-1}$,
and is invertible mod~$N$.
Moreover, we have
$U:=(M\ \mathrm{mod}\ N)^{-1}\in\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$, and
\begin{equation}\label{eq:general}
f(\tau)^{[\mathfrak{a}]} =
f^U (M\tau)
= f^U (\tau').
\end{equation} \end{theorem} For the computation of suitable $B$ and $C$ (and hence $\tau'$, $M$ and $U$), see Section~\ref{sec:basis} (second and fourth paragraph).
We implemented the complete computation of $\tau'$, $M$ and $U$ in \cite{cmcode} as $$\verb!tau.Shimura_reciprocity(a, N, period_matrix=True)!.$$ For the computation of $f^U$, see Section~\ref{sec:actioncompute}. This makes $f^U(\tau')$ an explicit expression for $f(\tau)^{[\mathfrak{a}]}$ that is suitable for computation.
\subsection{The class fields generated by complex multiplication}
Fix a CM-point $\tau$ and let the notation be as above. The field \[\cmext{N} = K^{\mathrm{r}}\big(f(\tau) : f\in\mathcal{F}_N\ \mbox{s.t.}\ f(\tau)\not=\infty\big)\subset \mathbf{C}.\]
is an abelian extension of~$K^{\mathrm{r}}$, and
we now describe the corresponding ideal group.
Let $F$ be the smallest positive integer satisfying $F\mathcal{O}_K\subset\mathcal{O}$. For~$x\in K$, we write $x\equiv 1\ \mathrm{mod}^\units\ N\mathcal{O}$ to mean
$x=a/b$ where $a$ and $b$ are elements of~$\mathcal{O}$ that are invertible modulo~$NF\mathcal{O}$
and congruent to each other modulo~$N\mathcal{O}$. For various equivalent definitions, see Definition~\ref{def:equivalence}. This is equivalent to standard definitions in the case $\mathcal{O}=\mathcal{O}_K$.
\begin{theorem}\label{thm:idealgroup} The extension $\cmext{N}/K^{\mathrm{r}}$ is abelian and of conductor dividing~$NF$. Its Galois group is isomorphic via the Artin isomorphism to the quotient group $I(NF) / \cmgp{N}$, where $I(NF)$ is the group of fractional $\mathcal{O}_{K^{\mathrm{r}}}$-ideals with numerator and denominator coprime to $NF$, and \begin{equation}\label{eq:HPhiO} \cmgp{N} = \left\{ \mathfrak{a}\in \IKr{NF} : \exists \mu \in K\ \text{with}\ \begin{array}{c} N_{\Phi^{\mathrm{r}}}(\mathfrak{a}) = \mu\mathcal{O}_K\\ \mu\overline{\mu} = N(\mathfrak{a})\in\mathbf{Q}\\ {\mu\equiv 1\ \mathrm{mod}^\units\ N\mathcal{O}}\end{array}\right\}. \end{equation}
\end{theorem} \begin{remark}A similar result exists for fields of definition of torsion points on normalized Kummer varieties (Main Theorem 2 in \S16 of \cite{shimura-taniyama, shimura};
Main Theorem 3 in \S17 of~\cite{shimura-taniyama}).
A similar result in adelic language exists for fields of moduli of abelian varieties with torsion structure (Corollary 5.16 of~\cite{shimuraAF}; Corollary 18.9 of~\cite{shimura}).
We give a proof for values of modular functions directly in the language of the fields $\mathcal{F}_N$ using the reciprocity laws, see Section~\ref{sec:proofidealgroup}. \end{remark}
Note that this theorem implies that $\cmext{N}$ depends only on $\mathcal{O}$ and~$\Phi$, not on~$\tau$.
\begin{definition}\label{def:mu} For $\mathfrak{a}\in \cmgp{1}$, we write $\mu(\mathfrak{a})$ to denote an element of $K^{\times}$ as in Theorem~\ref{thm:idealgroup}. \end{definition}
Note that $\mu(\mathfrak{a})$ is uniquely defined up to multiplication by roots of unity in~$K$.
\begin{algorithm}[Computing $\mu(\mathfrak{a})$]\label{alg:mu}
\\
\textbf{Input:} $\Phi^{\mathrm{r}}$ and a fractional ideal $\mathfrak{a}$ of $\mathcal{O}_{K^{\mathrm{r}}}$.\\
\textbf{Output:} The list of all
elements $\mu\in K^{\times}$ such that $N_{\Phi^r}(\mathfrak{a}) = \mu\mathcal{O}_K$
and $\mu\overline{\mu} \in\mathbf{Q}$.\\
\textbf{Algorithm:}
\begin{enumerate}
\item Compute the class group and unit group of $K$. Compute the maximal totally real
subfield $K_0$ of $K$ and its unit group $\mathcal{O}_{K_0}^{\times}$.
Compute the quotient $\mathcal{O}_{K_0}^{\times}/N_{K/K_0}(\mathcal{O}_{K}^{\times})$.
This can be done using e.g.~the algorithms of \cite{cohen},
or the software Magma~\cite{magma} or Pari~\cite{pari}.
Pari~\cite{pari} can be used through SageMath~\cite{sage}.
\item Compute $N_{\Phi^r}(\mathfrak{a})$ and test whether it is principal.
\begin{enumerate}
\item If it is, then let $\beta\in K^{\times}$ be a generator.
\item Otherwise return an empty list.
\end{enumerate} \item Let $u = \beta\overline{\beta} / N(\mathfrak{a}) \in \mathcal{O}_{K_0}^{\times}$ and test whether $u \in N_{K/K_0}(\mathcal{O}_{K}^{\times})$.
\begin{enumerate}
\item If it is, then take $v\in\mathcal{O}_K^{\times}$ such that $v\overline{v}=u$.
\item Otherwise return an empty list.
\end{enumerate} \item Return $\{w\beta/v \ :\ w\in(\mathcal{O}_{K}^{\times})^{\mathrm{tors}}\}$.
\end{enumerate} \noindent We implemented this as \verb!a_to_mus(Phir, a)! in~\cite{cmcode}. \end{algorithm} \begin{proof}[Proof of Algorithm~\ref{alg:mu}]
It is clear that every $\mu =
w
\beta /v$ in the output generates $N_{\Phi^r}(\mathfrak{a})$
and satisfies $\mu\overline{\mu}=N(\mathfrak{a})\in\mathbf{Q}$.
Conversely, suppose that $N_{\Phi^r}(\mathfrak{a})=\mu\mathcal{O}_K$
and $\mu\overline{\mu} \in\mathbf{Q}$. Then $\mu\overline{\mu} = N(\mathfrak{a})$
and $N_{\Phi^r}(\mathfrak{a})$ is principal, so $\beta$ exists.
Let $r =\beta/\mu\in\mathcal{O}_{K}^{\times}$.
Then $r\overline{r} = \beta\overline{\beta} / N(\mathfrak{a}) = u$,
hence $v$ exists.
Let $w = v/r\in\mathcal{O}_{K}^{\times}$. Then $w\overline{w}=1$, so $w$
is a root of unity. Therefore, $\mu = \beta/r = w\beta/v$ is listed by the algorithm. \end{proof}
Using Algorithm~\ref{alg:mu} and standard algorithms for computing ray class groups and computing quotients of groups, we can compute the group $H_{\Phi,\mathcal{O}}(N)/P(NF)$ as a subset of the ray class group $\mathrm{Cl}(NF) = I(NF)/P(NF)$ and in turn compute the group $\mathrm{Gal}(\cmext{N}/K^{\mathrm{r}}) = I(NF) / H_{\Phi,\mathcal{O}}(N)$. For an efficient and detailed algorithm, see Asuncion~\cite{jaredpaper1}.
\subsection{Class invariants and a special case of the main theorem}\label{ssec:statementbeforelast}
The reciprocity law (Theorem~\ref{thm:general}) gives the Galois action of $\mathrm{Gal}(\cmext{N})/K^{\mathrm{r}})$ on $f(\tau)$. In order to decide whether $f(\tau)$ is in the field $\cmext{1}$ generated by the values of Igusa invariants at~$\tau$, we need in particular the Galois action of the subgroup $\mathrm{Gal}(\cmext{N})/\cmext{1})$. For that particular subgroup, we have a simpler version of the reciprocity law as follows.
From Theorem~\ref{thm:idealgroup}, we have $\mathrm{Gal}(\cmext{N}/\cmext{1})= (\IKr{NF}\cap \cmgp{1}) / \cmgp{N}$. For any fractional ideal $\mathfrak{a}\in \IKr{NF}\cap\cmgp{1}$, we get $\mu=\mu(\mathfrak{a})\in K$ with $\mu\overline{\mu}=N(\mathfrak{a})\in\mathbf{Q}$ and $N_{\Phi^r}(\mathfrak{a})=\mu\mathcal{O}_K$ (cf.~Definition~\ref{def:mu}).
\newcommand{\wasepsilon}[1]{(\mult{#1}{B}{B})^{\mathrm{t}}}
\begin{theorem}\label{thm:special}
Let $\tau=\tau(\Phi, \mathfrak{b}, \xi, B)\in\mathbf{H}_g$ be a primitive CM point,
let $N$ be a positive integer and let $f\in\mathcal{F}_N$
be a function that does not have a pole at $\tau$.
For any $\mathfrak{a}\in \IKr{NF}\cap\cmgp{1}$,
we have
$$f(\tau)^{[\mathfrak{a}]} = f^{\wasepsilon{\mu(\mathfrak{a})}}(\tau).$$ \end{theorem}
The theorem defines a multiplicative map \label{sec:statementspecific} \begin{align}\label{eq:wascalledg} r : \frac{\IKr{NF}\cap\cmgp{1}}{\cmgp{N}} \quad \longrightarrow& \quad \GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})/S\\ [\mathfrak{a}]\quad \longmapsto & \quad\wasepsilon{\mu(\mathfrak{a})},\nonumber\end{align} where $S$ is the image in $\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ of the stabilizer $\mathrm{Stab}_\tau\subset \Spspecific{2g}(\mathbf{Z})$ of~$\tau$. Then we have $$f(\tau)^{[\mathfrak{a}]} = f^{r(\mathfrak{a})}(\tau).$$ \begin{remark}If $\mathfrak{a}$ is principal, then the reciprocity map becomes even more explicit: \begin{equation}\label{eq:moreexplicit}r((\alpha)) = \wasepsilon{N_{\Phi^{\mathrm{r}}}(\alpha))}\quad\text{for}\quad \alpha\inK^{\mathrm{r}}\ \text{with}\ (\alpha)\in \IKr{NF}. \end{equation} \end{remark}
We now get the following way to look for class invariants. Given $\tau = \tau(\Phi, \mathfrak{b}, \xi, B)$, we compute the image $r(X)$ for a set of generators $X$ of the domain of~$r$. Then $f(\tau)$ is a class invariant if and only if $f$ is fixed by $r(X)$.
\begin{algorithm}[Computing the image of $r$]\label{alg:imageofr}
\\
\textbf{Input:} $N$, $F$, $\Phi$, $\mathfrak{b}$, $\xi$, $B$.\\
\textbf{Output:} the image $r(X)$ of a set of generators $X$ of the domain of $r$.\\
\textbf{Algorithm:}
\begin{enumerate}
\item Compute $G = (I(NF)\cap H_{\Phi,\mathcal{O}}(1)) / P(NF)\subset \mathrm{Cl}(NF)$.
\item Let $X$ be a set of generators of $G$.
\item For every element of $X$, choose a representative $\mathfrak{a}$,
take an arbitrary $\mu$ in the output of Algorithm~\ref{alg:mu}
and compute $\wasepsilon{\mu}$.
Return the list of matrices $\wasepsilon{\mu}$ computed in this way. \end{enumerate} We implemented this algorithm as \verb!reciprocity_map_image(tau, N)! in~\cite{cmcode}. \end{algorithm}
We give an example in Section~\ref{ssec:detailedexample}.
\subsection{Complex conjugation}\label{ssec:conjugationstatement}
Now assume $f(\tau)$ is a class invariant, that is, is in $\cmext{1}$.
The coefficients of its minimal polynomial $H_f$ over~$K^{\mathrm{r}}$ are elements of~$K^{\mathrm{r}}$.
We now give a new result, which in many cases
gives us a way to make sure the coefficients are totally real
and hence in the maximal totally real subfield $\reflexfield_0\subsetneq K^{\mathrm{r}}$.
Let $\mathcal{M}:= \mathbf{Q}(f(\tau) : f \in \Fcal{1})$
be the \emph{field of moduli} of the
principally polarized abelian variety corresponding to~$\tau$,
and let $\mathcal{M}_0 = \mathcal{M}\reflexfield_0\subset \mathcal{M}K^{\mathrm{r}}=\cmext{1}$.
We give two results. Lemma~\ref{lem:complexconjugation} says that often
$\mathcal{M}_0$ is strictly smaller than $\cmext{1}$.
And if this is the case, then
Proposition~\ref{prop:complexconjugation} gives a criterion for
the minimal polynomial of $f(\tau)$ over $K^{\mathrm{r}}$
to have coefficients in $\reflexfield_0$.
\begin{lemma}\label{lem:complexconjugation}
Suppose $\tau$ corresponds to a pair $(\mathfrak{b},\xi)$. \begin{enumerate}
\item The degree of $\cmext{1}/\mathcal{M}_0$ equals~$2$
if and only if there is an ideal $\mathfrak{a}\in I(F)$
and an element $\mu\in K^\times$ such that
$N_{\Phi^{\mathrm{r}},\mathcal{O}}(\mathfrak{a})\overline{\mathfrak{b}} = \mu \mathfrak{b}$
and $\mu\overline{\mu} \in\mathbf{Q}$. \item If $g\leq 2$, $\mathfrak{b}$ is coprime to $F\mathcal{O}$, and $\Phi$ is a primitive CM-type, then the conditions in part~1 are satisfied and we can take \begin{enumerate} \item $g=1$, $\mathfrak{a}=N_{\Phi}(\mathfrak{b}\overline{\mathfrak{b}}^{-1}\mathcal{O}_K)$ and $\mu=1$; or
\item $g=2$, $\mathfrak{a}=N_{\Phi}(\mathfrak{b}\mathcal{O}_K)$ and~$\mu=N(\mathfrak{b})$. \end{enumerate} \item If $\mathfrak{b}=\mathcal{O}$, then the conditions in part 1 are satisfied and we can take $\mathfrak{a}=\mathcal{O}_{K^{\mathrm{r}}}$ and~$\mu=1$. \end{enumerate}
\end{lemma}
\begin{proposition}\label{prop:complexconjugation}
Given $\tau = \tau(\Phi, B)$ and $f\in \mathcal{F}_N$,
suppose
$\deg \cmext{1}/\mathcal{M}_0=2$
and $f(\tau)\in\cmext{1}$.
Let $(\mathfrak{a}, \mu)$ be as in Lemma~\ref{lem:complexconjugation}(1)
and assume without loss of generality that $\mathfrak{a}$ is coprime to~$N$.
Write $B=(b_1,\ldots,b_g,b_{g+1},\ldots,b_{2g})$, consider the $\mathbf{Q}$-basis
$C=(\mu^{-1} \overline{b_1},\ldots,
\mu^{-1}\overline{b_g},
\mu^{-1}\overline{b_{g+1}},\ldots,
\mu^{-1}\overline{b_{2g}})$ of~$K$,
and let $M=(\change{C}{B})^{\mathrm{t}}$.
Then~$M$ is invertible modulo~$N$ and has an inverse
$U\in \GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$.
Moreover, the following are equivalent:
\begin{enumerate}
\item $f(\tau)\in \mathcal{M}_0$,
\item $f^{U}({\tau})=f({\tau})$.
\end{enumerate}
If these conditions are satisfied, then
the minimal polynomial of $f(\tau)$ over
$K^{\mathrm{r}}$ has coefficients in~$\reflexfield_0$.
\end{proposition}
\begin{example}
Suppose $g=1$ and $\mathfrak{b} = \mathbf{Z}[\sqrt{D}]$.
Then we can take $b_1=\sqrt{D}$ and $b_2=1$, and by Lemma~\ref{lem:complexconjugation}
also $\mu=1$, so $c_1=-b_1$ and $c_2=b_2$, hence
$M$ is the diagonal $2\times 2$ matrix $(-1,0;0,1)$ and so is~$U$.
As the matrix $-I\in\mathrm{SL}_2(\mathbf{Z})$ acts trivially on every $\tau\in\mathbf{H}_1$,
we find that $U$ acts exactly as $i(-1\ \mathrm{mod}\ N)$ does, which
is as complex conjugation of the coefficients of~$f$.
The condition \ref{prop:complexconjugation}(2) then translates
to $f$ having only real coefficients in its $q$-expansion. \end{example}
We will prove these results in Section~\ref{sec:conjugationproof}.
These results show that, if we restrict to~$f$ that satisfy~$f^{U} = f$,
then the minimal polynomial of $f(\tau)$
over~$K^{\mathrm{r}}$ is defined over~$\reflexfield_0$.
We implemented the computation of the matrix $U$ in~\cite{cmcode} as
\verb!tau.complex_conjugation_symplectic_matrix(N)!.
\section{The ad\`elic version} \label{sec:ad}
In Section~\ref{sec:proof}, we give a proof of the restults
stated in Sections \ref{ssec:statementfirst}--\ref{ssec:statementbeforelast}
(including the reciprocity law).
For this, we use Shimura's own formulation of his reciprocity law,
which we state in Section~\ref{sec:ad}.
In Section~\ref{sec:conjugationproof}, we prove the results
about complex conjugation stated in Section~\ref{ssec:conjugationstatement}.
The reader who is not interested in the proof, or would
like to see the applications first, is advised
to skip ahead and read Sections~\ref{sec:theta} (Theta constants),
\ref{sec:examples} (Examples) and \ref{sec:applications}
(Applications)
first. They are independent of Sections~\ref{sec:ad}--\ref{sec:conjugationproof}.
Shimura developed his reciprocity laws for various types of multivariate modular functions, modular forms, and theta functions in a series of articles \cite{shimura-models-I, shimura-models-II, shimura-arithmetic, shimura-fourier, shimura-theta-cm, shimura-reciprocity-theta}. See also the textbook \cite[26.10]{shimura}. Rather than reproving the reciprocity law in our setting, we will quote a streamlined version stated by Shimura in the language of id\`eles
and rework it (in Section~\ref{sec:proof}) into a version with ideals and a more explicit group action. This means that our proof
will not be the most direct proof, as the ad{\`e}lic statement mashes all levels~$N$ together, and we take them apart again; and Shimura's original series of articles
starts with theta functions, while we give them as a special case afterwards (Section~\ref{sec:theta}). However, our approach does allow us to give both the computationally practical and the elegant ad{\`e}lic statement, explain how they are related, and keep the proofs short at the same time.
The reader who would rather see a direct proof of our explicit version of the reciprocity law should see Yang~\cite{yang-shimura}. Yang, inspired by our explicit statements, gives a direct proof of our explicit version of Shimura's reciprocity law and uses that to prove the ad\`elic statement.
We start by citing Shimura's ad\`elic action of~$\GSpspecific{2g}$, and linking it to the actions of Proposition~\ref{prop:groupaction}.
Let $\mathbf{A}$ be the ring of ad\`eles of~$\mathbf{Q}$ and call an element of its unit group
\emph{positive} if its $\mathbf{R}$-component is positive. Let $\widehat{\ZZ}=\lim_{\leftarrow} \mathbf{Z}/N\mathbf{Z}$ be the ring of finite integral ad\'eles, so $\mathbf{A} = (\widehat{\ZZ}\otimes\mathbf{Q}) \times \mathbf{R}$. Let $\mathcal{F}_\infty = \cup_{N}\mathcal{F}_N$. \begin{proposition}\label{prop:biggroupaction} Let $\mathrm{Aut}(\mathcal{F}_\infty)$ be the automorphism group of the field~$\mathcal{F}_\infty$. There is a unique homomorphism $\GSpspecific{2g}(\mathbf{A})^+\rightarrow \mathrm{Aut}(\mathcal{F}_\infty)$ satisfying \begin{enumerate} \item for~$x\in\mathbf{A}^\times$ and $f\in\mathcal{F}_\infty$, we define $f^{i(x)}$ as the function obtained from $f$ by acting with $x$ on the $q$-expansion coefficients,
\item for~$A\in \GSpspecific{2g}(\mathbf{Q})^+$, $f\in\mathcal{F}_\infty$, $\tau\in\mathbf{H}$, we have $f^A(\tau) = f(A \tau)$,
\item for any~$N$, the group $T{} = \{A\in \GSpspecific{2g}(\widehat{\ZZ}) : A \equiv 1\ \mathrm{mod}^\units\ N\}\times \GSpspecific{2g}(\mathbf{R})^+$ acts trivially on any~$f\in \mathcal{F}_N$, where we write $A\equiv 1\ \mathrm{mod}^\units\ N$ iff for all $p\mid N$ we have $A_p\in 1 + N\matrixring{2g}{\mathbf{Z}_p}$.
\end{enumerate}
\end{proposition} \begin{proof}
Existence is a special case of~\cite[Thm.~5(v,vi,vii)]{shimura-fourier}. Uniqueness follows from the proof of~\cite[Proposition~1.3]{shimura-reciprocity-theta}. \end{proof} \begin{remark}\label{rem:aboutproof} Our reference for existence in Proposition~\ref{prop:biggroupaction} is directly applicable to our situation, but
does not contain the proof. Therefore, we give some pointers for the proof. The action is constructed in~\cite[Section 2.7]{shimura-models-I} for a field $k_S(V_S)$.
The field $k_S(V_S)$ is defined without $q$-expansions, hence that reference only contains a weak version of~(1), but (2) is \cite[(2.7.2)]{shimura-models-I} and (3) follows immediately from \cite[{(2.5.3$_\mathrm{a}$)}]{shimura-models-I}. Our stronger version of~(1), as well as the link between $\mathcal{F}_\infty$ and~$k_S(V_S)$, is given in~\cite{shimura-fourier}. Both that reference and \cite[\S6]{shimura-arithmetic} claim that the proof is exactly the same as in the Hilbert modular case, which is~\cite{shimura-arithmetic}. \end{remark}
The following corollary proves exactly Proposition~\ref{prop:groupaction}.
\begin{corollary}\label{cor:zzmod} The action of Proposition~\ref{prop:biggroupaction} has the following property: \begin{enumerate} \setcounter{enumi}{3} \item For any positive integer $N$, any $f\in\mathcal{F}_N$, and any $$A=(A_{\mathrm{f}}, A_{\infty})\in\GSpspecific{2g}(\widehat{\ZZ})\times \GSpspecific{2g}(\mathbf{R})^+ \subset \GSpspecific{2g}(\mathbf{A})^+,$$ we have $f^A\in\mathcal{F}_N$, and $f^A$ depends only on $(A_{\mathrm{f}}\ \mathrm{mod}\ N)\in\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$. Moreover, the induced action of~$\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ on $\mathcal{F}_N$ is exactly as in Proposition~\ref{prop:groupaction}.
\end{enumerate} \end{corollary} \begin{proof} The fact $f^A\in\mathcal{F}_N$ follows from the construction of the action (see \cite[2.7 and (2.5.3)]{shimura-models-I} and Remark~\ref{rem:aboutproof} above). That~$f^A$ depends only on $(A_{\mathrm{f}} \ \mathrm{mod}\ N)$ is Proposition~\ref{prop:biggroupaction}(3). It follows that the action induces an action of~$\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ on~$\mathcal{F}_N$. To prove that this action is as in Proposition~\ref{prop:groupaction}, it remains only to compute this action for~$B\in \Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ and for~$B=i(t)$ with $t\in(\mathbf{Z}/N\mathbf{Z})^\times$.
In case of~$B\in\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$, we lift $B$ to $A'\in \Spspecific{2g}(\mathbf{Z})$ (possible by \cite[Thm.~VII.21]{newman}). As we have $$\Spspecific{2g}(\mathbf{Z}) = \GSpspecific{2g}(\mathbf{Q})^+ \cap (\GSpspecific{2g}(\widehat{\ZZ})\times \GSpspecific{2g}(\mathbf{R})^+),$$ we can then apply Proposition~\ref{prop:biggroupaction}(2) to find that the action of $B$ is as in Proposition~\ref{prop:groupaction}. In case of~$B=i(t)$ with $t\in(\mathbf{Z}/N\mathbf{Z})^\times$, we lift~$t$ to $\widehat{\ZZ}^\times$ and apply Proposition~\ref{prop:biggroupaction}(1). \end{proof}
Let $\tau\in \mathbf{H}$ correspond to a simple principally polarized abelian variety
$\mathbf{C}^g / (\tau\mathbf{Z}^g+\mathbf{Z}^g)$ with complex multiplication by~$K$.
Assume the CM-type~$\Phi$ to be primitive.
The type norm $N_{\Phi^{\mathrm{r}}}$ and the map $\epsilon : a\mapsto (\mult{a}{B}{B})^{\mathrm{t}}$ induce ad\`elic maps $N_{\Phi^{\mathrm{r}}}:K^{\mathrm{r}\units}_{\mathbf{A}}\rightarrow K^\times_\mathbf{A}$ and $\epsilon: K^\times_\mathbf{A}\rightarrow \mathrm{GL}_{2g}(\mathbf{A})$ and the composite map sends $K^{\mathrm{r}\units}_{\mathbf{A}}$ to $\GSpspecific{2g}(\mathbf{A})^+$.
Shimura gives the following reciprocity law, stated in a very sleek manner using the action of Proposition~\ref{prop:biggroupaction}. \begin{theorem}[{Shimura's reciprocity law for Siegel modular functions \cite{shimura-reciprocity-theta}}]\label{thm:shimrecipad} Let $\tau$ and the notation be as above.
Then for any $f\in\mathcal{F}_\infty$ such that~$f(\tau)$ is finite and any $x\in K^{\mathrm{r}\units}_{\mathbf{A}}$, we have $$f(\tau) \in K^{\mathrm{r}}_{\mathrm{ab}}\quad\mbox{and}\quad f(\tau)^x = f^{\matrixofShimura{x}^{-1}}(\tau).$$
\end{theorem}
\begin{proof}
This is exactly equation (3.43) of \cite[p.~57]{shimura-reciprocity-theta}. (Actually, that reference assumes that the abelian variety $A=\mathbf{C}^g/(\tau\mathbf{Z}^n+\delta \mathbf{Z}^n)$ for an integer $\delta\geq 3$ has CM, but that variety has CM by $K$ if and only if ours has, so that the result for~$\delta=1$ follows.) The matrix $\epsilon(a)$ is defined less explicitly in~\cite{shimura-reciprocity-theta}, namely for $a\in K$ by \begin{equation}\label{eq:epsilonShim}\rho(a)(\tau, 1) = (\tau, 1)\epsilon(a)^{\mathrm{t}} \end{equation} where $\rho(a)\in\matrixring{g}{\mathbf{C}}$ is the matrix of multiplication by $a$ with respect to the standard basis of~$\mathbf{C}^g$.
Our matrix $\epsilon(a) = (\mult{a}{B}{B})^{\mathrm{t}}$ satisfies the definition \eqref{eq:epsilonShim}.
\end{proof} \begin{remark} As in Remark~\ref{rem:aboutproof}, see also \cite[(2.7.3)]{shimura-models-I} (equivalently \cite[(6.2.3)]{shimura-models-II}). See also \cite[26.8(4)]{shimura} for a textbook version. \end{remark}
\section{Proof of the explicit reciprocity law} \label{sec:proof}
In this section, we prove our explicit version
of Shimura's reciprocity law,
using Shimura's ad{\`e}lic version (Theorem~\ref{thm:shimrecipad}).
Given $f\in\mathcal{F}_N$ and the image $[\mathfrak{a}]$
in the mod-$NF$ ray class group
of an id\`ele~$x$,
we want to compute $f(\tau)^{[\mathfrak{a}]} = f(\tau)^x$, which
by Theorem~\ref{thm:shimrecipad}
is $f^{\matrixofShimura{x}^{-1}}(\tau)$.
To do so, we
write $\matrixofShimura{x}^{-1}
= S\ U\ M$ with $M\in\GSpspecific{2g}(\mathbf{Q})^+$, $U\in\GSpspecific{2g}(\widehat{\ZZ})$, $S\in\mathrm{Stab}_f$,
and both $M$ and $(U\ \text{mod}\ N)$ explicit in terms of~$\mathfrak{a}$.
Then we can conclude $f^{\mathfrak{a}}(\tau) =
f^{(U\ \text{mod}\ N)}(M\tau)$,
by
Theorem~\ref{thm:shimrecipad}.
\begin{remark}
The strong approximation theorem for~$\GSpspecific{2g}(\mathbf{A})$
in fact tells us (\cite[Lemma~1.1]{shimura-reciprocity-theta})
that such a decomposition always exists, even with $U\in i(\smash{\widehat{\ZZ}^\times})$.
However, as in the genus-one case (Gee and Stevenhagen~\cite{gee-stevenhagen}),
we will be satisfied with having only
$U\in \GSpspecific{2g}(\smash{\widehat{\ZZ}})$.
In fact, by allowing $U\in\GSpspecific{2g}(\smash{\widehat{\ZZ}})$, we can make sure that $M\tau$
is in a fundamental domain for $\Spspecific{2g}(\mathbf{Z})$, which improves the speed of convergence when $f^{(U\ \text{mod}\ N)}$ is expressed in terms of theta constants.
\end{remark}
\subsection{Coprimality and congruence for fractions}
To help in translating ad{\`e}lic statements to more concrete statements, we first state some well-known equivalent definitions of $\ \mathrm{mod}^\units\ {}$ that we will use.
\newcommand{\mathcal{O}_{(p)}}{\mathcal{O}_{(p)}} Let $\mathcal{O}$ be an order in $K$ and let $F\in\mathbf{Z}$ be the smallest positive integer such that $\mathcal{O}\supset F\mathcal{O}_K$. For any prime number $p\in\mathbf{Z}$, let \[\mathcal{O}_{(p)} = \{a/b\in K : a\in\mathcal{O}, b\in\mathbf{Z}\setminus p\mathbf{Z}\}.\] In this section, for $a\in\mathcal{O}$, we use the notation $\overline{a}=(a\ \mathrm{mod}\ NF\mathcal{O})\in (\mathcal{O}/NF\mathcal{O})$.
\begin{definition}\label{def:equivalence}
For a positive integer $N$ and for $x\in K^{\times}$,
we say that $x$ is \emph{coprime to $NF$ with respect to $\mathcal{O}$} if one of the following equivalent
conditions holds (equivalence is proven below):
\begin{enumerate}
\item $x=a/b$ for some $a,b\in\mathcal{O}$
with $\overline{a}, \overline{b} \in(\mathcal{O}/NF\mathcal{O})^{\times}$
and $b\not=0$,
\item $x = a/b$ for some $a\in\mathcal{O}$
and $b\in\mathbf{Z}\setminus\{0\}$ with
$\overline{a}\in(\mathcal{O}/NF\mathcal{O})^{\times}$
and
$b\in 1 + NF\mathbf{Z}$,
\item for all prime numbers $p\mid NF$, we have $x \in \mathcal{O}_{(p)}^{\times}$,
\item
$x\mathcal{O} = \mathfrak{a}\mathfrak{b}^{-1}$
for non-zero $\mathcal{O}$-ideals $\mathfrak{a}$ and $\mathfrak{b}$
that are coprime to $NF$ in the sense that
$\mathfrak{a}+NF\mathcal{O}=\mathfrak{b}+NF\mathcal{O}=\mathcal{O}$.
\end{enumerate}
We write $x \equiv 1\ \mathrm{mod}^\units\ N\mathcal{O}$ to mean
that one of the following equivalent conditions hold,
which are stronger than the conditions above
(equivalence is proven below):
\begin{itemize}
\item[(1')] as in (1) above, with additionally
$a-b\in N\mathcal{O}$,
\item[(2')] as in (2) above, with additionally $a-1\in N\mathcal{O}$,
\item[(3')] as in (3) above, with additionally
$x -1 \in N\mathcal{O}_{(p)}$ for all prime numbers $p\mid N$.
\end{itemize}
In terms of $p$-adic numbers, both
$\mathcal{O}\otimes \mathbf{Z}_p$
and $K$ are subrings of $\mathcal{O}\otimes \mathbf{Q}_p$,
and their intersection is exactly $\mathcal{O}_{(p)}$.
In particular, the conditions (3) can equivalently
be written with $(\mathcal{O}\otimes\mathbf{Z}_p)$ instead of $\mathcal{O}_{(p)}$.
\end{definition}
\begin{proof}[Proof of equivalence in Definition~\ref{def:equivalence}] We start with the equivalence of (1)--(4).\\
$(2)\Rightarrow (1)$ is obvious.\\ $(1)\Rightarrow (2)$. Let $N(b) = \#(\mathcal{O}/b\mathcal{O})$. We start by showing that $NF$ is coprime to $N(b)$ and that $N(b)$ is an $\mathcal{O}$-multiple of~$b$.
We have $1\in b\mathcal{O}+NF\mathcal{O}$, so $NF$ is a unit modulo $b\mathcal{O}$, hence multiplication by $NF$ is invertible on the group $(\mathcal{O}/b\mathcal{O})$ of order $N(b)$, so $NF$ is coprime to~$N(b)$. Note that $N(b)$ annihilates the group $\mathcal{O}/b\mathcal{O}$, hence $N(b)\in b\mathcal{O}$, so $N(b)$ is a multiple of~$b$.
Let $c\in\mathbf{Z}$ be such that $b':=cN(b)\equiv 1\pmod{NF}$. We get $x=a'/b'$ with $a'=acN(b)/b\in\mathcal{O}$ and $\overline{a'}\in(\mathcal{O}/NF\mathcal{O})^\times$.
\\ $(1)\Rightarrow (3)$. Suppose that $x$ satisfies~$(1)$. Then $x^{-1}$ also satisfies~$(1)$. By ``$(1)\Rightarrow (2)$'', we then get that both $x$ and $x^{-1}$ satisfy~$(2)$. By the definition of $\mathcal{O}_{(p)}$, we then get $x, x^{-1}\in\mathcal{O}_{(p)}$, hence $x\in\mathcal{O}_{(p)}^{\times}$.\\ $(3)\Rightarrow (4)$. For every prime $p\mid NF$, write $x = a_p/b_p$ and $x^{-1} = c_p/d_p$ with $a_p, c_p\in\mathcal{O}$ and $b_p,d_p\in\mathbf{Z}\setminus p\mathbf{Z}$. Let $\mathfrak{b} = \sum_{p} b_p\mathcal{O}\subset\mathcal{O}$,
$\mathfrak{d} = \sum_{p} d_p\mathcal{O}\subset\mathcal{O}$, $\mathfrak{a} = \mathfrak{b}x = \sum_{p} a_p\mathcal{O}\subset \mathcal{O}$, and $\mathfrak{c} = \mathfrak{d}x^{-1} = \sum_{p} c_p\mathcal{O}\subset \mathcal{O}$. We get $\mathfrak{a}\mathfrak{c} = \mathfrak{b}\mathfrak{d}$. The ideals $\mathfrak{b}$ and $\mathfrak{d}$ are coprime to all prime numbers $p\mid NF$ because of $b_p\in \mathfrak{b}$ and $d_p\in \mathfrak{d}$. It follows that $\mathfrak{a}\mathfrak{c} = \mathfrak{b}\mathfrak{d}$ is coprime to $NF\mathcal{O}$. In particular, both $\mathfrak{a}$ and $\mathfrak{b}$ are coprime to $NF\mathcal{O}$, hence $\mathfrak{b}$ is invertible and we have $x\mathcal{O} = \mathfrak{a}\mathfrak{b}^{-1}$.\\ $(4)\Rightarrow (1)$. Suppose $x = \mathfrak{a}\mathfrak{b}^{-1}$
with $\mathfrak{a}$ and $\mathfrak{b}$ ideals of
$\mathcal{O}$ coprime to $NF\mathcal{O}$.
Then $\mathfrak{a}$ and $\mathfrak{b}$ are both invertible
and we have $x\mathfrak{b}=\mathfrak{a}$.
Choose $b\in\mathfrak{b}$ with $b\equiv 1\ (\mathrm{mod}\ NF\mathcal{O})$, which is possible by
definition of coprimality of ideals.
Let $a = bx$. Then $a\mathcal{O}=(b\mathfrak{b}^{-1})\mathfrak{a}$
is coprime to $NF\mathcal{O}$.
This proves~(1).
We have now proved that (1)--(4) are equivalent. It remains to prove that (1')--(3') are equivalent. Note that each ($n$') is strictly stronger than ($n$), so we may and will assume that (1)--(4) hold. So
write $x=a/b$ as in (1) and $x=a'/b'$ as in~(2). We have $ab'=a'b$, hence $b'(a-b) = b(a'-b')$. As $b$ and $b'$ are invertible in $\mathcal{O}/NF\mathcal{O}$, we get $(1')\Leftrightarrow b'(a-b)\in N\mathcal{O} \Leftrightarrow b(a'-b')\in N\mathcal{O}\Leftrightarrow (2')$.
Next, we have $b'(x-1) = (a'-1) - (b'-1)$. Here $b'\in\mathcal{O}_{(p)}^{\times}$ and $b'-1\in NF\mathbf{Z}\subset N\mathcal{O}$. In particular, we have (3') if and only if for all $p\mid N$ we have $a'-1\in N\mathcal{O}_{(p)}$. We also have $a'-1\in\mathcal{O}$ and $\cap_{p\mid N} N\mathcal{O}_{(p)}\cap \mathcal{O} = N\mathcal{O}$, o (3')$\Leftrightarrow$(2'). \end{proof}
\subsection{The conductor}\label{sec:conductor}
The first statement in the main theorem (Theorem~\ref{thm:general}) is that $f(\tau)$ lies in the ray class field for the modulus $NF$. We now prove this.
As in Section~\ref{sec:statement}, let $\mathfrak{b}$ be a fractional $\mathcal{O}$-ideal with $\mathrm{End}(\mathfrak{b})=\mathcal{O}$ and let $F$ be the smallest positive integer such that $F\mathcal{O}_K\subset\mathcal{O}$. Let $B$ be a $\mathbf{Z}$-basis of $\mathfrak{b}$.
\begin{lemma}\label{lem:epsiloninteger} For~$a\in K^\times$, we have $a\in\mathcal{O}$ if and only if $\mult{a}{B}{B}\in\matrixring{2g}{\mathbf{Z}}$. \end{lemma} \begin{proof} We have $a\in\mathcal{O}$ if and only if $a\mathfrak{b}\subset\mathfrak{b}$, which is equivalent to $\mult{a}{B}{B}\in\matrixring{2g}{\mathbf{Z}}$. \end{proof} \begin{lemma}\label{lem:epsilonmodstar} For~$a\in K$, we have $a\equiv 1\ \mathrm{mod}^\units\ N\mathcal{O}$ if and only if the following two conditions hold: \begin{enumerate}
\item we have $\mult{a}{B}{B}\in \mathrm{GL}_{2g}(\mathbf{Z}_p)$
for all $p\mid NF$, and
\item when taking the reduction modulo $N$ of $\mult{a}{B}{B}$ (which is possible, as by the previous point the denominators of the entries and the numerator of the determinant are coprime to~$N$), we get the identity matrix.
\end{enumerate} \end{lemma} \begin{proof} Lemma~\ref{lem:epsiloninteger} stays valid when considered locally at a prime number~$p$, that is, replacing $\mathcal{O}$ by
$\mathcal{O}_{(p)}$ and $\mathbf{Z}$ by $\mathbf{Z}_{(p)}$ for a prime~$p$. By Definition~\ref{def:equivalence}(3'), we have $a\equiv 1\ \mathrm{mod}^\units\ N\mathcal{O}$ if and only if $(a-1)/N\in
\mathcal{O}_{(p)}$ for all~$p|N$
and $a\in\mathcal{O}_{(p)}^\times$ for all $p|N$. The corollary follows if we apply the lemma to $(a-1)/N$ locally at all primes dividing~$N$ and to $a$ and $a^{-1}$ at all primes dividing~$NF$.
\end{proof}
Recall the abelian extension $\cmext{N}=K^{\mathrm{r}}(f(\tau) : f\in\mathcal{F}_N)$ of~$K^{\mathrm{r}}$. The first statement in Theorem~\ref{thm:general} is the following. \begin{proposition}\label{prop:conductor} The conductor of $\cmext{N}$ divides~$NF$. \end{proposition} \begin{proof} What we need to prove is equivalent to the statement that the subgroup $W_{NF}=\{x\in K^{\mathrm{r}\units}_{\mathbf{A}} :
x\equiv 1 \ \mathrm{mod}^\units\ NF\mathcal{O}_{K^{\mathrm{r}}}\}$ acts trivially on all $f\in \mathcal{F}_N$. So take any $x\in W_{NF}$, and let $y=(\mult{N_{\Phi^{\mathrm{r}}}(x)}{B}{B})^{-1}$. Then Theorem~\ref{thm:shimrecipad} tells us $f(\tau)^x = f^y(\tau)$.
We have $N_{\Phi^{\mathrm{r}}}(x)\equiv 1\ \mathrm{mod}^\units\ NF\mathcal{O}_K$, hence $N_{\Phi^{\mathrm{r}}}(x)\equiv 1\ \mathrm{mod}^\units\ N\mathcal{O}$. By Lemma~\ref{lem:epsilonmodstar}, we find that~$y$ is in the set $T{} = \{A\in \GSpspecific{2g}(\widehat{\ZZ}) : A \equiv 1\ \mathrm{mod}^\units\ N\}\times \GSpspecific{2g}(\mathbf{R})^+$, which acts trivially on $f$ by Proposition~\ref{prop:biggroupaction}. So we get $f^y=f$, hence $f(\tau)^x = f^y(\tau) = f(\tau)$. \end{proof}
\subsection{Changes of symplectic bases}\label{sec:mundane}
Theorem~\ref{thm:general} claims that the matrix~$M=(\change{B}{C})^{\mathrm{t}}$ is in the group~$\GSpspecific{2g}(\mathbf{Q})^+$. The purpose of the current section is to prove this.
Recall from \eqref{eq:matrixofmult} that $M\in\mathrm{GL}_{2g}(\mathbf{Q})$ satisfies $C=BM^{\mathrm{t}}$ where $B, C\in K^{2g}$ are
symplectic bases of the lattices~$\mathfrak{b}$ and $N_{\Phi^{\mathrm{r}},\mathcal{O}}(\mathfrak{a})^{-1}\mathfrak{b}$ with respect to polarizations $E_{\xi}$ and $E_{N(\mathfrak{a})\xi}$. \begin{lemma}\label{lem:mundane} Let $M$ be as above. Then we have $M\in\GSpspecific{2g}(\mathbf{Q})^+$ with $\nu(M) = N(\mathfrak{a})^{-1}$. \end{lemma} \begin{proof} This follows by taking $y=N(\mathfrak{a})$ in the following lemma. \end{proof} \begin{lemma}\label{lem:transpose} Let $(\mathbf{C}^g/\Lambda, E)$ be a principally polarized abelian variety and $B$ a symplectic basis of~$\Lambda$ for~$E$.
Given $M\in\mathrm{GL}_{2g}(\mathbf{Q})$, let $C$ be the $\mathbf{R}$-basis of $\mathbf{C}^g$ such that $\change{C}{B} = M^{\mathrm{t}}$ and let $\Lambda'$ be the lattice in $\mathbf{C}^g$ generated by~$C$.
Then the following are equivalent: \begin{enumerate} \item there exists $y\in\mathbf{Q}^\times$ such that~$yE$ is a principal polarization for~$\mathbf{C}^g/\Lambda'$ and $C$ is a sympletic basis for $yE$,
\item $M\in\GSpspecific{2g}(\mathbf{Q})^+$. \end{enumerate} Moreover, if this is the case, then the point in $\mathbf{H}$ corresponding to $C$ is $\tau' = M\tau$, and we have $y=\nu(M)^{-1}$. \end{lemma} \begin{proof} By \cite[Lemma 8.2.1]{birkenhake-lange} the group $\GSpspecific{2g}(\mathbf{Q})^+$ is stable under transposition,
so (2) is equivalent to $N := M^{\mathrm{t}} = \change{C}{B}\in\GSpspecific{2g}(\mathbf{Q})^+$. The matrix of $E$ with respect to $B$ is~$\Omega$, so the matrix of $yE$ with respect to $C$ is $yN^{\mathrm{t}} \Omega N$.
If (1) holds, then positive-definiteness of $(u,v)\mapsto yE(iu,v)$ implies $y>0$ and we get $yN^{\mathrm{t}} \Omega N = \Omega$, hence $M\in\GSpspecific{2g}(\mathbf{Q})^+$ with $\nu(M) = y^{-1}$.
Conversely, if (2) holds, then with $y = \nu(M)^{-1}$, we have a polarisation $yE$ with symplectic basis~$C$.
This proves that (1) and (2) are equivalent, and gives $y=\nu(M)^{-1}$.
Finally,
take the $g\times 2g$ matrix $\mathcal{B}=(B_1|B_2)$ whose columns are the elements of~$B$ and write $M = (a,b; c, d)$ in terms of $g\times g$ blocks $a,b,c,d$. Note that we have $\tau = B_2^{-1}B_1$
and $\mathcal{B}M^{\mathrm{t}} = (B_1a^{\mathrm{t}}+B_2b^{\mathrm{t}}|B_1c^{\mathrm{t}}+B_2d^{\mathrm{t}})$, hence $\tau' = (B_1c^{\mathrm{t}}+B_2d^{\mathrm{t}})^{-1} (B_1a^{\mathrm{t}}+B_2b^{\mathrm{t}})$. As $\tau$ and $\tau'$ are symmetric, we get $\tau = B_1^{\mathrm{t}} (B_2^{\mathrm{t}})^{-1}$ and $\tau' = (aB_1^{\mathrm{t}}+bB_2^{\mathrm{t}})(cB_1^{\mathrm{t}}+dB_2^{\mathrm{t}})^{-1} = (a\tau + b) (c\tau + d)^{-1} = M\tau$. \end{proof}
\subsection{Decomposing \texorpdfstring{$\matrixofShimura{x}$}{\matrixofShimurapdfstring{x}} modulo the stabilizer}\label{ssec:decomp}
The bridge between ad\`elic and ideal theoretic class field theory is the surjection
\[K^{\mathrm{r}\units}_{\mathbf{A}}/ K^{\mathrm{r}\units}
\rightarrow \IKr{NF} / \PKr{NF}\] that maps the class of an id\`ele $x\equiv 1\ \mathrm{mod}^\units\ NF$ to the class of the ideal $\mathfrak{a}$ with $\mathrm{ord}_{\mathfrak{p}}(\mathfrak{a}) = \mathrm{ord}_{\mathfrak{p}}(x_{\mathfrak{p}})$.
Let $\mathfrak{a}$ be a fractional $\mathcal{O}_{\reflexfield}$-ideal coprime to $NF$.
Let the notation be as in Theorem~\ref{thm:general}, and pick any id\`ele $x=(x_v)_v\in K^{\mathrm{r}\units}_{\mathbf{A}}$ such that \begin{enumerate} \item for every finite prime $\mathfrak{p}$ of~$K^{\mathrm{r}}$, we have $\mathrm{ord}_{\mathfrak{p}}(x_{\mathfrak{p}})=\mathrm{ord}_{\mathfrak{p}}(\mathfrak{a})$, and \item for every valuation $v$ of~$K^{\mathrm{r}}$ with $v(NF)>0$, we have $x_{v} = 1$.
\end{enumerate}
Then we immediately have \begin{equation}\label{eq:trivialbutimportant} \matrixofShimura{x} \equiv 1_{2g} \ \mathrm{mod}^\units\ NF, \end{equation} with $\epsilon : a\mapsto \mult{a}{B}{B}$ as defined above Theorem~\ref{thm:shimrecipad}.
Let us recall the situation of Theorem~\ref{thm:general}: we have a symplectic basis $B = (b_1,\ldots,b_g)$ of $\mathfrak{b}$ with respect to $E_{\xi}$ and a symplectic basis $C = (c_1,\ldots,c_g) = BM^{\mathrm{t}}$ of $N_{\Phi^{\mathrm{r}},\mathcal{O}}(\mathfrak{a})^{-1}\mathfrak{b}$ with respect to $E_{N(\mathfrak{a})\xi}$. Here $M = (\change{C}{B})^{\mathrm{t}}$.
\begin{lemma}\label{lem:defofa} The matrix $A:=\matrixofShimura{x}^{-1}M^{-1}$ lies in $\GSpspecific{2g}(\widehat{\ZZ})\times \GSpspecific{2g}(\mathbf{R})^+$.
\end{lemma} \begin{proof} Note $\nu\circ \epsilon\circ N_{\Phi^{\mathrm{r}}} = N_{K^{\mathrm{r}}/\mathbf{Q}}$, and the fact that~$K^{\mathrm{r}}$ has no real embeddings implies $N_{K^{\mathrm{r}}/\mathbf{Q}}(K^{\mathrm{r}}\otimes \mathbf{R})\subset \mathbf{R}_{\geq 0}$, so $\matrixofShimura{x}_{\infty}\in\GSpspecific{2g}(\mathbf{R})^+$.
We also have $M\in\GSpspecific{2g}(\mathbf{Q})^+$ by Lemma~\ref{lem:mundane}, hence $A_\infty \in \GSpspecific{2g}(\mathbf{R})^+$. It now suffices
to prove for every prime number $p$ that~$A_p$ is in $\GSpspecific{2g}(\mathbf{Z}_p)$. For any number field~$L$ and~$x\in L_{\mathbf{A}}^\times$, write $x_p\in L \otimes\mathbf{Z}_p$ for the part corresponding to primes over~$p$.
We have the following identity of~$\mathbf{Z}_p$-submodules of~$K\otimes\mathbf{Z}_p$ of rank $2g$: $$(N_{\Phi^{\mathrm{r}}}(\mathfrak{a})^{-1}\mathfrak{b})\otimes \mathbf{Z}_p=N_{\Phi^{\mathrm{r}}}(x)_p^{-1} (\mathfrak{b}\otimes\mathbf{Z}_p)$$ (indeed, for $p\mid FN$, both sides are equal to $\mathfrak{b}\otimes\mathbf{Z}_p$, while for $p\nmid F$, the order is locally maximal and the identity follows from $\mathrm{ord}_{\mathfrak{p}}(\mathfrak{a})=\mathrm{ord}_{\mathfrak{p}}(x_{\mathfrak{p}})$). We have already chosen a basis $c_1,\ldots,c_{2g}$ of the left hand side.
We take the $\mathbf{Z}_p$-basis $N_{\Phi^{\mathrm{r}}}(x)_p^{-1} b_1, \ldots, N_{\Phi^{\mathrm{r}}}(x)_p^{-1} b_{2g}$ of the right hand side and notice that~$A_p^{\mathrm{t}}$ is the matrix that transforms one basis to the other. In particular, we have $A_p\in\mathrm{GL}_{2g}(\mathbf{Z}_p)$. As the basis on the left is symplectic for~$N(\mathfrak{a})\xi$ and the one on the right is symplectic for~$N(x)_p\xi$, we apply the localization of Lemma~\ref{lem:transpose} and find
$A_p\in\GSpspecific{2g}(\mathbf{Q}_p)$. As we already had $A_p\in\mathrm{GL}_{2g}(\mathbf{Z}_p)$, we conclude $A_p\in\GSpspecific{2g}(\mathbf{Z}_p)$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:general}] The fact that $f(\tau)$ is in the ray class field modulo~$NF$ is Proposition~\ref{prop:conductor}. We have $M\in\GSpspecific{2g}(\mathbf{Q})^+$, $\nu(M) = N(\mathfrak{a})^{-1}$, and $\tau'=M\tau$ by Lemma~\ref{lem:mundane}.
It remains to prove that $M$ is invertible modulo $N$ and that $U = (M\ \mathrm{mod}\ N)^{-1}$ is in $\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ and satisfies $f(\tau)^x = f^U(M\tau)$.
We have $\matrixofShimura{x}^{-1} = AM$ with $A\in \GSpspecific{2g}(\widehat{\ZZ})\times \GSpspecific{2g}(\mathbf{R})^+$ by Lemma~\ref{lem:defofa}. The reciprocity law (Thm.~\ref{thm:shimrecipad}) now tells us $f(\tau)^x = f^{AM}(\tau)$. By Corollary~\ref{cor:zzmod}, we find that $A$ acts on~$f$ as $(A \ \mathrm{mod}\ N)$ does. Moreover, we have $A \equiv M^{-1} \ \mathrm{mod}^\units\ NF$ by \eqref{eq:trivialbutimportant}, so the definition of $U$ in Theorem~\ref{thm:general} is equivalent to $U=(A\ \mathrm{mod}\ N)$. Conclusion: $f(\tau)^x = f^{U}(M\tau)$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:special}] Theorem~\ref{thm:special} is a special case of
Theorem~\ref{thm:general} as follows. Pick $C = \mu^{-1} B$ in Theorem~\ref{thm:general}, that is, $c_i = \mu^{-1} b_i$ for $i=1,\ldots, 2g$, so $M = (\mult{1}{C}{B})^{\mathrm{t}} = (\mult{\mu^{-1}}{B}{B})^{\mathrm{t}}$. Then $M\tau=\tau$ since multiplication by $\Phi(\mu)$ is a $\mathbf{C}$-linear isomorphism that transforms the symplectic basis $C$ into~$B$. At the same time,
$U = (M\ \mathrm{mod}~N)^{-1} = ((\mult{\mu}{B}{B})^{\mathrm{t}}\ \mathrm{mod}~N)$. \end{proof}
\subsection{Determining the ideal group} \label{sec:proofidealgroup}
Next, we prove Theorem~\ref{thm:idealgroup}, which states the ideal group of $\cmext{N}/K^{\mathrm{r}}$. A similar result exists in terms of fields of moduli of torsion points (Main Theorem 3 in \S17 of~\cite{shimura-taniyama}), but we give a new proof, directly in the language of the fields $\mathcal{F}_N$ using the reciprocity law.
\begin{proof}[Proof of Theorem~\ref{thm:idealgroup}] Note that Theorem~\ref{thm:special} and Lemma~\ref{lem:epsilonmodstar} already imply that $\cmgp{N}$ acts trivially on~$\cmext{N}$. It remains to prove that if $\mathfrak{a}\in \IKr{NF}$ acts trivially on $\cmext{N}$, then $\mathfrak{a}\in \cmgp{N}$. Here without loss of generality the ideal $\mathfrak{a}$ is integral, that is, we have $\mathfrak{a}\subset\mathcal{O}$.
So let $\mathfrak{a}\in \IKr{NF}$ be an integral ideal with $f(\tau)^{\mathfrak{a}}=f(\tau)$ for all~$f\in\mathcal{F}_N$. Let $U$ and $M$ be as in Theorem~\ref{thm:general}, so that for all $f\in\mathcal{F}_N$, we get $f(\tau) = f(\tau)^{\mathfrak{a}} = f^{U}(M\tau)$ with $U\in\GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ and $M\in\GSpspecific{2g}(\mathbf{Q})^+$ such that~$(M\ \mathrm{mod}\ NF)$ is finite and invertible with inverse~$U$. We claim that without loss of generality, we have $U=1$, $M\equiv 1\ \mathrm{mod}^\units\ N$ and $M\tau=\tau$.
Proof of the claim: By taking $f=\zeta_N$, we find $\zeta_N^{\nu(U)}=\zeta_N$, hence $U\in\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$. Then lift $U$ to $\Spspecific{2g}(\mathbf{Z})$, and use the lift to change the chosen basis $c_1,\ldots,c_g$ of Theorem~\ref{thm:special}. We find that without loss of generality, we have $U=1$, which implies $M\equiv 1\ \mathrm{mod}^\units\ N$. We now have $f(\tau) = f(M\tau)$ for all $f\in\mathcal{F}_N$, and by \cite[(2.5.1)]{shimura-models-I}, this implies $\tau\in\Gamma_N M\tau$, i.e., $\tau=\gamma M\tau$ for some $\gamma\in\Gamma_N$. We use $\gamma$ to change the basis $c_1,\ldots,c_g$ again, and conclude also $M\tau=\tau$. This proves the claim.
Let $X=M^{-1}$, so $X\equiv 1\ \mathrm{mod}^\units\ N$ and $X\tau=\tau$. Let $\mathfrak{c}=N_{\Phi^{\mathrm{r}}, \mathcal{O}}(\mathfrak{a})$. We have $\mathfrak{c}\subset\mathcal{O}$ and $X$ sends a basis $c_1,\ldots,c_{2g}$ of~$\mathfrak{c}^{-1}\mathfrak{b}$ to a basis $b_1,\ldots,b_{2g}$ of~$\mathfrak{b}\subset\mathfrak{c}^{-1}\mathfrak{b}$, hence $X\in\matrixring{2g}{\mathbf{Z}}$. The congruences on $M$ now tell us $X\in\Gamma_N$.
The fact $X\tau=\tau$ shows that there is an isomorphism $h:\mathbf{C}^g/\Phi(\mathfrak{c}^{-1}\mathfrak{b})\rightarrow \mathbf{C}^g/\Phi(\mathfrak{b})=A$ preserving symplectic bases. The identity map on $\mathbf{C}^g$ induces an isogeny the other way around, which scales the polarization by $N(\mathfrak{a})$. Their composite is some $\mu \in \mathrm{End}(A)=\mathcal{O}$,
which therefore satisfies $\mu^{-1}\mathfrak{b} = \mathfrak{c}^{-1}\mathfrak{b}$ and $\mu\overline{\mu} = N(\mathfrak{a})\in\mathbf{Q}$. This last identity shows that~$\mu$ is coprime to~$F$, so if we look at the coprime-to-$F$ part of $\mu^{-1}\mathfrak{b} = \mathfrak{c}^{-1}\mathfrak{b}$ and use that the coprime-to-$F$ part of~$\mathfrak{b}$ is invertible, then we find $\mu\mathcal{O} = \mathfrak{c}$.
By definition of $\mu$, $h$, and $X$, the endomorphism $\mu$ acts as $M^{\mathrm{t}}$ on the chosen symplectic bases (i.e., $\epsilon(\mu) = M$). Lemma~\ref{lem:epsilonmodstar} therefore shows $\mu\equiv 1\ \mathrm{mod}^\units\ N\mathcal{O}$. \end{proof}
\section{Complex conjugation} \label{sec:conjugationproof}
Next, we prove the results in Section~\ref{ssec:conjugationstatement}.
\begin{proof}[Proof of Lemma~\ref{lem:complexconjugation}]
Recall $\mathcal{M}_0 = \reflexfield_0(f(\tau) : f \in \Fcal{1})$,
and consider the extension $\cmext{1}=\mathcal{M}_0K^{\mathrm{r}} / \mathcal{M}_0$.
Part (1) of Lemma~\ref{lem:complexconjugation} states
that this extension has degree 2 if and only if there exist
$\mathfrak{a}\in I(F)$ and $\mu\in K^\times$
such that $N_{\Phi^{\mathrm{r}}, \mathcal{O}}(\mathfrak{a})\overline{\mathfrak{b}} = \mu \mathfrak{b}$
and $\mu\overline{\mu} \in \mathbf{Q}$.
We start by proving the `only if' part, so suppose that
$\cmext{1}/\mathcal{M}_0$ has degree~$2$.
The non-trivial automorphism $\gamma_0$ of this extension
restricts to complex conjugation on~$K^{\mathrm{r}}$,
so $\gamma:x\mapsto \overline{x^{\gamma_0}}$ is an element of $\mathrm{Gal}(\cmext{1}/ K^{\mathrm{r}})$.
Suppose that $\tau$ is obtained from $(\mathfrak{b}, \xi)$, and let $A$ be the corresponding principally polarized abelian variety.
As $\gamma$ and complex conjugation are equal on~$\mathcal{M}_0$, we get that $\gamma(A)$ and $\overline{A}$ are isomorphic.
Then by \cite[Prop.~3.5.5]{lang-cm}, the abelian variety
$\overline{A}$ corresponds to $(\overline{\mathfrak{b}}, \xi)$.
At the same time, the automorphism $\gamma$ corresponds via the Artin map to
the class of an ideal $\mathfrak{a}$ of $K^{\mathrm{r}}$.
The isomorphism between $\gamma(A)$ and $\overline{A}$ then gives
an element $\mu\in K^\times$ such that we have
$N_{\Phi^{\mathrm{r}}}(\mathfrak{a})\overline{\mathfrak{b}} = \mu \mathfrak{b}$
and $N(\mathfrak{a}) = \mu\overline{\mu}$.
This proves the `only if' of~(1).
Conversely, if $\mathfrak{a}$ exists, by scaling $\mathfrak{a}$ (and scaling $\mu$ accordingly), we can assume $\mathfrak{a}$ to be coprime to $NF$. Then take the corresponding $\gamma\in\mathrm{Gal}(\cmext{1}/K^{\mathrm{r}})$ and let $\gamma_0:x\rightarrow \overline{x^\gamma}$, which is in $\mathrm{Gal}(\cmext{1}/\mathcal{M}_0)$ and is non-trivial as it restricts to complex conjugation on~$K^{\mathrm{r}}$.
For part~(2), in case $g=1$ and $\mathfrak{b}$ is coprime to $F\mathcal{O}$, we
simply take
$\mathfrak{a} = N_{\Phi}(\mathfrak{b}/\overline{\mathfrak{b}})$ and $\mu=1$
as~$N_{\Phi^{\mathrm{r}}}$ is an isomorphism with inverse~$N_{\Phi}$.
If $g=2$ and $\mathfrak{b}$ is coprime to $F\mathcal{O}$,
take $\mathfrak{a} = N_{\Phi}(\mathfrak{b}\mathcal{O}_K)$ and $\mu=N(\mathfrak{b})$.
We claim $N_{\Phi^{\mathrm{r}}}N_{\Phi}(\mathfrak{b}\mathcal{O}_K)=
N(\mathfrak{b}) \mathfrak{b}\overline{\mathfrak{b}}^{-1}\mathcal{O}_K$,
which implies part~(2).
Proof of the claim: let $\phi:\overline{K}\rightarrow \overline{K^{\mathrm{r}}}$
extend an element of~$\Phi$, and let $\sigma\in\mathrm{Aut}(\overline{K})$
be such that $(\phi\circ\sigma)_{|K}$ is the other element.
Then $N_{\Phi^{\mathrm{r}}}N_{\Phi}(\mathfrak{b})=\mathfrak{b}^2
\sigma(\mathfrak{b})\sigma^{-1}(\mathfrak{b})$.
On the other hand, as~$\Phi$ is a primitive CM-type,
we see from~\cite[Example~8.4(2)]{shimura-taniyama}
that the embeddings of $K$ into $\overline{K}$
are exactly $\mathrm{id}$, $\sigma$, $\sigma^{-1}$,
and complex conjugation, so $N_{\Phi^{\mathrm{r}}}N_{\Phi}(\mathfrak{b})=
N(\mathfrak{b}\mathcal{O}_K) \mathfrak{b}\overline{\mathfrak{b}}^{-1}$.
Finally, if $\mathfrak{b}=\mathcal{O}$, then
$\overline{\mathfrak{b}}=\overline{\mathcal{O}}=\mathcal{O}$,
so $\mathfrak{a}=1$ and $\mu=1$ suffice.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:complexconjugation}]
Assume that $\cmext{1}/\mathcal{M}_0$
is an extension of degree~$2$,
so $\mathfrak{a}$, $\mu$ and $\gamma_0$ as in the proof of Lemma~\ref{lem:complexconjugation} exist
with $\mathfrak{a}\in I(NF)$.
Let $f(\tau)$ be any class invariant with $f\in\mathcal{F}_N$.
Now $f(\tau)$ is in $\mathcal{M}_0$ if and only if $f(\tau)^{\gamma_0} = f(\tau)$,
i.e., if and only if $\overline{f(\tau)^{[\mathfrak{a}]}} = f(\tau)$.
The action of complex conjugation on $f(\tau)$ is easy to describe.
Note that $f^{i(-1\ \mathrm{mod}\ N)}$ is $f$
with its Fourier coefficients replaced by their
complex conjugates. Since complex conjugation is continuous
on~$\mathbf{C}$, we get by \eqref{eq:fourier}
\begin{equation}\label{eq:cconj}\overline{f(\tau)} =
f^{i(-1\ \mathrm{mod}\ N)}(-\overline{\tau}).
\end{equation}
Let's look at the action of $[\mathfrak{a}]$ via the reciprocity
law Theorem~\ref{thm:general}.
Let
$b_1,\ldots,b_{2g}$ be the symplectic basis of $\mathfrak{b}$ corresponding
to $\tau$
and consider the symplectic basis
\[C' = (\mu^{-1}\overline{b_1},\ldots,\mu^{-1}\overline{b_g},-\mu^{-1}\overline{b_{g+1}},\ldots,-\mu^{-1}\overline{b_{2g}})\]
of $\mu^{-1}\overline{\mathfrak{b}} = N_{\Phi^{\mathrm{r}},\mathcal{O}}(\mathfrak{a})^{-1}\mathfrak{b}$
with respect
to
$\mu\overline{\mu}\xi = N(\mathfrak{a})\xi$.
The period matrix corresponding to this symplectic basis $C'$ is~$-\overline{\tau}$.
Let $M' = (\change{C'}{B})^{\mathrm{t}}$ and let $U' = (M'\ \mathrm{mod}~N)^{-1}$
(we will confirm later that this reduction and inverse exist).
This basis $C'$ differs from the $\mathbf{Q}$-basis $C$ of~$K$ in
Proposition~\ref{prop:complexconjugation} by multiplying the final $g$
vectors by $-1$,
so we have $\change{C'}{C} = i(-1)$).
In particular,
we have ${M'}^{\mathrm{t}} = \change{C'}{B} =\change{C}{B} \change{C'}{C} = M^{\mathrm{t}} i(-1)$,
hence $M' = i(-1) M$ and $U' = U i(-1)$.
We apply Theorem~\ref{thm:general}
to~$C'$.
This gives that $M'$
is in $\GSpspecific{2g}(\mathbf{Q})^+$ and
indeed invertible modulo~$N$ and that we have
$f(\tau)^{[\mathfrak{a}]} = f^{U'}(M'\tau) = f^{Ui(-1)}(-\overline{\tau})$.
Combining this with \eqref{eq:cconj}, we find
$\overline{f(\tau)^{[\mathfrak{a}]}} = f^{U}(\tau)$.
We conclude that indeed $f^{U}(\tau) = f(\tau)$
if and only if $f(\tau)\in\mathcal{M}_0$.
Now let again $\gamma_0$ be the
non-trivial automorphism of $\cmext{1}$ that fixes $\mathcal{M}_0$.
Note that $\gamma_0$ restricts to complex conjugation on~$K^{\mathrm{r}}$.
Let $P\in K^{\mathrm{r}}[X]$ be the minimal polynomial of $\alpha = f(\tau)$ over~$K^{\mathrm{r}}$.
Then $\overline{P} = \gamma_0(P)$ is the minimal polynomial of $\gamma_0{\alpha}$ over~$K^{\mathrm{r}}$.
In the case $\alpha\in\mathcal{M}_0$, we have $\gamma_0(\alpha) = \alpha$, hence $\overline{P}=P$,
so $P$ has coefficients in~$\reflexfield_0$.
This finishes the proof of Proposition~\ref{prop:complexconjugation}.
\end{proof}
\section{Theta constants} \label{sec:theta}
\newcommand{\prm}{}
\newcommand{\transpose}{^{\mathrm{t}}}
\newcommand{t_{\mathrm{inv}}}{t_{\mathrm{inv}}}
For $c_1,c_2\in\mathbf{Q}^g$, the \emph{theta constant} with characteristic $c_1,c_2$ is
\begin{equation}\label{eq:deftheta}\theta [ c_1,c_2](\tau)= \sum_{v\in \mathbf{Z}^g} \text{exp}(\pi i (v+c_1) \tau (v+c_1)^{\text{t}}+2\pi i (v+c_1) c_2^{\text{t}}). \end{equation} We often restrict to theta constants with $c_i\in [0,1)^g$, because we have \begin{align}\label{eq:thetamodz}\theta[c_1+n_1, c_2+n_2] &= \exp(2\pi i c_1 n_2^{\mathrm{t}})\theta[c_1, c_2]& \mbox{for}\quad n_1,n_2\in\mathbf{Z}^g. \end{align}
Theta constants have a very explicit action, as the folowing result shows. The result itself is not suprising, but the author is unaware of an equally explicit version in the literature: directly working for $\GSpspecific{2g}$
instead of only $\Spspecific{2g}$ and working with arbitrary coefficient-wise lifts instead of having to lift to $\Spspecific{2g}(\mathbf{Z})$.
\begin{proposition}\label{prop:theta} Given $D\in 2\mathbf{Z}$ and $c_1,c_2,c_1',c_2'\in D^{-1}\mathbf{Z}^{g}$, we have $$\frac{\theta[c_1,c_2]}{\theta[c_1',c_2']}\in\mathcal{F}_{2D^2}.$$ Moreover, the action of $A\in \GSpspecific{2g}(\mathbf{Z}/2D^2\mathbf{Z})$ is as follows.
Take lifts $$\tbt{a & b \\ c & d}\in\matrixring{2g}{\mathbf{Z}} \quad \mbox{and}\quad t_{\mathrm{inv}}\in\mathbf{Z}$$ of $A$ and $\nu(A)^{-1}$.
Define $$\left(\genfrac{}{}{0pt}{0}{d_1^{\prm}}{d_2^{\prm}}\right) = A^{\mathrm{t}} \left(\genfrac{}{}{0pt}{0}{ c_1^{\prm}-\frac{1}{2}\ t_{\mathrm{inv}}\ \mathrm{diag}(cd^{\mathrm{t}})}{ c_2^{\prm} - \frac{1}{2}
\ t_{\mathrm{inv}}\ \mathrm{diag}(ab^{\mathrm{t}})}\right)\quad\mbox{and}
$$ $$r^{\prm}=\frac{1}{2}(t_{\mathrm{inv}} (dd_1^{\prm}-cd_2^{\prm})^{\mathrm{t}} (-bd_1^{\prm}+ad_2^{\prm}+\mathrm{diag}(ab^{\mathrm{t}})) - d_1^{\transpose} d_2^{\prm}),$$ and define $d_1'$, $d_2'$, $r'$ analogously. Then we have \begin{equation} \label{eq:theta} \left(\frac{\theta[c_1,c_2]}{\theta[c_1',c_2']}\right)^A =
\exp(2\pi i (r-r')) \frac{\theta[d_1,d_2]}{\theta[d_1',d_2']}. \end{equation}
\end{proposition}
\begin{remark} It is known that the field generated by all quotients as in Proposition~\ref{prop:theta} (for all $D$) equals the field~$\mathcal{F}_\infty$ (see e.g. \cite[27.15]{shimura}).
\end{remark}
To prove Proposition~\ref{prop:theta} we use the following lemma giving the action of $\Spspecific{2g}(\mathbf{Z})$.
\begin{lemma}\label{lem:theta} Given $A\in\Spspecific{2g}(\mathbf{Z})$, there is a holomorphic $\rho =\rho_A: \mathbf{H}_g\rightarrow \mathbf{C}^{\times}$ such that for all $c_1,c_2\in\mathbf{Q}^g$, we have $$\theta[c_1,c_2](A\tau) = \rho(\tau) \exp(2\pi i r)\theta[d_1,d_2](\tau),$$ where $d_1$, $d_2$, $r$ are as in the formulas of Proposition~\ref{prop:theta} with $t=1$. \end{lemma} \begin{proof}
This follows with some algebraic manipulation when substituting our $d$ for the $c$ in Formula~8.6.1 of~\cite{birkenhake-lange} (see \cite[Lemma~8.4.1(b)]{birkenhake-lange} for the definition of $M[d]$).
\end{proof} \begin{remark}
The function $\rho_A$ is as follows.
Choose a holomorphic branch of $\sqrt{\det(c\tau+d)}$.
Then there exists $\kappa(A)\in\langle\zeta_8\rangle$
for which
$\rho_A(\tau)=\kappa(A)\sqrt{\det(c\tau+d)}$.
Formulas for $\kappa(A)^2$ are
given in \cite[Exercise 8.11(9)]{birkenhake-lange}. \end{remark}
\begin{proof}[{Proof of Proposition~\ref{prop:theta}}]
Let $N=2D^2$.
We start by showing that the right hand side of \eqref{eq:theta} is independent
of the choices of lifts.
Note that a change of lift changes $A^{\mathrm{t}}$
at most by adding elements of $2D^2\mathbf{Z}$ to the entries.
Similarly, it changes $c_i - \frac{1}{2}\ t_{\mathrm{inv}}\ \mathrm{diag}(\cdots)\in D^{-1}\mathbf{Z}^g$
at most by adding elements of $D^2\mathbf{Z}^g$.
In particular, it changes $d_1$, $d_2$, $d_1'$, and $d_2'$
at most by adding elements of $2D\mathbf{Z}^g$.
In turn, this means that $r$ changes at most by adding an element of~$\mathbf{Z}$.
Neither change has effect on the right hand side of \eqref{eq:theta}
by
\eqref{eq:thetamodz}.
Now that we know that \eqref{eq:theta} is independent
of the chosen lifts, we prove it
for $A\in\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$
by taking a lift in $\Spspecific{2g}(\mathbf{Z})$
and taking $t_{\mathrm{inv}} = 1$.
In that case, the result is Lemma~\ref{lem:theta}.
In particular, the equality \eqref{eq:theta} holds
for all $A\in\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$.
Now we show that $f = \theta[c_1,c_2]/\theta[c_1',c_2']$
is indeed in $\mathcal{F}_N$. Multiplying the numerator and denominator of~$f$ by $\theta[0,0]^7$ and using $\rho_A(\tau)^{8} = (\det c\tau+d)^4$, we thus find that $f$ is a quotient of modular forms of equal weight and of level~$N$ with Fourier coefficients in~$\mathbf{Q}(\zeta_{N})$. Note next that if $A\in\Spspecific{2g}(\mathbf{Z})$ is trivial in $\Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$, then \eqref{eq:theta} gives $f^A = f$ (by taking the alternative lift $1_{2g\times 2g}$ of $(A\ \mathrm{mod}\ N)=(1_{2g\times 2g}\ \mathrm{mod}\ N)$).
This proves that we have $f\in \mathcal{F}_N$.
\newcommand{\ch}[1]{#1^0}
Finally, any element $A\in \GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ can be written as~$A=\ch{A}M$. with $M=i(t)$, $t = \nu(A)\in\mathbf{Z}/N\mathbf{Z}$, and $\ch{A}\in\Spspecific{2g}{2g}(\mathbf{Z}/N\mathbf{Z})$. Choose a lift $\widetilde{t}\in\mathbf{Z}$ of $t$. Starting from \eqref{eq:theta} for~$\ch{A}$, we compare what happens when we either multiply $\ch{A}$ by~$M$ from the right, or act on the right hand side of \eqref{eq:theta} by~$M$.
The latter replaces $\zeta_N$ by $\zeta_N^{\widetilde{t}}$, which is equivalent (by the definition \eqref{eq:deftheta}) to changing $r$ into $\widetilde{t}r$ and $(d_1,d_2)$ into $(d_1,\widetilde{t}d_2)$.
Writing $\ch{A} = (\ch{a},\ch{b};\ch{c},\ch{d})$, we get $A = \ch{A}M = (\ch{a},\ch{b};\ch{c}t,\ch{d}t)$. It is straightforward to check that multiplying $\ch{b}$ and $\ch{d}$ by $\widetilde{t}$ and changing $\ch{t}_{\mathrm{inv}} = 1$ into $t_{\mathrm{inv}}$ changes $(d_1,d_2)$ into $(d_1, \widetilde{t}d_2)$ modulo $2D\mathbf{Z}^2$. In turn, this changes $r$ into $\widetilde{t}r$ modulo $\mathbf{Z}$. By \eqref{eq:thetamodz} this gives the same result as just changing $r$ into $\widetilde{t}r$ and $(d_1,d_2)$ into $(d_1,\widetilde{t}d_2)$. \end{proof}
For $N = 2D^2$, given a rational function $f\in\mathcal{F}_N$ that is expressed in terms of theta constants with characteristics in $D^{-1} \mathbf{Z}^{2g}$ with $N\mid 2D^2$, we can now evaluate the action of $A\in \Spspecific{2g}(\mathbf{Z}/N\mathbf{Z})$ on $f$ without the need of lifting to~$\Spspecific{2g}(\mathbf{Z})$.
\begin{remark}
In the case of functions $f\in \mathcal{F}_N$ expressed in terms
of theta constants with denominator $D$ with $N\neq 2D^2$,
we do have to lift to $\GSpspecific{2g}(\mathbf{Z}/2D^2\mathbf{Z})$.
However, instead we will just make sure in our applications
that we already start with matrices in $\GSpspecific{2g}(\mathbf{Z}/2D^2\mathbf{Z})$.
In any case, lifting from
$\GSpspecific{2g}(\mathbf{Z}/m\mathbf{Z})$ to $\GSpspecific{2g}(\mathbf{Z}/n\mathbf{Z})$
prime-by-prime
is a relatively simple matter of linear algebra over~$\mathbf{F}_p$. \end{remark}
If $f$ is a quotient of homogeneous polynomials of equal degree in the theta constants, then we can simply apply the formulas in Proposition~\ref{prop:theta} directly to the individual theta constants and don't really have to write $f$ as a rational function of quotients of the form $\theta[c_1,c_2]/\theta[c_1',c_2']$. For example, note that we have
\begin{equation}\label{eq:f} f = \frac {\theta[\frac{1}{2},0,0,\frac{1}{2}]} {\theta[\frac{1}{2},\frac{1}{2},0,0] + \theta[0,0,0,0]} \quad =\quad \frac {\frac{\theta[\frac{1}{2},0,0,\frac{1}{2}]}{\theta[c_1',c_2']}} {\frac{\theta\theta[\frac{1}{2},\frac{1}{2},0,0]}{\theta[c_1',c_2']} + \frac{\theta[0,0,0,0]}{\theta[c_1',c_2']}}, \end{equation} and the copies of $\exp(-2\pi i r')\theta[d_1',d_2']^{-1}$ in the numerator and denominator cancel in the end anyway.
We implemented the formulas of Proposition~\ref{prop:theta}
in \cite{cmcode}
as \verb!f^A! where $f$ as in \eqref{eq:f} can be constructed
using
\[ \verb!f = ThetaModForm([1/2,0,0,1/2]) / (ThetaModForm([1/2,1/2,0,0]) +!\]
\[\verb!ThetaModForm([0,0,0,0]))!.\]
\section{Finding class invariants and minimal polynomials}\label{sec:examples}
In this section, we work out examples in detail
to demonstrate how to use
the main results for finding class invariants.
We
refer to the results and algorithms
given before
and give some additional results
and algorithms as we need them.
Given an order~$\mathcal{O}$ in a CM-field~$K$
and a CM-type~$\Phi$ of~$K$,
a \emph{class invariant} is a value $f(\tau)$
with $f\in\mathcal{F}_\infty$ and $f(\tau)\in \cmext{1}$.
For example, if $K$ is quadratic and $\mathcal{O} = \mathbf{Z}+\tau\mathbf{Z}$, then $j(\tau)$ is a class invariant, and its minimal polynomial over~$K$ is called the~\emph{Hilbert class polynomial}~$H_{\mathcal{O}}\in\mathbf{Z}[X]$. Weber \cite{weber3} gave class invariants of imaginary quadratic orders with minimal polynomial that have much smaller coefficients than~$H_{\mathcal{O}}$ and from which $j(\tau)$ can be recovered. As mentioned in the introduction, we are interested in having such smaller class invariants, but not just for imaginary quadratic fields. For CM-fields of degree~$2g>2$, we can compare the class invariants with known generators of~$\Fcal{1}$, such as for~$g=2$ the values of absolute Igusa invariants~\cite{igusa}.
Given~$f\in\mathcal{F}_N$, we check the inclusion $K^{\mathrm{r}}(f(\tau))\subset\cmext{1}$ (equivalently~$f(\tau)\in\cmext{1}$) using Theorem~\ref{thm:special}. If~$f$ is sufficiently general, then the inclusion of fields $K^{\mathrm{r}}(f(\tau))\subset\cmext{1}$ is an equality, which can be verified numerically using~\eqref{eq:general}. Equation~\eqref{eq:general} also allows us to numerically determine the minimal polynomial of $f(\tau)$ over~$K^{\mathrm{r}}$.
\subsection{Finding a class invariant}\label{sec:findingclassinv}\label{ssec:detailedexample}
As an example, we will look for small~$f$ that are quotients of products of theta constants with~$c_1,c_2\in\{0,\frac{1}{2}\}^2$, that is, $g=2$, $D=2$,~$N=8$. We also include this example as a demonstration at the beginning of the file \verb!article.sage! at~\cite{cmcode}, so it could be followed step by step on a computer. The theta constants for which~$4c_1 c_2^{\mathrm{t}}$ is odd are identically zero, and we are left with~$10$ so-called \emph{even theta constants}, which happen to have Fourier coefficients in~$\mathbf{Z}$. Following~\cite{dupont}, we use the notation $\theta[(a,b), (c,d)] =: \theta_{16b + 8 a +4 d + 2c}$ for $a,b,c,d\in\{0,\frac{1}{2}\}$, so the even theta constants are $\theta_k$ for $k\in\{0,1,2,3,4,6, 8, 9, 12, 15\}$.
We take the quartic CM-field $K= \mathbf{Q}(\alpha) = \mathbf{Q}[X]/(X^4+27X^2+52)$, which also features in \cite[Example~III.3.2]{phdthesis}. Its real quadratic subfield is $K_0=\mathbf{Q}(\sqrt{521})$. Take the CM-type~$\Phi$ of~$K$ consisting of the two embeddings $K\rightarrow \mathbf{C}$ that map $\alpha$ to the positive imaginary axis. Let $w$ be the (positive) square root of~$13$. The real quadratic subfield of the reflex field~$K^{\mathrm{r}}$ is~$\mathbf{Q}(w)$.
We start by finding one pair $(\mathfrak{b}, \xi)$ and a corresponding~$\tau$ using the algorithms of Van Wamelen~\cite{vanwamelen} (see also~\cite{runtime}). In our case, this yields $\mathfrak{b}=\mathcal{O}$ and $\xi=2(22411531\alpha^3 + 46779315\alpha)^{-1}$. We compute a symplectic basis $B$ of $\mathfrak{b}$ as in Section~\ref{sec:basis} (second and fourth paragraph),
and get \begin{align*}
B = &\textstyle{\frac{1}{4}}(653 \alpha^{3} + 3414 \alpha^{2} + 1363 \alpha + 7126,\\ &\quad 401 \alpha^{3} + 2360 \alpha^{2} + 837 \alpha + 4926,\\ &\quad -653 \alpha^{3} + 1306 \alpha^{2} - 1363 \alpha + 2726,\\ &\quad 2108 \alpha^{2} + 4400)
\end{align*} We take~$\tau = \tau(\Phi, \mathfrak{b},\xi,B)$.
Next, we compute generators of the image of the map $$r : \frac{\IKr{N}\cap\cmgp{1}}{\cmgp{N}} \longrightarrow \GSpspecific{2g}(\mathbf{Z}/N\mathbf{Z})/S$$ from \eqref{eq:wascalledg} in Section~\ref{sec:statementspecific} using the command \verb!reciprocity_map_image(tau, 8)! of \cite{cmcode}, that is, using Algorithm~\ref{alg:imageofr}. This yields a list $R$ of $6$ matrices in $\GSpspecific{2g}(\mathbf{Z}/8\mathbf{Z})$, which we give explicitly in~\cite{cmcode} as \verb!r_image!.
In our particular example, the groups are a bit simpler, as explained by the following lemma and corollary.
\begin{lemma}\label{lem:oddclassnumber} In the case $g=2$, the square of every element $\mathfrak{a}$ of $\cmgp{1}$ is a fractional ideal of $\mathcal{O}_{\reflexfield_0}$ times an element of~${K^{\mathrm{r}}}^\times$. \end{lemma} \begin{proof} Given $\mathfrak{a}$, let $\mu$ be a generator of $N_{\Phi^{\mathrm{r}}}(\mathfrak{a})$ as in the definition of $\cmgp{1}$.
Let $\sigma$ be the non-trivial automorphism of $\reflexfield_0$. One can check $N_{\Phi}(N_{\Phi^{\mathrm{r}}}(\mathfrak{a})) =N(\mathfrak{a})\mathfrak{a}\overline{\mathfrak{a}}^{-1}$. Indeed, this (with $\Phi$ and $\Phi^{\mathrm{r}}$ swapped) is \cite[Lemma~I.8.4]{phdthesis}, and also appears in the proof of Lemma~\ref{lem:complexconjugation}.
Next, note $N(\mathfrak{a})= (\mathfrak{a}\overline{\mathfrak{a}})\cdot \sigma(\mathfrak{a}\overline{\mathfrak{a}})$, so we get $N_{\Phi}(\mu)\mathcal{O}_{K^{\mathrm{r}}}=\mathfrak{a}^2 / \sigma(N_{K/K_0}(\mathfrak{a}))$, which finishes the proof. \end{proof} \begin{corollary}\label{cor:principal}
If the class number of $K^{\mathrm{r}}$ is odd and the class number of $\reflexfield_0$
is~$1$, then $H_{\Phi,\mathcal{O}_K}(1)$ is the group of principal ideals. \end{corollary} \begin{proof}
Given $\mathfrak{a}\in H_{\Phi, \mathcal{O}_K}(1)$, the lemma
and the assumption $h_{\reflexfield_0}=1$ give that
$\mathfrak{a}^2$ is principal.
As $h_{K^{\mathrm{r}}}$ is odd, this implies that $\mathfrak{a}$ is principal. \end{proof} In the specific example we are treating right now, the class number of~$K^{\mathrm{r}}$ is~$7$ and the class number of $\reflexfield_0$ is~$1$, so the corollary applies. So when computing $r$, we restrict to principal ideals~$(\alpha)$ and have~$r((\alpha))=\epsilon(N_{\Phi^{\mathrm{r}}}(\alpha))$. We only need $(\alpha)$ up to $\cmgp{8}\supset P(8)$,
so we only need $\alpha$ modulo~$8$. So we can compute a set of generators of the group $(\mathcal{O}_{K^{\mathrm{r}}} / (8))^\times\cong C_{12}^2\times C_2^4$ and compute $N_{\Phi^{\mathrm{r}}}$ and $\epsilon$ with respect to the chosen symplectic basis of $\mathfrak{b}$. This would yield an alternative list of $6$ matrices in $\GSpspecific{2g}(\mathbf{Z}/8\mathbf{Z})$ that could be used instead of~$R$.
A function $f\in\Fcal{8}$ yields a class invariant if it is fixed by all elements of~$R$. Let us look at the action on quotients of theta constants of Proposition~\ref{prop:theta} more closely, starting with $8$th powers so that the $8$th root of unity factor $\exp(2\pi i (r-r'))$ vanishes. This action can be viewed as an action on the numerator and denominator separately. So this is an action of $\GSpspecific{2g}(\mathbf{Z}/8\mathbf{Z})$ on the set of 8th powers of the ten even theta constants. Under the action of the subgroup generated by~$R$, we compute that this set is partitioned into~4 orbits: $\{\theta_0^8, \theta_{1}^8, \theta_6^8\}$, $\{\theta_2^8, \theta_4^8, \theta_3^8\}$, $\{\theta_8^8,\theta_9^8,\theta_{15}^8\}$, $\{\theta_{12}^8\}$.
Let us restrict our search for class invariants to those functions~$f$ that are products of powers of the theta constants. To ensure $\mathrm{div}(f)$ is fixed by the image of~$r$, we use whole orbits, that is, write $$f= c
(\theta_0\theta_6\theta_1)^j (\theta_2\theta_3\theta_4)^k (\theta_8\theta_9\theta_{15})^l\theta_{12}^{m}$$ with $3j+3k+3l+m=0$ and $c\in\mathbf{Q}^{\mathrm{ab}}$.
There are various values of $(j,k,l,m)$ that one could try, but we prefer the minimal polynomial of $f(\tau)$ over $K^{\mathrm{r}}$ to have coefficients in~$\reflexfield_0$, so we also look at the action
of $U$ from Proposition~\ref{prop:complexconjugation}. It turns out that this action swaps the first two orbits, so we take~$j=k$. In fact, we like to use small products of theta constants, so we leave out these six theta constants, that is, we take $j=k=0$. We then get $3l=-m$, so with $n=-l$ we get
\[f = c f_0^n\quad\mbox{where}\quad f_0 = \frac{\theta_{12}^3}{\theta_8\theta_9\theta_{15}}.\] Note that if $8$ divides $n$ and $c\in\mathbf{Q}$, then we have $f(\tau)\in \cmext{1}$, but to let $f(\tau)$ have small height, we want to try smaller values of~$n$.
Explicitly computing the action of $R$ and $U$ on~$f$ for small $n$ and $c\in\langle \zeta_8\rangle$, we find that $n=2$, $c=i$ gives a function that is invariant under $R$ and $U$, that is, such that $f(\tau)\in\mathcal{M}_0$.
In greater generality, here is how to find $f$. \begin{algorithm}[Finding class invariants as quotients of theta constants]
\\
\textbf{Input:} A positive integer $D$ and a CM period matrix $\tau$.\\
\textbf{Output:} Generators for the group of functions $f$ that are a constant from $\mathbf{Q}(\zeta_{2D^2})^\times$
times a product of
powers of theta constants
$\theta[c_1,c_2]$ with $Dc_i\in\mathbf{Z}^g$ such that $f$ is fixed
by the image of $r$, up to constants.\\
\textbf{Algorithm:}
\begin{enumerate}
\item Let $R$ be the image of $r$
and let $N=2D^2$.
\item Let $H_1 = \{v\in\mathbf{Z}^{T} : \sum_{i} v_i = 0\}$,
where $T$ is a full set of representatives of the set of theta characteristics $c$ with $Dc\in\mathbf{Z}^{2g}$
up to $\{\pm 1\}$.
Interpret $v\in H_1$ as the function $f = \prod_{i = (c_1,c_2)\in T} \theta[c_1,c_2]^{v_i}\in\mathcal{F}_N$.
\item Let $H_2$ be the subgroup of $H_1$ consisting of those elements $f$ for which $f^N$
is fixed by the action
of $R$ (that is, $f$ is fixed up to roots of unity).
\item Let $H_3$ be the kernel of the map
\begin{align*} F : H_2 & \rightarrow H^1(\mathrm{Gal}(\mathcal{H}(N)/\cmext{1}), \mathbf{Q}(\zeta_N)^\times)\\
f &\mapsto (\sigma \mapsto f^\sigma / f).
\end{align*}
Here for the computation we only need the action of the set $R$ of generators of $\mathrm{Gal}(\mathcal{H}(N)/\cmext{1})$.
\item Take a basis of $H_3$. For each $f$ in that basis, compute $x\in\mathbf{Q}(\zeta_N)^\times$ such that
$F(f) = (\sigma \mapsto x^\sigma / x)$ and output $x^{-1} f$. \end{enumerate}
To make the output restrict to functions $f$ that are
also fixed by~$U$, consistently replace $R$ by $R\cup \{U\}$ and $\cmext{1}$ by $\mathcal{M}_0$
in the algorithm. \end{algorithm} Applying this algorithm to the example above with $R\cup \{U\}$, one gets $H_2 = \langle f_0\rangle$ and $H_3 = \langle f_0\rangle$.
\subsection{Computing the minimal polynomial} \label{sec:computingminpoly}
So now we have our class invariant $f(\tau)\in\cmext{1}$ and we would like to compute its minimal polynomial over~$K^{\mathrm{r}}$. The Galois group $\mathrm{Gal}(\cmext{1}/K^{\mathrm{r}})$ is the group of invertible fractional ideals of $K^{\mathrm{r}}$ modulo the ideal group $\cmgp{1}$ of Theorem~\ref{thm:idealgroup}. In this particular case, the group $\cmgp{1}$ is the group of principal ideals (Corollary~\ref{cor:principal}), so the Galois group is simply the class group of~$K^{\mathrm{r}}$.
For each of the~$7$ ideal classes of~$K^{\mathrm{r}}$, we compute~$U$ and~$\tau'$ as in Theorem~\ref{thm:general}. We make sure that the basis $C$ is such that $\tau'$ is \emph{reduced} for the action of~$\Spspecific{2g}(\mathbf{Z})$ (see the end of Section~\ref{sec:basis}) so that the theta constants can be numerically evaluated most efficiently.
Then we compute $f^U$ as in Algorithm~\ref{alg:thetatrans} and evaluate it numerically at~$\tau'$ to get a root of the minimal polynomial of $f(\tau)$ over~$K^{\mathrm{r}}$. This gives us a numerical approximation of the minimal polynomial $$H_f = \prod_{i=1}^{7} (X-f^{U_i}(\tau_i'))\in \reflexfield_0[X],$$ and we recognize its coefficients as elements of $\reflexfield_0\subset\mathbf{C}$ with the LLL-algorithm as in~\cite[Section~7]{lattices}. The entire calculation is at the beginning of the file \verb!article.sage! of~\cite{cmcode}.
We find that numerically with high precision, we have {\footnotesize \begin{align*} 3^8 101^2 H_f& = 66928761 X^7 + (21911488848 w - 76603728240) X^6 \\ &\quad + (-203318356742784 w + 733099844294784) X^5 \\ &\quad + (-280722122877358080 w + 1012158088965439488) X^4 \\ &\quad + (-2349120383562514432 w + 8469874588158623744) X^3 \\ &\quad + (-78591203121748770816 w + 283364613421131104256) X^2 \\ &\quad + (250917334141632512 w - 904696010264018944) X \\ &\quad - 364471595827200 w + 1312782658043904, \end{align*}}
\noindent which is significantly smaller than the smallest minimal polynomial obtained when using Igusa invariants, even with the small Igusa invariants from~\cite{runtime}: \footnotesize \begin{align*}
& 101^2 H_1 = 10201 X^7\\ &\qquad + (155205162116358647755w + 559600170220938887110)X^6\\ &\qquad + (152407687697460195175920750535594152550 w \\ &\qquad\quad + 549513732768094956258970636118192859400) X^5\\ &\qquad + \tfrac{1}{2}(2201909580030523730272623848434538048317834513875w\\ &\qquad\quad + 7939097894735431844153019089320973153011210882125)X^4\\ &\qquad + (1047175262927393182849164587480891367594710449395570625 w\\ &\qquad\quad + 3775644104882200832865729346429752069380200097845736875)X^3\\ &\qquad + \tfrac{1}{2}(90739291480049485513675299110604131111640471324738060
7234375w\\ &\qquad \quad + 327165168130591119268893142372375309476346120037916993
8284375)X^2\\ &\qquad +
(1501416604965651986004588022297124411339065052590506998
7454062500w\\
&\qquad\quad +
541343455503671907856059844455869398930835318514053659
78411062500)X\\ &\qquad + \tfrac{1}{2}(32085417029115132212877701052175189051312077050549053
7777676328984375w\\
&\qquad\quad + 115685616293120067038709321144324285012570966768326545
9917987279296875). \end{align*} \normalsize
As the first polynomial is so much smaller, we needed a much lower precision to reconstruct it from a numerical approximation. As our invariant $f$ is built up from the same theta constants as the absolute Igusa invariants (see \cite[Section~8]{runtime}), it takes the same time to evaluate it to any given precision, so saving precision in this way means saving time.
\subsection{More examples}\label{sec:engethomeexample}
We searched for class invariants with $D=g=2$ for a few more fields. For each of the fields we tried, the results were similar to Section~\ref{sec:findingclassinv}: an easily found product of powers of the ten even theta constants yielded a class invariant, which reduced the precision required for finding the class polynomials. We made such examples available online in \verb!article.sage! at~\cite{cmcode}.
We mention one of them in particular. Andreas Enge and Emmanuel Thom\'e, when demonstrating their implementation of a method for computing class polynomials~\cite{enge-thome}, presented at the GeoCrypt 2011 conference\footnote{\url{http://iml.univ-mrs.fr/ati/GeoCrypt2011/slides/thome.pdf}} a computation of the Igusa class polynomials of the maximal order~$\mathcal{O}_K$ of the field $K=\mathbf{Q}[X]/(X^4 + 310 X^2 + 17644)$ of class number~$3948$.
Following the steps of Section~\ref{sec:findingclassinv}, we found that the functions $$t=\frac{\theta_0\theta_8}{\theta_4\theta_{12}}\in \mathcal{F}_8,\quad u=\left(\frac{\theta_2\theta_8}{\theta_6\theta_{12}}\right)^2\in\mathcal{F}_2,\quad v = \left(\frac{\theta_0\theta_2}{\theta_4\theta_6}\right)^2\in\mathcal{F}_2$$ are class invariants for a certain $\tau$ with CM by~$\mathcal{O}_K$. These invariants would have significantly sped up the computation.
These invariants have the additional advantage that $(t^2, u, v)$ is a triple of \emph{Rosenhain invariants}, that is, the abelian variety corresponding to $\tau$ is the Jacobian of the curve $$y^2 = x(x-1)(x-t(\tau)^2)(x-u(\tau))(x-v(\tau)).$$ In particular, these invariants will be especially useful for constructing curves as in Section~\ref{ssec:cmmethod}
\section{Applications} \label{sec:applications}
\subsection{Class fields}
Class fields of number fields can be computed using Kummer theory \cite{cohen2, magma}, but that requires the use of auxiliary roots of unity, which make such computations computationally too costly for larger examples. Class fields of imaginary quadratic fields are often more efficiently computed using class invariants for $g=1$~\cite{pari,stevenhagen-computationalcft}. Whether this works for more general fields (with $g>1$) is currently being investigated by Jared Asuncion as part of his PhD thesis supervised by the author and Andreas Enge.
\subsection{Curves of genus two with prescribed Frobenius} \label{ssec:cmmethod}
In this section we show how class invariants give a practical improvement to the CM method for constructing curves of genus two. We start with a sketch of the CM method without class invariants (\ref{sssec:cm1}). Then we recall how class invariants are used in genus one (\ref{sssec:cm2}). And finally we explain how class invariants give an improvement in genus two (\ref{sssec:cm3}).
\subsubsection{The CM method} \label{sssec:cm1}
We would like to construct a $g$-dimensional abelian variety over a finite field with a prescribed characteristic polynomial $f$ of the Frobenius endomorphism~$\pi$. Indeed, when choosing $f$ appropriately, this yields curves of genus one or two with a prescribed number of points, or with good cryptographic properties~\cite{HEHCC,freeman-stevenhagen-streng,spallek}.
The idea of the CM construction is to start with an abelian variety $\widetilde{A}$ in characteristic zero with a nice endomorphism ring~$\mathcal{O}$, and take the reduction $A$ of $\widetilde{A}$ modulo a prime. The endomorphism ring of the reduction $A$ will contain both $\pi$ and the endomorphism ring~$\mathcal{O}$ of~$\widetilde{A}$. A `lack of space' in the endomorphism ring will then relate $\pi$ to $\mathcal{O}$, giving us the control that we need.
In more detail, assuming for simplicity that~$f$ is irreducible, this works as follows. The field $K=\mathbf{Q}[X]/(f)$ is a CM-field of degree $2g$, the constant coefficient $f(0) = p^{gm}$ is a prime power, and the root $\pi_0=(X\ \mathrm{mod}\ f)$ is a so-called \emph{Weil $p^{m}$-number}, that is, satisfies $\pi_0\overline{\pi_0}=p^m$.
Suppose that $\widetilde{A}$ is an abelian variety over a number field $k$
with $\mathrm{End}(\widetilde{A}_{\overline{k}})\cong \mathcal{O}_K$. Let~$\Phi$ be the CM-type of~$\widetilde{A}$ and assume $k\supset K^{\mathrm{r}}$, or equivalently, that the endomorphisms of $\widetilde{A}$ over $\overline{k}$ are defined over~$k$. Let $\mathfrak{P}/p$ be a prime of~$k$. Suppose $\widetilde{A}$ has good reduction at $\mathfrak{P}$ and let $A$ be the reduction. Let $\pi\in\mathrm{End}(A)$ be the Frobenius endomorphism of~$A$. Reduction modulo~$\mathfrak{P}$ gives an embedding $\mathcal{O}_K=\mathrm{End}(\widetilde{A})\subset \mathrm{End}(A)$ and we have the following result. \begin{theorem}[{Shimura-Taniyama formula~\cite[Thm.1 in \S13]{shimura-taniyama}}]\label{thm:shimura-taniyama} The endomorphism $\pi$ is an element of the ring $\mathcal{O}_K\subset\mathrm{End}(A)$ and generates the ideal $N_{\Phi^{\mathrm{r}}}(N_{k/K^{\mathrm{r}}}(\mathfrak{P}))$ of~$\mathcal{O}_K$. \end{theorem} This, together with the fact $\pi\overline{\pi}= \# (\mathcal{O}_k/\mathfrak{P})^g$ determines $\pi$ up to roots of unity. In fact, by taking $k$ to be minimal, we get $\pi=\pi_0$ up to roots of unity, that is, up to twists of~$A$.
This CM method can be made to be practical for at least $g=1$ \cite{atkin-morain, bbel, sutherlandCRT}, $g=2$ \cite{spallek, vanwamelen, dupont, runtime}, and $g=3$ \cite{weng-g3, koike-weng, bilv, lario-somoza, KLLRSS}, as well as for a certain class of curves with $g=5$ \cite{somoza-thesis}.
In all practical situations, one does not write down defining equations for the characteristic-zero abelian variety~$\widetilde{A}$, but only computes some elements of~$\Fcal{1}$ evaluated at~$\widetilde{A}$. In case $g=1$, it suffices to take the $j$-invariant, while in the cases $g\geq 2$, one chooses a tuple $i_1$, $i_2, \ldots, i_d$ of generators of~$\Fcal{1}$, which we call \emph{absolute invariants}. For example, for $g=2$, we take a triple of absolute Igusa invariants $i_1$, $i_2$, $i_3$.
In the case $g=1$, the elliptic curve $A$ can be reconstructed from $j(A) = (j(\widetilde{A})\ \mathrm{mod}\ \mathfrak{P})$ by a simple formula found in any textbook on elliptic curves. In the case $g=2$, for generic values of the Igusa invariants modulo $\mathfrak{P}$, one can reconstruct $\widetilde{A}$ as the Jacobian of a hyperelliptic curve using
Mestre's algorithm~\cite{mestre}. Similar constructions
are used in the references above for $g=3$ and $g=5$.
In the CM method for $g=1$, the value $j(\tau)$ is represented by its minimal polynomial, the Hilbert class polynomial. Reduction modulo $\mathfrak{P}$ is done by taking the reduction modulo~$p$ of the Hilbert class polynomial, and taking a root of that in~$\overline{\mathbf{F}_p}$.
In the case $g\geq 2$, one can take a minimal polynomial~$H_{i_1}$ of the first invariant, $i_1(\widetilde{A})$, over~$K^{\mathrm{r}}$, and let $i_2, \ldots, i_d$ be represented by polynomials $$ \widehat{H}_{i_1,i_n} = \sum_\gamma i_n(\widetilde{A})^\gamma \prod_{\sigma} (X - i_n(\widetilde{A})^\sigma) \in \reflexfield_0[X],$$ where sum and product range over $\mathrm{Gal}(\cmext{1}/K^{\mathrm{r}})$ (see~\cite{ghkrw-2adic}). Reducing~$H_{i_1}$ modulo a prime $\mathfrak{p}_0$ of $\reflexfield_0$ and taking any root is equivalent to reducing $i_1(\widetilde{A})$ modulo a prime over~$\mathfrak{p}_0$. Changing $\mathfrak{P}$ without changing $\mathfrak{p}_0$ will change the ideal in Theorem~\ref{thm:shimura-taniyama} at most by complex conjugation, which will not affect the characteristic polynomial of Frobenius. We can then find $i_2(A),\ldots, i_d(A)$ by computing $$i_n(A) = \frac{\widehat{H}_{i_1,i_n}(i_1(A))}{H_{i_1}'(i_1(A))}$$ if $p$ is sufficiently large.
This is how the CM method works, and now we would like to use class invariants for efficiency.
\subsubsection{Class invariants for genus one} \label{sssec:cm2}
In the case $g=1$, the use of class invariants in the CM method is standard. Let~$H_f$ be the minimal polynomial of a class invariant~$f(\tau)$, and let $\Phi_{f,j}(X,Y)\in\mathbf{Q}(X)[Y]$ be such that $\Phi_{f,j}(j, Y)\in \mathbf{Q}(j)[Y]$ is the minimal polynomial of $f\in \mathcal{F}_N$ over $\Fcal{1}=\mathbf{Q}(j)$. Then $\Phi_{f,j}(j(\tau), f(\tau)) = 0$, so we can find $j(\tau)$ by solving for $X$ in $\Phi_{f,j}(X, f(\tau))=0$.
For the CM-method, we now only compute $H_f$ and~$\Phi_{f,j}$. Here $\Phi_{f,j}$ can be precomputed once for every function~$f\in\mathcal{F}_N$ that we would like to use for class invariants, and $H_f$ is much smaller than the Hilbert class polynomial~$H_j$, hence needs less precision. We compute $f(\tau)$ modulo a prime over $\mathfrak{P}$ by taking a root of $H_f$ modulo~$p$ (we get $f(A)$). Then we compute $j(A)$ by solving for $X$ in $\Phi_{f,j}(X, f(A))=0$. There may be multiple solutions to try here, but at least one of them gives $j(A)$.
\subsubsection{Class invariants in general} \label{sssec:cm3}
For genus $g\geq 2$, we give three methods for using class invariants.
\textbf{Using modular polynomials as in $g=1$.} For $g\geq 2$, modular polynomials are much harder to compute \cite{gruenewald-thesis, broker_lauter_modpol, martindale-modpol, milio-robert}, and the higher-dimensional analogue of solving $\Phi_{f,j}(X, f(A))=0$ means finding a Groebner basis, which can be hard as well, but for some choices of invariants this may be doable.
\textbf{A modular interpretation of the class invariants.} Some class invariants themselves give rise to models of curves in a direct way, without the need of invariants from $\Fcal{1}$. For example, the modular functions $t$, $u$, $v\in\Fcal{8}$ of Section~\ref{sec:engethomeexample} give rise to the curve $y^2 = x(x-1)(x-t^2)(x-u)(x-v)$. In particular, a curve of genus $2$ can be computed directly from these invariants, without the need of Igusa invariants.
\textbf{Computing interpolating polynomials.} In general, we can work with polynomials $$ \widehat{H}_{f,i_n} = \sum_\gamma i_n(\widetilde{A})^\gamma \prod_{\sigma} (X - f(\widetilde{A})^\sigma) \in K^{\mathrm{r}}[X].$$ In fact, if the conditions of Proposition~\ref{prop:complexconjugation} are satisfied, then these polynomials are in $\reflexfield_0[X]$. We find $f(A)$ as in the elliptic case, and compute $i_n(A)$ for $n$ from it by the formula $$i_n(A) = \frac{\widehat{H}_{f,i_n}(f(A))}{H_{f}'(f(A))}.$$
We do need to compute $4$ polynomials instead of $3$ when using the class invariant~$f$, compared to when only using $i_1$, $i_2$, and $i_3$, but as the size is dominated by the first invariant, which is now $f$ instead of $i_1$, the total size of the polynomials still goes down.
For the example from Sections~\ref{ssec:detailedexample}--\ref{sec:computingminpoly}, we have computed the polynomials $H_{f}, \widehat{H}_{f,i_n}$ and made them available online (close to line 200 of the file \verb!article.sage! of~\cite{cmcode}). These four polynomials together take up 15\% the three polynomials $H_{i_1}, \widehat{H}_{i_1,i_n}$, and the largest coefficient (which determines the precision at which theta constants need to be evaluated, the dominant step in the computation)
is 40\%
\end{document} | arXiv |
Three positive integers $a$, $b$, and $c$ satisfy $a\cdot b\cdot c=8!$ and $a<b<c$. What is the smallest possible value of $c-a$?
Our goal is to divide the factors of 8! into three groups in such a way that the products of the factors in each group are as close together as possible. Write $8!$ as $8\cdot 7 \cdot 6 \cdot 5\cdot 4\cdot 3\cdot 2$. Observe that $30^3<8!<40^3$, so the cube root of $8!$ is between $30$ and $40$. With this in mind, we group $7$ and $5$ to make one factor of $35$. We can also make a factor of $36$ by using $6$ along with $3$ and $2$. This leaves $8$ and $4$ which multiply to give $32$. The assignment $(a,b,c)=(32,35,36)$ has the minimum value of $c-a$, since $31$, $33$, $34$, $37$, $38$, and $39$ contain prime factors not present in $8!$. Therefore, the minimum value of $c-a$ is $\boxed{4}$. | Math Dataset |
\begin{document}
\begin{abstract} Since the 1980s, it has been known that essential surfaces in alternating link complements can be isotoped to be transverse to the link diagram almost everywhere, with the exception of some well-understood intersections, and described combinatorially as a result. This was called standard position for surfaces and has had numerous applications. However, the original techniques only apply to classical alternating links projected onto the 2-sphere inside the 3-sphere. In this paper, we prove that standard position for surfaces can be extended to a broader class of 3-manifolds, namely weakly generalized alternating link complements. Such link complements include all classical prime non-split alternating links in 3-sphere, but also many complements of links that are alternating on higher genus surfaces, or lie in manifolds besides the 3-sphere. As an application, we show that all such links are prime, and that under mild restrictions, essential Conway spheres in such links interact with the diagram exactly as in classical alternating links. \end{abstract}
\maketitle
\section{Introduction}\label{Sec:Intro}
Essential embedded surfaces in a 3-manifold can provide useful information about the topology of that manifold. For example, Haken manifolds, which contain an orientable closed essential surface, have been an object of significant research since they were introduced in the 1960's and have many important properties \cite{Haken, Waldhausen}. To work with essential embedded surfaces, it is often useful to put them in a certain topological form that relates to properties of the ambient 3-manifold. When the ambient 3-manifold is an alternating link complement, Menasco and Thistlethwaite showed that an essential surface could be isotoped into \emph{standard position} with respect to the link projection surface~\cite{Menasco1984, MenascoThistlethwaite}. For more general 3-manifolds, an essential surface may be isotoped into \emph{normal form} with respect to a triangulation of a 3-manifold~\cite{Haken}, or similarly normal form with respect to polyhedral or other decompositions~\cite{lac00, fg09}. In \cite{HowiePurcell}, Howie and Purcell extended normal form to broad generalizations of alternating links, called weakly generalized alternating links, and used this form to determine geometric information, generalizing several of the above results.
In this paper, we take this generalization further. Menasco in~\cite{Menasco1984} and Menasco and Thistlethwaite in~\cite{MenascoThistlethwaite} proved that essential surfaces in alternating link complements could be isotoped so that they closely interact with the combinatorics of the link diagram. Here, we generalize these results to weakly generalized alternating links.
As a corollary, we show that all weakly generalized alternating links are prime, extending a result of Howie and Purcell~\cite[Corollary~4.7]{HowiePurcell}, who showed primeness of such links with some restrictions.
We also classify essential Conway spheres in weakly generalized alternating links satisfying mild restrictions, showing that they either intersect the diagram in a single curve meeting four punctures (a ``visible'' sphere), or intersect it in two well-behaved curves (a ``hidden'' sphere). The visible--hidden dichotomy is analogous to a similar result for classical alternating links, discovered by Menasco~\cite{Menasco1984}, named and refined by Thistlethwaite~\cite{Thistlethwaite}, and used to study alternating tangles by others, for example in other collaborations by the authors: Hass, Thompson, and Tsvietkova~\cite{HTT3}, and Champanerkar, Kofman, and Purcell~\cite{CKP:RAK}.
\subsection{Alternating links in the 3-sphere and beyond} Normal and standard form for surfaces in alternating links in the 3-sphere have had many applications. Historically, Menasco used it to prove a number of results for alternating link complements in the 3-sphere, such as classifying when the link is prime, split and hyperbolic \cite{Menasco1984}, and determining when a surface is incompressible \cite{Menasco1985}. Together with Thistlethwaite, they used it to show that the only reducible Dehn fillings are the expected ones on $(2,q)$-torus knots~\cite{MenascoThistlethwaite}, and thus to prove the cabling conjecture for all alternating knots. Lackenby used normal form to describe exceptional Dehn fillings \cite{lac00}. Howie used it to show that essential embedded surfaces with boundary can be used to detect classical alternating links \cite{Howie2017}. Hass, Thompson and Tsvietkova used it to give universal polynomial bounds for the number of embedded surfaces in alternating link complements \cite{HTT1, HTT2}. These are just some examples of numerous applications to alternating links in 3-sphere.
In recent years, there has been significant interest in the study of alternating projections onto surfaces besides the 2-sphere, for example due to Adams~\cite{Adams:ToroidAlt}, Hayashi~\cite{Hayashi}, Ozawa~\cite{oza06}, Howie~\cite{how15t}, and others. There has also been interest in alternating knots within manifolds besides the 3-sphere, such as virtual knots, for example by Adams~\cite{Adams:ThickenedSfces}, Champanerkar, Kofman, and Purcell~\cite{CKP:Biperiodic}, and Howie and Purcell~\cite{HowiePurcell}. The weakly generalized alternating links of this paper match those of Howie and Purcell. They include classical alternating knots on a 2-sphere in the 3-sphere, and also include many of the other generalizations above, including families of toroidally alternating knots and alternating knots on other Heegaard surfaces, alternating knots in thickened surfaces, i.e.\ virtual alternating knots, and further examples still. The full definition of such links is given in \refsec{WGA}.
\subsection{Meridianal surfaces in meridianal form} One of the key results in this paper is to show that essential surfaces meeting a knot in meridians can be simultaneously isotoped into normal form without distorting the boundary curves. We call this \emph{meridianal form}, defined in \refsec{SBP}.
Surfaces meeting knots in meridians arise frequently. Menasco showed that any closed essential surface embedded in a non-split prime alternating link complement in $S^3$ contains a closed curve isotopic to the link meridian~\cite{Menasco1984}. This sometimes is referred to as Meridian Lemma. By compressing along a meridianal annulus, it allows one to reduce the study of closed essential surfaces to the study of essential surfaces with meridianal boundary. A meridian lemma also holds for certain weakly generalized alternating links due to Howie and Purcell~\cite[Lemma~4.9]{HowiePurcell}, and has been announced for weakly generalized alternating links that are also virtual alternating links by Wei Lin~\cite{WeiLin}. A meridian lemma holds for other knots, such as algebraic knots~\cite{Ozawa:Algebraic}. For knots that satisfy such a lemma, a meridian compression yields a surface that can be put into meridianal form. This was used, for example, in the proof of a polynomial upper bound on the number of closed surfaces in alternating links by Hass, Thompson and Tsvietkova~\cite{HTT1}, and in Lozano and Przytycki work on 3-braid links~\cite{LP}.
Surfaces that meet a knot in a meridian also include meridianal annuli. This is an essential annulus with both boundary components forming meridians of the knot.
By definition, if a link complement admits an essential meridianal annulus, it is not prime. Studying such surfaces in the past led to results that a classical alternating link is prime if and only if its diagram is prime, due to Menasco~\cite{Menasco1984}. In this paper, we show that all weakly generalized alternating links must be prime in \refthm{Prime}. This is analogous to Menasco's result, since weakly generalized alternating links have a diagrammatic primeness condition built into the definition. Howie and Purcell proved a similar result for weakly generalized alternating links with an additional condition (namely $\hat{r}>4$; see \refsec{WGA})~\cite[Corollary~4.7]{HowiePurcell}. Howie and Purcell's result already implies that all virtual alternating links that are checkerboard colorable and have a weakly prime diagram must also be prime. For example this is true for virtual alternating links whose diagrams have all complementary regions disks, as studied by Adams \emph{et al}~\cite{Adams:ThickenedSfces}. However, the results in this paper extend primeness to broader families of weakly generalized alternating links, for example on projection surfaces that are compressible, which do not satisfy Howie and Purcell's stronger requirement.
Note that in the virtual setting, other notions of primeness also appear in the literature, for example recent work of Kindred~\cite{Kindred:Prime}.
Finally, surfaces meeting a knot in a meridian include 4-punctured spheres, which separate a knot into tangles. Examining such surfaces has allowed the detection and study of prime tangles, for example by Menasco and Thistlethwaite~\cite{Menasco1984, Thistlethwaite}, and more recently by Hass, Thompson and Tsvietkova~\cite{HTT3}, and Champanerkar, Kofman, and Purcell~\cite{CKP:RAK}. The tools of this paper also allow us to extend such results to larger families of links. We consider essential 4-punctured spheres in weakly generalized alternating links satisfying an extra condition (namely $\hat{r}(\pi(L),\Pi)>4$), and we show that they satisfy the same constraints as in classical alternating links; see \refthm{EssentialTangles}. In particular, they result in either a ``visible'' tangle (one $PPPP$ curve in standard position in Menasco's language), or a ``hidden'' tangle (two $PSPS$ curves); see \refsec{ConwaySpheres} for definitions. We expect this to lead to further study of tangles for knots projected onto surfaces beyond the 2-sphere.
\subsection{Cyclic words associated to a surface}
We extend standard position to closed surfaces and meridianal surfaces, but also to surfaces with non-meridianal boundary.
One of the benefits of standard position for classical alternating links is that it allows us to translate topological properties of surfaces into combinatorial properties of their curves of intersection with the (slightly modified) projection surface for a link. Such intersections were encoded in \cite{Menasco1984} by letters $P$ (for link intesections) and $S$ (for crossing ball intersection). Every curve of intersection was put in correspondence with a cyclic word in such letters. Then possible words were investigated. In \cite{HTT1}, this was extended further to surfaces with boundary, now with letters $B$ (standing for arriving at the boundary of the surface) and $S$. We generalize this to weakly generalized alternating links in this paper.
This combinatorial approach played important role in several results for alternating links in 3-sphere, including many noted above. In addition, it has been used to classify genus two surfaces \cite{Thistlethwaite} and to find incompressible surfaces in alternating links in \cite{Menasco1985}.
In \refsec{BBBB}, we prove that a collection of some of the intersection curves determines the isotopy classes of a surface with non-meridianal boundary. We also investigate how disks with a small number of intersections with the link or crossing balls behave. This is a generalization of the fact proved in \cite{HTT2}. There, it allows to see, for example, how many embedded surfaces with different pattern of intersection with the (slighty modified) projection plane end up being isotopic.
In a subsequent paper, we apply this work to the problem of bounding the total number of embedded essential surface in these link complements~\cite{PT}. We apply these results together with combinatorial and geometric arguments to give a polynomial upper bound on the number of embedded surfaces in a wide class of cusped 3-manifolds, namely weakly generalized alternating link complements and their Dehn fillings. No other universal polynomial bounds for the number of surfaces in cusped 3-manifolds beyond alternating links in $S^3$ are known, as in \cite{HTT1, HTT2}; in other existing work, the bound depends on the 3-manifold.
\subsection{Organization}
In \refsec{WGA} we recall the definition of weakly generalized alternating links. Their primary feature is recalled in \refsec{Chunks}, namely, they have a decomposition into checkerboard ``chunks'' that extend much of the useful decomposition of alternating links into topological polyhedra via checkerboard decomposition, which Thurston linkened to gears of a machine~\cite{Thurston:notes}. We recall the definition of normal surfaces in chunks in \refsec{NormalSfc}. The labelling of intersections of such surfaces with chunk boundary by letters $P$, $B$, and $S$ is described in \refsec{SBP}.
Our first main result, on meridianal form for normal surfaces, is proved in \refsec{Meridianal}. In \refsec{Standard}, we compile a few results on surfaces in standard form, including a result that shows that such a surface cannot meet the same saddle twice and bound a disk.
In \refsec{Area}, we recall the notion of combinatorial area, introduced in this setting by Howie and Purcell~\cite{HowiePurcell}. Combining combinatorial area results with the labelling of boundary curves by letters $P$, $B$, and $S$, we find that the only zero area surfaces in a weakly generalized alternating link with a cellular diagram are disks with certain words on their boundaries. The next sections are devoted to ruling out instances of zero area disks, or controlling when they occur. These are Sections~\ref{Sec:SSSS}, \ref{Sec:PPPP}, and~\ref{Sec:BBBB}. As a corollary of the work of \refsec{PPPP}, we prove cellular weakly generalized alternating links are prime.
Finally, in \refsec{ConwaySpheres}, we apply our results to essential Conway spheres to reproduce a result of Menasco~\cite{Menasco1984}, that such spheres are either ``visible'' or ``hidden'', with terminology due to Thistlethwaite~\cite{Thistlethwaite}.
\subsection{Acknowledgments} We thank an anonymous referee whose suggestions have greatly improved the paper. Purcell was partially supported by the Australian Research Council, grants DP160103085 and DP210103136. Tsvietkova was partially supported by the National Science Foundation (NSF) of the United States, grants DMS-2142487 (CAREER), DMS-1664425 (previously 1406588) and DMS-2005496, the Institute of Advanced Study under NSF grant DMS-1926686, and the Okinawa Institute of Science and Technology.
\section{Weakly generalized alternating link complements}\label{Sec:WGA}
In this section, we follow Howie and Purcell~\cite{HowiePurcell} to introduce a broad class of links that have a diagram that is alternating on a closed surface $\Pi$ embedded in a 3-manifold $Y$, possibly with boundary. As noted in the introduction, this is a very broad class that includes classical alternating knots, but also alternating knots on a Heegaard torus or more general Heegaard surface, originally studied by Adams~\cite{Adams:ToroidAlt} and Hayashi~\cite{Hayashi}, virtual alternating knots~\cite{Adams:ThickenedSfces, CKP:Biperiodic}, and general alternating diagrams on any embedded surface in any compact orientable 3-manifold.
\begin{assumption}
Throughout this paper, we will always require the 3-manifold $Y$ to be compact, orientable, and irreducible. The projection surface $\Pi$ is required to be a closed, orientable surface. If $Y$ has boundary, we will require $\partial Y$ to be incompressible in $Y-N(\Pi)$. \end{assumption}
\subsection{Generalized projection}
A \emph{generalized projection surface} $\Pi$ is a (possibly disconnected) oriented surface embedded in $Y$ so that $Y-\Pi$ is irreducible. Let $N(\Pi) = \Pi\times (-1,1)$ denote a regular neighborhood.
The connected components of $\Pi$, denoted $\Pi_1, \dots, \Pi_p$ are closed two-sided (orientable) surfaces. Since $Y-\Pi$ is irreducible, if such a component is a 2-sphere, then $\Pi$ is homeomorphic to $S^2$, and $Y$ is homeomorphic to $S^3$. Let $L$ be a link that can be projected onto $\Pi$ in general position. That is, $L$ can be isotoped through $Y$ to lie in $N(\Pi)$ so that the image of the projection $\pi(L)$ consists of crossings and arcs between them on the surface $\Pi$. We assume that there is at least one crossing of the link on each $\Pi_i$. We call $\pi(L)$ a \emph{generalized diagram}.
Whenever in the paper we mention $\Pi$ or $\pi(L)$, we mean a generalized projection surface and the generalized diagram of $L$ on $\Pi$. We will also use the following terms. \begin{itemize} \item An arc of $\pi(L)$ (on $\Pi$) between two crossings is called an \emph{edge} of the diagram. \item A \emph{crossing arc} is a simple arc in the exterior of $L$ running from an overpass to an underpass of a crossing. \item A \emph{region} of the diagram is a complementary region of the projection of $\pi(L)$ to $\Pi$. It is bounded by edges of the diagram. \end{itemize}
Every knot has a trivial (no crossings) generalized diagram on the torus boundary of a regular neighborhood of the knot. To ensure our diagrams are non-trivial, we introduce the notion of representativity. Let $N(\Pi)$ be $\Pi\times (-1,1)$. Because every connected component $\Pi_i$ of $\Pi$ is 2-sided, for each $i$, $Y- N(\Pi)$ has two of its boundary components homeomorphic to $\Pi_i$, namely $\Pi_i^{\pm}=\Pi_i\times\{\pm 1\}$. Denote $\bigcup \Pi_i^+$ by $\Pi^+$, and similarly for $\Pi^-$. Define $r^{\pm}(\pi(L), \Pi_i)$ to be the minimum number of intersections between the projection of $\pi(L)$ onto $\Pi_i^{\pm}$ and the boundary of any essential compressing disk for $\Pi_i^{\pm}$ in $Y- \Pi$. If there are no essential compressing disks for $\Pi_i^{\pm}$ in $Y- \Pi$, then set $r^{\pm}(\pi(L), \Pi_i)=\infty$. The \emph{representativity} $r(\pi(L), \Pi)$ is the minimum of all values of $r^-(\pi(L), \Pi_i)$ and $r^+(\pi(L), \Pi_i)$, over all $i$. By definition, a sphere $S^2$ embedded in $S^3$ admits no essential compressing disk, so a usual alternating diagram has representativity $\infty$.
\begin{example} \reffig{CheckColorable}, left, which is modified from a figure in \cite{HowiePurcell}, shows a diagram $\pi(L)$ on a torus $\Pi$. The representativity of the diagram depends on the manifold $Y$ and the embedding of $\Pi$ into $Y$. For example, first let $Y=S^3$, and embed the torus $\Pi$ as the standard Heegaard torus for $S^3$, with the vertical red curve shown mapping to a meridian of one solid torus in the Heegaard splitting of $S^3$ and the horizontal red curve mapping to a meridian of the other solid torus. Then $r^+(\pi(L),\Pi)=3$ and $r^-(\pi(L),\Pi)=0$, so $r(\pi(L),\Pi)=0$. \begin{figure}
\caption{Left: An example of an alternating diagram on a torus. The representativity will depend on the embedding of the torus into $Y$. In any case, the diagram is not checkerboard colorable. Right: A checkerboard colorable diagram.}
\label{Fig:CheckColorable}
\end{figure}
If instead we let $Y=T^2\times {\mathbb{R}}$, the thickened torus, and we embed the torus $\Pi$ as the surface $T^2\times\{0\}$, then $\Pi$ admits no essential compressing disk, so $r^+(\pi(L),\Pi) = r^-(\pi(L),\Pi)=\infty$. \end{example}
Finally, define the hat-representativity, $\hat{r}(\pi(L), \Pi)$, to be the minimum of \[ \bigcup_i \max\{ r^-(\pi(L), \Pi_i), r^+(\pi(L), \Pi_i) \}. \] Thus for example a surface with no compressing disks on one side will have infinite hat-representativity.
A generalized diagram $\pi(L)$ is said to be \emph{alternating} if for each region of $\Pi {\smallsetminus}\pi(L)$, each boundary component of the region is alternating, i.e.\ it can be given an orientation such that crossings run from under to over in the direction of orientation. An alternating generalized diagram $\pi(L)$ is said to be \emph{checkerboard colorable} if each region of $\Pi {\smallsetminus}\pi(L)$ can be oriented so that the induced orientation on each region's boundary is alternating: crossings run from under to over in the direction of orientation. Given a checkerboard colorable diagram, regions on opposite sides of an edge of $\pi(L)$ will have opposite orientations. We can color all regions with one orientation white, and all regions with the opposite orientation shaded.
\begin{example} \reffig{CheckColorable}, left, shows a diagram that is alternating on a torus. For the outer annular region $A$ of the diagram, the figure shows an orientation assigned to its boundary. The crossings run from under to over with this orientation. Hence the region is alternating. However, this orientation is not induced by an orientation on $A$. If we choose an orientation on $A$, and then take the induced orientation on $\partial A$, one boundary component will be oriented so that crossings run under to over, and the other will be oriented so that crossings run over to under. Hence this example is not checkerboard colorable. Note it will not be checkerboard colorable regardless of the manifold $Y$ it is embedded within.
However, \reffig{CheckColorable}, right, shows another diagram of an alternating link on a torus, embedded in $Y=T^2\times I$. This link is checkerboard colorable. \end{example}
Most results in the literature on alternating links require a reduced or simplified diagram; for example \cite{Menasco1984, HTT1, HTT2}. For a classical alternating link on $S^2\subset S^3$, the condition we need is diagrammatic primeness: if an essential curve intersects the diagram exactly twice, then it bounds a region of the diagram containing a single embedded arc. Diagrammatic primeness rules out connected sums of knots and nugatory crossings. Analogously, a generalized diagram $\pi(L)$ on generalized projection surface $\Pi = \bigcup \Pi_i$ is \emph{weakly prime} if whenever $D\subset \Pi_i$ is a disk with $\partial D$ intersecting $\pi(L)$ transversely exactly twice, either the disk $D$ contains a single embedded arc, or $\Pi_i$ is a 2-sphere and there is a single embedded arc on $\Pi_i-D$.
\subsection{Weakly generalized alternating link}\label{Sec:WGAKnots}
The diagram $\pi(L)$ on $\Pi$ is said to be a \emph{weakly generalized alternating link diagram} if \begin{enumerate} \item\label{Itm:Alternating} $\pi(L)$ is alternating on $\Pi$, \item $\pi(L)$ is weakly prime, \item\label{Itm:Connected} $\pi(L)\cap \Pi_i \neq \emptyset$ for each component $\Pi_i$ of $\Pi$. \item\label{Itm:slope} each component of $L$ projects to at least one crossing in $\pi(L)$. \item\label{Itm:CheckCol} $\pi(L)$ is checkerboard colorable, and \item\label{Itm:Rep} the representativity $r(\pi(L), \Pi)\geq 4$. \end{enumerate}
Note these conditions were originally enumerated by Howie~\cite{how15t}, and in some sense are as general as possible to obtain prime alternating diagrams on embedded surfaces in $S^3$ in Howie's setting. Perhaps the most mysterious condition is item~\refitm{Rep}, the representativity condition. The restriction on representativity means that small surfaces such as compressing disks and meridianal annuli must be split by the projection surface $\Pi$ into disks parallel to $\Pi$, and thus they interact with the diagram on $\Pi$; they cannot ``hide'' as compression disks. The representativity condition is used throughout Howie--Purcell~\cite{HowiePurcell}, and consequently used throughout this paper. Recall as well that classical alternating knots and virtual alternating knots have infinite representativity; the representativity condition only needs to be checked when $\Pi$ is compressible, for example if it is a Heegaard surface.
From now on, every link we consider will be weakly generalized alternating. Note that a classical reduced, prime, alternating diagram of a link $L$ on $\Pi=S^2$ in $S^3$ is an example of a weakly generalized alternating link. The example of \reffig{CheckColorable}, right, is also a weakly generalized alternating link on $\Pi = T^2\times\{0\}$ in $Y=T^2\times{\mathbb{R}}$.
These conditions are enough to guarantee that the link exteriors are irreducible and boundary irreducible~\cite[Corollary~3.16]{HowiePurcell}, which we will use below.
\section{Decomposition of a link complement into chunks}\label{Sec:Chunks}
Knot and link complements alternating on a projection plane in $S^3$ have a well-known decomposition into topological polyhedra, suggested by W.~Thurston and described by Menasco \cite{men83}; see also \cite{lac00} or \cite[Section~11.1.1]{Purcell:HKT}. A more general decomposition into angled blocks was defined by Futer and Gu{\'e}ritaud \cite{fg09}. This was generalized further by Howie and Purcell in \cite{HowiePurcell} for weakly generalized alternating links. We review this generalization in this section.
\subsection{A decomposition of weakly generalized alternating link complements}\label{Sec:ChunkDecomp} A checkerboard colorable diagram admits two checkerboard surfaces, one white and one shaded. The white checkerboard colored surface is obtained by taking a disk corresponding to each white region of the diagram, and connecting these disks by twisted bands at crossings. Similarly for the shaded surface. The decomposition of weakly generalized alternating links is obtained by cutting along the white and shaded checkerboard surfaces. Since the disk regions of these surfaces tile $\Pi$, this cuts $Y-N(L)$ into components $Y-N(\Pi)$. The checkerboard surfaces intersect exactly at crossing arcs; these become ideal edges lying on $\partial(N(\Pi))$ after cutting. Strands of the knot corresponding to overcrossings become ideal vertices; we contract each of these to lie at an overcrossing of the diagram.
Propositions~3.1 and~3.3 of \cite{HowiePurcell} prove that the decomposition has the following properties.
The components of the decomposition, which are called \emph{chunks}, are connected components of $Y- N(\Pi)$. These are 3-manifolds with boundary, where the boundary components are components of $\partial Y$, along with $\Pi_i^-$ and $\Pi_i^+$. Note that each $\Pi_i$ lies on the boundary of one or two chunks, appearing as $\Pi_i^+$ and $\Pi_i^-$. When $\Pi$ is connected, we have one or two chunks total, depending on whether $\Pi$ is separating or not. For example if $\Pi$ is the usual plane of projection $S^2$ in a classical alternating link diagram, there are two chunks, each homeomorphic to a ball component of $S^3-N(S^2)$.
Decorate the surfaces $\Pi_i^-$ and $\Pi_i^+$ with: \begin{enumerate} \item A copy of the edges of the link diagram on $\Pi_i$. These will be called \emph{interior edges} of a chunk, and correspond to crossing arcs in $Y-N(L)$. \item Interior edges meet at crossings of the diagram, which become \emph{ideal vertices}. That is, crossings become vertices, which are removed (ideal). \item Regions bounded by edges and vertices will be called \emph{faces} of the chunk. They are not necessarily simply connected. The faces correspond to regions of the diagram, and will be checkerboard colored, white and shaded. For example, the outer region of \reffig{CheckColorable} is an annulus, so gives rise to a face that is an annulus. \end{enumerate} An example of the chunk decomposition of the link of \reffig{CheckColorable}, right, is shown in \reffig{ChunkDecomp1}. The two chunks are the components of the complement of $T^2\times(-\epsilon, \epsilon)$ in $T^2\times{\mathbb{R}}$ for some $\epsilon>0$. Hence they are homeomorphic to $T^2\times(-\infty, -\epsilon]$ and $T^2\times[\epsilon,\infty)$. Edges, faces, and ideal vertices are marked on $T^2\times\{-\epsilon\}$ and $T^2\times\{\epsilon\}$, respectively, with faces shown checkerboard colored, white and shaded.
\begin{figure}
\caption{An example of a chunk decomposition. On the left is a manifold homeomorphic to $T^2\times(-\infty,-\epsilon]$ with faces, edges, and ideal vertices marked on $T^2\times\{-\epsilon\}$. On the right is a manifold homeomorphic to $T^2\times[\epsilon,\infty)$ with faces, edges, and ideal vertices marked on $T^2\times\{+\epsilon\}$.}
\label{Fig:ChunkDecomp1}
\end{figure}
\subsection{Gluing}\label{Sec:Gluing} To obtain $Y- L$, each face of $\Pi_i^-$ should be glued to the corresponding face of $\Pi_i^+$. A face $F^-$ of $\Pi_i^-$ corresponds to a face $F^+$ of $\Pi_i^+$ if the same region of $\pi(L)$ gave rise to $F^-$ and $F^+$. The gluing is by the identity away from boundary components of the faces, i.e. away from the edges of the chunk(s). In a neighborhood of a boundary component of the face, the gluing rotates the ideal edges by one notch either clockwise or counterclockwise, depending on whether the face is white or shaded. That is, an ideal edge in a white face will rotate to the next ideal edge of the same face in the clockwise direction for the gluing, and similarly for shaded faces in the counterclockwise direction. The arrows on the faces of the chunk in \reffig{ChunkDecomp1} indicate the gluing.
Under the gluing, four interior edges glue to a single \emph{crossing arc} in $Y- L$. The crossing arc is identified to two edges each on $\Pi^-$ and $\Pi^+$, with the edges meeting as opposite edges at a vertex. This is illustrated in \reffig{CrossingArc}.
\begin{figure}
\caption{Four edges are identified to a crossing arc, with two on each of $\Pi^-$ and $\Pi^+$. The edges meet as opposite edges at a vertex.}
\label{Fig:CrossingArc}
\end{figure}
Now truncate the ideal vertices of chunks: this replaces an ideal vertex with a quadrilateral \emph{truncation face}. We call an edge bordering a truncation face a \emph{truncation edge}. The truncation faces correspond to crossings of the link diagram, and tile the boundary torus of the link with a \emph{harlequin tiling}. See \reffig{HarlequinTiling}. Note the slightly different terminology here: such faces and edges are called boundary faces and edges in \cite{HowiePurcell}.
\begin{figure}
\caption{On the left is shown a single truncation face (shaded) as it appears embedded in the link complement. On the right, the boundary torus of the link has been unrolled into an annulus; truncation faces are shown.}
\label{Fig:HarlequinTiling}
\end{figure}
A chunk with truncated ideal vertices is called a \emph{truncated chunk}. Below, if we refer to faces of a chunk, we mean both truncation faces and other faces. Similarly, if we refer to an edge of a chunk, we mean either a truncation edge or an interior edge. We note that under the gluing described above, the truncation faces that are adjacent to an interior face $I$ are also rotated one notch around $I$, either clockwise or counterclockwise, together with the boundary of $\partial I$.
\section{Normal surfaces}\label{Sec:NormalSfc}
Both normal and standard position begin with a surface being transverse: normal surfaces are transverse to faces and edges of tetrahedra in a triangulation of a 3-manifold, and surfaces in standard position are transverse to a projection surface of a link away from crossings. Here we review surfaces that are normal with respect to a chunk introduced in \cite{HowiePurcell}, which generalizes both of the above.
\begin{assumption} Throughout, we use the topological definition of incompressible surfaces: \begin{itemize} \item A surface $Z$ that is neither a disk nor a 2-sphere is incompressible in a 3-manifold $M$ if any disk $D$ with interior embedded in $M-N(Z)$ with boundary on $Z$ satisfies $\partial D$ bounds a disk in $Z$. A surface that is not incompressible admits an essential compression disk, namely a disk $D$ with interior embedded in $M-N(Z)$ with $\partial D$ an essential curve on $Z$. \item A 2-sphere is incompressible if and only if it does not bound a 3-ball. \item By convention, disks will be neither incompressible nor compressible. \item Finally, note that we allow $Z$ to be orientable or nonorientable, with or without boundary. \end{itemize} \end{assumption}
\subsection{Normal surface in a chunk}
First, consider a surface $Z'$, possibly with boundary, properly embedded in a truncated chunk $C$, with $\partial Z'\subset \partial C$.
\begin{definition}\label{Def:NormalSurface} The surface $Z'$ is \emph{normal} with respect to the chunk $C$ if it satisfies the following. \begin{enumerate} \item[(0)] Each non-disk subsurface of $Z'$ is incompressible in $C$. \item[(1)] $Z'$ and $\partial Z'$ are transverse to all faces and edges of $C$. \item[(2)] If a component of $\partial Z'$ lies entirely in a face of $C$, then it does not bound a disk in that face. \item[(3)] If an arc $\gamma$ of $\partial Z'$ in a face of $C$ has both endpoints on the same edge, then the arc $\gamma$ along with an arc of the edge cannot bound a disk in that face. \item[(4)] If an arc $\gamma$ of $\partial Z'$ in an interior face of $C$ has one endpoint on a truncation edge and the other on an adjacent interior edge, then the union of $\gamma$ as well as adjacent arcs of the two edges cannot bound a disk in the interior face. \end{enumerate} \end{definition}
Example of arcs that make a surface fail to be normal are shown in \reffig{NonNormal}.
\begin{figure}
\caption{Shown are arcs of $Z'\cap C$ that make the surface $Z'$ fail to be normal. On the left, the surface fails (2), in the middle it fails (3), on the right it fails (4).}
\label{Fig:NonNormal}
\end{figure}
\subsection{Normal surface with respect to a chunk decomposition} Given a chunk decomposition of $Y- L$, a surface $Z$ embedded in $Y- L$ is \emph{normal} with respect to the chunk decomposition if for every chunk $C$, the intersection $Z\cap C$ is a (possibly disconnected) normal surface in $C$.
Here and further we subdivide a surface $Z$ in subsurfaces $Z_i$ cut out by faces of the chunks. We assume that each $Z_i$ is connected, closed or with boundary, and possibly with multiple boundary components. All our surfaces and subsurfaces from now on will be embedded. Moreover, we will assume that if the surface $Z$ has boundary and $Y$ has boundary, then the boundary of $Z$ lies on $N(L)$ and is disjoint from $\partial Y$. This includes surfaces of any slope on $L$, as well as closed surfaces.
It was shown in Theorem~3.8 of \cite{HowiePurcell} that any essential surface in a 3-manifold with a chunk decomposition can be put into normal form with respect to the chunk decomposition; see also \cite[Theorem~2.8]{fg09}. We will revisit this proof below in \refthm{Normal} to show that the process of putting surfaces into normal form preserves other desirable features of the surface.
In fact, we note that item~(0) of \refdef{NormalSurface} is slightly stronger than the definition in \cite[Definition~3.7]{HowiePurcell}, in that we require all non-disk subsurfaces $Z_i$ to be incompressible and not just closed components. However, we will see in \refthm{Normal} below that any surface that is normal with respect to the definition of \cite{HowiePurcell} can be isotoped to be normal with respect to \refdef{NormalSurface}.
\section{Saddles, meridian punctures, and boundary of the surface}\label{Sec:SBP}
For normal subsurfaces $Z_j$ of the surface $Z$, each boundary component of $\partial Z_j$ runs over truncation edges and interior edges of the respective chunks. In this section we will describe a labeling of the components of $\partial Z_j$ by letters $P$, $S$, and $B$, analogous to similar labelings in \cite{Menasco1984, HTT1, HTT2}.
In these other papers, an essential surface $Z$ in an alternating link complement in $S^3$ is studied by considering its curves of intersection with the projection plane and with small balls around crossings called \emph{crossing balls}. The surface in standard position from \cite{Menasco1984, HTT1, HTT2} intersects the projection plane where $Z$ has \emph{saddles} inside crossing balls. See \reffig{Saddle} left. In \cite{Menasco1984, HTT1, HTT2}, each saddle is labeled with the letter $S$.
For a weakly generalized alternating link, if a surface $Z$ has a saddle, that saddle intersects a crossing arc exactly once. Recall from subsection~\ref{Sec:Gluing} that each interior edge of the chunk decomposition is identified to exactly one crossing arc, with four interior edges in total identified to a single crossing arc. Thus saddles meet interior edges as in \reffig{Saddle}, right. Shown in that figure are the four edges that glue up to the crossing arc (red), and the way the saddle surface meets them. For a surface in normal form, we consider how the surface intersects interior edges and truncation edges. Because interior edges are identified to crossing arcs, and each such arc runs through exactly one saddle, analogous to the notation of \cite{Menasco1984}, we label each intersection of $\partial Z_j$ and an interior edge with an $S$.
\begin{figure}
\caption{Left: A saddle runs between arcs of the diagram at a crossing. Right: This leads to $Z$ intersecting interior edges of the chunk diagram. Shown are intersections of interior edges on $\Pi^+$ and $\Pi^-$.}
\label{Fig:Saddle}
\end{figure}
Recall that a surface $Z$ in a link exterior $Y-N(L)$ is \emph{meridianally compressible} if there is a disk $D$ embedded in $Y$ such that $D\cap Z = \partial D$, the interior of $D$ intersects $L$ exactly once transversely, and the boundary $\partial D\subset Z$ does not bound an annulus $A$ on $Z$ with its other boundary a meridian of $\partial N(L)$. Otherwise $Z$ is \emph{meridianally incompressible}. For a meridianally compressible surface, performing surgery along $D-N(L)$ yields a new surface with boundary forming a meridian of $N(L)$; this is called a \emph{meridianal compression} of $Z$.
Both for classical alternating links in $S^3$~\cite{Menasco1984} and for weakly generalized alternating links under certain conditions~\cite[Lemma~4.9]{HowiePurcell}, a closed surface $Z'$ is meridianally compressible. After meridianal compressions the resulting surface $Z$ meets the diagram in meridians. In \cite{Menasco1984, HTT1} for links in $S^3$, each intersection of $Z$ with a meridian in the projection plane is labeled with a $P$, which stands for a meridianal \textit{puncture}. For weakly generalized alternating links, when $Z$ again meets the diagram in meridians, such a meridian will intersect two truncation faces. We also wish to label intersections with a $P$, but the setup will be slightly different. To describe our labelling, we first need the following definition.
\begin{definition}\label{Def:MeridianalForm} Isotope $Z$ to meet the diagram $\pi(L)$ transversely away from crossings. After this isotopy, in the chunk decomposition each meridianal curve of $\partial Z$ meets exactly two truncation faces, which are quadrilaterals. It runs through adjacent truncation edges on each face, cutting off a single corner of each of two quads in the harlequin tiling of the boundary; see \reffig{MeridianalForm}. Moreover, note that opposite vertices of a truncation face are always identified (they lie on the same endpoint of a crossing arc). The meridian does not cut off such a vertex, but rather cuts off one of the other two vertices of a truncation face. We say that $Z$ is in \emph{meridianal form} if each component of $\partial Z$ \begin{enumerate} \item meets exactly two truncation faces, one on each side of the projection surface; \item runs between adjacent edges in each such truncation face, cutting off a vertex that is not identified to another corner on the same truncation face. \end{enumerate} \noindent See \reffig{MeridianalForm}. The curve that is the meridianal boundary of $Z$ is said to be in meridianal form as well. \end{definition}
\begin{figure}
\caption{Left: Isotope $Z$ with meridianal boundary to meet $N(L)$ transversely away from crossings. Right: In the chunk decomposition, such a curve meets exactly two truncation faces, and cuts off a single corner of each truncation face. Note that the corner cut off by a meridian is not one of the corners that are identified.
}
\label{Fig:MeridianalForm}
\end{figure}
Suppose $Z$ is a surface in meridianal form. Assign a label $P$ to each intersection of a component of $\partial Z$ with a truncation face. Note just as for a classical alternating link in $S^3$, a meridianal puncture in $Z$ corresponds to a single $P$ above the diagram on $\Pi^+$, and a single $P$ below on $\Pi^-$.
In Hass, Thompson, and Tsvietkova~\cite{HTT2}, for a spanning surface $Z$ of an alternating link in $S^3$, the letters $B$ are introduced. A letter $B$ indicates where $Z$ intersects the link transversally on the projection plane. In the case of weakly generalized alternating links, a surface $Z$ with boundary will have normal subsurfaces $Z_j$ with $\partial Z_j$ meeting truncation faces. When the corresponding curve on $\partial Z$ is not necessarily a meridianal curve in meridianal form, we label each intersection of $\partial Z_j$ with a truncation edge by $B$. Note that above, we labeled intersections of $\partial Z_j$ and truncation faces by $P$. For surfaces with non-meridianal boundary, we consider intersections with truncation edges rather than faces. The letter $B$ is a reminder that the surface might have boundary components that are not meridianal.
The labeling of components of $\partial Z_i$ (which are curves) by letters $S$, $B$, or $P$ associates a cyclic word to every such curve. An example is shown in \reffig{SBPCurves}.
\begin{figure}
\caption{On the left is shown an example of a $BBSSS$ curve from \cite{HTT2}. On the right is shown the corresponding $BBSSS$ curve in a chunk of the decomposition of the same link. The curve on the right meets the same saddles (interior edges) and link overcrossings (truncation faces) as the curve on the left.}
\label{Fig:SBPCurves}
\end{figure}
\begin{remark}\label{Rem:BBReplacesP}
In fact, the arguments below for surfaces with labels $B$ on truncation edges work equally well when we replace an instance of $P$ with two instances of $B$, and so we allow curves with $B$ labels also to be in meridianal form. This will be useful for links, for example, when some components of $\partial Z$ are meridians and some are not. In this mixed case, with some meridian boundary components and some non-meridians, we will label all intersections with truncation edges by $B$. However, when a surface has strictly meridianal boundary, we obtain more information using only labels $P$. We do not allow words in both $P$ and $B$. \end{remark}
For classical alternating links, the curves of intersection subdivide the surface into disks lying in topological 3-balls above and below the link. The curves of intersection and the position of the disks are uniquely determined by the associated words in $S$, $P$, and $B$ and by the position of each letter on the link diagram. The disks are then glued along their boundaries, and the gluing pattern is also uniquely determined by the these words in $S$, $P$ and $B$ and by the position of each letter on the link diagram. For weakly generalized alternating links, there are also words associated to boundary components of $Z_i$, and the position of each letter on the knot projection. But the boundary curves subdivide the surface $Z$ into subsurfaces $Z_i$ that might have positive genus, and multiple boundary components. Moreover, the subsurfaces are not in topological 3-balls anymore: rather, they are in chunks which might have complicated topology themselves.
\section{Normal form, saddles, and meridians}\label{Sec:Meridianal}
We call the number of intersections of a surface $Z$ with interior edges (saddles) the \emph{weight} of the embedding with respect to a chunk decomposition. In this section, we ensure that a surface can be isotoped into normal form while preserving meridianal form for the surface, by an isotopy that does not increase weight. Note that there are many (isotopic) ways to put a surface in normal form or standard position, and in applications, often the one that minimizes weight, as well as the number of punctures and curves of intersection with $\Pi$ is usually chosen.
For convenience, we collect assumptions here. They will be used throughout the rest of the paper, and will be a part of hypothesis in each lemma, proposition and theorem we prove.
\begin{assumption} \begin{itemize} \item Let $Y$ be a compact, orientable, irreducible 3-manifold. If $Y$ has boundary, we require $\partial Y$ to be incompressible in $Y-N(\Pi)$. We assume that $\pi(L)$ is a weakly generalized alternating projection of a link $L$ onto an orientable projection surface $\Pi$ in $Y$.
\item We consider $(Z,\partial Z)$ to be an essential surface (either orientable or nonorientable) embedded in $(Y-N(L), \partial N(L))$, where the boundary of $Z$ is possibly empty. Moreover, we assume that if the surface $Z$ has boundary and $Y$ has boundary, then the boundary of $Z$ lies on $N(L)$ and is disjoint from $\partial Y$.
\item We also assume that there is a chunk decomposition of $Y-N(L)$ as in subsection~\ref{Sec:ChunkDecomp}, and $Z_i$ are connected subsurfaces of $Z$ cut out by chunks, closed or with boundary, and possibly with multiple boundary components. That is $Z=\bigcup Z_i$. \end{itemize} \end{assumption}
\begin{theorem}\label{Thm:Normal} A surface $Z$ in $Y-N(L)$ can be isotoped into normal form such that: \begin{enumerate} \item[(a)] No step of the isotopy increases the weight, i.e.\ the number of intersections of $Z$ with interior edges (saddles). \item[(b)] If $Z$ begins in meridianal form, then after isotopy into normal form, $Z$ remains in meridianal form. \end{enumerate} \end{theorem}
\begin{proof} The proof that an isotopy exists putting the surface into normal form is a standard innermost disk / outermost arc argument that is very similar to arguments that appear elsewhere (see \cite{HowiePurcell, fg09}), but we walk through it to verify (a) and (b).
We need to step through the requirements of \refdef{NormalSurface}. Since $Z$ is essential, condition~(0) automatically holds for closed surfaces within the chunk, without any isotopy. If $D$ is an essential compressing disk for $Z\cap C$ within $C$, then because $Z$ is incompressible, $D$ bounds a disk $E$ in $Z$, and by irreducibility of $Y$, $D\cup E$ bounds a ball. Hence $E$ can be isotoped through the ball and past $E$ to remove the compressing disk. Such an isotopy only decreases intersections with interior edges~(a), and does not affect the boundary of $Z$ at all for~(b).
A small isotopy of $Z$ ensures transversality conditions~(1). We may ensure this isotopy does not increase the number of intersections with any interior edge, for~(a), and does not affect meridianal form for~(b).
For condition~(2) of \refdef{NormalSurface}, suppose some $Z_k$ has boundary $\partial Z_k$ lying in a face of $C$ and bounding a disk in that face. Take an innermost such disk $D$. It is not an essential compressing disk for $Z$ in $Y$ because $Z$ is incompressible. Because $Y$ is irreducible, there is a ball $B$ with $D\subset \partial B$ and $\partial B-D\subset Z$. Isotope $Z$ through $B$ and slightly further, removing all intersections of $Z$ with faces and interior edges that lie within the ball $B$, and removing the intersection of $\partial Z_k$ on a face of $C$, all without changing $Z$ outside a small neighborhood of $B$. Note this move cannot increase the number of intersections of $Z$ with interior edges, for~(a), and avoids the boundary of $Z$ entirely, so does not affect meridianal form, for~(b).
For condition~(3) of \refdef{NormalSurface}, suppose for some $Z_k$, an arc of $\partial Z_k$ lies in a single face with endpoints on the same edge, and together with a part of the edge cuts off a disk $D$ in that face. Assume $D$ is innermost with this property in that face, meaning its interior does not intersect $Z$.
There are three cases depending on the type of edge and the type of face.
Suppose first that the arc of $\partial Z_k$ has endpoints on the same interior edge. Then use a regular neighborhood of the disk $D$ to isotope $Z$ past that edge, strictly reducing the number of intersections with an interior edge, for~(a). This move does not affect $\partial Z$, for~(b).
Next suppose that the arc of $\partial Z_k$ lies in a truncation face, with both endpoints on the same truncation edge. Then again use a regular neighborhood of $D$ in $Y-N(L)$ to isotope $\partial Z$ along this truncation face and past the edge, removing two intersections with truncation edges. This move does not affect intersections with interior edges, giving~(a). Moreover, such an arc is not in meridianal form, so does not arise for~(b).
Finally suppose the arc $\alpha$ of $\partial Z_k$ has both endpoints on the same truncation edge but lies in an interior face. Then the disk $D$ has the form of a boundary compression disk for $Z$. Since $Z$ is boundary incompressible, there must be an arc $\beta\subset \partial Z$ whose endpoints agree with those of $\alpha$, and such that $\alpha\cup \beta$ bounds a disk $D'$ in $Z$. Then $D\cup D'$ forms a disk with boundary on $\partial N(L)$. Because $Y- N(L)$ is boundary irreducible, by \cite[Corollary~3.16]{HowiePurcell}, the disk $D\cup D'$ must co-bound a ball with a disk of $\partial N(L)$. Use this ball to isotope $D'$ in $Z$ past $D$ to remove the two intersections with the truncation face, as well as any other intersections of $Z$ with edges and faces within the ball. For~(a), the number of intersections with saddles does not increase. As for~(b), the arc $\beta$ of $\partial Z$ cannot be in meridianal form, since both its endpoints are on the same truncation edge, so this situation does not arise for surfaces in meridianal form.
Condition~(4) of \refdef{NormalSurface} requires the most care, since it could arise for surfaces in meridianal form. Nevertheless, we show that an isotopy eliminates this case while preserving meridianal form and decreasing weight. The argument requires careful consideration of the combinatorics of the chunk decomposition around a truncation face and an adjacent interior edge. We step through it using a number of figures below.
Suppose an arc $\gamma$ of $\partial Z_k$ in an interior face of a chunk $C$ has an endpoint on a truncation edge and an endpoint on an adjacent interior edge, cutting off a disk $D$ with these two edges. Isotope $Z$ through a regular neighborhood of $D$, by sliding $\partial Z$ along the adjacent truncation faces and past the interior edge, then sliding the rest of $Z$ through $D$ to follow. An example of the effect of this move on the harlequin tiling of the boundary is shown in \reffig{Condition4-BdryView}. This removes an intersection of $Z$ with an interior edge, giving~(a). We may have introduced arcs in truncation faces with both endpoints on the same truncation edge, as in \reffig{Condition4-BdryView}. Such edges can be eliminated as above, without affecting weight. Thus if the surface is not in meridianal form, we are done at this point.
\begin{figure}
\caption{A curve of $\partial Z$ forming a meridian on the harlequin tiling of the boundray is shown on the left. If we isotope across a disk that violates condition~(4), the curve is changed as shown on the right. Note further isotopy is required to put it into normal form, meridianal form.}
\label{Fig:Condition4-BdryView}
\end{figure}
To complete the proof of~(b), we need to analyse the isotopy of $Z$ through $N(D)$ more carefully, ensuring that if $D$ is adjacent to a truncation face that $Z$ meets in meridianal form, then after the isotopy the result is still in meridianal form.
Suppose $Z$ is in meridianal form. We set up some notation. The arc of $\partial Z_k$ that runs from the interior edge to the truncation edge must continue through the truncation face in meridianal form. That is, it cuts off a single vertex of the truncation face. Say it cuts off $D\subset \Pi^+$. Because the interior face containing $D$ is glued to another interior face on $\Pi^-$, there is another such arc bounding a disk $D'\subset \Pi^-$, and that arc extends to run through another truncation face in meridianal form. Indeed, these two arcs in meridianal form together form a meridian boundary component of $\partial Z$. Finally, on each of $\Pi^+$ and $\Pi^-$ there is another interior edge identified to the one meeting $D$ or $D'$. The surface $Z$ must also run through that interior edge. The setup must therefore appear as in \reffig{NormalMerid}.
\begin{figure}
\caption{When $Z$ is in meridianal form, and an arc runs from a truncation edge to an adjacent interior edge, $Z$ must intersect $\Pi^+$ and $\Pi^-$ as shown on the left (possibly with roles of $\Pi^+$, $\Pi^-$ switched). The right shows the saddle and the adjacent meridian intersection in $Y-N(L)$.}
\label{Fig:NormalMerid}
\end{figure}
That is, there are arcs $ab$ and $cd$ of $Z_k\cap \Pi^+$ meeting interior edges identified to the crossing arc of a saddle, and a subarc of $ab$ is an arc of the boundary of the disk $D$. There are additional arcs $a'd'$ and $b'c'$ in $\Pi^-$, with an arc of $a'd'$ forming an arc of the boundary of disk $D'$. Moreover, $a$ is glued to $a'$, $b$ to $b'$, $c$ to $c'$ and $d$ to $d'$.
For the leftmost and middle pictures in \reffig{NormalMerid}, note that the endpoints of arcs labeled by $a, b,c,d$ are in the same quadrants as $a', b', c', d'$ respectively. This is because two interior faces that are identified correspond to the same region of the link diagram. Also note that the arcs $ab, cd, a'd', b'c'$, etc. and their intersection pattern with interior edges and truncation edges of the chunks correspond to rotating the boundary of every interior face under the gluing one notch, as described in \refsec{Gluing}.
From the middle picture in \reffig{NormalMerid}, rotate quadrant I one notch counterclockwise and identify it with quadrant I in the leftmost picture. Similarly, rotate quadrant III one notch counterclockwise and identify it with quadrant III in the leftmost picture. Here interior edges are rotating to the next interior edges in the counterclockwise direction, and truncation edges are rotating to the next truncation edge in the counterclockwise direction. Rotate quadrants II and IV one notch clockwise to the leftmost picture.
Now we isotope $Z$ across the disk $D$, to remove intersections with the interior edge identified to the crossing arc. Note this simultaneously isotopes across $D'$ in the other chunk. This isotopes arcs $cd$ and $b'c'$ to run through the adjacent truncation faces; this is shown in \reffig{NormalMeridIsotope1}. Away from these arcs, the intersection of the surface $Z$ with $\Pi^+$ and $\Pi^-$ is unchanged.
\begin{figure}
\caption{The isotopy through $D$ and $D'$ has the effect shown.}
\label{Fig:NormalMeridIsotope1}
\end{figure}
Note that the surface is not in normal form; on the left of \reffig{NormalMeridIsotope1}, an arc of $\partial Z'$ in the truncation face meets the same truncation edge twice. We perform an isotopy to slide the arc off the truncation face. It will slide across the truncation face on $\partial Z$ into the truncation face in the centre on the right of \reffig{NormalMeridIsotope1} on $\Pi^-$. It also adjusts the pattern of arcs on truncation faces, adjusting \reffig{Condition4-BdryView} to remove the arc with both endpoints on the same truncation edge (part of the arc $ab$ in \reffig{NormalMeridIsotope1}, left), and thereby connecting two arc components on one truncation face into one (part of the arc $d'c'$ in \reffig{NormalMeridIsotope2}). The final result of this isotopy, in $\Pi^-$, $\Pi^+$ and in $Y-N(L)$, is shown in \reffig{NormalMeridIsotope2}.
\begin{figure}
\caption{Isotoping further into normal form yields surface in meridianal form.
}
\label{Fig:NormalMeridIsotope2}
\end{figure}
The surface is now again in meridianal form. In the link, the entire procedure swept a meridian through a saddle, removing the intersection with that saddle. Thus~(b) also holds. \end{proof}
\begin{assumption} From now on, when we put a surface in normal position, we always make two assumptions: \begin{enumerate} \item we do it so that the surface is in meridianal form; \item out of all ways to do it so that the surface is in meridianal form, we choose one with least weight. \end{enumerate} \end{assumption}
\section{Generalizing standard position}\label{Sec:Standard}
The next proposition generalizes properties of standard position for closed surfaces from \cite{Menasco1984}, and for surfaces with non-meridianal boundary from Propositions 2.1-2.2 of \cite{MenascoThistlethwaite}.
\begin{proposition}\label{Prop:TruncationEdgesPairs} The following holds for a surface $Z$ in $Y-N(L)$:
\begin{enumerate}
\item[(1)] There is no meridianal compression of $Z$ to a component of $\partial N(L)$ that meets $\partial Z$ such that $\partial Z$ forms a non-meridianal slope.
\item[(2)] $Z$ has no connected components that are spheres or projective planes.
\item[(3)] If $Z$ is in normal position, a curve of $\partial Z_i$ meets truncation edges in pairs. Thus labels $B$ occur in pairs.
\item[(4)] If $Z$ is meridianally incompressible and in normal form, $\partial Z$ does not meet the same saddle twice while cutting off a disk. That is, let $C$ be a chunk, and let $e$ be an interior edge of $C$ that is identified to an interior edge $e'$ of $C$. Then no arc $\alpha$ of $Z\cap \partial C$ runs from $e$ back to $e$ so that $\alpha$ and an arc on $e$ co-bound a disk on the projection surface $\Pi$. Similarly, no arc $\alpha$ of $Z\cap \partial C$ runs from $e$ to $e'$, co-bounding a disk with an arc between $e$ and $e'$. See \reffig{NoSaddleTwice_2Cases}.
\end{enumerate} \end{proposition}
\begin{proof} For (1), suppose $Z$ is a surface with non-meridianal boundary that admits a meridianal compression. Such a compression defines an embedded annulus $A$ with one boundary component on $Z$ and one on the link, with interior disjoint from $Z$. But a meridian intersects any non-meridianal closed curve on the link boundary torus. If $\partial Z$ meets this component of $\partial N(L)$ and is non-meridianal, $\partial Z$ must intersect the annulus $A$, and hence $Z$ meets the interior of $A$. This contradicts the definition of $A$.
For (2): Recall that $\Pi$ was chosen such that $Y- \Pi$ is irreducible, thus any chunk is irreducible. Therefore a spherical component cannot bound anything but a ball. But a 2-sphere is incompressible if it does not bound a ball. Hence item (0) of \refdef{NormalSurface} implies that the intersection of a normal surface with a chunk has no spherical components. Similarly, because $Y-\Pi$ is orientable, the regular neighborhood of any embedded incompressible projection plane would be an essential 2-sphere, which cannot exist in the irreducible $Y-\Pi$.
For (3): Every time $\partial Z_i$ meets a truncation edge, it runs into a truncation face and then must meet another truncation edge as it runs out of that truncation face.
For (4): By contradiction, suppose an arc of intersection $\alpha$ of $Z$ with the boundary of a chunk $C$ has both endpoints on interior edges that are identified to the same crossing arc in $Y-N(L)$. Suppose further that there is a disk $D$ in the projection surface $\Pi$ with $\partial D = \alpha\cup \beta$, where $\beta$ is an arc with $\partial \beta = \partial \alpha$ and $\beta$ is either contained in an interior edge, or runs from one interior edge across a truncation face to an opposite interior edge that is identified to the same crossing arc. See \reffig{NoSaddleTwice_2Cases}.
\begin{figure}
\caption{The two possible cases if an arc $\alpha$ of $Z\cap \partial C$ has endpoints on interior edges that are identified to the same crossing arc.}
\label{Fig:NoSaddleTwice_2Cases}
\end{figure}
We will show that $Z$ can be isotoped into normal form in a way that strictly decreases the number of intersections with interior edges, contradicting the assumption that $Z$ has least weight. The proof is similar to \cite[Lemma~1(ii)]{Menasco1984}; we use the disk in $D\subset \Pi$ to reproduce Menasco's argument.
Suppose first that $\alpha$ has both endpoints on the same interior edge $e$, as on the left of \reffig{NoSaddleTwice_2Cases}. Suppose $\alpha$ and the arc $\beta$ of $e$ together bound a disk $D$, and suppose $D$ is innermost with that property.
If $\alpha$ happens to lie in only one face, then $Z$ is not normal and we have already seen that $Z$ can be put into normal form in a way that removes $\alpha$ (\refthm{Normal}).
But it could be the case that $D$ intersects multiple faces and edges. In this case, we will isotope $Z$ to remove two intersections of $Z$ with $e$, illustrated in \reffig{NoSaddleTwice}.
\begin{figure}
\caption{Shows how we isotope $Z$ through the disk $D'$ to remove intersections with $e$, in 3-dimensions and a 2-dimensional cross section. In the leftmost and rightmost figures, $e$ is the edge in red, with $\beta$ slightly darker red on $e$. In the center, a cross-section is depicted, and $e$ becomes a single point. }
\label{Fig:NoSaddleTwice}
\end{figure}
Carefully, push the disk $D$ slightly into the interior of the chunk, obtaining a new disk $D'$, shown in darker grey on the left of \reffig{NoSaddleTwice}. Then isotope $Z$ in a small regular neighborhood of $D'$, removing two intersections with $e$, as shown on the right of the figure. Note that within $Y-N(L)$, the edge $e$ is identified to other edges to obtain a crossing arc that contains $\beta$, and $\beta$ runs between two saddles intersecting the crossing arc. This move eliminates the intersections of $Z$ at endpoints of $\beta$ by sliding past. Note \reffig{NoSaddleTwice} does not show what happens in $Y-N(L)$, only the chunk decomposition, prior to the edge identifications.
Alternatively, the isotopy is equivalent to the following move. Take a small product neighborhood of $D'$. This is a ball $B=D'\times(-\epsilon,\epsilon)$, for some sufficiently small $\epsilon>0$. The boundary $\partial B$ consists of a disk $E$ on $Z$ of the form $E=\alpha'\times(-\epsilon,\epsilon)$, and a second disk $F=\partial B - E$. Replace the surface $Z$ by removing the disk $E$ from $Z$ and replacing it with the disk $F$. Push slightly through the boundary of the chunk. The resulting surface is isotopic to $F$, but meets the edge $e$ two fewer times.
The resulting surface is still essential and meridianally incompressible, so it can be isotoped further into normal form in a manner that does not increase intersections with interior edges. This is a contradiction, since $Z$ was assumed to have minimal weight.
Second, $\alpha$ may have endpoints on two distinct interior edges that are glued to form a crossing arc of $Y-N(L)$. In this case, the endpoints of $\alpha$ are separated by a truncation face. The disk $D$ has boundary on $\alpha$, and on an arc $\beta$. The arc $\beta$ consists of an arc of one interior edge, an arc cutting across a truncation face running between opposite vertices, and an arc on another interior edge, shown on the right of \reffig{NoSaddleTwice_2Cases}. Again we may assume $\alpha$ is innermost with this property. Then glue interior edges to obtain a crossing arc. Because $\alpha$ is innermost, the two endpoints of $\alpha$ on interior edges are then identified. The arc on the truncation face forms a meridian. Thus under gluing, $\alpha\subset Z$ bounds a meridian compression disk. Because $Z$ is meridianally incompressible, $D$ must be parallel into $Z$. That is, $\alpha$ bounds an annulus in $Z$ whose other boundary component on $\partial Z$ is a meridian in $\partial Z$, and together $D$ and this annulus bound a thickened annulus. Isotope $Z$ through this thickened annulus, past $D$, removing all intersections of $Z$ with faces, edges, and truncation edges within the thickened annulus. The number of intersections of $Z$ with interior edges strictly decreases. This contradicts the minimum weight assumption. \end{proof}
\section{Angled chunks and combinatorial area}\label{Sec:Area}
So far, we have isotoped essential surfaces into normal form, and have considered some of the resulting subsurfaces in chunks and their boundary curves. In this section, we recall an important tool to reduce the options for such subsurfaces, namely combinatorial area.
In its most general form, combinatorial area can be defined for surfaces in any chunk with dihedral angles assigned to interior edges that satisfy certain conditions; this is called an \emph{angled chunk decomposition}, and it is described in full generality in \cite{HowiePurcell}. In our setting, label each interior edge with angle $\pi/2$. By \cite[Proposition~3.15]{HowiePurcell}, the decomposition satisfies the requirements to be an angled chunk decomposition.
We now review a few consequences.
\subsection{Combinatorial area}
Let $Z$ be in normal form with respect to the chunk decomposition of $Y-N(L)$. Write $Z=\bigcup_{j=1}^m Z_j$, where each $Z_j$ is a connected normal surface embedded in a chunk. Consider $\partial Z_j$. Each component of $\partial Z_j$ meets interior edges a total of $n_i$ times; this is the number of instances of $S$ on the words decorating $\partial Z_j$. It meets truncation edges a total of $n_t$ times, where $n_t$ must be even, since $Z_j$ enters and exits each truncation face by meeting a truncation edge. In case $Z$ is meridianal, $n_t/2$ is the total number of instances of $P$ in the words decorating $\partial Z_j$. Otherwise, $n_t$ is the total number of instances of $B$.
The \emph{combinatorial area} of $Z_j$ is defined to be \begin{equation}\label{Eqn:SubArea}
a(Z_j) = n_i(\pi/2) + n_t(\pi/2) - 2\pi\chi(Z_j). \end{equation} Denote the number of $S$'s in $Z_j$ by $\#S$, and the number of $P$'s by $\#P$. If $Z$ is meridianal, we rewrite this as \begin{equation}
a(Z_j) = \frac{\pi}{2}(\# S) + \pi(\# P) - 2\pi\chi(Z_j). \end{equation} The \emph{combinatorial area} of $Z$ is defined to be \begin{equation}
a(Z) = \sum_{i=1}^n a(Z_i). \end{equation}
The combinatorial area satisfies a Gauss--Bonnet formula~\cite[Proposition~3.12]{HowiePurcell}: \begin{equation}\label{Eqn:GB} a(S) = -2\pi\chi(Z). \end{equation}
We also have the following results that follow from~\cite{HowiePurcell}.
\begin{lemma}\label{Lem:Areas} Let $Z_i$ be a normal surface in a chunk. The combinatorial area of $Z_i$ satisfies the following. \begin{enumerate} \item[(1)] If $\chi(Z_i)< 0$, then $a(Z_i)\geq 2\pi$. \item[(2)] If $\chi(Z_i)\geq 0$, then either $a(Z_i)\geq \pi/2$, or $a(Z_i)=0$. \item[(3)] Additionally, in the case when $a(Z_i)=0$, $Z_i$ is either:
\begin{enumerate}
\item[(a)] an essential torus or Klein bottle embedded in a chunk, hence $Z=Z_i$ is a torus or Klein bottle,
\item[(b)] an annulus or M\"obius band with boundary meeting no edges, or
\item[(c)] a disk such that $\partial Z_i$ meets exactly four edges of the chunk decomposition.
\end{enumerate} \end{enumerate} \end{lemma}
\begin{proof} Equation (\ref{Eqn:SubArea}) in the definition of combinatorial area implies the first item. Proposition~3.11 in~\cite{HowiePurcell} asserts that in the orientable case, $a(Z_i)\geq 0$, and proves that if $a(Z_i)=0$, then the third item holds.
It remains to prove the nonorientable case, and that if $a(Z_i)>0$, then $a(Z_i)\geq \pi/2$.
Note that since $Z_i$ is a connected surface, with or without boundary, and $Z_i$ is never a sphere or projective plane by~\refprop{TruncationEdgesPairs}~(2), we have $\chi(Z_i) \leq 1$.
If $\chi(Z_i)=0$, then $Z_i$ is either an annulus, M\"obius band, torus, or Klein bottle. If $Z_i$ is an annulus or M\"obius band and $\partial Z_i$ meets an edge, then $a(Z_i)\geq \pi/2$ by equation~\refeqn{SubArea}. If $Z_i$ is a torus or Klein bottle, it lies in the interior of the chunk, so meets no edges, and $a(Z_i)=0$. Similarly if it is an annulus or M\"obius band such that $\partial Z_i$ meets no edges, then $a(Z_i)=0$.
If $\chi(Z_i)=1$, then $Z_i$ is a disk. Then $a(Z_i) = k\pi/2 -2\pi$ by formula \refeqn{SubArea}, where $k$ is the number of edges (interior or truncation) met by $\partial Z_i$. Thus if $k>4$, then $a(Z_i)\geq \pi/2$. If $k=4$, then $a(Z_i)=0$ and~(3)(c) holds. Finally, the cases $k=0,1,2,3$ are ruled out by~\cite[Proposition~3.11]{HowiePurcell}. \end{proof}
The Gauss--Bonnet formula, equation~\refeqn{GB}, together with \reflem{Areas} immediately implies that weakly generalized alternating link complements are irreducible and boundary irreducible: an essential sphere or disk would have negative combinatorial area, which is impossible by \reflem{Areas}. This was observed by Howie and Purcell~\cite[Corollary~3.16]{HowiePurcell}. Additionally, that paper studies surfaces with combinatorial area zero, namely annuli and tori, to analyze when weakly generalized alternating knots are hyperbolic~\cite[Section~4]{HowiePurcell}. By contrast, the work in this paper allows us to study more surfaces: higher genus surfaces, punctured spheres, etc. For a surface with fixed Euler characteristic, the Gauss--Bonnet formula \refeqn{GB} restricts potential normal subsurfaces to those with combinatorial area no more than the original. This paper helps us analyze the structure of potential normal subsurfaces, and how they interact with the diagram. We will see that of key importance are subsurfaces with zero combinatorial area.
The next lemma motivates our study of disks whose boundaries are as in Sections~\ref{Sec:SSSS}, \ref{Sec:PPPP}, and \ref{Sec:BBBB}. In the lemma, we restrict to links with a \emph{cellular} diagram, i.e.\ a diagram for which the complementary regions of the diagram graph are disks on the projection surface $\Pi$.
\begin{lemma}\label{Lem:0AreaEnumerated} In a weakly generalized alternating link with a cellular diagram, the only normal subsurfaces within a chunk decomposition that have zero combinatorial area are disks with boundaries labeled $PP$, $PSS$, $SSSS$, $BBBB$, $BBSS$, and $PBB$. \end{lemma}
In the last case, recall that we do not allow both $P$ and $B$ labels in the same word. Thus we will regard $PBB$ disks as instances of $BBBB$ disks, as in \refrem{BBReplacesP}. Thus they are treated with $BBBB$ disks below.
\begin{proof}[Proof of \reflem{0AreaEnumerated}] By \reflem{Areas}, a zero area disk meets four edges. If these are all interior edges, the disk is labeled $SSSS$. If it meets truncation edges, it must do so in pairs, thus either two adjacent $B$s or a single $P$ (encoding two truncation edges for a curve in meridianal form). Thus the possibilities are as claimed. \end{proof}
\section{Eliminating $SSSS$ disks}\label{Sec:SSSS}
In this section, we restrict disks of the form $SSSS$.
\begin{theorem}\label{Thm:NoSSSS}
Suppose $Z$ is meridianally incompressible surface in normal form with respect to the chunk decomposition of $Y-N(L)$. Then any $SSSS$ disk is an essential compression disk for $\Pi$ meeting the diagram $\pi(L)$ exactly four times. Thus if the representativity satisfies $r(\pi(L),\Pi)>4$, there are no $SSSS$ disks. \end{theorem}
\begin{proof} Suppose $Z_i$ is a normal disk meeting exactly four interior edges and suppose $Z_i$ is not a compressing disk for $\Pi$. Then $Z_i$ is parallel into the boundary surface $\Pi^+$ or $\Pi^-$ of a chunk, without loss of generality say $\Pi^+$, so the curve $\partial Z_i$ bounds a disk on $\Pi^+$ (meeting edges, faces, etc. in its interior).
\begin{figure}
\caption{Left: Disk with boundary SSSS in $Y-N(L)$. Middle: form in chunk decomposition with boundary $\Pi^+$. Right: glued to $\Pi^-$ as shown.}
\label{Fig:SSSS}
\end{figure}
The form of $Z_i$ in the diagram $\pi(L)$ is shown on the left of \reffig{SSSS}, where the curve $\partial Z_i \cap \Pi$ is shown in blue along with four saddles. In particular, if one follows the component of $\partial Z_i\cap \Pi$, the overpasses of $\pi(L)$ alternate between being on the right and on the left due to the diagram $\pi(L)$ being alternating. The form of $\partial Z_i$ in one chunk is shown in the middle, where $\partial Z_i$ lies on some component $\Pi_j^+$ of $\Pi^+$. Denote the four arcs of $\partial Z_i$ between the intersection points with edges by $A, B, C, D$ as on the middle figure. The surface $\Pi_j^+$ is glued to $\Pi_j^-$ via a gluing that rotates the boundaries of each face as in subsection~\ref{Sec:Gluing}. Thus the arcs $A$, $B$, $C$, $D$ are glued to four arcs $A'$, $B'$, $C'$, $D'$ as shown on the right. The blue arcs in the right figure are arcs of boundary curves of other normal subsurfaces $Z_k$ in the chunk; thus they are portions of simultaneously embedded closed curves.
First, observe that while $A'$, $B'$, $C'$, and $D'$ no longer connect to bound a disk in $\Pi^-$, the union of the arcs $A'$, $B'$, $C'$, and $D'$ along with meridianal arcs through the four truncation faces of the chunk and pieces of interior edges between them all bound a disk on $\Pi_j^-$. This is almost a copy of the disk bounded by $A\cup B\cup C\cup D$ in $\Pi^+$ transferred to the homeomorphic surface $\Pi^-$: the difference is that we have moved the disk slightly with respect to the truncation faces.
Label the blue arcs that run into this disk in $\Pi^-$ by 1, 2, 3, 4 as in \reffig{SSSS}, right. Because the arcs connect to disjoint embedded closed curves (the boundaries of normal surfaces in the chunk), the arcs $1$, $2$, $3$, and $4$ must exit the disk region shown. They might exit either by connecting to each other, e.g.\ $1$ might connect to $2$, or they might exit by running through the region between $1$ and $2$, between arcs $B'$ and $C'$, between $3$ and $4$, or between arcs $D'$ and $A'$. We call these regions zones $a$, $b$, $c$, and $d$, and they are shown in \reffig{SSSS}, right, as well.
The arcs labeled 1 and 2 cannot connect to each other or exit through zone $a$ by \refprop{TruncationEdgesPairs}~(4), since they each meet the interior edge in that region once already. Similarly, the arcs 3 and 4 cannot connect with each other or exit through zone $c$. Note also that arcs 1 and 4 cannot run to zone $d$ by the same lemma, since arc $A'$ already meets the edge of this region, and similarly for $D'$. Similarly arcs 2 and 3 cannot run through zone $b$.
It follows that the arc labeled $1$ runs into zone $b$ or $c$. If $b$, then the arc labeled $2$ must run into zone $a$ or $b$ in order to remain disjoint from the arc labeled $1$. But this is impossible: these are exactly the zones that $2$ is not allowed to enter. Hence the arc labeled $1$ runs to zone $c$. But then the arc labeled $4$ must run into zone $c$ or $d$ in order to remain disjoint from $1$. Again this is impossible: these are exactly the zones that $4$ is not allowed to enter. It follows that there can be no $SSSS$ disk that is not a compressing disk for $\Pi$. \end{proof}
\section{Eliminating disks with words in the letter $P$} \label{Sec:PPPP}
In this section, we prove that if $Z$ has meridianal boundary, there are no disks whose boundaries are curves of intersection labeled with two or three letters in $P$ and $S$. That is, there are no words of type $SS$, $PP$, $PS$, $PPP$, $PSS$, $SSS$, or $PPS$. The proof for $SS$ and $SSS$ holds in more generality, and does not require meridianal boundary.
We also prove that for any subsurface $Z_i$ of $Z$, there are no boundary components $\partial Z_i$ labeled with an odd number of instances of $P$ and $S$. Finally, we use this to show all weakly generalized alternating links are prime.
Note that for classical alternating links, i.e.\ links on $S^2$ in $S^3$, 3-letter words were not possible due to a simple argument: two closed curves, one representing a component of the link, and one from the surface $Z$, must intersect an even number of times. This argument does not work when the projection surface $\Pi$ might have positive genus, so the proofs here are different from the classical case.
\begin{theorem}\label{Thm:No2LetterWords}
Suppose $Z$ is in normal form with respect to the chunk decomposition of $Y-N(L)$. Then no normal subsurface of $Z$ is a disk meeting exactly two truncation faces in meridianal form (no disk with $PP$ boundary curve), exactly one truncation face in meridianal form and one interior edge (no $PS$ disk), or exactly two interior edges (no $SS$ disk). \end{theorem}
\begin{proof} Recall that the boundary of a chunk is decorated by the diagram graph, where interior edges corresond to diagram edges, and truncation faces correspond to neighborhoods of crossings. Therefore the boundary of a disk labeled by two letters $PP$, $PS$, or $SS$ gives a curve on the diagram graph $\pi(L)\subset \Pi$ meeting the diagram in diagram edges or at crossings.
In the case of a label $P$, there is a corresponding arc through a truncation face in meridianal form. We isotope this curve very slightly off the crossing in a direction determined by the side of the truncation face that is met by the curve. See \reffig{IsotopeP}, where an example is shown of a boundary $PP$. The isotopy moves $\partial Z_i$ slightly to the curve $\gamma$, which we think of as lying on $\Pi$ meeting two diagram edges of the diagram graph $\pi(L)$.
\begin{figure}
\caption{If $\partial Z_i$ is labeled $PP$, it determines a curve $\gamma$ meeting the diagram exactly twice.}
\label{Fig:IsotopeP}
\end{figure}
Then in all cases, the disk $Z_i$ determines a curve $\gamma$ on $\Pi$ meeting the diagram exactly twice transversely in diagram edges. Because $Z_i$ is a disk, $\gamma$ must also bound a disk. Because the diagram has representativity at least $4$, by definition of weakly generalized alternating, the disk must be parallel to $\Pi$. Then the fact that the diagram is weakly prime implies that $\gamma$ bounds a disk that does not meet any crossings.
In the case that the curve is of the form $SS$ or $PS$, this gives an immediate contradiction: the curve $\partial Z_i$ is not in normal form, violating condition~(3) or~(4) of \refdef{NormalSurface}, respectively.
In the case the curve is of the form $PP$, then it has the form shown in \reffig{IsotopeP}. Both arcs are in meridianal form, thus they cut off a corner that is not identified to another corner of the truncation face. However, consider the interior edge encircled by the curve $\partial Z_i$. At one of its endpoints, this edge is identified to another edge across the corresponding truncation face, as described in \refsec{Gluing}; see also \reffig{CrossingArc}. Thus one of its endpoints meets a corner of a truncation face that is identified to another corner of the same truncation face. But this is a contradiction: because $\partial Z_i$ is in meridianal form, neither endpoint of the edge can have this property. \end{proof}
\begin{theorem}\label{Thm:No3LetterWords} Suppose $Z$ is in normal form with respect to the chunk decomposition of $Y-N(L)$. Then no normal subsurface of $Z$ is a disk meeting exactly three truncation faces (no $PPP$ disk), exactly two truncation faces and an interior edge (no $PPS$ disk), exactly one truncation face and two interior edges (no $PSS$ disk), or exactly three interior edges (no $SSS$ disk).
More generally, no normal subsurface $Z_i$ of $Z$ has a boundary component meeting an odd number of letters $S$ and $P$. \end{theorem}
\begin{proof} As in the proof of \refthm{No2LetterWords}, a curve $\partial Z_i$ determines an embedded curve $\gamma$ on the diagram graph $\pi(L) \subset \Pi$, with letters $S$ running transversely through diagram edges, and letters $P$ running through a diagram edge immediately to the left or right of a crossing, determined by the position of the arc on the truncation face in meridianal form as in \reffig{IsotopeP}.
In each case, consider the checkerboard coloring of the diagram. Passing through a letter $P$ or $S$ changes the color of the face meeting $\partial Z_i$. If the word labelling $\partial Z_i$ is made up of exactly three letters, then the color must change exactly three times. But this is impossible: if we start in a white face and traverse $\partial Z_i$, it changes to shaded when it meets the first letter, to white when it meets the second letter, to shaded when it meets the third, and then it must close up, implying that the starting and ending face is both shaded and white. \begin{figure}
\caption{A $PSS$ disk should have three arcs in three faces of distinct colors, but this is impossible for a checkerboard colored diagram.}
\label{Fig:No3LetterWords}
\end{figure} This contradiction is illustrated in the case $PSS$ in \reffig{No3LetterWords}.
More generally, it is impossible for a curve of $\partial Z_i$ to meet an odd number of letters $S$ and $P$, since again the existence of such a curve would contradict the checkerboard coloring of the diagram. \end{proof}
The above theorem gives us a quick way to prove the fact that weakly generalized alternating links with cellular diagram are prime. When the hat-representativity satisfies $\hat{r}(\pi(L), \Pi)>4$, this result was originally proved in Howie--Purcell~\cite[Corollary~4.7]{HowiePurcell}. We can extend now to all weakly generalized alternating links without the additional hat-representativity condition.
\begin{theorem}\label{Thm:Prime}
A weakly generalized alternating link is prime. \end{theorem}
\begin{proof} Suppose not. Then there exists an essential meridianal annulus $Z$. Put it into normal form with respect to the chunk decomposition. We may assume it decomposes into normal subsurfaces that are meridianal by \refthm{Normal}, and have zero combinatorial area by the Gauss--Bonnet formula, equation~\refeqn{GB}. Because it meets two meridians of the link, at least one of the subsurfaces $Z_i$ must have a boundary curve $\partial Z_i$ that meets an instance of $P$. Because both boundary components are meridianal, there will be no instances of $B$. By \reflem{Areas}, $Z_i$ must be a disk meeting exactly four edges of the chunk decomposition. By \reflem{0AreaEnumerated}, the possibilities are disks with boundaries labeled $PP$ or $PSS$. Disks labeled $PP$ are ruled out by \refthm{No2LetterWords}. Disks labeled $PSS$ are ruled out by \refthm{No3LetterWords}. This gives a contradiction. \end{proof}
\section{Disks with $BBBB$ and $BBSS$ words}\label{Sec:BBBB}
This section concerns disks whose boundaries are $BBBB$ or $BBSS$ words. As mentioned in \refrem{BBReplacesP}, an instance of $P$ may be replaced with two instances of $B$, and the results for $BB$ go through. Hence the results for $BBBB$ disks immediately apply to $PBB$ disks, and we will use this in the sequel.
For a surface $Z$ with non-meridianal boundary, consider a normal disk $Z_i$ that meets exactly four truncation edges, and no interior edges. Such a subsurface is a disk that corresponds to a $BBBB$ word. For an example, see \reffig{BBBBExample}, left. Similarly, a normal disk that meets exactly two truncation edges and exactly two interior edges corresponds to a $BBSS$ word. An example is shown in \reffig{BBBBExample}, right.
\begin{figure}
\caption{Left: an example of a normal disk of type $BBBB$. Right: an example of type $BBSS$.}
\label{Fig:BBBBExample}
\end{figure}
As described in \cite{HTT2}, topologically the label $B$ means that $\partial Z_i$ meets the link boundary torus. An arc $BB$ can be a part of $\partial Z$, where $\partial Z$ is on the link torus. In a chunk decomposition, such an arc travels between two truncation edges in a truncation face. An arc $BB$ can also connect two $BB$ arcs of the previous type, and then it lies in an interior face of a chunk. Each $BBBB$ disk is hence a quadrilateral with two opposite sides on truncation faces and the other pair of opposite sides on interior faces. Since a $BB$-arc that lies in a truncation face is a part of $\partial Z$, it is not glued to any other arc of $\partial Z_i$, for any $i$. On the other hand, a $BB$-arc that lies inside an interior face, as well as a $BS$-arc, is not a part of $\partial Z$, and is rather in the interior of $Z$. Such an arc therefore must be glued to a similar arc of some $Z_j$.
We say that two $BBBB$ disks are \emph{connected} if they share a common arc on an interior face. A collection of such disks is \emph{connected} if every disk is glued to another one in the collection along common arcs. A collection of connected disks is \emph{maximal} if it is not a strict subset of a connected collection. Two $BBSS$ disks are \emph{connected} if they share a common arc, and a collection of such disks is \emph{connected} if they are glued end to end along common arcs.
\begin{lemma}\label{Lem:BBBB} Assume that the hat-representativity $\hat{r}(\pi(L),\Pi)>4$, and that $\pi(L)$ is not a string of bigons on $\Pi$. Let $Z$ be in normal position, with a non-meridianal boundary component. Then a maximal connected collection of \begin{enumerate} \item $BBBB$ disks for $Z$ or \item $BBSS$ disks for $Z$ \end{enumerate} forms a disk. \end{lemma}
\begin{proof} As explained above, the $BBBB$ or $BBSS$ disks are connected only along arcs on interior faces, since the arcs on truncation faces are a part of the surface boundary $\partial Z$.
For (1), a maximal connected collection is therefore glued end to end along opposite interior faces. The only way it could not form a disk is if it forms an essential annulus, made entirely of $BBBB$ disks. But in this case, it was shown in \cite[Theorem~4.6]{HowiePurcell} that $\pi(L)$ is a string of bigons on $\Pi$ (here we use the hypothesis $\hat{r}(\pi(L),\Pi)>4$).
For (2), arcs in the interior faces have the form $BS$ or $SS$, and they must be glued to arcs of the form $BS$ or $SS$ respectively. All arcs $BB$ lie on truncation faces, and are unglued, because they lie on $\partial Z$. If the connected collection of $BBSS$ disks is not a disk, it will contain as a subset a normal annulus made up of $BBSS$ disks, at least one of which is parallel into $\Pi$ because $\hat{r}(\pi(L),\Pi)>4$. But then by~\cite[Lemma~4.5]{HowiePurcell}, the diagram $\pi(L)$ is a string of bigons on $\Pi$, contradicting our hypotheses. \end{proof}
\begin{proposition}\label{Prop:NoBBBBandBBSS} For a surface $Z$ in normal position, no $BBBB$ region is connected to a $BBSS$ region. \end{proposition}
\begin{proof} Two normal subsurfaces $Z_i, Z_j$ in a chunk can only be connected across arcs in interior faces, not across arcs in truncation faces, which are left unglued and become the boundary of the normal surface. The arcs in the interior faces for a $BBBB$ region have both endpoints on a truncation edge. The arcs in the interior faces for a $BBSS$ region have one end on a truncation edge and one end on an interior edge, or both ends on interior edges. Because interior edges glue to interior edges, each arc of a $BBSS$ region must glue to an arc with an endpoint on an interior edge. Thus it cannot glue to a $BBBB$ region. \end{proof}
\begin{proposition}\label{Prop:MaxBBBBDiffersBBSS} Assume that the hat-representativity $\hat{r}(\pi(L),\Pi)>4$, and that $\pi(L)$ is not a string of bigons on $\Pi$. For a surface $Z$ in normal position, the boundary of a maximal connected collection of $BBBB$ disks is a closed curve labeled $BB\dots B$. The boundary of a maximal connected collection of $BBSS$ disks is a closed curve with at least one $S$ label. Thus the boundary curves of a maximal connected collection of $BBBB$ disks cannot agree with those of a maximal connected collection of $BBSS$ disks. \end{proposition}
\begin{proof} Because all words for disks in a maximal connected collection of $BBBB$ disks are in letters $B$, the word for the boundary of such maximal region also has only letters $B$.
As for a maximal connected collection of $BBSS$ disks, any $BBSS$ disks can only be glued along edges labeled $BS$ or edges labeled $SS$, as the $BB$ edges are located in truncation faces. If the boundary component of a maximal connected collection of $BBSS$ region is labeled only with $B$'s, then there is a sequence of $BBSS$ regions glued end to end along their $BS$ edges, with their $BB$ edges on the boundary of the connected collection. But this must form an annulus with the other boundary component labeled only with $S$. Because $\hat{r}(\pi(L),\Pi)>4$, one of these disks will be parallel to $\Pi$. Then~\cite[Lemma~4.5]{HowiePurcell} implies that this is impossible unless the diagram is a string of bigons. \end{proof}
\begin{theorem}\label{Thm:NonBBBBdeterminesZ} Assume that the hat-representativity $\hat{r}(\pi(L),\Pi)>4$, and $\pi(L)$ is not a string of bigons on $\Pi$. Let $Z$ be in normal form. Then all subsurfaces $Z_i$ that are neither $BBBB$ disks nor $BBSS$ disks, together with the link $L$, determine the surface $Z$ up to isotopy. \end{theorem}
\begin{proof} By \reflem{BBBB}~(1), maximal connected collections of $BBBB$ disks form disjoint disks. Similarly for $BBSS$ disks by \reflem{BBBB}~(2). By \refprop{NoBBBBandBBSS}, no such disks are connected. Therefore, if we remove maximal connected collections of $BBBB$ and $BBSS$ regions, we are left with a surface with a number of disjoint disks removed.
Suppose $Z'$ is another surface in normal form, whose normal subsurfaces agree with those of $Z$ away from $BBBB$ and $BBSS$ disks. Then $Z$ and $Z'$ are both obtained by adding disks to the same surface with boundary, along the same boundary components. The link $L$ has irreducible complement by~\cite[Theorem~3.14]{HowiePurcell}. Therefore, the two surfaces disagree except in disks that can be isotoped to bound balls. Thus the surfaces are isotopic. \end{proof}
\section{Essential Conway spheres}\label{Sec:ConwaySpheres}
In this section, we apply our results above to generalize the work of Menasco~\cite{Menasco1984} that allows the characterization of essential 4-punctured spheres, or essential Conway spheres, in alternating links as ``visible'' and ``hidden'', with terminology due to Thistlethwaite~\cite{Thistlethwaite}. We call an essential Conway sphere \textit{visible} if its intersection with $\Pi^{\pm}$ consists of only one $PPPP$ curve, and \textit{hidden} if consists of exactly two $PSPS$ curves. See \reffig{VisibleHidden} and compare to~\cite[Figure~3(i)-(ii)]{Thistlethwaite}.
Note that in the classical alternating case, the regions exterior to the dashed line in \reffig{VisibleHidden} will also be disks, hence will contain an alternating tangle. In the weakly generalized alternating setting, this region contains an arbitrary alternating diagram on a surface possibly with higher genus.
\begin{figure}
\caption{Left: a visible essential Conway sphere is made up of two disks labeled $PPPP$, one on either side of the projection surface. Right: a hidden Conway sphere is made up of four disks labeled $PSPS$, two on either side of the projection plane. Here we show how they meet one side of the diagram. The gray disk denotes a portion of the diagram contained in a disk in the projection surface. }
\label{Fig:VisibleHidden}
\end{figure}
\begin{theorem}\label{Thm:EssentialTangles} Let $L$ be a weakly generalized alternating link on a connected projection surface $\Pi$, with a cellular diagram (i.e.\ where all regions are disks), and hat-representativity $\hat{r}(\pi(L),\Pi)>4$. Then any essential Conway sphere has one of two forms: visible or hidden. \end{theorem}
We prove the theorem by considering how essential Conway spheres decompose in a chunk decomposition, ruling out certain subsurfaces and labels on their boundaries. The proof will follow from a few lemmas.
\begin{lemma}\label{Lem:4PunctToDisks} Under the hypotheses of \refthm{EssentialTangles}, suppose $Z=\bigcup Z_i$ is an essential Conway sphere in normal form with respect to the chunk decomposition. Then all the subsurfaces $Z_i$ must be disks. \end{lemma}
\begin{proof} An essential Conway sphere is an essential 4-punctured sphere, hence it will have combinatorial area $a(Z) = -2\pi\chi(S) = 4\pi$. Because it meets the knot in meridianal punctures, it will be in meridianal form. It decomposes into normal subsurfaces within chunks that are genus-zero surfaces. Also, on each of $\Pi^{\pm}$ it must meet four instances of the letter $P$, so eight instances of $P$ total.
The only genus-zero surfaces that might arise are disks, annuli, and 3-holed spheres; all other subsurfaces will contribute too much area by equations~\refeqn{SubArea} and~\refeqn{GB}.
In a cellular link diagram, no boundary component of a genus-zero surface in normal position can lie entirely in a face of the diagram. This means that each boundary component of an annulus or 3-holed sphere must meet instances of $S$ or $P$, and by \refthm{No3LetterWords}, it must meet at least two such instances. We will prove that any annulus or 3-holed sphere subsurface contributes too much area, and rule these out.
The smallest possible area contribution from an annulus has area $2\pi$, meeting two instances of $S$ on each boundary component. But we still need eight instances of the letter $P$, and the total area of all subsurfaces must add up to $4\pi$. Since each $P$ contributes area $\pi$, annuli have too much area. Similarly for 3-holed spheres. Thus all subsurfaces must be disks. \end{proof}
\begin{lemma}\label{Lem:4PunctDiskBdry} Under the hypotheses of \refthm{EssentialTangles}, suppose $Z=\bigcup Z_i$ is an essential Conway sphere in normal form with respect to the chunk decomposition. Then each of the disks $Z_i$ must be labeled $SSSS$, $PSSS$, $PPSS$, $PSPS$, $PPPS$, or $PPPP$. \end{lemma}
\begin{proof} Consider the possible boundary words for disks with area at most $2\pi$.
Lemma~\ref{Lem:0AreaEnumerated}, along with the results of \refsec{PPPP}, rules out all zero area disks $PP$ and $PSS$. Any odd number of letters is ruled out by \refthm{No3LetterWords}. If the representativity of the diagram is 4, then 0-area $SSSS$ disks are possible, not ruled out by \refthm{NoSSSS}, although they will be ruled out for higher representativity.
The only disk contributing area $\pi/2$ has boundary $PSSS$. The possibilities contributing area $\pi$ are $PPSS$, $PSPS$, and $SSSSSS$. Those contributing area $3\pi/2$ are $PPPS$ and $PSSSSS$. Those contributing area $2\pi$ include $PPPP$, as well as other words with at most three instances of $P$ and some instances of $S$.
Each instance of $P$ in the disks enumerated above contributes at least $\pi/2$ to the sum. Using the fact that there must be eight instances of $P$ in total, in fact each of the disks with positive area must contain a letter $P$ and the area of the disk will be $\pi/2$ times the number of instances of $P$. Because there must be four instances of $P$ on each side of $\Pi$, the only possibilities for disks overall are 0-area $SSSS$ disks, and positive area $PSSS$, $PPSS$, $PSPS$, $PPPS$, and $PPPP$ disks. \end{proof}
By the assumption that $\hat{r}(\pi(L),\Pi)>4$, the boundaries of compression disks on one side of $\Pi$ meet the diagram in more than four points. So on this side of $\Pi$, any $PSSS$, $PPSS$, or $SSSS$ curve bounds a disk parallel to the projection surface. Without loss of generality, say this side is $\Pi^+$. We note that unlike in the classical case of alternating links in $S^3$, here the proofs for $\Pi^+$ and $\Pi^-$ from this moment are not analogous.
We say that a $PPSS$ or $PSSS$ disk is \emph{innermost} on $\Pi^+$ if the boundary curve bounds a disk on the surface $\Pi^+$ that contains no other intersections with $Z$. We say it is \emph{outermost} if it bounds a disk $D$ on $\Pi^+$ such that $\Pi^+-D$ is a surface with boundary containing no intersections with $Z$.
\begin{lemma}\label{Lem:PSSSInnerOuter} Under the hypotheses of \refthm{EssentialTangles}, if there exists a $PPSS$ disk or a $PSSS$ disk with boundary on $\Pi^+$, then there exists one that is innermost or one that is outermost with boundary on $\Pi^+$. \end{lemma}
\begin{proof} Suppose first that there is a $PPSS$ disk with boundary on $\Pi^+$. Because any disk with boundary meeting $\Pi^+$ at most four times is parallel to $\Pi^+$ (by the assumption on $\hat{r}$), we may assume the $PPSS$ curve bounds a disk in $\Pi^+$. Since there are four instances of $P$ on each side of $\Pi$, there will either be another $PPSS$
disk, or a $PSPS$ disk, or two $PSSS$ disks with boundary on $\Pi^+$ by Lemma~\ref{Lem:4PunctDiskBdry}. In the first two cases, the second disk lies on one side of the $PPSS$ disk and so there are no curves $\partial Z_i$ on the other side. In the last case, there may be a $PSSS$ disk on both sides, but then such a $PSSS$ disk is innermost. This proves the lemma when there is a $PPSS$ disk.
So now suppose there is a $PSSS$ disk with boundary on $\Pi^+$. Again, its boundary curve encloses a disk on $\Pi^+$. Then within the same chunk there could be a single $PPPS$ disk, which means the $PSSS$ disk is innermost or outermost. There could be another $PSSS$ disk and a disk with two instances of $P$ (namely $PSPS$ or $PPSS$), in which case one of the two $PSSS$ disks will be innermost or outermost. Or there could be three additional $PSSS$ disks, in which case one is innermost or outermost.
\end{proof}
The following lemma can be viewed as an extension of a lemma of Menasco~\cite[Lemma~2]{Menasco1984} in our setting.
\begin{lemma}\label{Lem:NoPSSSorPPSS} Under the hypotheses of \refthm{EssentialTangles}, suppose $Z=\bigcup Z_i$ is an essential Conway sphere in normal form with respect to the chunk decomposition. Then there are no $PSSS$, $PPSS$, $PPPS$, or $SSSS$ disks. \end{lemma}
\begin{proof} We will first rule out $PPSS$ and $PSSS$ disks with their boundaries in $\Pi^+$. Suppose there is such a disk. Then \reflem{PSSSInnerOuter} implies that there is either an innermost or outermost such disk.
The portion of the boundary in the $PPSS$ or $PSSS$ disk that runs between two instances of $S$ must lie in a single region of the link diagram; here we are using the fact that the edges and ideal vertices on $\Pi^+$ come from $\pi(L)$ as in Subsection~\ref{Sec:ChunkDecomp}. Because the diagram is alternating, it must meet the first instance of $S$ in a saddle with the link on one side, and in the second instance the link lies on the other side; see \reffig{NoPSSS} (a) and (b).
\begin{figure}
\caption{The $SS$ portion of a $PSSS$ or $PPSS$ disk, shown in (a) with saddles in the link complement, and in (b) on the chunk decomposition. In (c) and (d), in fact, it must be a $PSSS$ disk, either with $P$ away from the arc between identified edges as in (c), or with $P$ meeting this arc as in (d).}
\label{Fig:NoPSSS}
\end{figure}
Each instance of $S$ lies on an interior edge of a chunk that is glued to another interior edge. Hence there must be at least one other curve of $\partial Z_i$ running through that glued edge, as shown in \reffig{NoPSSS} (b). But this gives two curves lying on opposite sides of the $PSSS$ or $PPSS$ curve that we consider. Because our disk is innermost or outermost, this is possible only if one of these two curves is actually still part of our original $PPSS$ or $PSSS$ boundary. If the arc of the $PSSS$ disk between the two identified interior edges cuts off a disk on the projection surface $\Pi$ with those edges, then we have a contradiction to \refprop{TruncationEdgesPairs}(4). This was Menasco's argument in~\cite{Menasco1984}. In our setting, it may not be the case that this arc cuts of a disk with the edge, and so \refprop{TruncationEdgesPairs}(4) may not apply.
So assume that the arc does not cut off a disk on $\Pi$. There is a third instance of $S$ on the boundary curve. It follows that the original disk is a $PSSS$ disk.
Suppose first that the instance of $P$ does not lie between the two instances of $S$ on identified edges. Hence there is an arc, say $R$, on the boundary of the $PSSS$ disk meeting a single (disk) face of the chunk decomposition, with endpoints on interior edges that are identified under the gluing. Denote the crossing of $\pi(L)$ that correpsonds to these interior edges by $O$. By assumption, the disk on $\Pi$ bounded by the $PSSS$ curve does not contain the crossing $O$, or else the arc $R$ would cut off a disk on $\Pi$ that contradicts \refprop{TruncationEdgesPairs}(4). Because of checkerboard coloring, the endpoints of the arc $R$ must lie on opposite sides of the crossing $O$, as shown in \reffig{NoPSSS}(c). But then the union of this arc $R$ and another arc running between the endpoints of $R$ through the crossing $O$ forms a closed curve on the surface $\Pi$ that meets the arc $R$ only once (after slight isotopy). This curve is shown by the thin green line in \reffig{NoPSSS}(c). This is impossible: either this curve meets the boundary of the $PSSS$ disk only once on $\Pi$, which is impossible for a disk boundary, or it meets it again as the $PSSS$ disk boundary exits through one of the identified edges at the crossing $O$. But in the latter case, the disk bounded by the $PSSS$ curve in $\Pi$ would then again give a contradiction to \refprop{TruncationEdgesPairs}(4).
So the instance of $P$ lies between these two instances of $S$; see \reffig{NoPSSS}(d). However, then consider the two arcs between the two identified instances of $S$ and the $P$, and in particular, the arcs that are identified to them on $\Pi^-$. Because the surface $Z$ is in meridianal form, the instance of $P$ on $\Pi^+$ is glued across truncation edges to an instance of $P$ in $\Pi^-$; see \reffig{MeridianalForm}. In particular, the arc in $\Pi^+$ shown in dashed lines in \reffig{NoPSSS}(b), running from $S$ to $P$ and back to an identified $S$, is mapped to a single arc in $\Pi^-$ running from some $S$ to $P$ and back to some $S$. However, on $\Pi^-$, the two endpoints of that arc must be identified together; this is due to the fact that the gluing of chunks rotates the boundary of each face in the clockwise direction in white faces, and the counterclockwise direction in shaded faces. Thus this arc in $\Pi^+$ is identified to a closed curve in $\Pi^-$. It must be the boundary of a disk $Z_i$ by \reflem{4PunctToDisks}. Then this is a $PS$ disk meeting $\Pi^-$. But by \refthm{No2LetterWords}, there are no $PS$ disks, and we have a contradiction. It follows that there are no $PPSS$ or $PSSS$ disks with boundaries in $\Pi^+$.
Now consider $SSSS$ disks. There is no such disk on $\Pi^+$, by \refthm{NoSSSS} and the fact that any compressing disk meets $\Pi^+$ more than four times. Observe that we have now ruled out $SSSS$, $PPSS$, and $PSSS$ disks on $\Pi^+$. By \reflem{4PunctDiskBdry}, there are no further options for disks with two adjacent instances of $S$.
Since the portion of $\partial Z_i$ running between two instances of $S$ must be glued to another boundary curve running between two instances of $S$, there cannot be a disk with two adjacent instances of $S$ on the other side of $\Pi$ as well. This rules out $SSSS$, $PSSS$, and $PPSS$ disks on $\Pi^-$.
Finally, since there are no $PSSS$ disks, there can be no $PPPS$ disks, because there must be exactly four instances of $P$ on each side of the projection surface, and there are no options for disks meeting only one instance of $P$ to pair with $PPPS$. \end{proof}
\begin{proof}[Proof of \refthm{EssentialTangles}] Suppose $Z$ is an essential Conway sphere. By \reflem{4PunctToDisks}, it meets the diagram in normal disks. By \reflem{4PunctDiskBdry} and \reflem{NoPSSSorPPSS}, the only possible disks are labeled $PPPP$ or $PSPS$.
In the case of a $PPPP$ disk, the fact that there must be four instances of $P$ on each side of $\Pi$ implies that there are exactly two $PPPP$ disks, lying on opposite sides of the projection surface. They cut off a visible essential Conway sphere.
In the case of a $PSPS$ disk, there must be four such disks, with one $PSPS$ disk meeting another across each saddle $S$ on the same side of the projection surface. It must additionally meet another two disks at the saddle $S$ on the opposite side of the projection surface. The only possibility is that the four disks fit together with two on one side as in \reffig{VisibleHidden} right. This is a hidden essential Conway sphere. \end{proof}
\end{document} | arXiv |
Fernique's theorem
Fernique's theorem is a result about Gaussian measures on Banach spaces. It extends the finite-dimensional result that a Gaussian random variable has exponential tails. The result was proved in 1970 by Xavier Fernique.
Statement
Let (X, || ||) be a separable Banach space. Let μ be a centred Gaussian measure on X, i.e. a probability measure defined on the Borel sets of X such that, for every bounded linear functional ℓ : X → R, the push-forward measure ℓ∗μ defined on the Borel sets of R by
$(\ell _{\ast }\mu )(A)=\mu (\ell ^{-1}(A)),$
is a Gaussian measure (a normal distribution) with zero mean. Then there exists α > 0 such that
$\int _{X}\exp(\alpha \|x\|^{2})\,\mathrm {d} \mu (x)<+\infty .$
A fortiori, μ (equivalently, any X-valued random variable G whose law is μ) has moments of all orders: for all k ≥ 0,
$\mathbb {E} [\|G\|^{k}]=\int _{X}\|x\|^{k}\,\mathrm {d} \mu (x)<+\infty .$
References
• Fernique, Xavier (1970). "Intégrabilité des vecteurs gaussiens". Comptes Rendus de l'Académie des Sciences, Série A-B. 270: A1698–A1699. MR0266263
• Da Prato, Giuseppe; Zabczyk, Jerzy (1992). Stochastic equations in infinite dimension. Cambridge University Press. Theorem 2.7. ISBN 0-521-38529-6.
| Wikipedia |
\begin{document}
\makeatletter \definecolor{brick}{RGB}{204,0,0} \def\@cite#1#2{{
\m@th\upshape\mdseries[{\small\sffamily #1}{\if@tempswa, \small\sffamily
\color{brick} #2\fi}]}} \newcommand{\sandor}{{\color{blue}{S\'andor \mdyydate\today}}} \newenvironment{refmr}{}{} \newcommand{\biblio}{
}
\setitemize[1]{leftmargin=*,parsep=0em,itemsep=0.125em,topsep=0.125em}
\newcommand\james{M\hskip-.1ex\raise .575ex \hbox{\text{c}}\hskip-.075ex Kernan}
\renewcommand\thesubsection{\thesection.\Alph{subsection}}
\renewcommand\subsection{
\renewcommand{phv}{phv}
\@startsection{subsection}
{2}{0pt}{-\baselineskip}{.2\baselineskip}{\raggedright
\sffamily\itshape\small
}} \renewcommand\section{
\renewcommand{phv}{phv}
\@startsection{section}
{1}{0pt}{\baselineskip}{.2\baselineskip}{\centering
\sffamily
\scshape
}}
\setlist[enumerate]{itemsep=3pt,topsep=3pt,leftmargin=2em,label={\rm (\roman*)}}
\newlist{enumalpha}{enumerate}{1} \setlist[enumalpha]{itemsep=3pt,topsep=3pt,leftmargin=2em,label=(\alph*\/)}
\newcounter{parentthmnumber} \setcounter{parentthmnumber}{0} \newcounter{currentparentthmnumber} \setcounter{currentparentthmnumber}{0} \newcounter{nexttag} \newcommand{\setnexttag}{
\setcounter{nexttag}{\value{enumi}}
\addtocounter{nexttag}{1} } \newcommand{\placenexttag}{ \tag{\roman{nexttag}} }
\newenvironment{thmlista}{ \label{parentthma} \begin{enumerate} }{ \end{enumerate} } \newlist{thmlistaa}{enumerate}{1} \setlist[thmlistaa]{label=(\arabic*), ref=\autoref{parentthm}\thethm(\arabic*)}
\newcommand*{\parentthmlabeldef}{
\expandafter\newcommand
\csname parentthm\the\value{parentthmnumber}\endcsname } \newcommand*{\ptlget}[1]{
\romannumeral-`\x
\ltx@ifundefined{parentthm\number#1}{
\ltx@space
\parentthmundefined
}{
\expandafter\ltx@space
\csname mymacro\number#1\endcsname
} } \newcommand*{\parentthmundefined}{\textbf{??}} \parentthmlabeldef{parentthm}
\newenvironment{thmlistr}{ \label{parentthm}
\begin{thmlistrr}}{ \end{thmlistrr}} \newlist{thmlistrr}{enumerate}{1} \setlist[thmlistrr]{label=(\roman*), ref=\autoref{parentthm}(\roman*)}
\newcounter{proofstep} \setcounter{proofstep}{0} \newcommand{\pstep}[1]{
\noindent
\emph{{\sc Step \arabic{proofstep}:} #1.}\addtocounter{proofstep}{1}}
\newcounter{lastyear}\setcounter{lastyear}{\the\year} \addtocounter{lastyear}{-1}
\newcommand\sideremark[1]{ \normalmarginpar \marginpar [ \hskip .45in \begin{minipage}{.75in} \tiny #1 \end{minipage} ] { \hskip -.075in \begin{minipage}{.75in} \tiny #1 \end{minipage} }} \newcommand\rsideremark[1]{ \reversemarginpar \marginpar [ \hskip .45in \begin{minipage}{.75in} \tiny #1 \end{minipage} ] { \hskip -.075in \begin{minipage}{.75in} \tiny #1 \end{minipage} }}
\newcommand\Index[1]{{#1}\index{#1}} \newcommand\inddef[1]{\emph{#1}\index{#1}} \newcommand\noin{\noindent} \newcommand\hugeskip{
} \newcommand\smc{\sc} \newcommand\dsize{\displaystyle} \newcommand\sh{\subheading} \newcommand\nl{\newline}
\newcommand\get{\input /home/kovacs/tex/latex/} \newcommand\Get{\Input /home/kovacs/tex/latex/} \newcommand\toappear{\rm (to appear)} \newcommand\mycite[1]{[#1]} \newcommand\myref[1]{(\ref{#1})} \newcommand{\parref}[1]{\eqref{\bf #1}} \newcommand\myli{
\newline
\noindent{$\bullet$}\quad} \newcommand\vol[1]{{\bf #1}\ } \newcommand\yr[1]{\rm (#1)\ }
\newcommand\cf{cf.\ \cite} \newcommand\mycf{cf.\ \mycite} \newcommand\te{there exist\xspace} \newcommand\st{such that\xspace} \newcommand\CM{Cohen-Macaulay\xspace}
\newcommand\myskip{3pt} \newtheoremstyle{bozont}{3pt}{3pt}
{\itshape}
{}
{\bfseries}
{.}
{.5em}
{\thmname{#1}\thmnumber{ #2}\thmnote{ #3}}
\newtheoremstyle{bozont-sub}{3pt}{3pt}
{\itshape}
{}
{\bfseries}
{.}
{.5em}
{\thmname{#1}\ \arabic{section}.\arabic{thm}.\thmnumber{#2}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-named-thm}{3pt}{3pt}
{\itshape}
{}
{\bfseries}
{.}
{.5em}
{\thmname{#1}\thmnumber{#2}\thmnote{ #3}}
\newtheoremstyle{bozont-named-bf}{3pt}{3pt}
{}
{}
{\bfseries}
{.}
{.5em}
{\thmname{#1}\thmnumber{#2}\thmnote{ #3}}
\newtheoremstyle{bozont-named-sf}{3pt}{3pt}
{}
{}
{\sffamily}
{.}
{.5em}
{\thmname{#1}\thmnumber{#2}\thmnote{ #3}}
\newtheoremstyle{bozont-named-sc}{3pt}{3pt}
{}
{}
{\scshape}
{.}
{.5em}
{\thmname{#1}\thmnumber{#2}\thmnote{ #3}}
\newtheoremstyle{bozont-named-it}{3pt}{3pt}
{}
{}
{\itshape}
{.}
{.5em}
{\thmname{#1}\thmnumber{#2}\thmnote{ #3}}
\newtheoremstyle{bozont-sf}{3pt}{3pt}
{}
{}
{\sffamily}
{.}
{.5em}
{\thmname{#1}\thmnumber{ #2}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-sc}{3pt}{3pt}
{}
{}
{\scshape}
{.}
{.5em}
{\thmname{#1}\thmnumber{ #2}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-remark}{3pt}{3pt}
{}
{}
{\scshape}
{.}
{.5em}
{\thmname{#1}\thmnumber{ #2}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-subremark}{3pt}{3pt}
{}
{}
{\scshape}
{.}
{.5em}
{\thmname{#1}\ \arabic{section}.\arabic{thm}.\thmnumber{#2}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-def}{3pt}{3pt}
{}
{}
{\bfseries}
{.}
{.5em}
{\thmname{#1}\thmnumber{ #2}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-reverse}{3pt}{3pt}
{\itshape}
{}
{\bfseries}
{.}
{.5em}
{\thmnumber{#2.}\thmname{ #1}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-reverse-sc}{3pt}{3pt}
{\itshape}
{}
{\scshape}
{.}
{.5em}
{\thmnumber{#2.}\thmname{ #1}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-reverse-sf}{3pt}{3pt}
{\itshape}
{}
{\sffamily}
{.}
{.5em}
{\thmnumber{#2.}\thmname{ #1}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-remark-reverse}{3pt}{3pt}
{}
{}
{\sc}
{.}
{.5em}
{\thmnumber{#2.}\thmname{ #1}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-def-reverse}{3pt}{3pt}
{}
{}
{\bfseries}
{.}
{.5em}
{\thmnumber{#2.}\thmname{ #1}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-def-newnum-reverse}{3pt}{3pt}
{}
{}
{\bfseries}
{}
{.5em}
{\thmnumber{#2.}\thmname{ #1}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-def-newnum-reverse-plain}{3pt}{3pt}
{}
{}
{}
{}
{.5em}
{\thmnumber{\!(#2)}\thmname{ #1}\thmnote{ \rm #3}}
\newtheoremstyle{bozont-number}{3pt}{3pt}
{}
{}
{}
{}
{0pt}
{\thmnumber{\!(#2)} }
\newtheoremstyle{bozont-step}{3pt}{3pt}
{\itshape}
{}
{\scshape}
{}
{.5em}
{$\boxed{\text{\sc \thmname{#1}~\thmnumber{#2}:\!}}$}
\theoremstyle{bozont}
\ifnum \value{are-there-sections}=0 {
\basetheorem{proclaim}{Theorem}{} } \else {
\basetheorem{proclaim}{Theorem}{section} } \fi
\maketheorem{thm}{proclaim}{Theorem} \maketheorem{mainthm}{proclaim}{Main Theorem} \maketheorem{cor}{proclaim}{Corollary} \maketheorem{cors}{proclaim}{Corollaries} \maketheorem{lem}{proclaim}{Lemma} \maketheorem{prop}{proclaim}{Proposition} \maketheorem{conj}{proclaim}{Conjecture} \basetheorem{subproclaim}{Theorem}{proclaim} \maketheorem{sublemma}{subproclaim}{Lemma} \newenvironment{sublem}{ \setcounter{sublemma}{\value{equation}} \begin{sublemma}} {\end{sublemma}}
\theoremstyle{bozont-sub}
\maketheorem{subthm}{equation}{Theorem} \maketheorem{subcor}{equation}{Corollary}
\maketheorem{subprop}{equation}{Proposition} \maketheorem{subconj}{equation}{Conjecture}
\theoremstyle{bozont-named-thm} \maketheorem{namedthm}{proclaim}{}
\theoremstyle{bozont-sc} \newtheorem{proclaim-special}[proclaim]{\specialthmname} \newenvironment{proclaimspecial}[1]
{\def\specialthmname{#1}\begin{proclaim-special}}
{\end{proclaim-special}}
\theoremstyle{bozont-subremark}
\maketheorem{subrem}{equation}{Remark} \maketheorem{subnotation}{equation}{Notation} \maketheorem{subassume}{equation}{Assumptions} \maketheorem{subobs}{equation}{Observation} \maketheorem{subexample}{equation}{Example} \maketheorem{subex}{equation}{Exercise} \maketheorem{inclaim}{equation}{Claim}
\maketheorem{subquestion}{equation}{Question}
\theoremstyle{bozont-remark} \basetheorem{subremark}{Remark}{proclaim} \makeremark{subclaim}{subremark}{Claim} \maketheorem{rem}{proclaim}{Remark} \maketheorem{claim}{proclaim}{Claim} \maketheorem{notation}{proclaim}{Notation} \maketheorem{assume}{proclaim}{Assumptions} \maketheorem{obs}{proclaim}{Observation} \maketheorem{example}{proclaim}{Example} \maketheorem{examples}{proclaim}{Examples} \maketheorem{complem}{equation}{Complement} \maketheorem{const}{proclaim}{Construction} \maketheorem{ex}{proclaim}{Exercise} \newtheorem{case}{Case} \newtheorem{subcase}{Subcase} \newtheorem{step}{Step} \newtheorem{approach}{Approach} \maketheorem{Fact}{proclaim}{Fact} \newtheorem{fact}{Fact}
\newtheorem*{SubHeading*}{\SubHeadingName} \newtheorem{SubHeading}[proclaim]{\SubHeadingName} \newtheorem{sSubHeading}[equation]{\sSubHeadingName} \newenvironment{demo}[1] {\def\SubHeadingName{#1}\begin{SubHeading}}
{\end{SubHeading}} \newenvironment{subdemo}[1]{\def\sSubHeadingName{#1}\begin{sSubHeading}}
{\end{sSubHeading}} \newenvironment{demo-r}[1]{\def\SubHeadingName{#1}\begin{SubHeading-r}}
{\end{SubHeading-r}} \newenvironment{demor}[1]{\def\SubHeadingName{#1}\begin{SubHeading-r}}
{\end{SubHeading-r}} \newenvironment{subdemo-r}[1]{\def\sSubHeadingName{#1}\begin{sSubHeading-r}}
{\end{sSubHeading-r}} \newenvironment{demo*}[1]{\def\SubHeadingName{#1}\begin{SubHeading*}}
{\end{SubHeading*}}
\maketheorem{defini}{proclaim}{Definition} \maketheorem{defnot}{proclaim}{Definitions and notation} \maketheorem{question}{proclaim}{Question} \maketheorem{terminology}{proclaim}{Terminology} \maketheorem{crit}{proclaim}{Criterion} \maketheorem{pitfall}{proclaim}{Pitfall} \maketheorem{addition}{proclaim}{Addition} \maketheorem{principle}{proclaim}{Principle}
\maketheorem{condition}{proclaim}{Condition} \maketheorem{exmp}{proclaim}{Example} \maketheorem{hint}{proclaim}{Hint} \maketheorem{exrc}{proclaim}{Exercise} \maketheorem{prob}{proclaim}{Problem} \maketheorem{ques}{proclaim}{Question} \maketheorem{alg}{proclaim}{Algorithm} \maketheorem{remk}{proclaim}{Remark} \maketheorem{note}{proclaim}{Note} \maketheorem{summ}{proclaim}{Summary} \maketheorem{notationk}{proclaim}{Notation} \maketheorem{warning}{proclaim}{Warning} \maketheorem{defn-thm}{proclaim}{Definition--Theorem} \maketheorem{convention}{proclaim}{Convention} \maketheorem{hw}{proclaim}{Homework} \maketheorem{hws}{proclaim}{\protect{${\mathbb\star}$}Homework}
\newtheorem*{ack}{Acknowledgment} \newtheorem*{acks}{Acknowledgments}
\theoremstyle{bozont-number} \newtheorem{say}[proclaim]{} \newtheorem{subsay}[equation]{} \theoremstyle{bozont-def} \maketheorem{defn}{proclaim}{Definition} \maketheorem{subdefn}{equation}{Definition}
\theoremstyle{bozont-reverse}
\maketheorem{corr}{proclaim}{Corollary} \maketheorem{lemr}{proclaim}{Lemma} \maketheorem{propr}{proclaim}{Proposition} \maketheorem{conjr}{proclaim}{Conjecture}
\theoremstyle{bozont-remark-reverse} \newtheorem{SubHeading-r}[proclaim]{\SubHeadingName} \newtheorem{sSubHeading-r}[equation]{\sSubHeadingName} \newtheorem{SubHeadingr}[proclaim]{\SubHeadingName} \newtheorem{sSubHeadingr}[equation]{\sSubHeadingName}
\theoremstyle{bozont-reverse-sc} \newtheorem{proclaimr-special}[proclaim]{\specialthmname} \newenvironment{proclaimspecialr}[1] {\def\specialthmname{#1}\begin{proclaimr-special}} {\end{proclaimr-special}}
\theoremstyle{bozont-remark-reverse} \maketheorem{remr}{proclaim}{Remark} \maketheorem{subremr}{equation}{Remark} \maketheorem{notationr}{proclaim}{Notation} \maketheorem{assumer}{proclaim}{Assumptions} \maketheorem{obsr}{proclaim}{Observation} \maketheorem{exampler}{proclaim}{Example} \maketheorem{exr}{proclaim}{Exercise} \maketheorem{claimr}{proclaim}{Claim} \maketheorem{inclaimr}{equation}{Claim} \maketheorem{definir}{proclaim}{Definition}
\theoremstyle{bozont-def-newnum-reverse} \maketheorem{newnumr}{proclaim}{} \theoremstyle{bozont-def-newnum-reverse-plain} \maketheorem{newnumrp}{proclaim}{}
\theoremstyle{bozont-def-reverse} \maketheorem{defnr}{proclaim}{Definition} \maketheorem{questionr}{proclaim}{Question} \newtheorem{newnumspecial}[proclaim]{\specialnewnumname} \newenvironment{newnum}[1]{\def\specialnewnumname{#1}\begin{newnumspecial}}{\end{newnumspecial}}
\theoremstyle{bozont-step} \newtheorem{bstep}{Step}
\newcounter{thisthm} \newcounter{thissection}
\newcommand{\ilabel}[1]{
\newcounter{#1}
\setcounter{thissection}{\value{section}}
\setcounter{thisthm}{\value{proclaim}}
\label{#1}} \newcommand{\iref}[1]{
(\the\value{thissection}.\the\value{thisthm}.\ref{#1})}
\newcounter{lect} \setcounter{lect}{1} \newcommand\resetlect{\setcounter{lect}{1}\setcounter{page}{0}} \newcommand\lecture{
\centerline{\sffamily\bfseries Lecture \arabic{lect}} \addtocounter{lect}{1}} \newcounter{topic} \setcounter{topic}{1} \newenvironment{topic} {\noindent{\sc Topic \arabic{topic}:\ }}{\addtocounter{topic}{1}\par}
\counterwithin{equation}{proclaim} \counterwithin{figure}{section} \newcommand\equinsect{\numberwithin{equation}{section}} \newcommand\equinthm{\numberwithin{equation}{proclaim}} \newcommand\figinthm{\numberwithin{figure}{proclaim}} \newcommand\figinsect{\numberwithin{figure}{section}}
\newenvironment{sequation}{ \setcounter{equation}{\value{thm}} \numberwithin{equation}{section} \begin{equation} }{ \end{equation} \numberwithin{equation}{proclaim} \addtocounter{proclaim}{1} }
\newcommand{\arabic{section}.\arabic{proclaim}}{\arabic{section}.\arabic{proclaim}}
\newenvironment{pf}{
\noindent {\sc Proof. }}{\qed
}
\newenvironment{enumerate-p}{
\begin{enumerate}}
{\setcounter{equation}{\value{enumi}}\end{enumerate}} \newenvironment{enumerate-cont}{
\begin{enumerate}
{\setcounter{enumi}{\value{equation}}}}
{\setcounter{equation}{\value{enumi}}
\end{enumerate}}
\let\lenumi\labelenumi \newcommand{\rmlabels}{\renewcommand{\labelenumi}{\rm \lenumi}} \newcommand{\rmlabelsoff}{\renewcommand{\labelenumi}{\lenumi}}
\newenvironment{heading}{\begin{center} \sc}{\end{center}}
\newcommand\subheading[1]{
\noindent{{\bf #1.}\ }}
\newlength{\swidth} \setlength{\swidth}{\textwidth} \addtolength{\swidth}{-,5\parindent} \newenvironment{narrow}{
\noindent
\begin{minipage}{\swidth}}
{\end{minipage}
}
\newcommand\nospace{\hskip-.45ex} \newcommand{\sffamily\bfseries}{\sffamily\bfseries} \newcommand{\sffamily\bfseries\small}{\sffamily\bfseries\small} \newcommand{\textasciitilde}{\textasciitilde} \makeatother
\newcounter{stepp} \setcounter{stepp}{0} \newcommand{\nextstep}[1]{
\addtocounter{stepp}{1}
\begin{bstep}
{#1}
\end{bstep}
\noindent } \newcommand{\resetsteps}{\setcounter{stepp}{0}}
\title {Deformations of log canonical and $F$-pure singularities} \author{J\'anos Koll\'ar and S\'andor J Kov\'acs} \date{\today} \thanks{J\'anos Koll\'ar was supported in part by NSF Grant DMS-1362960. \\ \indent S\'andor Kov\'acs was supported in part by NSF Grant DMS-1565352 and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics.} \address{JK: Department of Mathematics, Princeton University, Fine
Hall, Washington Road, Princeton, NJ 08544-1000, USA} \email{[email protected]} \urladdr{http://www.math.princeton.edu/$\sim$kollar\xspace} \address{SK: University of Washington, Department of Mathematics, Box 354350, Seattle, WA 98195-4350, USA} \email{[email protected]\xspace} \urladdr{http://www.math.washington.edu/$\sim$kovacs\xspace}
\begin{abstract}
We introduce a lifting property for local cohomology, which leads to a unified
treatment of the dualizing complex for flat morphisms with semi-log-canonical,
Du~Bois or $F$-pure fibers. As a consequence we obtain that, in all 3 cases, the
cohomology sheaves of the relative dualizing complex are flat and commute with base
change. We also derive several consequences for deformations of
semi-log-canonical, Du~Bois and $F$-pure singularities. \end{abstract}
\maketitle \setcounter{tocdepth}{1} \tableofcontents
\newcommand{liftable local cohomology\xspace}{liftable local cohomology\xspace} \newcommand{$F$-pure\xspace}{$F$-pure\xspace} \newcommand{$F$-anti-nilpotent\xspace}{$F$-anti-nilpotent\xspace}
\section{Introduction}\label{sec:introduction}
\noindent One of the difficulties of higher dimensional birational geometry and moduli theory is that the occuring singularities are frequently not Cohen-Macaulay.
On a proper Cohen-Macaulay scheme we have a dualizing \emph{sheaf} $\omega_X$ and Serre duality. By contrast, on an arbitrary proper scheme we have a dualizing \emph{complex} $\dcx X$ and the isomorphism of Serre duality is replaced by a spectral sequence of Grothendieck duality.
The ``most important'' cohomology sheaf of the dualizing complex $\dcx X$ is \[ {\sf h}^{-\dim X}(\dcx X)\simeq \omega_X, \] and $X$ is Cohen-Macaulay if and only if the other cohomology sheaves ${\sf h}^{-i}(\dcx X)$ are all zero, cf.\ \cite[3.5.1]{Conrad00}. Thus these ${\sf h}^{-i}(\dcx X)$ measure ``how far'' $X$ is from being Cohen-Macaulay; see \autoref{prop:S_n-via-hi} for a more precise claim. Our main result implies that in flat families $X\to B$ with log canonical or $F$-pure fibers, the cohomology sheaves ${\sf h}^{-i}(\dcx{X/B})$ are flat over $B$ and commute with base change. In particular, being \CM is a deformation invariant property for such singularities. Note that for flat families \CM is always an open condition but usually not a closed one.
One of the puzzles of higher dimensional singularity theory is that while the singularities of the Minimal Model Program (log terminal, log canonical, Du~Bois, etc.) and of positive characteristic commutative algebra ($F$-pure, $F$-injective, etc.) are very closely related, the methods to study them are completely different. Here we isolate the following quite powerful common property for some of these classes.
\begin{defini}\label{def:liftable-cohom}
Let $A$ be a noetherian ring, and $(T,\mathfrak n)$ a noetherian local
$A$-algebra. We say that $T$ has \emph{liftable local cohomology\xspace over $A$} if for any noetherian
local $A$-algebra $(R,\mathfrak{m})$ and nilpotent ideal $I\subset R$ such that
$R/I\simeq T$, the natural morphism on local cohomology
\[
\xymatrix{
H^i_\mathfrak{m}(R) \ar@{->>}[r] & H^i_{\mathfrak n}(T) }
\]
is surjective for all $i$.
We say that $T$ has \emph{liftable local cohomology\xspace} if it has liftable local cohomology\xspace over $\mathbb{Z}$. \end{defini}
\begin{rem}\label{rem:inheriting-top-loc-cohs}
Notice that, using the above notation, if $\phi:A'\to A$ is a ring homomorphism
from another noetherian ring $A'$ then if $T$ has liftable local cohomology\xspace over $A'$, then it
also has liftable local cohomology\xspace over $A$. In particular, if $T$ has liftable local cohomology\xspace over $\mathbb{Z}$,
then it has liftable local cohomology\xspace over any noetherian ring $A$ justifying the above
terminology.
Furthermore, if $A=k$ is a field of characteristic $0$ then the notions of having
liftable local cohomology\xspace over $k$ and over $\mathbb{Z}$ are equivalent. This follows in one direction
by the above and in the other by the Cohen structure theorem
\cite[\href{https://stacks.math.columbia.edu/tag/032A}{Tag 032A}]{stacks-project}. \end{rem}
\begin{rem}
A closely related notion, a ring being \emph{cohomologically full}, is defined in
\cite{DSM18}. This notion and results of this article are used in
\cite{ConcaVarbaro18} to settle a conjecture by Herzog on ideals with square-free
initial ideals. \end{rem}
We prove in \autoref{DB.toploccohs} that {Du\thinspace\nolinebreak Bois}\xspace singularities have liftable local cohomology\xspace. On the Frobenius side, the right concept seems to be $F$-anti-nilpotent singularities, a notion introduced in \cite{MR2460693}, that lies between $F$-pure and $F$-injective \cite{MR3271179,ma_quy_2017}. We are very grateful to L.~Ma and K.~Schwede for pointing out that, by a result of Ma--Schwede--Shimomoto \cite{MSS17}, $F$-anti-nilpotent singularities also have liftable local cohomology\xspace over their ground field. We discuss this in \autoref{prop:F-anti-nilp-has-liftable-cohs}.
With this definition (cf.~\autoref{def:top-loc-cohs-for-schemes}), our main technical theorem is the following.
\begin{thm}[=~\autoref{thm:main-strong}]\label{thm:main.new}
Let $f:X\to B$ be a flat morphism of schemes that is essentially of finite type and
let $b\in B$
such that $X_{b}$ has liftable local cohomology\xspace over $B$. Then there exists an open neighborhood
$X_b\subset U\subset X$ such that ${\sf h}^{-i}(\dcx{U/B})$ is flat over $B$ and
commutes with base change for each $i\in\mathbb{Z}$. \end{thm}
\noin For applications the following consequences are especially important.
\begin{cor}\label{cor:main-db-implies-slc}
Let $f:X\to B$ be a flat morphism of schemes, essentially of finite type over
a field $k$. Let $b\in B$ be a point. Assume that
\begin{enumerate}
\item\label{item:7} either $\operatorname{char} k=0$ and $X_{b}$ is {Du\thinspace\nolinebreak Bois}\xspace, e.g.,
semi-log-canonical,
\item\label{item:8} or $\operatorname{char} k>0$ and $X_{b}$ is $F$-anti-nilpotent\xspace, e.g., $F$-pure\xspace.
\end{enumerate}
Then there exists an open neighborhood $X_b\subset U\subset X$ such that
${\sf h}^{-i}(\dcx{U/B})$ is flat over $B$ and commutes with base change for each
$i\in \mathbb{Z}$. \end{cor}
\autoref{cor:main-db-implies-slc}\autoref{item:7} can be viewed as a generalization of the following \autoref{cor:cm-defo-implies-cm}\autoref{item:19}, proved in \cite{MR2629988} for projective morphisms and in \cite{MSS17} in general (cf.~\cite{KS13}).
\begin{cor}\label{cor:cm-defo-implies-cm}
Let $(X,x)$ be a local scheme, essentially of finite type over a field and assume
that
\begin{enumerate}
\item\label{item:19} either $\operatorname{char} k=0$ and $X$ is {Du\thinspace\nolinebreak Bois}\xspace, e.g.,
semi-log-canonical,
\item\label{item:20} or $\operatorname{char} k>0$ and $X$ is $F$-anti-nilpotent\xspace, e.g., $F$-pure\xspace.
\end{enumerate}
If $(X,x)$ admits a flat deformation whose generic fiber is \CM then $(X,x)$ is
also \CM. \end{cor}
In many cases this is quite sharp, see \autoref{cone.exmp.5} and \autoref{thm:cm-defo-of-db} for some stronger versions. This also gives the following immediate corollary.
\begin{cor}[(cf.~\autoref{cor:cone-over-abelian-s3})]
\label{cor:cone-over-abelian-is-not-smoothable}
Let $X$ be an abelian variety of dimension at least $2$ defined over a field $k$.
If $\kar k>0$ assume that $X$ is ordinary. Then the cone over an arbitrary
projective embedding of $X$ is not smoothable. \end{cor}
Note that a special case of \autoref{cor:cone-over-abelian-is-not-smoothable} over $\mathbb{C}$, the non-smoothability of the projective cone over an abelian variety of dimension at least $2$, was proved in \cite{MR522037}. See \autoref{cor:cone-over-abelian-s3} for a stronger version.
\begin{demor}{The organization of the paper}
In \autoref{sec:examples} we give examples and show some applications of the main
results. In \autoref{sec:local-cohomology-db} we prove that a {Du\thinspace\nolinebreak Bois}\xspace local scheme has
\emph{liftable local cohomology\xspace}. In \autoref{sec:fp-sings} we recall a few basic notions about
singularities defined by the behaviour of the Frobenius morphism in positive
characteristic and recall that $F$-anti-nilpotent singularities\xspace have liftable local cohomology\xspace over
their ground field. In \autoref{sec:filtrations} and \autoref{sec:db-families-over}
we study infinitesimal deformations of schemes with liftable local cohomology\xspace and prove the main
result for families over Artinian bases. In \autoref{sec:flatness-base-change} we
prove a rather general flatness and base change criterion, see
\autoref{thm:generalized-flat-and-base-change}, which may be of independent
interest and derive \autoref{thm:main.new} as relatively easy consequences of this
and the results of \autoref{sec:db-families-over}. In
\autoref{sec:degenerations-cm-db} we prove a criterion for $S_n$ singularities\xspace in terms of
the dualizing complex, see \autoref{prop:S_n-via-hi}, and use this and
\autoref{thm:main.new} to prove \autoref{cor:cm-defo-implies-cm}. \end{demor}
\begin{demor}{Dualizing complex and its relatives}\label{def-and-not}
The \emph{(normalized) dualizing complex} of $X$ is denoted by $\omega_X^\kdot$
and if $X$ is of pure dimension $n$ the \emph{canonical sheaf} of $X$ is defined as
$\omega_X\colon\!\!\!= {\sf h}^{-n}(\omega_X^\kdot)$. Note that if $X$ is not normal, then
this is not necessarily the push-forward of the canonical sheaf from the
non-singular locus.
We will work with \emph{three} closely related, but generally different
\emph{objects}:
\begin{itemize}
\item the dualizing complex; $\dcx {X}$,
\item the canonical sheaf; $\omega_{X}= {\sf h}^{-n}(\omega_X^\kdot)$, and
\item the object defined by $\uldcx X\colon\!\!\!={\mcR\!}\sHom_X (\dbcx X, \dcx X)$.
\quad (See \autoref{sec:local-cohomology-db} for a description of $\dbcx X$).
\end{itemize}
Note that one has a natural morphism $\uldcx X\to \dcx X$ dual to
$\eta: \mathscr{O}_X\to \dbcx X$.
For a morphism $f:X\to B$, the \emph{(normalized) relative dualizing complex} of
$f$ will be denoted by $\dcx{X/B}$ and if $f$ has equidimensional fibers of
dimension $n$, then the \emph{relative canonical sheaf} of $f$ is
$\omega_{X/B}\colon\!\!\!= {\sf h}^{-n}(\dcx{X/B})$. If $B$ consists of a single (closed)
point, then these notions reduce to the ones discussed above. For more details on
relative dualizing complexes see
\cite[\href{http://stacks.math.columbia.edu/tag/0E2S}{Tag 0E2S}]{stacks-project} \end{demor}
\end{ack}
\section{Examples}\label{sec:examples}
\subsection{Characteristic zero} In this section we review rational, \CM, and {Du\thinspace\nolinebreak Bois}\xspace singularities of cones in characteristic zero and demonstrate some consequences of the main results.
\begin{exmp}[\emph{Deformations of cones}]\label{cone.exmp.5}
Let $X$ be a projective variety and $\mathscr{L}$ an ample line bundle on $X$. Assume for
simplicity that $X$ has rational singularities. Let
\[
C_a(X,\mathscr{L}):=\Spec_k \bigoplus_{r=0}^{\infty} H^0(X, \mathscr{L}^r)
\]
be the affine cone over $X$ with conormal bundle $\mathscr{L}$ and vertex $v$. Then the
singularity $v\in C_a(X,\mathscr{L})$ is
\begin{enumerate}[leftmargin=3.2em,label=(\ref{cone.exmp.5}.\arabic*)]
\item\label{item:21} rational $\Leftrightarrow$ $H^i(X, \mathscr{L}^r)=0$ for every
$i>0, r\geq 0$,
\item\label{item:22} \CM $\Leftrightarrow$ $H^i(X, \mathscr{L}^r)=0$ for every
$\dim X>i>0, r\geq 0$ and
\item\label{item:23} {Du\thinspace\nolinebreak Bois}\xspace $\Leftrightarrow$ $H^i(X, \mathscr{L}^r)=0$ for every $i>0, r> 0$;
\end{enumerate}
see \cite[3.11, 3.13]{SingBook} and \cite[2.5]{GK14} for proofs.
Let $D\subset X$ be an effective divisor with rational singularities such that
$\mathscr{L}\simeq \mathscr{O}_X(D)$. Set $\mathscr{L}_D:=\mathscr{L}|_D$. There is a natural morphism
$C_a(D, \mathscr{L}_D)\to C_a(X,\mathscr{L})$ which is an embedding if and only if
$ H^0(X, \mathscr{L}^r)\twoheadrightarrow H^0(D, \mathscr{L}^{r}_D)$ is surjective for every $r\geq 0$,
equivalently, if and only if $H^1(X, \mathscr{L}^r)\hookrightarrow H^1(X, \mathscr{L}^{r+1})$ is injective
for every $r\geq 0$. As in \cite[3.10]{SingBook}, using Serre vanishing we get that
\begin{enumerate}[resume,leftmargin=3.2em,label=(\ref{cone.exmp.5}.\arabic*)]
\item $C_a(D, \mathscr{L}_D)$ is a Cartier divisor of $C_a(X,\mathscr{L})$ $\Leftrightarrow$
$ H^1(X, \mathscr{L}^{r})=0$ for every $r\geq 0$.
\end{enumerate}
If this holds then $C_a(D, \mathscr{L}_D)$ has a deformation whose generic fiber is
$X\setminus D$. So $C_a(D, \mathscr{L}_D)$ is smoothable if $X\setminus D$ is smooth.
By looking at the cohomology of the sequences
\setcounter{equation}{4}
\begin{equation}
\label{eq:1}
0\to \mathscr{L}^{r-1}\to \mathscr{L}^r\to \mathscr{L}^r_D\to 0
\end{equation}
we see that
\begin{enumerate}[leftmargin=3.2em,label=(\ref{cone.exmp.5}.\arabic*)]
\setcounter{enumi}{5}
\item $C_a(D,\mathscr{L}_D)$ is Du~Bois $\Leftrightarrow$
$H^1(X, \mathscr{L}^r)\twoheadrightarrow H^1(X, \mathscr{L}^{r+1})$ is surjective for every $r\geq 0$ and
$H^i(X, \mathscr{L}^r)=0$ for every $i>1, r\geq 0$.
\end{enumerate}
Putting all these together we see that if $C_a(D,\mathscr{L}_D)$ is Du~Bois and has a flat
deformation to $X\setminus D$ then $v\in C_a(X,\mathscr{L})$ is a rational singularity. In
particular, $C_a(D,\mathscr{L}_D)$ is \CM.
This is actually stronger than \autoref{cor:cm-defo-implies-cm}, but here we also
assumed that $X\setminus D$ has rational singularities. See also \cite{KS13} for
closely related results. \end{exmp}
By \autoref{cor:cm-defo-implies-cm}, if a local, {Du\thinspace\nolinebreak Bois}\xspace scheme $(X,x)$ is smoothable, then it is \CM. The next example shows that this is close to being optimal for some cones.
\begin{exmp}
Let $(S, H)$ be a polarized K3 surface and set $X:=S\times \mathbb{P}^2$. Fix $a,b\geq 1$
and set $\mathscr{L}(a,b):=\pi_1^*\mathscr{O}_S(aH)\otimes \pi_2^*\mathscr{O}_{\mathbb{P}^2}(b)$ and let
$D(a,b)\subset X$ be a smooth member of the associated linear system.
The affine cone $C_a\bigl(D(a,b), \mathscr{L}(a,b)|_{D(a,b)}\bigr)$ is a hyperplane section
of the cone $C_a\bigl(X, \mathscr{L}(a,b)\bigr)$, hence smoothable. It is not \CM since
$H^2\bigl(D(a,b),\mathscr{O}_{D(a,b)}\bigr)=1$ and also not {Du\thinspace\nolinebreak Bois}\xspace since
$H^1\bigl(D(a,b), \mathscr{L}(a,b)|_{D(a,b)}\bigr)=1$.
However, for any $a'>a, b'>b$ the cone
$C_a\bigl(D(a,b), \mathscr{L}(a',b')|_{D(a,b)}\bigr)$ is {Du\thinspace\nolinebreak Bois}\xspace but still not \CM. Thus the
cones $C_a\bigl(D(a,b), \mathscr{L}(a',b')|_{D(a,b)}\bigr)$ are not smoothable.
More generally, one gets similar examples starting with any smooth variety $X$ for
which $H^1(X, \mathscr{O}_X)=0$ but $H^i(X, \mathscr{O}_X)\neq 0$ for some $2\leq i\leq \dim X-2$. \end{exmp}
\begin{exmp}[\emph{Singularities of cones I}]\label{cor:cone-over-abelian}
Let $X$ be a smooth, projective variety such that $K_X\equiv 0$. Kodaira vanishing
and (\ref{cone.exmp.5}.2--3) show that $C_a(X, \mathscr{L})$ is {Du\thinspace\nolinebreak Bois}\xspace. It is \CM iff
$H^i(X,\mathscr{O}_X)=0$ for $0<i<\dim X$. This and \autoref{cor:cm-defo-implies-cm} imply
that if $C_a(X, \mathscr{L})$ is smoothable then $H^i(X,\mathscr{O}_X)=0$ for $0<i<\dim X$. In
particular, if $X$ is an abelian variety then $C_a(X,\mathscr{L})$ is not smoothable.
In \autoref{cor:cone-over-abelian-s3} we prove that if $X$ is an abelian variety,
then $C_a(X,\mathscr{L})$ cannot be deformed even to an $S_3$ scheme. \end{exmp}
\subsection{All characteristics}
\begin{exmp}[\emph{Singularities of cones II}]\label{exmp:sings-of-cones}
Let us use the notation introduced in \autoref{cone.exmp.5} and first note that
\autoref{item:21} and \autoref{item:22} remain true in all characteristics:
\begin{enumerate}[leftmargin=3.2em,label=(\ref{exmp:sings-of-cones}.\arabic*)]
\item $C_a(X,\mathscr{L})$ is rational $\Leftrightarrow$ $H^i(X, \mathscr{L}^r)=0$ for every
$i>0, r\geq 0$, and
\item $C_a(X,\mathscr{L})$ is \CM $\Leftrightarrow$ $H^i(X, \mathscr{L}^r)=0$ for every
$\dim X>i>0, r\geq 0$.
\end{enumerate}
\noin For future reference we add a more sophisticated version of
\autoref{item:22}:
\begin{enumerate}[resume,leftmargin=3.2em,label=(\ref{exmp:sings-of-cones}.\arabic*)]
\item\label{item:24} $C_a(X,\mathscr{L})$ is $S_n$ for some $n\in\mathbb{N}$ $\Leftrightarrow$
$H^i(X, \mathscr{L}^r)=0$ for every $n-1>i>0, r\geq 0$.
\end{enumerate}
The reader may find a proof, for instance, in \cite[4.3]{MR3123642},
cf.~\cite[3.11]{SingBook}. \end{exmp}
\subsection{Positive characteristic}
For the definition of $F$-singularities\xspace appearing in this section, please refer to \autoref{sec:fp-sings}.
\begin{exmp}\label{exmp:F-pure}
Let $k$ be a field of characteristic $p>0$ and let $Y$ be the curve coinsisting of
the three coordinate axes in $\mathbb{A}^3_k$, i.e., let $Y=\Spec k[x,y,z]/(xy,xz,yz)$.
Then $Y$ is $F$-pure\xspace by \cite[5.38]{MR0417172} and hence it is also $F$-anti-nilpotent\xspace by
\cite{MR3271179}. \end{exmp}
A frequently used way to show that a class of singularities is invariant under small deformation is to show the following two conditions: \begin{enumerate} \item\label{item:13} The class of singularities\xspace in question satisfies an \emph{inversion of
adjunction type} property, i.e., if a Cartier divisor $Y\subseteq X$ belongs to
this class, then so does $X$. \item\label{item:14} The class of singularities\xspace in question satisfies a \emph{Bertini type}
property, i.e., if $X$ belongs to this class and $Y\subseteq X$ is a general member
of a very ample linear system, then $Y$ also belongs to this class. \end{enumerate} It is easy to see that these two conditions imply that if in a flat family a fiber belongs to the given class of singularities\xspace, then so do nearby fibers.
This method indeed proves that $\mathbb{Q}$-Gorenstein $F$-pure\xspace singularities\xspace defined over an algebraically closed field are invariant under small deformation. Property \autoref{item:13}, the inversion of adjunction type property, for Gorenstein $F$-pure\xspace singularities\xspace holds by \cite[3.4(2)]{MR701505} and property \autoref{item:14}, the Bertini type property, at least over an algebraically closed field holds by \cite{MR3108833}. The $\mathbb{Q}$-Gorenstein case of \autoref{item:13} can be proved using \cite[7.2]{MR2587408}.
Similarly, $F$-injective singularities with liftable local cohomology\xspace satisfy \autoref{item:13} \cite{MR3263925}.
Without the $\mathbb{Q}$-Gorenstein assumption $F$-pure\xspace singularities\xspace do not satisfy \autoref{item:13} \cite{MR701505,MR1693967}. In contrast, $F$-anti-nilpotent\xspace singularities\xspace satisfy \autoref{item:13} \cite{ma_quy_2017}, but it is not known at the moment whether they satisfy \autoref{item:14}.
The fact that $F$-anti-nilpotent\xspace singularities\xspace satisfy \autoref{item:13}, but $F$-pure\xspace singularities\xspace in general do not, leads to simple examples of $F$-anti-nilpotent\xspace singularities\xspace that are not $F$-pure\xspace:
\begin{exmp}\label{exmp:F-anti-nilp}\cite{MR701505,MR1693967,MR3649223,ma_quy_2017}
Let $X=\Spec k[x,y,z,t]/(xy,xz,y(z-t^2)$ and $Y=(t=0)\subseteq X$. Then
$Y\simeq \Spec k[x,y,z]/(xy,xz,yz)$ and hence it is $F$-pure\xspace by
\autoref{exmp:F-pure}. Furthermore, then $X$ is also $F$-anti-nilpotent\xspace by
\cite[4.2]{ma_quy_2017}, but it is not $F$-pure\xspace by \cite[3.2]{MR1693967}. \end{exmp}
\begin{exmp}[\emph{Singularities of cones III}]\label{exmp:sing-of-cones-ii}
We have seen in \autoref{cor:cone-over-abelian} that in characteristic $0$ a cone
over an abelian variety has {Du\thinspace\nolinebreak Bois}\xspace singularities\xspace. We have a similar statement in positive
characteristic: Let $X$ be an ordinary abelian variety over a field of positive
characteristic and $\mathscr{L}$ an ample line bundle on $X$. Then $C_a(X,\mathscr{L})$ has $F$-pure\xspace
singularities by \cite[Lemma~1.1]{MR916481}
and hence $F$-anti-nilpotent\xspace singularities\xspace by \cite{MR3271179}. \end{exmp}
\section{Filtrations on modules over Artinian local rings}\label{sec:filtrations}
\noin We will use the following notation throughout.
\begin{demo-r}{Maximal filtrations}\label{notation}
Let $(S,\mathfrak{m}, k)$ be an Artinian local ring and $N$ a finite $S$-module with a
filtration
$N= N_0\supsetneq N_1\supsetneq \dots \supsetneq N_{q} \supsetneq N_{q+1}=0$ such
that $\factor {N_{j}}{N_{j+1}} \simeq k$ as $S$-modules for each $j=0,\dots,q$.
Further let $f:(X,x)\to (\Spec S,\mathfrak{m})$ be a flat local morphism and denote the
fiber of $f$ over $\mathfrak{m}$ by $X_{\mathfrak{m}}$. It follows that then for each
$j=0,\dots,q$,
\begin{equation}
\label{eq:4}
f^*\left( \factor {N_{j}}{N_{j+1}} \right) \simeq \mathscr{O}_{X_{\mathfrak{m}}}.
\end{equation} \end{demo-r} \begin{demo-r}{Filtering $S$}\label{filt-S}
In particular, considering $S$ as a module over itself, we choose a filtration of
$S$ by ideals
$S=I_0\supsetneq I_1\supsetneq \dots \supsetneq I_{q} \supsetneq I_{q+1}=0$ such that
$\factor {I_{j}}{I_{j+1}} \simeq k$ as $S$-modules for all $0\leq j\leq q$.
Observe that in this case $I_{1}=\mathfrak{m}$ and for every $j$ there exists a
$t_j\in I_j$ such that the composition
$\xymatrix{S\ar[r]^-{t_j \cdot} & I_j \ar[r] & \factor {I_j}{I_{j+1}}}$ induces an
isomorphism $\factor S\mathfrak{m}\simeq \factor {I_j}{I_{j+1}}$. In particular,
$\ann\left(\factor {I_j}{I_{j+1}}\right) =\mathfrak{m}$. Finally, let
$S_j:=\factor S{I_j}$. Note that $S_1=\factor S\mathfrak{m}$ and $S_{q+1}=S$. \end{demo-r} \begin{demo-r}{Filtering $\omega_S$}\label{filt-o}
Applying Grothendieck duality to the closed embedding given by the surjection
$S\twoheadrightarrow S_j$ implies that $\omega_{S_j} \simeq \Hom_S(S_j, \omega_S)$ and we obtain
injective $S$-module homomorphisms
$\varsigma_j: \omega_{S_j} \hookrightarrow \omega_{S_{j+1}}$ induced by the natural
surjection $S_{j+1}\twoheadrightarrow S_j$. Using the fact that $\omega_S$ is an injective
$S$-module and applying the functor $\Hom_S(\dash{1em}, \omega_S)$ to the short exact
sequence of $S$-modules
\[
\xymatrix{
0 \ar[r] & \factor{I_j}{I_{j+1}} \ar[r] & S_{j+1} \ar[r] & S_j \ar[r] & 0,
}
\]
we obtain another short exact sequence of $S$-modules:
\begin{equation}
\label{eq:10}
\xymatrix{
0 \ar[r] & \omega_{S_j}
\ar[r]^-{\varsigma_j} & \omega_{S_{j+1}} \ar[r] & \Hom_S\left(k,
\omega_S\right )\simeq k \ar[r] & 0.
}
\end{equation}
Therefore we obtain a filtration of $N=\omega_S$ by the submodules
$N_j:=\omega_{S_{q+1-j}}$ as in \eqref{notation} where
$q+1=\length_S(S)=\length_S(\omega_S)$. The composition of the embeddings in
\autoref{eq:10} will be denoted by
$\varsigma:=\varsigma_{q}\circ\dots\circ\varsigma_1 :\omega_{S_1}\hookrightarrow
\omega_{S_{q+1}}= \omega_{S}$. \end{demo-r}
Recall that the \emph{socle} of a module $M$ over a local ring $(S,\mathfrak{m},k)$ is \begin{equation}
\label{eq:16}
\Soc M :=(0:\mathfrak{m})_M = \{x\in M \mid \mathfrak{m}\cdot x =0 \} \simeq \Hom_S(k, M). \end{equation} $\Soc M$ is naturally a $k$-vector space and $\dim_k\Soc\omega_S=1$ by the definition of the canonical module. In particular, $\Soc\omega_S\simeq k$ and this is the only $S$-submodule of $\omega_S$ isomorphic to $k$.
\begin{lem}\label{lem:I-times-omega}
Using the notation from \eqref{filt-S} and \eqref{filt-o}, we have that
\begin{equation}
\label{eq:12}
\im \varsigma = \Soc \omega_{S} = I_{q} \omega_{S}.
\end{equation}
\end{lem}
\begin{subrem}
Note that we are not simply stating that these modules in \autoref{eq:12} are
isomorphic, but that they are equal as submodules of $\omega_{S}$. \end{subrem}
\begin{proof}
Since $S_{1}\simeq \factor S\mathfrak{m}\simeq k$, and hence $\omega_{S_{1}}\simeq k$, it
follows that the image of the embedding $\varsigma :\omega_{S_1}\hookrightarrow \omega_{S}$
maps $\omega_{S_1}$ isomorphically onto $\Soc \omega_{S}$:
\begin{equation}
\label{eq:18}
\xymatrix{
\im \varsigma
= \Soc \omega_{S}.
}
\end{equation}
As $\omega_{S}$ is a dualizing sheaf, $I_{q}\omega_{S}\neq 0$, and since
$I_{q}\simeq \factor S\mathfrak{m}$ it follows that
\[
0\neq I_{q}\omega_{S} \subseteq (0:\mathfrak{m})_{\omega_{S}} =\Soc\omega_{S}\simeq k.
\]
Since $k$ is a simple $S$-module, this implies that
$I_{q}\omega_{S} =\Soc\omega_{S}$ which proves \autoref{eq:12}.
\end{proof}
\section{Families over Artinian local rings}\label{sec:db-families-over}
\noin We will frequently use the following notation.
\begin{notation}\label{not:top-loc-cohs}
Let $A$ be a noetherian ring, $(R,\mathfrak{m})$ a noetherian local $A$-algebra,
$I\subset R$ a nilpotent ideal and $(T,\mathfrak{n})\colon\!\!\!= (R/I,\mathfrak{m}/I)$ with natural
morphism $\alpha:R\twoheadrightarrow T$.
\end{notation}
\begin{defini}\label{def:top-loc-cohs-for-schemes}
Recall from \autoref{def:liftable-cohom} that we say that $(T,\mathfrak{n})$ has
\emph{liftable local cohomology\xspace over $A$} if for any $(R,\mathfrak{m})$ as in \autoref{not:top-loc-cohs},
the induced homomorphism on local cohomology $H^i_\mathfrak{m}(R) \twoheadrightarrow H^i_{\mathfrak{n}}(T)$ is
surjective for all $i$.
We extend this definition to schemes: Let $(X,x)$ be a local scheme over a
noetherian ring $A$. Then we say that $(X,x)$ has \emph{liftable local cohomology\xspace over $A$} if
$\mathscr{O}_{X,x}$ has liftable local cohomology\xspace over $A$. If $f:X\to B$ is a morphism of
schemes then we say that $X$ has \emph{liftable local cohomology\xspace over $B$} if $(X,x)$ has
liftable local cohomology\xspace over $A$ for each $x\in X$ and for each $\Spec A\subseteq B$ open
affine neighbourhood of $f(x)\in B$. \end{defini}
\begin{rem}\label{rem:top-loc-cohs-is-hereditary}
A simple consequence of the definition is that if $X$ has liftable local cohomology\xspace over a scheme
$Z$, then for any morphism $g:X\to B$ of $Z$-schemes $X$ has liftable local cohomology\xspace over $B$
as well. In particular, if $X$ has liftable local cohomology\xspace over a field $k$, then it has
liftable local cohomology\xspace over any other $k$-scheme to which it admits a map. In addition if
$\kar k=0$, then $X$ has liftable local cohomology\xspace by \autoref{rem:inheriting-top-loc-cohs}. \end{rem}
\noin Next we need a simple lemma regarding liftable local cohomology\xspace:
\begin{lem}\label{lem:surj-for-R-implies-surj-for-M}\label{loc-coh-surj-2}
Using \autoref{not:top-loc-cohs} let $M$ be an $R$-module such that there exists a
surjective morphism $M\twoheadrightarrow T$. Assume that the induced natural homomorphism
$H^i_\mathfrak{m}(R) \twoheadrightarrow H^i_\mathfrak{n}(T)$ is surjective for some $i\in\mathbb{N}$. Then the induced
homomorphism on local cohomology
\begin{equation}
\label{eq:32}
\xymatrix{
H^i_\mathfrak{m}(M) \ar@{->>}[r] & H^i_\mathfrak{m}(T)\simeq H^i_\mathfrak{n}(T) }
\end{equation}
is surjective for the same $i$. In particular, if $(T,\mathfrak{n})$ has liftable local cohomology\xspace over
$A$, then the homomorphism in \autoref{eq:32} is surjective for every $i\in\mathbb{N}$. \end{lem}
\begin{proof}
Let $t\in M$ be such that $\alpha(t)=1\in T$ and let $\beta: R\to M$ be defined by
$1\mapsto t$. Then $\alpha\circ\beta=\alpha :R \to T$ is the natural quotient
morphism, hence the surjective morphism $H^i_\mathfrak{m}(R) \twoheadrightarrow H^i_\mathfrak{m}(T)$ factors
through $H^i_\mathfrak{m}(M)$ which proves the statement. \end{proof}
\begin{prop}\label{thm:loc-coh-inj}
Let $(S,\mathfrak{m},k)$ be an Artinian local ring and $f:(X,x)\to (\Spec S,\mathfrak{m})$ a flat
local morphism. Let $N$ be a finite $S$-module with a filtration as in
\eqref{notation} and assume that $(X_{\mathfrak{m}},x)$ has liftable local cohomology\xspace over $S$. Then for
each $i, j$, the natural sequence of morphisms induced by the embeddings
$N_{j+1}\hookrightarrow N_j$ forms a short exact sequence,
\[
\xymatrix{
0 \ar[r] & H^i_x(f^*N_{j+1}) \ar[r] & H^i_x(f^*N_{j}) \ar[r] & H^i_x\left(
f^*\left(\factor {N_{j}}{N_{j+1}}\right) \right)\simeq
H^i_x\left(\mathscr{O}_{X_{\mathfrak{m}}} \right) \ar[r] & 0. }
\] \end{prop}
\begin{proof}
Since $\ann\left(\factor {N_{j}}{N_{j+1}}\right) =\mathfrak{m}$, there is a natural
surjective morphism
\[
f^*N_{j}\otimes \mathscr{O}_{X_{\mathfrak{m}}}\twoheadrightarrow f^*\left(\factor {N_{j}}{N_{j+1}}\right).
\]
By \autoref{loc-coh-surj-2} and \autoref{eq:4}, the natural homomorphism
\begin{equation}
\label{eq:5}
\xymatrix{
H^i_x(f^*N_{j}) \ar@{->>}[r] & H^i_x\left(
f^*\left(\factor {N_{j}}{N_{j+1}}\right)
\right)\simeq
H^i_x\left(\mathscr{O}_{X_{\mathfrak{m}}} \right) }
\end{equation}
is surjective for all $i$.
Since $f$ is flat, we have a short exact sequence for every $j>0$:
\[
\xymatrix{
0 \ar[r] & f^*N_{j+1} \ar[r] & f^*N_{j} \ar[r] & f^*\left(\factor
{N_{j}}{N_{j+1}}\right) \ar[r] & 0,
}
\]
and hence the statement follows from \autoref{eq:5}. \end{proof}
\begin{demo-r}{The exceptional inverse image of the structure
sheaves}\label{uppershriek}
Let $(S,\mathfrak{m},k)$ be an Artinian local ring with a filtration by ideals as in
\eqref{filt-S}. Further let $f:X\to \Spec S$ be a flat morphism that is essentially
of finite type and $f_j=f\resto{X_j}:X_j:=X\times_{\Spec S}\Spec S_j\to \Spec S_j$
where $S_j=S/I_j$ as defined in \eqref{filt-S}, e.g., $X_{q+1}=X$ and
$X_{1}=X_{\mathfrak{m}}$, the fiber of $f$ over the closed point of $S$. By a slight abuse
of notation we will denote $\omega_{\Spec S}$ with $\omega_S$ as well, but it will
be clear from the context which one is meant at any given time.
Using the description of the exceptional inverse image functor via the
residual/dualizing complexes \cite[(3.3.6)]{Conrad00} (cf.\cite[3.4(a)]{RD},
\cite[\href{http://stacks.math.columbia.edu/tag/0E9L}{Tag 0E9L}]{stacks-project}):
\begin{equation}
\label{eq:21}
f^! = {\mcR\!}\sHom_X({\mcL\!} f^*{\mcR\!}\sHom_S(\dash{1em}, \dcx S), \dcx X)
\end{equation}
and the facts that $S$ is Artinian and $f$ is flat, we have that
\begin{equation*}
\dcx{X_j/S_j}\simeq f_j^!\mathscr{O}_{\Spec S_j}\simeq
{\mcR\!}\sHom_{X_j}(f_j^*\omega_{S_j}, \dcx {X_j}).
\end{equation*}
By Grothendieck duality
\[
{\mcR\!}\sHom_{X_j}(f_j^*\omega_ {S_j}, \dcx {X_j}) \simeq
{\mcR\!}\sHom_{X}(f_j^*\omega_ {S_j}, \dcx {X}),
\]
and as $f_j^*\omega_ {S_j}=f^*\omega_ {S_j}$ and
$\omega_{S_j}\simeq \Hom_S(S_j, \omega_S)\simeq {\mcR\!}\sHom_S(\mathscr{O}_{\Spec S_j}, \dcx
S)$ we obtain that
\begin{equation}
\label{eq:20}
\dcx{X_j/S_j}\simeq {\mcR\!}\sHom_X(f^*\omega_ {S_j}, \dcx X)\simeq f^!\mathscr{O}_{\Spec S_j},
\end{equation}
in particular, that
\begin{equation}
\label{eq:9}
\dcx{X_{\mathfrak{m}}}\simeq f^!k \simeq {\mcR\!}\sHom_X(f^*\Hom_S(k,\omega_ {S}), \dcx X)
\simeq {\mcR\!}\sHom_X(\mathscr{O}_{X_\mathfrak{m}}, \dcx X).
\end{equation} \end{demo-r}
\begin{demo-r}{Natural morphisms of dualizing complexes}\label{nat-morph}
We will continue using the notation from \eqref{uppershriek}. Applying $f^!$ to
the natural surjective morphism $\xymatrix{S_{j+1} \ar@{->>}[r] & S_j}$ gives a
natural morphism
\begin{equation}
\label{eq:17}
\xymatrix{
\varrho_j: \dcx {X_{j+1}/S_{j+1}} \ar[r] & \dcx {X_{j}/S_{j}}. }
\end{equation}
Notice that $\varrho_j$ is Grothendieck dual to $f^*\varsigma_j$ defined in
\eqref{filt-S}. Indeed, $\varsigma_j$ is obtained by applying
$\Hom_S(\dash{1em},\omega_S)$ to the morphism $\xymatrix{S_{j+1} \ar@{->>}[r] & S_j}$,
and then $\varrho_j$ is obtained by applying $f^*$ and then
${\mcR\!}\sHom_X(\dash{1em},\dcx X)$. Notice further that ${\sf h}^{-i}(\varrho_j)$ factors
through the natural base change morphism of \autoref{prop:base-change-map} for each
$i\in\mathbb{Z}$.
The composition of the surjective morphisms $\xymatrix{S_{j+1} \ar@{->>}[r] & S_j}$
for all $j$ is the natural surjective morphism
$\xymatrix{S \ar@{->>}[r] & \factor S{\mathfrak{m}} \simeq k}$, and hence the composition
of the $\varrho_j$'s gives the natural morphism
\begin{equation}
\label{eq:19}
\xymatrix{
\varrho:= \varrho_1\circ\dots \circ\varrho_q
: \dcx {X/S} \ar[r] & \dcx {X_{1}/S_{1}} =\dcx{X_\mathfrak{m}},}
\end{equation}
which is then Grothendieck dual to
$f^*\varsigma:=f^*(\varsigma_q\circ\dots\circ\varsigma_1)$ and
${\sf h}^{-i}(\varrho_j)$ factors through the natural base change morphism of
\autoref{prop:base-change-map} for each $i\in\mathbb{Z}$. \end{demo-r}
\noin In the rest of this section we will use the following notation and assumptions.
\begin{assume}\label{ass:DB}
Let $(S,\mathfrak{m},k)$ be an Artinian local ring and $f:(X,x)\to (\Spec S,\mathfrak{m})$ a flat
local morphism that is essentially of finite type. Assume that $(X_{\mathfrak{m}}, x)$,
where $X_\mathfrak{m}$ is the fiber of $f$ over the closed point of $\Spec S$, has
liftable local cohomology\xspace over $S$. Note that by definition $x\in X_{\mathfrak{m}}$ and that we will keep
using the notation introduced in \autoref{eq:17} and \autoref{eq:19}. \end{assume}
\begin{thm}\label{thm:surjectivity}
For each $i,j\in\mathbb{N}$,
\begin{enumerate}
\item\label{item:1} the natural morphism $\xymatrix{
{\sf h}^{-i}(\varrho_j) : {\sf h}^{-i}(\dcx {X_{j+1}/S_{j+1}}) \ar@{->>}[r] &
{\sf h}^{-i}(\dcx {X_{j}/S_{j}}) }$ is surjective,
\item\label{item:2} the natural morphism $\xymatrix{
{\sf h}^{-i}(\varrho) : {\sf h}^{-i}(\dcx {X/S}) \ar@{->>}[r] & {\sf h}^{-i}(\dcx
{X_{\mathfrak{m}}}) }$ is surjective,
\item\label{item:3} the natural morphisms induced by $\varrho_j$ form a short exact sequence\xspace,
\[
\xymatrix@C4em{
0 \ar[r] & {\sf h}^{-i}(\dcx {X_\mathfrak{m}}) \ar[r] & {\sf h}^{-i}(\dcx {X_{j+1}/S_{j+1}})
\ar[r]^-{{\sf h}^{-i}(\varrho_j)} & {\sf h}^{-i}(\dcx {X_j/S_j}) \ar[r] & 0,}
\]
\item\label{item:5}
$\ker {\sf h}^{-i}(\varrho_j) = I_{j}{\sf h}^{-i}(\dcx {X_{j+1}/S_{j+1}}) \simeq
\factor {I_{j}{\sf h}^{-i}(\dcx {X/S})}{I_{j+1}{\sf h}^{-i}(\dcx {X/S})}$,
\item\label{item:4}
${\sf h}^{-i}(\dcx {X_{j}/S_{j}})\simeq \factor {{\sf h}^{-i}(\dcx {X/S})}
{I_{j}{\sf h}^{-i}(\dcx {X/S})}\simeq {{\sf h}^{-i}(\dcx {X/S})} \otimes_{\mathscr{O}_X}
\mathscr{O}_{X_j}$, and
\item\label{item:11} $\ker {\sf h}^{-i}(\varrho) = \mathfrak{m}{\sf h}^{-i}(\dcx {X/S})$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $N=\omega_S$ and consider the filtration on $N$ given by
$\omega_{S_j}=N_{{q+1-j}}$, cf.~\eqref{filt-o}, \autoref{eq:10}.
Further let
$(\ )\widehat{\ }$ denote the completion at $x$ (the closed point of $X$). Then by
\autoref{thm:loc-coh-inj}, for all $i, j\in\mathbb{N}$, there exists a short exact
sequence
\begin{equation}
\label{eq:8}
\xymatrix{
0 \ar[r] & H^i_x(f^*\omega_{S_{j}}) \ar[r] & H^i_x(f^*\omega_{S_{j+1}}) \ar[r] &
H^i_x\left(f^*\left(\factor {\omega_{S_{j+1}}}{\omega_{S_{j}}}\right)\right)
\ar[r] & 0. }
\end{equation}
Applying local duality \cite[Corollary~V.6.5]{RD} to \autoref{eq:8} gives the short
exact sequence
\[
\[email protected]{
0 \ar[r] & \sExt^{-i}_{X} \left(f^*\left(\factor
{\omega_{S_{j+1}}}{\omega_{S_{j}}}\right),
\dcx{X} \right) \widehat{\vphantom{\big)}} \ar[r] &
\sExt^{-i}_{X}(f^*\omega_{S_{j+1}}, \dcx{X} )\ \widehat{} \ar[r] &
\sExt^{-i}_{X}(f^*\omega_{S_{j}}, \dcx{X} )\ \widehat{} \ar[r] & 0. }
\]
for all $i, j\in\mathbb{N}$. Since completion is faithfully flat
\cite[\href{http://stacks.math.columbia.edu/tag/00MC}{Tag 00MC}]{stacks-project},
this implies that there are short exact sequences
\begin{align}
\label{eq:11}
\begin{split}
\xymatrix{
0 \ar[r] & \sExt^{-i}_{X} \left(f^*\left(\factor
{\omega_{S_{j+1}}}{\omega_{S_{j}}}\right),
\dcx{X} \right) \ar[r] & \hskip6em& \hskip6em } \\ \xymatrix{
&& & \ar[r] & \sExt^{-i}_{X}\left(f^*\omega_{S_{j+1}}, \dcx{X} \right)
\ar[r] & \sExt^{-i}_{X}\left(f^*\omega_{S_{j}}, \dcx{X} \right) \ar[r] &
0. }
\end{split}
\end{align}
Recall that
$\sExt^{-i}_{X}\left(f^*\omega_{S_{j}}, \dcx{X} \right)\simeq {\sf h}^{-i}(\dcx
{X_{j}/S_{j}})$ for each $i,j$,
by \autoref{eq:20}.
Further observe that the surjective morphism in \autoref{eq:11} is the
$-i^\text{th}$ cohomology sheaf of the Grothendieck dual of $f^*\varsigma_{j}$ and
hence via the above isomorphisms, it corresponds to
${\sf h}^{-i}(\varrho_{j})$. Therefore \autoref{eq:11} implies \autoref{item:1}.
By \autoref{eq:10}
$f^*\left(\factor {\omega_{S_{j+1}}}{\omega_{S_{j}}}\right)\simeq \mathscr{O}_{X_\mathfrak{m}}$,
and hence
$\sExt^{-i}_{X} \left(f^*\left(\factor {\omega_{S_{j+1}}}{\omega_{S_{j}}}\right),
\dcx{X} \right)\simeq {\sf h}^{-i}(\dcx {X_\mathfrak{m}})$,
so \autoref{eq:11} also implies \autoref{item:3}. Composing the surjective
morphisms in \autoref{eq:11} for all $j$ implies that the natural morphism
\[
\xymatrix@C4em{
{\sf h}^{-i}(\dcx {X/S})\simeq \sExt^{-i}_{X}\left(f^*\omega_S, \dcx{X} \right)
\ar@{->>}[r]^-{{\sf h}^{-i}(\varrho)} & \sExt^{-i}_{X}\left(f^*\omega_{S_q}, \dcx{X}
\right) \simeq {\sf h}^{-i}(\dcx {X_{\mathfrak{m}}}) }
\]
is surjective and hence \autoref{item:2} follows as well.
Similarly, composing the injective maps in \autoref{eq:8} for all $j$ shows that
the embedding $\varsigma: \omega_{S_1}\hookrightarrow \omega_S$ induces an embedding on local
cohomology:
\begin{equation}
\label{eq:15}
H^i_x(f^*\omega_{S_1}) \subseteq
H^i_x(f^*\omega_{S}).
\end{equation}
Next we prove \autoref{item:5} for $j=q$ first. Since
${\sf h}^{-i}(\dcx {X_{q}/S_{q}})$ is supported on $X_{q}$ it follows that
\[
I_{q}{\sf h}^{-i}(\dcx {X/S})\subseteq K:= \ker {\sf h}^{-i}(\varrho_{q})
\]
Recall from \eqref{filt-S} that there exists a $t_{q}\in I_{q}$ such that
$I_{q}=St_{q}\simeq \factor S\mathfrak{m}$ and from \autoref{lem:I-times-omega} that
$I_{q}\omega_S=\Soc \omega_S$. It follows that for $j=q$ the short exact sequence
of \autoref{eq:10} takes the form
\begin{equation}
\label{eq:6}
\xymatrix@C3em{
0 \ar[r] & \omega_{S_{q}} \ar[r] & \omega_{S} \ar[r]^-{\tau} &
\Soc\omega_S \ar[r] & 0, }
\end{equation}
where $\tau:\omega_S\twoheadrightarrow\Soc\omega_S\subset \omega_S$ may be identified with
multiplication by $t_{q}$ on $\omega_S$. Applying $f^*$ and taking local
cohomology we obtain the short exact sequence
\begin{equation}
\label{eq:14}
\xymatrix@C3em{
0 \ar[r] & H^i_x(f^*\omega_{S_{q}}) \ar[r] & H^i_x(f^*\omega_{S})
\ar[r]^-{H^i_x(\tau)} & H^i_x\left(f^*\Soc\omega_S \right) \ar[r] & 0, }
\end{equation}
which is of course just \autoref{eq:8} for $j=q$. Clearly,
the morphism $H^i_x(\tau)$ may also be identified with multiplication by $t_{q}$ on
$H^i_x(f^*\omega_S)$. By \autoref{lem:I-times-omega} and \autoref{eq:15}, the
natural morphism
$H^i_x(\varsigma): H^i_x\left(f^*\Soc\omega_S \right) =H^i_x(I_qf^*\omega_S) =
H^i_x(f^*\omega_{S_1}) \to H^i_x(f^*\omega_S)$
is injective. Since $H^i_x(\tau)$, i.e., multiplication by $t_q$ on
$H^i_x(f^*\omega_{S})$, is surjective onto $H^i_x\left(f^*\Soc\omega_S \right)$, it
follows that
\begin{equation}
\label{eq:25}
\xymatrix@R5em{
H^i_x\left(f^*\Soc\omega_S \right) \ar[r]_-{H^i_x(\varsigma)}^-{\simeq} &
\im H^i_x(\varsigma) = I_{q} H^i_x(f^*\omega_{S}) \ar@{^(->}[r] &
H^i_x(f^*\omega_{S}),
}
\end{equation}
i.e., $H^i_x\left(f^*\Soc\omega_S \right)$ coincides with
$I_{q} H^i_x(f^*\omega_{S})$ as submodules of $H^i_x(f^*\omega_{S})$.
Next let $E$ be an injective hull of $\kappa(x)=\factor{\mathscr{O}_{X,x}}{\mathfrak{m}_{X,x}}$ and
consider a morphism $\phi: H^i_x(f^*\Soc\omega_S)\to E$. As $E$ is injective,
$\phi$ extends to a morphism $\widetilde\phi: H^i_x(f^*\omega_S)\to E$. If
$a\in H^i_x(f^*\omega_S)$, then
$t_{q}a\in I_{q}H^i_x(f^*\omega_S)=H^i_x\left(f^*\Soc\omega_S \right)$, so
\[
t_{q}\widetilde\phi(a)=\widetilde\phi(t_{q}a)=\phi(t_{q}a)=\left(\phi\circ
H^i_x(\tau)\right)(a)
\]
Therefore, $\phi\circ H^i_x(\tau)= t_{q}\widetilde\phi$.
Similarly, if $\psi: H^i_x(f^*\omega_S)\to E$ is an arbitrary morphism, then
setting $\phi=\psi\resto{H^i_x(f^*\Soc\omega_S)}: H^i_x(f^*\Soc\omega_S)\to E$ and
applying the same computation as above, with $\widetilde\phi$ replaced by $\psi$, shows
that $\phi\circ H^i_x(\tau)= t_{q}\psi$.
It follows that the embedding induced by $H^i_x(\tau)$,
\begin{equation}
\label{eq:28}
\alpha: \Hom_{\mathscr{O}_{X,x}}(H^i_x(f^*\Soc\omega_S), E)\hookrightarrow
\Hom_{\mathscr{O}_{X,x}}(H^i_x(f^*\omega_S), E)
\end{equation}
identifies $\Hom_X(H^i_x(f^*\Soc\omega_S), E)$ with
$I_{q}\Hom_X(H^i_x(f^*\omega_S), E)$. By local duality this implies that
\[
\left(\factor{\ker \left[ {\sf h}^{-i}(\varrho_{q}): {\sf h}^{-i}(\dcx {X/S})\twoheadrightarrow
{\sf h}^{-i}(\dcx {X_{q}/S_{q}}) \right]}{I_{q}{\sf h}^{-i}(\dcx {X/S})}\right)
\otimes \widehat{\mathscr{O}}_{X,x}=0
\]
and hence, since completion is faithfully flat, this implies \autoref{item:5} in
the case $j=q$.
Running through the same argument with $S$ replaced by $S_{j+1}$ gives the equality
in \autoref{item:5} for all $j$. In addition, \autoref{item:5} for $j=q$ also
implies \autoref{item:4} for $j\geq q$.
Assuming that \autoref{item:4} holds for $j=r+1$ implies the isomorphism in
\autoref{item:5} for $j=r$. In turn, the entire \autoref{item:5} for $j=r$,
combined with \autoref{item:4} for $j=r+1$, implies \autoref{item:4} for
$j=r$. Therefore, \autoref{item:5} and \autoref{item:4} follow by descending
induction on $j$ and then \autoref{item:11} follows from \autoref{item:5} and the
definition of $\varrho$. \end{proof}
\noin Next we need a simple lemma.
\begin{lem}\label{lem:simple}
Let $R$ be a ring. $M$ an $R$-module, $t\in R$ and $J=(t)\subseteq R$. Assume that
$(0:J)_M=(0:J)_R\cdot M$. Then the natural morphism
$\xymatrix{J\otimes_R M\ar[r]^-\simeq & JM}$ is an isomorphism. \end{lem}
\begin{proof}
This natural morphism is always surjective. Suppose $m\in M$ is such that
$t\otimes m\mapsto 0$ via this morphism. In other words such that $tm=0$. This
means, by definition, that $m\in (0:J)_M$ and hence by assumption there exist
$y\in (0:J)_R\subseteq R$ and $m'\in M$ such that $m=ym'$. Then
$t\otimes m=t\otimes ym'=yt\otimes m'=0$, since $yt=0$. This proves the claim. \end{proof}
\begin{prop}\label{prop:tensor}
Using the same notation as above,
\begin{enumerate}
\item\label{item:10}
$I_j\otimes {\sf h}^{-i}(\dcx {X/S})\simeq I_{j}{\sf h}^{-i}(\dcx {X/S})$,
\item\label{item:6} for any $l\in\mathbb{N}$,
$\factor{I_j}{I_{j+l}}\otimes {\sf h}^{-i}(\dcx {X/S})\simeq \factor
{I_{j}{\sf h}^{-i}(\dcx {X/S})}{I_{j+l}{\sf h}^{-i}(\dcx {X/S})}$, and
\item\label{item:12} for any $l\in\mathbb{N}$,
$\factor{\mathfrak{m}^l}{\mathfrak{m}^{l+1}}\otimes {\sf h}^{-i}(\dcx {X/S})\simeq \factor
{\mathfrak{m}^l{\sf h}^{-i}(\dcx {X/S})}{\mathfrak{m}^{l+1}{\sf h}^{-i}(\dcx {X/S})}$.
\end{enumerate} \end{prop}
\begin{proof}
Notice that since $H^i_x(f^*\Soc\omega_S)$ is both a quotient and a submodule of
$H^i_x(f^*\omega_S)$, there are two natural maps between
$\Hom_{\mathscr{O}_{X,x}}(H^i_x(f^*\Soc\omega_S), E)$ and
$\Hom_{\mathscr{O}_{X,x}}( H^i_x(f^*\omega_S), E)$. Regarding $H^i_x(f^*\Soc\omega_S)$ a
quotient module via $H^i_x(\tau)$ we get the embedding
$\alpha=(\dash{1em})\circ H^i_x(\tau)$ in \autoref{eq:28}, and considering it a
submodule the restriction map
\begin{equation*}
\xymatrix@R0em{
\beta: \Hom_{\mathscr{O}_{X,x}}( H^i_x(f^*\omega_S), E) \ar[r] &
\Hom_{\mathscr{O}_{X,x}}(H^i_x(f^*\Soc\omega_S), E). \\
\phi \ar@{|->}[r] & \phi\resto{H^i_x(f^*\Soc\omega_S)} }
\end{equation*}
These maps are of course not inverses to each other. In fact, we have already
established (cf.~\autoref{eq:28}) that
$\phi\resto{H^i_x(f^*\Soc\omega_S)}\circ H^i_x(\tau)= t_{q}\phi$ and hence the
composition $\alpha\circ\beta$ is just multiplication by $t_q$:
\begin{equation}
\label{eq:24}
\begin{aligned}
\xymatrix{
&
\ar@{}[l]^(.675){}="a"
\ar@{} "a";[d]^(.05){}="b"
\ar[rd]^(.55){\alpha\circ\beta}
\phi\in \Hom_{\mathscr{O}_{X,x}}(H^i_x(f^*\omega_S), E) \ar[r]^-\beta &
\Hom_{\mathscr{O}_{X,x}}(H^i_x(f^*\Soc\omega_S), E) \ar[d]^-\alpha_-\simeq \\
&&
\ar@{}[l]^(.45){}="c"
\ar@{} "c";[u]^(.15){}="d"
\ar@/_1em/@{|->} "b";"c"
t_q \phi\in I_q\Hom_{\mathscr{O}_{X,x}}(H^i_x(f^*\omega_S), E). }
\end{aligned}
\end{equation}
This implies, (cf.~\autoref{eq:15} and \autoref{eq:25}), that ${\sf h}^{-i}(\varrho)$
may be identified with multiplication by $t_q$ on ${\sf h}^{-i}(\dcx {X/S})$.
Together with \autoref{thm:surjectivity}\autoref{item:11} this implies that
\[
(0:I_q)_{{\sf h}^{-i}(\dcx {X/S})} =\ker{\sf h}^{-i}(\varrho)= \mathfrak{m} {\sf h}^{-i}(\dcx {X/S})
= (0:I_q)_S\cdot {\sf h}^{-i}(\dcx {X/S}),
\]
and hence the natural morphism
\begin{equation}
\label{eq:26}
\xymatrix{
I_q\otimes_S {\sf h}^{-i}(\dcx {X/S}) \ar[r]^-\simeq & I_q{\sf h}^{-i}(\dcx {X/S})
}
\end{equation}
is an isomorphism by \autoref{lem:simple}.
Now assume, by induction, that \autoref{item:10} holds for $S_q$ in place of
$S$. In particular, keeping in mind that $S_q=\factor S{I_q}$, the natural map
\begin{equation}
\label{eq:30}
\xymatrix{
\factor{I_{j}}{I_{q}}\otimes_{S_q}
{\sf h}^{-i}(\dcx {X_q/S_q})\ar[r]^-\simeq &
\left(\factor{I_{j}}{I_{q}}\right){\sf h}^{-i}(\dcx {X_q/S_q})
}
\end{equation}
is an isomorphism for all $j$.
Consider the short exact sequence\xspace (cf.~\autoref{thm:surjectivity}\autoref{item:4}),
\begin{equation*}
\xymatrix{
0 \ar[r] & I_q {\sf h}^{-i}(\dcx {X/S})
\ar[r] & {\sf h}^{-i}(\dcx {X/S}) \ar[r] &
{\sf h}^{-i}(\dcx {X_q/S_q}) \ar[r] & 0
}
\end{equation*}
and apply $\factor{I_{j}}{I_{q}}\otimes_{S} (\dash{1em})$. The image of
$\factor{I_{j}}{I_{q}}\otimes_{S} I_q {\sf h}^{-i}(\dcx {X/S})$ in
$\factor{I_{j}}{I_{q}}\otimes_{S} {\sf h}^{-i}(\dcx {X/S})$ is $0$ and hence by
\autoref{eq:30} the natural map
\begin{multline*}
\xymatrix{
\boxed{\factor{I_{j}}{I_{q}}\otimes_{S} {\sf h}^{-i}(\dcx {X/S})} \simeq
\factor{I_{j}}{I_{q}}\otimes_{S_q} {\sf h}^{-i}(\dcx {X_q/S_q}) \ar[r]^-\simeq &
\left(\factor{I_{j}}{I_{q}}\right) {\sf h}^{-i}(\dcx{X_q/S_q})
\simeq }\\
\xymatrix{
\ar@{}[r] & \simeq \left(\factor{I_{j}}{I_{q}}\right)
\factor{{\sf h}^{-i}(\dcx{X/S})}{I_q {\sf h}^{-i}(\dcx {X/S})} \simeq
\boxed{\factor{I_j{\sf h}^{-i}(\dcx{X/S})}{I_q {\sf h}^{-i}(\dcx {X/S})}}. }
\end{multline*}
is an isomorphism. This, combined with \autoref{eq:26} and the 5-lemma, implies
\autoref{item:10}. Then \autoref{item:6} is a direct consequence of
\autoref{item:10} and the fact that tensor product is right exact.
Finally, recall, that the choice of filtration in \eqref{filt-S} was fairly
unrestricted. In particular, we may assume that the filtration $I_{{\,\begin{picture}(1,1)(-1,-2)\circle*{2}\end{picture}\,}}$ of $S$ is
chosen so that for all $l\in \mathbb{N}$, there exists a $j(l)$ such that
$I_{j(l)}=\mathfrak{m}^l$. Applying \autoref{item:6} for this filtration implies
\autoref{item:12}. \end{proof}
\noin The following theorem is an easy combination of the results of this section.
\begin{thm}\label{thm:key}
Let $(S,\mathfrak{m},k)$ be an Artinian local ring and $f:(X,x)\to \Spec S$ a flat local
morphism that is essentially of finite type. If $(X_{\mathfrak{m}},x)$ has liftable local cohomology\xspace
over $S$, then ${\sf h}^{-i}(\dcx {X/S})$ is flat over $\Spec S$ for each $i$. \end{thm}
\begin{proof}
This follows directly from \autoref{prop:tensor}\autoref{item:12} and
\cite[\href{http://stacks.math.columbia.edu/tag/0AS8}{Tag 0AS8}]{stacks-project}. \end{proof}
\section{Flatness and base change}\label{sec:flatness-base-change}
\noindent In this section we prove a rather general flatness and base change theorem for the cohomology sheaves of the relative dualizing complex. The main essential assumption is that the relative dualizing complex exists.
\begin{defnot}\label{def-and-not.2}\label{def:embeddable}
For morphisms $f:X\to B$ and $\vartheta: Z\to B$, the symbol $X_Z$ will denote
$X\times_B Z$ and $f_Z:X_Z\to Z$ the induced morphism. In particular, for $b\in B$
we write $X_b = f^{-1}(b)$.
Let $f:X\to B$ be a morphism of locally noetherian schemes. Then $f$ is
\emph{embeddable into a smooth morphism of dimension $N$} if there exists a smooth
morphism
$\pi: P\to B$ of pure relative dimension $N$ over $B$ and a closed embedding
$\jmath: X\hookrightarrow P$ such that $f=\pi\circ\jmath$.
Furthermore, $f$ is \emph{locally embeddable into a smooth morphism} if each
$x\in X$ has a neighbourhood $x\in U_x\subseteq X$ such that $f\resto{U_x}$ is
{embeddable into a smooth morphism of dimension $N$} for some $N\in\mathbb{N}$.
Note that if $f:X\to B$ is a flat morphism that is essentially of finite type then
it is locally embeddable into a smooth morphism and that if $f$ is flat and locally
embeddable into a smooth morphism then it admits a relative dualizing complex by
\cite[\href{http://stacks.math.columbia.edu/tag/0E2X}{Tag 0E2X}]{stacks-project}. \end{defnot}
\begin{lem}\label{lem:D-duality}
Let $(B,b)$ be a local scheme and $f:X\to B$ a flat morphism embeddable into a
smooth morphism $P\to B$ of relative dimension $N$. Then
\[
{\sf h}^{-i}(\dcx{X/B})\simeq \sExt^{N-i}_{P}(\mathscr{O}_X,\omega_{P/B}) \quad\text{ and } \quad
{\sf h}^{-i}(\dcx{X_b})\simeq \sExt^{N-i}_{P_b}(\mathscr{O}_{X_b},\omega_{P_b}).
\] \end{lem}
\begin{proof}
Since $P/B$ is an $N$-dimensional smooth morphism, $\dcx{P/B}= \omega_{P/B}[N]$ is
a relative dualizing complex.
By Grothendieck duality \cite[VII.3.4]{RD},
\begin{equation*}
{\sf h}^{-i}(\dcx{X/B})\simeq
{\sf h}^{-i}({\mcR\!}\sHom_{P}(\mathscr{O}_X,\omega_{P/B}[N])) \simeq
\sExt^{N-i}_{P}(\mathscr{O}_X,\omega_{P/B}).
\end{equation*}
The same argument implies the equivalent statement for ${\sf h}^{-i}(\dcx{X_b})$. \end{proof}
\noindent The following statement is standard. We include it for ease of reference.
\begin{prop}\label{prop:base-change-map}
Let $f:X\to B$ be a flat morphism of schemes that admits a relative dualizing
complex and let $Z\to B$ be a morphism. Then for each $i\in\mathbb{Z}$ there exists a
natural base change morphism,
\[
\varrho^{-i}_Z: {\sf h}^{-i}(\dcx{X/B})\otimes_B\mathscr{O}_Z \longrightarrow
{\sf h}^{-i}(\dcx{X_Z/Z}).
\] \end{prop}
\begin{proof}
For any complex $\cmx{\sf A}$, tensoring with an object induces a natural morphism,
\[
{\sf h}^i(\cmx{\sf A})\otimes {\sf M} \longrightarrow {\sf h}^i(\cmx{\sf A}\lotimes {\sf M}).
\]
Applying this to the dualizing complex gives a natural map
\[
\varrho^{-i}_Z: {\sf h}^{-i}(\dcx{X/B})\otimes_B\mathscr{O}_Z \longrightarrow
{\sf h}^{-i}\big(\dcx{X/B}\lotimes_B\mathscr{O}_Z\big).
\]
But $\dcx{X/B}\lotimes_B\mathscr{O}_Z\simeq \dcx{X_Z/Z}$ by the base change property of
dualizing complexes \cite[\href{http://stacks.math.columbia.edu/tag/0E2Y}{Tag
0E2Y}]{stacks-project}, so the statement follows. \end{proof}
\begin{terminology}\label{terminology:base-change}
Let $f:X\to B$ be a flat morphism of schemes and $\vartheta:Z\to B$ a
morphism. Then for an $i\in\mathbb{Z}$, we will say that ${\sf h}^{-i}(\dcx{X/B})$
\emph{commutes with base change to $Z$} if the natural base change morphism
$\varrho_Z^{-i}$ of \autoref{prop:base-change-map} is an isomorphism. \end{terminology}
\begin{rem}\label{rem:commuting-is-local}
Since the base change morphism is defined naturally, it can be checked locally
whether it is an isomorphism. In other words, if ${\sf h}^{-i}(\dcx{X/B})$ commutes
with base change to $Z$ locally on $X$, then it commutes with base change to $Z$. \end{rem}
\begin{rem}
A simple case when the condition in \eqref{terminology:base-change} holds is if
$f:X\to B$ has \CM fibers. In that case the only non-zero cohomology sheaf of the
relative dualizing complex is ${\sf h}^{-m}(\dcx{X/B})\simeq \omega_{X/B}$ where
$m=\dim X-\dim B$ by \cite[3.5.1]{Conrad00} and it commutes with base change by
\cite[3.6.1]{Conrad00}. In moduli theory typically one has to deal with
non-Cohen-Macaulay fibers. The next example shows that for these not even
$\omega_{X/B}$ commutes with base change. However, we see in
\autoref{lem:completion} that the ${\sf h}^{-i}(\dcx{X/B})$ commute with inverse
limits. \end{rem}
\begin{exmp}\label{exmp:not-commuting-with-base-change}
Let $Y$ be a normal quasi-projective threefold with isolated singularities and a
trivial canonical divisor. Assume that $Y$
is $S_2$, but not $S_3$. For instance, a non-\CM normal threefold such as a cone
over an abelian surface in characteristic $0$ satisfies these conditions
cf.~\autoref{cor:cone-over-abelian}. Consider a general projection of $Y$ to a line
and resolve the indeterminacies of the projection map. Let $X$ denote the blow-up
of $Y$ on which this rational map becomes a morphism and let $\pi:X\to \mathbb{A}^1$
denote the resulting morphism. Note that since the projection was general we may
assume that the birational morphism $X\to Y$ is locally isomorphic near their
singular points. In particular, we may assume that $X$ is a normal affine threefold
with isolated singularities and a trivial canonical divisor, which is $S_2$, but
not $S_3$. Observe that then
${\sf h}^{-2}(\dcx{X/\mathbb{A}^1})\simeq \omega_{X/\mathbb{A}^1}\simeq \mathscr{O}_X$ by construction. Next
let $z\in\mathbb{A}^1$ be the image of a non-$S_3$ point of $X$. Then $X_z$ and hence
$\mathscr{O}_{X_z}$ is not $S_2$, since otherwise $X$ would be $S_3$ along $X_z$. At the
same time ${\sf h}^{-2}(\dcx{X_z/\{z\}})\simeq\omega_{X_z}$ is an $S_2$ sheaf
(cf.\cite[5.69]{KM98}, \cite[3.7.5]{Kovacs17b}) and hence cannot be isomorphic to
$\mathscr{O}_{X_z}$. This implies that ${\sf h}^{-2}(\dcx{X/\mathbb{A}^1})$ does not commute with
base change for the morphism $\pi:X\to \mathbb{A}^1$. \end{exmp}
\begin{notation}\label{not:mod-out-by-m-to-the-q}
Let $f:(X,x)\to (B,b)=(\Spec S,\mathfrak{m})$ be a local morphism. Let $q\in\mathbb{N}$,
$S_q:=S/\mathfrak{m}^q$, $\mathfrak{m}_q=\mathfrak{m}/\mathfrak{m}^q$ its (unique) maximal ideal, $B_q=\Spec S_q$,
$X_q:=X\times_BB_q$, and $f_q:(X_q,x)\to (B_q,b)$ the induced local morphism.
Further let $\widehat B:=\Spec (\invlim S_q)$, the completion of $B$ at $b$ and
$\widehat X:=X\times _B\widehat B$. \end{notation}
\begin{lem}\label{lem:completion}
Let $f:(X,x)\to (B,b)$ be a flat local morphism that admits a relative dualizing
complex. Fix an $i\in\mathbb{Z}$ and assume that the inverse system
$\left({\sf h}^{-i-1} (\dcx{X_q/B_q})\right)$ satisfies the Mittag-Leffler condition
\cite[\href{http://stacks.math.columbia.edu/tag/0595}{Tag 0595}]{stacks-project}.
Then the natural base change morphism (cf.~\autoref{prop:base-change-map}) induces
an isomorphism:
\[
\invlim\left({\sf h}^{-i}(\dcx{X/B})\otimes_X\mathscr{O}_{X_q}\right)
\overset\simeq\longrightarrow
\invlim{\sf h}^{-i}(\dcx{X_q/B_q}),
\] \end{lem}
\begin{rem}
If the local scheme $(X_b,x)$ has liftable local cohomology\xspace over $B$, then the inverse system
$\left({\sf h}^{-i-1} (\dcx{X_q/B_q})\right)$ satisfies the Mittag-Leffler condition
by \autoref{thm:surjectivity}\autoref{item:1}. \end{rem}
\begin{proof}
By the base change property of dualizing complexes
\cite[\href{http://stacks.math.columbia.edu/tag/0E2Y}{Tag 0E2Y}]{stacks-project}
there exist natural restricting morphisms,
\[
\dcx{X_{q+1}/B_{q+1}}\longrightarrow \dcx{X_{q+1}/B_{q+1}}
\lotimes_{X_{q+1}}\mathscr{O}_{X_q} \simeq \dcx{X_q/B_q},
\]
so $(\dcx{X_q/B_q})$ forms an inverse system in $D^b(X)$ and hence
${\mcR\!}\lim\dcx{X_q/B_q}$, the derived limit of the inverse system $(\dcx{X_q/B_q})$,
exists \cite[\href{http://stacks.math.columbia.edu/tag/0CQD}{Tag
0CQD}]{stacks-project}.
Since
the inverse system $\left({\sf h}^{-i-1} (\dcx{X_q/B_q})\right)$ satisfies the
Mittag-Leffler condition, ${\mcR\!}^1\lim {\sf h}^{-i-1} (\dcx{X_q/B_q})=0$ by
\cite[\href{http://stacks.math.columbia.edu/tag/091D}{Tag
091D(3)}]{stacks-project}. Combined with
\cite[\href{http://stacks.math.columbia.edu/tag/0CQE}{Tag 0CQE}]{stacks-project}
this implies that the natural base change morphism of
\autoref{prop:base-change-map} induces an isomorphism
\[
{\sf h}^{-i}({\mcR\!}\lim \dcx{X_q/B_q}) \overset\simeq\longrightarrow \invlim{\sf h}^{-i}
(\dcx{X_q/B_q}).
\]
The base change property of dualizing complexes
also applies to $\dcx{X/B}$ and hence the natural restricting morphisms induce
isomorphisms,
\[
\dcx{X/B}
\lotimes_{X}\mathscr{O}_{X_q} \simeq \dcx{X_q/B_q}.
\]
Then the derived completion of $\dcx{X/B}$ with respect to the ideal
$\mathscr{J}:=f^*\mathfrak{m}_{B,b}=\mathscr{I}_{X_b\subseteq X}\subseteq \mathscr{O}_{X,x}$
\cite[\href{http://stacks.math.columbia.edu/tag/0BKH}{Tag 0BKH}]{stacks-project} is
isomorphic to ${\mcR\!}\lim\dcx{X_q/B_q}$ constructed above. Then the statement
follows by \cite[\href{http://stacks.math.columbia.edu/tag/0A06}{Tag 0A06}
]{stacks-project}. \end{proof}
\begin{rem}
In the proof above it is important to consider the derived limit
${\mcR\!}\lim\dcx{X_q/B_q}$ as a derived completion over $X$ and not over $B$, because
for the cited results the ${\sf h}^{-i}(\dcx{X/B})$ need to be finite modules. They
are finite over $\mathscr{O}_{X,x}$ but not necessarily over $\mathscr{O}_{B,b}$. \end{rem}
\noindent Next we prove our main flatness and base change statement.
\begin{thm}\label{thm:generalized-flat-and-base-change}
Let $X\to B$ be a flat morphism locally embeddable into a smooth morphism.
Fix an $i\in\mathbb{Z}$ and assume that for any Artinian scheme $Z$ and morphism $Z\to B$,
the sheaf ${\sf h}^{-i}(\dcx{X_Z/Z})$ is flat over $Z$ and commutes with any
base change to a closed subscheme of $Z$. Then ${\sf h}^{-i}(\dcx{X/B})$ is flat over
$B$ and commutes with arbitrary base change. \end{thm}
\begin{proof}
Since the base change morphism $\varrho_Z^{-i}$ is natural, the statement is local
on $B$, so we may replace $B$ with a local scheme $(B,b)$. Furthermore, since both
flatness and whether or not $\varrho_Z^{-i}$ is an isomorphism can be tested
locally on $X$, we may also replace $X$ with a local scheme $(X,x)$, use the
notation established in \autoref{not:mod-out-by-m-to-the-q}, assume that
$f:(X,x)\to (B,b)$ is embeddable into a smooth morphism and apply
\autoref{lem:D-duality}.
Let $M_q:={\sf h}^{-i}\big(\dcx {X_q/B_q}\big)$. Then by assumption $M_q$ is flat over
$B_q$ for every $q\in\mathbb{N}$ and the natural base change morphism is an isomorphism:
\[
M_{q+1}\otimes_{B_{q+1}}\mathscr{O}_{B_q} \overset\simeq\longrightarrow M_q.
\]
In particular, the induced natural morphism $M_{q+1}\twoheadrightarrow M_q$ is surjective and
hence $(M_q)$ satisfies the Mittag-Leffler condition and $\invlim M_q$ is flat over
$B$ by the first statement of
\cite[\href{http://stacks.math.columbia.edu/tag/0912}{Tag 0912}]{stacks-project}.
Furthermore, let $Q=\mathscr{O}_{X_j}$ for a fixed $j\in\mathbb{N}$. Then
$M_q\otimes_XQ\simeq M_j$ for any $q\geq j$ by assumption and hence
$\invlim(M_q\otimes_XQ)\simeq M_j$. Then by the second statement of
\cite[\href{http://stacks.math.columbia.edu/tag/0912}{Tag 0912}]{stacks-project},
\begin{equation}
\label{eq:37}
(\invlim M_q) \otimes_X\mathscr{O}_{X_j} =(\invlim M_q) \otimes_XQ \simeq
\invlim(M_q\otimes_XQ)\simeq M_j.
\end{equation}
On the other hand,
$\invlim M_q \simeq\invlim\big({\sf h}^{-i}(\dcx{X/B})\otimes_X\mathscr{O}_{X_q}\big)$ by
\autoref{lem:completion} and so by
\cite[\href{http://stacks.math.columbia.edu/tag/031C}{Tag 031C}]{stacks-project},
\begin{equation}
\label{eq:38}
(\invlim M_q)\otimes_X\mathscr{O}_{X_j}\simeq {\sf h}^{-i}(\dcx{X/B})\otimes_X\mathscr{O}_{X_j}.
\end{equation}
Comparing \autoref{eq:37} and \autoref{eq:38} shows that ${\sf h}^{-i}(\dcx{X/B})$
commutes with base change to $B_q$ for every $q\in\mathbb{N}$ and then
${\sf h}^{-i}(\dcx{X/B})$ commutes with arbitrary base change by
\autoref{lem:D-duality} and \cite[1.9]{MR555258}.
Using that $M_q={\sf h}^{-i}(\dcx{X_q/B_q})$ is flat over $B_q$ for every $q\in\mathbb{N}$,
it follows that ${\sf h}^{-i}(\dcx{X/B})$ is flat over $B$ by
\cite[\href{http://stacks.math.columbia.edu/tag/0523}{Tag 0523}]{stacks-project}. \end{proof}
\begin{cor}\label{cor:flatness}
Let $f:(X,x)\to (B,b)$ be a flat local morphism that is essentially of finite type.
If $(X_{b},x)$ has liftable local cohomology\xspace over $B$ then ${\sf h}^{-i}(\dcx{X/B})$ is flat over
$B$ and commutes with arbitrary base change for each $i\in\mathbb{Z}$. \end{cor}
\begin{proof}
By \autoref{thm:key}, \autoref{thm:surjectivity}\autoref{item:1} and
\autoref{thm:surjectivity}\autoref{item:4} $f$ satisfies the assumptions of
\autoref{thm:generalized-flat-and-base-change} and hence the statement follows from
\autoref{thm:generalized-flat-and-base-change}. \end{proof}
\noin Now we are ready to prove \autoref{thm:main.new}.
\begin{thm}[=~\autoref{thm:main.new}]\label{thm:main-strong}
Let $f:X\to B$ be a flat morphism of schemes that is essentially of finite type and
let $b\in B$
such that $X_{b}$ has liftable local cohomology\xspace over $B$. Then there exists an open neighborhood
$X_b\subset U\subset X$ such that ${\sf h}^{-i}(\dcx{U/B})$ is flat over $B$ and
commutes with base change for each $i\in\mathbb{Z}$. \end{thm}
\begin{proof}
Let $x\in X_b$ and temporarily replace $f:X\to B$ with the induced local morphism
$(X,x)\to (B,b)$. Then ${\sf h}^{-i}(\dcx{X/B})$ is flat over $B$ and commutes with
arbitrary base change by \autoref{cor:flatness}.
Since localization is an exact functor, we obtain that for the original $f:X\to B$
and any $x\in X_b$ the localized cohomology sheaves ${\sf h}^{-i}(\dcx{X/B})_x$ are
flat over $B$ and commute with base change for each $i\in\mathbb{Z}$. Both of these
properties are open on $X$ and hence there is an open neighbourhood of $x$ where
they hold. The union of these neighbourhoods for all $x\in X_b$ provide an open
neighbourhood of $X_b$ where these properties hold.
\end{proof}
Now \autoref{thm:main-strong} and \autoref{prop:loc-coh-surj-1} implies \autoref{cor:main-db-implies-slc}\autoref{item:7} and \autoref{thm:main-strong} and \autoref{prop:F-anti-nilp-has-liftable-cohs} implies \autoref{cor:main-db-implies-slc}\autoref{item:8}.
\section{Du\thinspace Bois singularities}\label{sec:local-cohomology-db}
\noin In characteristic 0, the optimal setting for deformation invariance of cohomology seems to be the class of {Du\thinspace\nolinebreak Bois}\xspace singularities, introduced by Steenbrink \cite{Steenbrink83}. For a proper complex variety with {Du\thinspace\nolinebreak Bois}\xspace singularities the natural morphism \begin{equation*}
\label{eq:star}
\xymatrix{
H^i(X,\mathbb{C}_X)\ar@{->>}[r] & H^i(X,\mathscr{O}_X)
}
\tag{$\star$} \end{equation*} is surjective, and one should think of {Du\thinspace\nolinebreak Bois}\xspace singularities as the largest class for which this holds, cf.~\cite{Kovacs11d}. This surjectivity enables one to use topological arguments to control the sheaf cohomology groups $H^i(X,\mathscr{O}_X) $ in flat families as in \cite{MR0376678}.
The proof of \autoref{cor:cm-defo-implies-cm} for projective morphisms in \cite{MR2629988} very much relied on global duality, hence properness. Our first hope was that one can localize the proofs by replacing \autoref{eq:star} with the analogous map between local cohomology groups \begin{equation*}
\xymatrix{
H^i_x(X,\mathbb{C}_X)\ar@{->>}[r] & H^i_x(X,\mathscr{O}_X).
} \end{equation*} However, this turned out to be too simplistic, one needs to consider instead the map \begin{equation*}
\xymatrix{
H^i_x(X,\mathbb{C}_X)\ar@{->>}[r] & \mathbb{H}^i_x(X,\dbcx X),
} \end{equation*} where $\dbcx X$ denotes the $0^\text{th}$ associated graded Du~Bois complex of $X$. For the construction of the Du~Bois complex see \cite{DuBois81,GNPP88} and for its relevance to higher dimensional geometry see \cite[\S 6]{SingBook}. The surjectivity in \autoref{eq:star} seems simple, but it is a key element of Kodaira type vanishing theorems \cite{Kollar87b}, \cite[\S 12]{Kollar95s},\cite{Kovacs00c},\cite{MR2646306} and leads to various results on deformations of {Du\thinspace\nolinebreak Bois}\xspace schemes \cite{MR0376678,MR2629988,KS13}. Eventually we understood that for our purposes the key property is liftable local cohomology\xspace.
\begin{thm}\label{DB.toploccohs}\label{prop:loc-coh-surj-1}
Let $X$ be a scheme, essentially of finite type over a field of characteristic $0$.
Assume that $X$ is {Du\thinspace\nolinebreak Bois}\xspace. Then $X$ has liftable local cohomology\xspace.
\end{thm}
For the definition of {Du\thinspace\nolinebreak Bois}\xspacesingularities\xspace the reader is referred to \cite[\S 6]{SingBook}. A scheme defined over a field of characteristic $0$ is said to have \emph{{Du\thinspace\nolinebreak Bois}\xspace singularities\xspace} if its base extension to $\mathbb{C}$ does. In addition to the properties mentioned above recall that {Du\thinspace\nolinebreak Bois}\xspace singularities\xspace are invariant under small deformation by \cite[4.1]{Kovacs-Schwede11b}.
Let us start the proof of \autoref{DB.toploccohs} by recalling the following statement.
\begin{thm}[\cite{Kovacs-Schwede11b,KS13,MSS17}]
\label{thm:key-injectivity}
Let $X$ be a scheme, essentially of finite type over
a field of characteristic $0$. Then the natural morphism
\[
{\sf h}^i(\uldcx X) \hookrightarrow {\sf h}^i(\dcx X)
\]
is injective for every $i\in\mathbb{Z}$. \end{thm}
\begin{rem}
\autoref{thm:key-injectivity} was first proved in
\cite[Theorem~3.3]{Kovacs-Schwede11b}.
A version for pairs, essentially with the same proof, was given in
\cite[Theorem~B]{KS13}. Both of these were stated for reduced schemes even though
the proof does not need that assumption. This was noticed and carefully confirmed
in \cite[Lemma~3.2]{MSS17} where the proof is carried out in detail for the
not-necessarily-reduced case. \end{rem}
\begin{cor}\label{cor:key-surjectivity}
Let $X$ be a scheme, essentially of finite type over a field of characteristic $0$
and $x\in X$ a closed
point. Then the natural morphism
\[
\xymatrix{
H^i_x(\mathscr{O}_X) \ar@{->>}[r] & \mathbb{H}^i_x(\dbcx X)
}
\]
is surjective for each $i\in\mathbb{Z}$. \end{cor}
\begin{proof}
Let $E$ be an injective hull of the residue field $\kappa(x)$
as an $\mathscr{O}_{X,x}$-module. Then by local duality \cite[Theorem~V.6.2]{RD} there
exists a commutative diagram where the vertical maps are isomorphisms:
\begin{equation}
\label{eq:2}
\begin{aligned}
\xymatrix{
{\mcR\!}\Gamma_x(\mathscr{O}_X)\ar[r] \ar[d]_\simeq & {\mcR\!}\Gamma_x(\dbcx X) \ar[d]^\simeq \\
{\mcR\!}\!\Hom_{\mathscr{O}_{X,x}} ( \dcx X, E) \ar[r] & {\mcR\!}\!\Hom_{\mathscr{O}_{X,x}} ( \uldcx
X, E). }
\end{aligned}
\end{equation}
Since $E$ is injective, the functor $\Hom_{\mathscr{O}_{X,x}} ( \dash{1em}, E)$ is exact and
hence it commutes with taking cohomology. Thus one has that
\[
{\sf h}^i( {\mcR\!}\!\Hom_{\mathscr{O}_{X,x}} ( \dcx X, E) ) \simeq \Hom_{\mathscr{O}_{X,x}} ( {\sf h}^i(\dcx
X), E)
\]
and
\[
{\sf h}^i( {\mcR\!}\!\Hom_{\mathscr{O}_{X,x}} ( \uldcx X, E) ) \simeq \Hom_{\mathscr{O}_{X,x}} (
{\sf h}^i(\uldcx X), E).
\]
It follows that by taking cohomology of the diagram in \autoref{eq:2} one obtains
for each $i$ the commutative diagram
\begin{equation}
\label{eq:3}
\begin{aligned}
\xymatrix{
H^i_x(\mathscr{O}_X)\ar[r] \ar[d]_\simeq & \mathbb{H}^i_x(\dbcx X) \ar[d]^\simeq \\
\Hom_{\mathscr{O}_{X,x}} ( {\sf h}^i(\dcx X), E) \ar[r] & \Hom_{\mathscr{O}_{X,x}} (
{\sf h}^i(\uldcx X), E). }
\end{aligned}
\end{equation}
Again, since $\Hom_{\mathscr{O}_{X,x}} ( \dash{1em}, E)$ is exact, it follows from
\autoref{thm:key-injectivity} that the bottom homomorphism is surjective which
implies the desired statement. \end{proof}
\begin{rem}
An important aspect of \autoref{cor:key-surjectivity} is that the local cohomology
of $\mathscr{O}_X$ depends on the non-reduced structure, while that of $\dbcx X$ does
not. Essentially, the left hand side reflects the algebraic structure, while the
right hand side behaves as if it only depended on the topology (this is not
entirely true!).
This behavior allows us to prove \autoref{DB.toploccohs}. The proof is based on the
interplay between the non-reduced and reduced data. It is similar in spirit to the
proofs of \cite[Lemme 1]{MR0376678}, \cite[Theorem~5.1]{KS13}, and
\cite[Lemma~3.3]{MSS17}. \end{rem}
\begin{proof}[Proof of \autoref{DB.toploccohs}]
Using \autoref{not:top-loc-cohs} assume that $A=k$ is a field of characteristic $0$
and that $X=\Spec T$ has Du~Bois singularities. We need to prove that the induced
morphism on local cohomology $H^i_\mathfrak{m}(R) \twoheadrightarrow H^i_\mathfrak{m}(T)$ is surjective for each
$i$.
Consider the following diagram:
\[
\xymatrix@C=3em@R=3em{
H^i_\mathfrak{m}(R) \ar@{->>}[d]_\xi \ar[rr]^-\chi
&
& H^i_\mathfrak{m}(T) \ar[d]_\simeq^\vartheta \\
H^i_\mathfrak{m}(\dbcx R) \ar[rr]^-\simeq_-\zeta & & H^i_\mathfrak{m}(\dbcx {T}) }
\]
Using the notation of the diagram, one has that $\xi$ is surjective by
\autoref{cor:key-surjectivity}, $\zeta$ is an isomorphism because
$\dbcx R\simeq \dbcx {T}$, and $\vartheta$ is an isomorphism, because $\Spec T$ has
{Du\thinspace\nolinebreak Bois}\xspace singularities\xspace. It follows then that
$\chi$ is surjective. \end{proof}
\section{$F$-pure\xspace singularities\xspace}\label{sec:fp-sings}
\noin There is an intriguing correspondence between singularities of the minimal model program in characteristic $0$ and singularities defined by the action of the Frobenius morphism in positive characteristic. For more on this correspondence the reader may consult \cite[App.~C]{MR2932591} or \cite[\S8.4]{SingBook}. Our goal here is to show that $F$-pure, or more generally $F$-anti-nilpotent singularities have liftable local cohomology\xspace over their ground field.
\begin{defini}
Let $(R,\mathfrak{m})$ be a noetherian local ring of characteristic $p>0$ with the
Frobenius endomorphism $F:R\to R$; $x\mapsto x^p$.
Recall that a homomorphism of $R$-modules $M\to M'$ is called \emph{pure} if for
every $R$-module $N$ the induced homomorphism $M\otimes_RN\to M'\otimes_RN$ is
injective. $R$ is called \emph{$F$-pure} if the Frobenius endomorphism is pure. $R$
is called \emph{$F$-finite} if $R$ is a finitely generated $R$-module via the
Frobenius endomorphism $F$. For instance, if $R$ is essentially of finite type over
a field, then it is $F$-finite. Further note that if $R$ is $F$-finite or complete
then it is $F$-pure if and only if the Frobenius endomorphism $F:R\to R$ has a left
inverse \cite[5.3]{MR0417172}.
$R$ is called \emph{$F$-injective} if the induced Frobenius action on $H^i_\mathfrak{m}(R)$
is injective for all $i\in \mathbb{N}$. This holds for example if $R$ is $F$-pure by
\cite[2.2]{MR0417172} and if $R$ is Gorenstein then it is $F$-pure if and only if
it is $F$-injective \cite[3.3]{MR701505}.
A strengthening of the notion of $F$-injective was recently introduced in
\cite{MR2460693}: Consider the induced Frobenius action
$F:H^i_\mathfrak{m}(R)\to H^i_\mathfrak{m}(R)$. A submodule $M\subseteq H^i_\mathfrak{m}(R)$ is called
\emph{$F$-stable} if $F(M)\subseteq M$ and $R$ is called \emph{$F$-anti-nilpotent}
if for any $F$-stable submodule $M\subseteq H^i_\mathfrak{m}(R)$, the induced Frobenius
action on the quotient $H^i_\mathfrak{m}(R)/M$ is injective. If $R$ is $F$-anti-nilpotent,
then it is $F$-injective, since $\{0\}\subseteq H^i_\mathfrak{m}(R)$ is an $F$-stable
submodule. Furthermore, if $R$ is $F$-pure, then it is $F$-anti-nilpotent by
\cite[3.8]{MR3271179}. So we have the following implications:
\begin{equation}
\label{eq:13}
\xymatrix{
\text{$F$-pure } \ar@{=>}[r] & \text{ $F$-anti-nilpotent } \ar@{=>}[r] &
\text{ $F$-injective}.
}
\end{equation}
Let $(X,x)$ be a local scheme. Then we say that $X$ has \emph{$F$-pure},
resp.~\emph{$F$-anti-nilpotent}, resp.~\emph{$F$-injective} singularities if the
local ring $\mathscr{O}_{X,x}$ has the corresponding property. An arbitrary scheme $X$ of
equicharacteristic $p>0$ has \emph{$F$-pure}, resp.~\emph{$F$-anti-nilpotent},
resp.~\emph{$F$-injective} singularities if the local scheme $(X,x)$ has the
corresponding property for each $x\in X$. \end{defini}
These singularities are related to the singularities of the minimal model program. Normal $\mathbb{Q}$-Gorenstein $F$-pure singularities are log canonical by \cite{MR1874118} and it is conjectured that in some form the converse also holds. Similarly, $F$-injective singularities correspond to {Du\thinspace\nolinebreak Bois}\xspace singularities\xspace: If $X$ is essentially of finite type over a field of characteristic $0$ and its reduction mod $p$ is $F$-injective for infinitely many $p$'s, then $X$ has {Du\thinspace\nolinebreak Bois}\xspace singularities\xspace by \cite{MR2503989} and the converse of this is also conjectured to hold. So the (outside) implication in \autoref{eq:13} is analogous to that log canonical singularities\xspace are {Du\thinspace\nolinebreak Bois}\xspace \cite{MR2629988}.
Curiously, we have this additional notion, $F$-anti-nilpotent, in between the more familiar $F$-pure and $F$-injective notions. It turns out that $F$-anti-nilpotent, and hence $F$-pure, rings have liftable local cohomology\xspace, but $F$-injective in general do not. This suggests that possibly $F$-anti-nilpotent is a better analog of {Du\thinspace\nolinebreak Bois}\xspace singularities\xspace in positive characteristic than $F$-injective. Of course, this is far from conclusive evidence, and this issue will not be settled here.
These singularities, defined by the action of Frobenius, have been studied extensively through their local cohomology. So it is no surprise that the fact that $F$-anti-nilpotent singularities have liftable local cohomology\xspace is a relatively simple consequence of known results. The following statement is essentially proved in \cite[Remark~3.4]{MSS17}, although their statement is slightly different, so we include a proof for completeness.
\begin{prop}[(Ma-Schwede-Shimomoto)]\label{prop:F-anti-nilp-has-liftable-cohs}
$F$-anti-nilpotent singularities have liftable local cohomology\xspace over their ground field. \end{prop}
\begin{proof}
(Following the argument in \cite[Remark~3.4]{MSS17}).
Using \autoref{not:top-loc-cohs} assume that $A=k$ is a field of characteristic
$p>0$ and that $(R,\mathfrak{m})$ has $F$-anti-nilpotent\xspace singularities\xspace. We need to prove that the induced
morphism on local cohomology $H^i_\mathfrak{m}(R) \twoheadrightarrow H^i_\mathfrak{m}(T)$ is surjective for each
$i$ (cf.~\autoref{def:liftable-cohom}).
Since the statement is about local cohomology we may assume that $R$ is complete
and hence $R\simeq R'/J$ where $R'$ is a complete regular local ring and
$J\subseteq R'$ is an ideal. Note that denoting the pre-image of $I\subset R$ in
$R'$ by $I'$, we have that $T\simeq R'/I'$ is also a quotient of $R'$.
Let $M\colon\!\!\!= \im\left[H^i_\mathfrak{m}(R)\to H^i_\mathfrak{m}(T)\right]$, which is an $F$-stable
submodule of $H^i_\mathfrak{m}(T)$. Then $M$ contains $F^e(H^i_\mathfrak{m}(T))$ for some $e>0$ by
\cite[Lemma~2.2]{MR2197409} and hence the Frobenius action on $H^i_\mathfrak{m}(T)/M$ is
nilpotent. In particular it is injective only if this quotient is $0$. Therefore,
if $R$ is $F$-anti-nilpotent, then $H^i_\mathfrak{m}(R)\twoheadrightarrow H^i_\mathfrak{m}(T)$ is surjective as
desired. \end{proof}
\begin{cor}[(Ma)]\label{cor:F-pure-has-liftable-cohs}
$F$-pure singularities have liftable local cohomology\xspace over their ground field. \end{cor}
\begin{proof}
$F$-pure singularities\xspace are $F$-anti-nilpotent by \cite[3.8]{MR3271179}, so this is a
direct consequence of \autoref{prop:F-anti-nilp-has-liftable-cohs}. \end{proof}
\section{Degenerations of \CM singularities\xspace with liftable local cohomology\xspace}\label{sec:degenerations-cm-db}
\begin{prop}\label{prop:S_n-via-hi}
Let $Z$ be a scheme that admits a dualizing complex $\dcx{Z}$, $z\in Z$ a
(not-necessarily-closed) point, and $n\in\mathbb{N}$. Then $Z$ is $S_n$ at $z$ if and only
if for all $i\in\mathbb{Z}$,
\begin{equation}
\label{eq:27}
{\sf h}^{-i}(\dcx Z)_z = 0 \text{ for $i<\min(n,\dim_z Z)+\dim z$}.
\end{equation}
In particular, if $Z$ is equidimensional, then $Z$ is $S_n$ if and only if for all
$i\in\mathbb{Z}$, $i<\dim Z$,
\begin{equation}
\label{eq:29}
\dim\supp{\sf h}^{-i}(\dcx Z)
\leq i-n.
\end{equation}
(In this statement we take $\dim \emptyset=-\infty$). \end{prop}
\begin{proof}
Since ${\sf h}^{-i}(\dcx Z)=\sExt^{-i}_Z(\mathscr{O}_Z,\dcx Z)$, \autoref{eq:27} follows
directly from \cite[Prop~3.2]{MR2918171}.
Next, let $i\in\mathbb{Z}$, $i<\dim Z$ be such that ${\sf h}^{-i}(\dcx Z)\neq 0$ and let
$z\in\supp{\sf h}^{-i}(\dcx Z)$ be a general point such that
$\dim z=\dim\supp{\sf h}^{-i}(\dcx Z)$.
If $Z$ is $S_n$, then $i\geq \min(n,\dim_z Z)+\dim z$ by \autoref{eq:27} and hence,
since $i<\dim Z$, we must have $\min(n,\dim_z Z)=n$ (this is where $Z$ being
equidimensional is used), so indeed $\dim\supp{\sf h}^{-i}(\dcx Z) \leq i-n$.
In order to prove the other implication let $i\in\mathbb{Z}$, $i<\dim Z$ be again such
that ${\sf h}^{-i}(\dcx Z)\neq 0$, but now choose an arbitrary point
$z\in\supp{\sf h}^{-i}(\dcx Z)$. In this case we only have that
$\dim z\leq \dim\supp{\sf h}^{-i}(\dcx Z)$, but this will be enough. If
$\dim\supp{\sf h}^{-i}(\dcx Z) \leq i-n<\dim Z -n$, then
$n< \dim Z - \dim\supp{\sf h}^{-i}(\dcx Z)\leq \dim Z -\dim z =\dim_zZ$, i.e.,
$\min(n,\dim_z Z)=n$. It also follows that
$i\geq n+ \dim\supp{\sf h}^{-i}(\dcx Z)\geq\min(n,\dim_z Z) +\dim z$, and hence $Z$ is
$S_n$ at $z$ by \autoref{eq:27}. We obtain that for all $i<\dim Z$,
$\supp{\sf h}^{-i}(\dcx Z)$ is contained in the $S_n$-locus of $Z$. However, $Z$ is
\CM and hence $S_n$ at every point in
$Z\setminus\bigcup_{i<\dim Z}\supp{\sf h}^{-i}(\dcx Z)$, which proves \autoref{eq:29}. \end{proof}
\begin{cor}
Let $Z$ be an equidimensional scheme that admits a dualizing complex $\dcx{Z}$. If
$Z$ is $S_n$ for some $n\in\mathbb{N}$, then ${\sf h}^{-i}(\dcx Z)=0$ for $i<n$. \end{cor}
\begin{thm}\label{cm-is-a-closed-prop}
Let $f:X\to B$ be a flat morphism with equidimensional fibers that is locally
embeddable into a smooth morphism.
Assume that there exists a $b_0\in B$ such that $X_{b_0}$ has liftable local cohomology\xspace over $B$.
If $X_{b_0}$ is not $S_n$ then there exists an open subset $b_0\in V\subseteq B$
such that $X_b$ is not $S_n$ for each $b\in V$. \end{thm}
\begin{proof}
By \autoref{thm:main-strong} there exists an open neighborhood
$X_b\subset U\subset X$ such that ${\sf h}^{-i}(\dcx{U/B})$ is flat over $B$ and
commutes with base change for each $i\in\mathbb{Z}$. Then
$\dim \supp{\sf h}^{-i}(\dcx{U_b})$ is a locally constant function on the set
$\{b\in B\skvert {\sf h}^{-i}(\dcx{X_b})\neq 0\}$, so the claim follows from
\autoref{eq:29}. \end{proof}
\subsection{Deformations of local schemes}
\begin{defini}
Let $(A,\mathfrak{m}_1,\dots,\mathfrak{m}_r)$ be a semi-local ring. Then $(X,x_1,\dots,x_r)$ is
called a \emph{semi-local scheme} where $X=\Spec A$ and
$x_1=\mathfrak{m}_1,\dots,x_r=\mathfrak{m}_r \in X$. If $r=1$ and $A$ is a local ring then
$(X,x_1)$ is a \emph{local scheme}.
A \emph{family of semi-local schemes} consists of a pair $(\mathchanc{X},\mathchanc{x})$ where
$\mathchanc{x}\subseteq \mathchanc{X}$ is a closed subscheme and a flat morphism $f:\mathchanc{X}\to B$ that
is essentially of finite type such that $f\resto{\mathchanc{x}}:\mathchanc{x}\to B$ is a dominant
finite morphism and for any $b\in B$, $(\mathchanc{X}_b,\mathrm{red}(\mathchanc{x}_b))$ is an equidimensional
semi-local scheme. By a slight abuse of notation this family of semi-local schemes
will be denoted by $f:(\mathchanc{X},\mathchanc{x})\to B$.
Let ${\sf P}$ be a local property of a scheme such as being {Du\thinspace\nolinebreak Bois}\xspace,
$F$-pure\xspace, $F$-anti-nilpotent,
$S_n$, or \CM. We will say that a semi-local scheme $(X,x_1,\dots,x_r)$ has
property ${\sf P}$ if $X$ is ${\sf P}$ at $x_1,\dots,x_r$. In particular, we will say
that $(X,x_1,\dots,x_r)$ is a {Du\thinspace\nolinebreak Bois}\xspace semi-local scheme, etc.
Similarly for ``local scheme'' in place of ``semi-local scheme''.
\end{defini}
\begin{thm}\label{thm:cm-defo-of-topcoh}\label{thm:cm-defo-of-db}
Let $f:(\mathchanc{X},\mathchanc{x})\to B$ be a family of semi-local schemes. Assume that
\begin{enumerate}
\item\label{item:15} $B$ is irreducible,
\item\label{item:16} the fibers of $f$ are equidimensional,
\item\label{item:17} there exists a $b_0\in B$ such that $\mathchanc{X}_{b_0}$ has
liftable local cohomology\xspace over $B$, and
\item\label{item:18} there exists a $b_1\in B$ and an $n\in\mathbb{N}$ such that
$\mathchanc{X}_{b_1}$ is $S_n$ (resp.~\CM).
\end{enumerate}
Then there exists an open set $b_0\in U\subseteq B$ such that $\mathchanc{X}_{b}$ is $S_n$
(resp.~\CM) for each $b\in U$. In particular, $\mathchanc{X}_{b_0}$ is $S_n$ (resp.~\CM). \end{thm}
\begin{proof}
It is enough to prove the statement for the $S_n$ property. Since $f\resto \mathchanc{x}$
is proper,
the set $U:=\{b\in B \skvert \mathchanc{X}_b \text{ is } S_n \}$ is open in $B$ by
\cite[12.1.6]{EGAIV3}. By \autoref{item:18} it is non-empty and hence it is dense
in $B$. Then it must contain $b_0$ by \autoref{cm-is-a-closed-prop}, which proves
the statement. \end{proof}
\begin{rem}
If $\mathchanc{X}_{b_0}$ has {Du\thinspace\nolinebreak Bois}\xspace singularities\xspace then we may even choose $U$ such that $\mathchanc{X}_b$ is
$S_n$ and has {Du\thinspace\nolinebreak Bois}\xspace singularities\xspace for all $b\in U$ by \cite[4.1]{Kovacs-Schwede11b}. As we
mentioned earlier, it is not known whether small deformations of $F$-anti-nilpotent\xspace singularities\xspace remain
$F$-anti-nilpotent\xspace. It is also an interesting question whether the condition of having
liftable local cohomology\xspace is invariant under small deformations. \end{rem}
It follows that \autoref{thm:cm-defo-of-topcoh} applies to families of semi-local schemes with {Du\thinspace\nolinebreak Bois}\xspace or $F$-anti-nilpotent singularities\xspace by \autoref{prop:loc-coh-surj-1} and \autoref{prop:F-anti-nilp-has-liftable-cohs} (cf.~\autoref{rem:top-loc-cohs-is-hereditary}).
\noin As a simple consequence we obtain a generalization of \autoref{cor:cone-over-abelian}.
\begin{cor}\label{cor:cone-over-abelian-s3}
Let $Z$ be a normal projective variety over a field $k$ such that $K_Z$ is
$\mathbb{Q}$-Cartier and numerically equivalent to $0$. Let $\mathscr{L}$ be an ample line bundle
on $Z$ and $X=C_a(Z,\mathscr{L})$ the affine cone over $Z$ with conormal bundle $\mathscr{L}$. If
$X$ has liftable local cohomology\xspace over $k$ and admits an $S_n$ deformation for some $n\in\mathbb{N}$,
then $H^i(Z,\mathscr{O}_Z)=0$ for $0<i<n-1$. In particular, a cone over an abelian variety
(ordinary, if $\kar k>0$) of dimension at least $2$ does not admit an $S_3$
deformation. \end{cor}
\begin{proof}
If $X$ admits an $S_n$ deformation for some $n\in\mathbb{N}$, then $X$ itself is $S_n$ by
\autoref{thm:cm-defo-of-topcoh} and the first statement follows from
\autoref{item:24}. Then the second statement follows from
\autoref{cor:cone-over-abelian} and \autoref{DB.toploccohs} in characteristic $0$
and from \autoref{exmp:sing-of-cones-ii} and \autoref{cor:F-pure-has-liftable-cohs}
in positive characteristic. \end{proof}
\begin{rem}
As noted in the introduction, it is proved in \cite{MR522037} that the projective
cone over an abelian variety of dimension at least $2$ over $\mathbb{C}$ is not smoothable
inside the ambient projective space. This is a strict special case of
\autoref{cor:cone-over-abelian-s3}. In general, it is possible that a
non-smoothable projective variety is locally smoothable. An example of that is
given in \cite[2.18]{CoughlanTaro16}. Furthermore,
\autoref{cor:cone-over-abelian-s3} is also valid in positive characteristic. \end{rem}
\def$'${$'$} \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def$'${$'$}
\def$'${$'$} \def$'${$'$} \def$'${$'$}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def$''${$''$}
\def$'${$'$} \def$'${$'$} \def$'${$'$} \def$'${$'$}
\def$'${$'$} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR}
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2}
\end{document} | arXiv |
Let $\triangle XOY$ be a right-angled triangle with $m\angle XOY = 90^{\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN=19$ and $YM=22$, find $XY$.
Let $OM = a$ and $ON = b$. Then $$
19^2 = (2a)^2 + b^2 \quad \text{and} \quad 22^2 = a^2 + (2b)^2.
$$ [asy]
unitsize(0.3cm);
pair X,Y,O,N,M;
X=(0,8);
O=(0,0);
Y=(13,0);
N=(6,0);
M=(0,4);
path a=X--Y--O--cycle;
path b=M--Y;
draw(a);
draw(X--N);
draw(shift((16,0))*a);
draw(shift((16,0))*b);
for (int i=0; i<2; ++i) {
label("$X$",shift((16*i,0))*X,W);
label("$O$",shift((16*i,0))*O,S);
label("$Y$",shift((16*i,0))*Y,S);
}
label("$N$",N,S);
label("$2a$",(0,4),W);
label("$b$",(3,0),S);
label("$2b$",(22,0),S);
label("$a$",(16,1.5),W);
label("19",(2,4),S);
label("22",(21,2.5),NE);
label("$M$",shift((16,0))*M,W);
[/asy] Hence $$
5(a^2+b^2) = 19^2 + 22^2 = 845.
$$ It follows that $$
MN = \sqrt{a^2 + b^2} = \sqrt{169}= 13.
$$ Since $\triangle XOY$ is similar to $\triangle MON$ and $XO=2\cdot MO$, we have $XY= 2 \cdot MN = \boxed{26}$. [asy]
pair X,M,O,N,Y;
O=(0,0);
Y=(24,0);
N=(12,0);
M=(0,5);
X=(0,10);
label("$X$",X,W);
label("$M$",M,W);
label("$O$",O,SW);
label("$N$",N,S);
label("$Y$",Y,S);
label("$a$",(0,2.5),W);
label("$a$",(0,7.5),W);
label("$b$",(6,0),S);
label("$b$",(18,0),S);
label("13",(4,4),E);
label("26",(12,7),E);
draw(X--Y--O--cycle);
draw(M--N);
[/asy] | Math Dataset |
Subsets and Splits